Deterministic and stochastic dynamics in spinodal decomposition of a binary system

A model for diffusion and phase separation, which takes into account hyperbolic relaxation of the solute diffusion flux, is developed. Such a ‘hyperbolic model’ provides analysis of ‘hyperbolic evolution’ of patterns in spinodal decomposition in systems supercooled below critical temperature. Analyt...

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Опубліковано в: :	Успехи физики металлов
Дата:	2009
Автори:	Kharchenko, D.O., Galenko, P.K., Lebedev, V.G.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут металофізики ім. Г.В. Курдюмова НАН України 2009
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/98091
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Цитувати:	Deterministic and stochastic dynamics in spinodal decomposition of a binary system / D.O. Kharchenko, P.K. Galenko, V.G. Lebedev // Успехи физики металлов. — 2009. — Т. 10, № 1. — С. 27-102. — Бібліогр.: 92 назв. — англ.

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spelling	Kharchenko, D.O. Galenko, P.K. Lebedev, V.G. 2016-04-08T20:11:11Z 2016-04-08T20:11:11Z 2009 Deterministic and stochastic dynamics in spinodal decomposition of a binary system / D.O. Kharchenko, P.K. Galenko, V.G. Lebedev // Успехи физики металлов. — 2009. — Т. 10, № 1. — С. 27-102. — Бібліогр.: 92 назв. — англ. 1608-1021 PACS numbers: 05.40.-a, 05.45.-a, 05.70.Fh, 05.70.Ln, 64.60.-i, 64.75.Nx, 81.30.-t https://nasplib.isofts.kiev.ua/handle/123456789/98091 A model for diffusion and phase separation, which takes into account hyperbolic relaxation of the solute diffusion flux, is developed. Such a ‘hyperbolic model’ provides analysis of ‘hyperbolic evolution’ of patterns in spinodal decomposition in systems supercooled below critical temperature. Analytical results for the hyperbolic model of spinodal decomposition are summarized in comparison with outcomes of classic Cahn−Hilliard theory. Numeric modelling shows that the hyperbolic evolution leads to sharper boundary between two structures of a decomposed system in comparison with prediction of parabolic equation given by the theory of Cahn and Hilliard. Considering phase separation processes in stochastic systems with a field-dependent mobility and an internal multiplicative noise, we study dynamics of spinodal decomposition for parabolic and hyperbolic models separately. It is that the domain growth law is generalized when internal fluctuations are introduced into the model. A mean field approach is carried out in order to obtain the stationary probability, bifurcation and phase diagrams displaying re-entrant phase transitions. We relate our approach to entropy-driven phase-transitions theory. Розвинуто модель дифузії та фазового розшарування, який враховує гіперболічну релаксацію дифузійного потоку. Такий «гіперболічний модель» призводить до «гіперболічного» рівнання щодо формування модульованих структур при спинодальнім розпаді в системах, охолоджених нижче критичної температури. Аналітичні результати для гіперболічного моделю спинодального розпаду порівнюються із відповідними результатами, що випливають з класичної теорії Кана—Хіллярда. За допомогою чисельного моделювання показано, що еволюція системи в гіперболічнім моделю призводить до різкої міжфазної межі у порівнянні з обчисленнями за параболічним модельом Кана−Хіллярда. З розглядом процесів фазового розшарування в стохастичних системах із залежною від поля концентрації рухливістю та внутрішнім мультиплікативним шумом вивчається динаміка спинодального розпаду для параболічного та гіперболічного моделів. Показано, що закон зростання розмірів зерен може бути узагальнений введенням у розгляд внутрішніх флюктуацій, залежних від поля концентрації. Для дослідження стаціонарної картини (функції розподілу, біфуркаційних та фазових діяграм) розвинуто теорію середнього поля, в рамках якої встановлено, що відповідні перетворення носять реверсивний характер. Показано, що опис процесу фазового розшарування у стохастичних системах із внутрішнім шумом забезпечується використанням теорії ентропійнокерованих фазових переходів. В работе развита модель для описания диффузии и фазового расслоения, которая учитывает гиперболическую релаксацию диффузионного потока. Такая «гиперболическая модель» приводит к гиперболическому уравнению описания формирования модулированных структур при спинодальном распаде в системах, охлажденных ниже критической температуры. Аналитические результаты для гиперболической модели спинодального распада сравниваются с соответствующими результатами, следующими из классической теории Кана—Хилларда. С помощью численного моделирования показано, что эволюция системы в гиперболической модели приводит к резким межфазным границам в сравнении с вычислениями согласно параболической модели Кана—Хилларда. При рассмотрении процессов фазового расслоения в стохастических системах с зависимой от поля концентрации подвижностью и внутренним мультипликативным шумом изучена динамика спинодального распада для параболической и гиперболической моделей. Показано, что закон роста размеров зерен может быть обобщен введением в рассмотрение внутренних флуктуаций, зависимых от поля концентрации. Для исследования стационарной картины (функции распределения, бифуркационных и фазовых диаграмм) развита теория среднего поля, в рамках которой установлено, что соответствующие превращения носят реверсивный характер. Показано, что описание процесса фазового расслоения в стохастических системах с внутренним шумом обеспечивается использованием теории энтропийноуправляемых фазовых переходов. We thank David Jou and Alexander Olemskoi for fruitful discussions and useful exchanges. Dmitrii Kharchenko acknowledges financial support from the Fundamental Research State Fund of Ukraine (No. GP/F26/0010). Peter Galenko acknowledges financial support from the German Research Foundation (DFG) under the Project No. HE 160/19 and DLR Agency under contract 50WM0736. Vladimir Lebedev acknowledges financial support from the Russian Foundation of Basic Research (RFBR) under the Project No. 08-02-91957. en Інститут металофізики ім. Г.В. Курдюмова НАН України Успехи физики металлов Deterministic and stochastic dynamics in spinodal decomposition of a binary system Детерміністична і стохастична динаміка в спинодальнім розпаді бінарної системи Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	Deterministic and stochastic dynamics in spinodal decomposition of a binary system
spellingShingle	Deterministic and stochastic dynamics in spinodal decomposition of a binary system Kharchenko, D.O. Galenko, P.K. Lebedev, V.G.
title_short	Deterministic and stochastic dynamics in spinodal decomposition of a binary system
title_full	Deterministic and stochastic dynamics in spinodal decomposition of a binary system
title_fullStr	Deterministic and stochastic dynamics in spinodal decomposition of a binary system
title_full_unstemmed	Deterministic and stochastic dynamics in spinodal decomposition of a binary system
title_sort	deterministic and stochastic dynamics in spinodal decomposition of a binary system
author	Kharchenko, D.O. Galenko, P.K. Lebedev, V.G.
author_facet	Kharchenko, D.O. Galenko, P.K. Lebedev, V.G.
publishDate	2009
language	English
container_title	Успехи физики металлов
publisher	Інститут металофізики ім. Г.В. Курдюмова НАН України
format	Article
title_alt	Детерміністична і стохастична динаміка в спинодальнім розпаді бінарної системи
description	A model for diffusion and phase separation, which takes into account hyperbolic relaxation of the solute diffusion flux, is developed. Such a ‘hyperbolic model’ provides analysis of ‘hyperbolic evolution’ of patterns in spinodal decomposition in systems supercooled below critical temperature. Analytical results for the hyperbolic model of spinodal decomposition are summarized in comparison with outcomes of classic Cahn−Hilliard theory. Numeric modelling shows that the hyperbolic evolution leads to sharper boundary between two structures of a decomposed system in comparison with prediction of parabolic equation given by the theory of Cahn and Hilliard. Considering phase separation processes in stochastic systems with a field-dependent mobility and an internal multiplicative noise, we study dynamics of spinodal decomposition for parabolic and hyperbolic models separately. It is that the domain growth law is generalized when internal fluctuations are introduced into the model. A mean field approach is carried out in order to obtain the stationary probability, bifurcation and phase diagrams displaying re-entrant phase transitions. We relate our approach to entropy-driven phase-transitions theory. Розвинуто модель дифузії та фазового розшарування, який враховує гіперболічну релаксацію дифузійного потоку. Такий «гіперболічний модель» призводить до «гіперболічного» рівнання щодо формування модульованих структур при спинодальнім розпаді в системах, охолоджених нижче критичної температури. Аналітичні результати для гіперболічного моделю спинодального розпаду порівнюються із відповідними результатами, що випливають з класичної теорії Кана—Хіллярда. За допомогою чисельного моделювання показано, що еволюція системи в гіперболічнім моделю призводить до різкої міжфазної межі у порівнянні з обчисленнями за параболічним модельом Кана−Хіллярда. З розглядом процесів фазового розшарування в стохастичних системах із залежною від поля концентрації рухливістю та внутрішнім мультиплікативним шумом вивчається динаміка спинодального розпаду для параболічного та гіперболічного моделів. Показано, що закон зростання розмірів зерен може бути узагальнений введенням у розгляд внутрішніх флюктуацій, залежних від поля концентрації. Для дослідження стаціонарної картини (функції розподілу, біфуркаційних та фазових діяграм) розвинуто теорію середнього поля, в рамках якої встановлено, що відповідні перетворення носять реверсивний характер. Показано, що опис процесу фазового розшарування у стохастичних системах із внутрішнім шумом забезпечується використанням теорії ентропійнокерованих фазових переходів. В работе развита модель для описания диффузии и фазового расслоения, которая учитывает гиперболическую релаксацию диффузионного потока. Такая «гиперболическая модель» приводит к гиперболическому уравнению описания формирования модулированных структур при спинодальном распаде в системах, охлажденных ниже критической температуры. Аналитические результаты для гиперболической модели спинодального распада сравниваются с соответствующими результатами, следующими из классической теории Кана—Хилларда. С помощью численного моделирования показано, что эволюция системы в гиперболической модели приводит к резким межфазным границам в сравнении с вычислениями согласно параболической модели Кана—Хилларда. При рассмотрении процессов фазового расслоения в стохастических системах с зависимой от поля концентрации подвижностью и внутренним мультипликативным шумом изучена динамика спинодального распада для параболической и гиперболической моделей. Показано, что закон роста размеров зерен может быть обобщен введением в рассмотрение внутренних флуктуаций, зависимых от поля концентрации. Для исследования стационарной картины (функции распределения, бифуркационных и фазовых диаграмм) развита теория среднего поля, в рамках которой установлено, что соответствующие превращения носят реверсивный характер. Показано, что описание процесса фазового расслоения в стохастических системах с внутренним шумом обеспечивается использованием теории энтропийноуправляемых фазовых переходов.
issn	1608-1021
url	https://nasplib.isofts.kiev.ua/handle/123456789/98091
citation_txt	Deterministic and stochastic dynamics in spinodal decomposition of a binary system / D.O. Kharchenko, P.K. Galenko, V.G. Lebedev // Успехи физики металлов. — 2009. — Т. 10, № 1. — С. 27-102. — Бібліогр.: 92 назв. — англ.
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fulltext	27 PACS numbers: 05.40.-a, 05.45.-a, 05.70.Fh, 05.70.Ln, 64.60.-i, 64.75.Nx, 81.30.-t Deterministic and Stochastic Dynamics in Spinodal Decomposition of a Binary System D. O. Kharchenko, P. K. Galenko,, and V. G. Lebedev** Institute of Applied Physics, N.A.S. of the Ukraine, 58 Petropavlivs’ka Str., 40030 Sumy, Ukraine Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany Institut für Festkörperphysik, Ruhr-Universität Bochum, 44780 Bochum, Germany *Udmurt State University, Department of Theoretical Physics, 426034 Izhevsk, Russia A model for diffusion and phase separation, which takes into account hy- perbolic relaxation of the solute diffusion flux, is developed. Such a ‘hy- perbolic model’ provides analysis of ‘hyperbolic evolution’ of patterns in spinodal decomposition in systems supercooled below critical temperature. Analytical results for the hyperbolic model of spinodal decomposition are summarized in comparison with outcomes of classic Cahn−Hilliard theory. Numeric modelling shows that the hyperbolic evolution leads to sharper boundary between two structures of a decomposed system in comparison with prediction of parabolic equation given by the theory of Cahn and Hil- liard. Considering phase separation processes in stochastic systems with a field-dependent mobility and an internal multiplicative noise, we study dy- namics of spinodal decomposition for parabolic and hyperbolic models sepa- rately. It is that the domain growth law is generalized when internal fluc- tuations are introduced into the model. A mean field approach is carried out in order to obtain the stationary probability, bifurcation and phase dia- grams displaying re-entrant phase transitions. We relate our approach to entropy-driven phase-transitions theory. Розвинуто модель дифузії та фазового розшарування, який враховує гі- перболічну релаксацію дифузійного потоку. Такий «гіперболічний мо- дель» призводить до «гіперболічного» рівнання щодо формування моду- льованих структур при спинодальнім розпаді в системах, охолоджених нижче критичної температури. Аналітичні результати для гіперболічно- го моделю спинодального розпаду порівнюються із відповідними резуль- Успехи физ. мет. / Usp. Fiz. Met. 2009, т. 10, сс. 27—102 Оттиски доступны непосредственно от издателя Фотокопирование разрешено только в соответствии с лицензией © 2009 ИМФ (Институт металлофизики им. Г. В. Курдюмова НАН Украины) Напечатано в Украине. 28 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV татами, що випливають з класичної теорії Кана—Хіллярда. За допомогою чисельного моделювання показано, що еволюція системи в гіперболічнім моделю призводить до різкої міжфазної межі у порівнянні з обчислення- ми за параболічним модельом Кана−Хіллярда. З розглядом процесів фа- зового розшарування в стохастичних системах із залежною від поля кон- центрації рухливістю та внутрішнім мультиплікативним шумом вивча- ється динаміка спинодального розпаду для параболічного та гіперболіч- ного моделів. Показано, що закон зростання розмірів зерен може бути узагальнений введенням у розгляд внутрішніх флюктуацій, залежних від поля концентрації. Для дослідження стаціонарної картини (функції розподілу, біфуркаційних та фазових діяграм) розвинуто теорію серед- нього поля, в рамках якої встановлено, що відповідні перетворення но- сять реверсивний характер. Показано, що опис процесу фазового розша- рування у стохастичних системах із внутрішнім шумом забезпечується використанням теорії ентропійнокерованих фазових переходів. В работе развита модель для описания диффузии и фазового расслоения, которая учитывает гиперболическую релаксацию диффузионного пото- ка. Такая «гиперболическая модель» приводит к гиперболическому уравнению описания формирования модулированных структур при спи- нодальном распаде в системах, охлажденных ниже критической темпе- ратуры. Аналитические результаты для гиперболической модели спино- дального распада сравниваются с соответствующими результатами, сле- дующими из классической теории Кана—Хилларда. С помощью числен- ного моделирования показано, что эволюция системы в гиперболической модели приводит к резким межфазным границам в сравнении с вычисле- ниями согласно параболической модели Кана—Хилларда. При рассмот- рении процессов фазового расслоения в стохастических системах с зави- симой от поля концентрации подвижностью и внутренним мультиплика- тивным шумом изучена динамика спинодального распада для параболи- ческой и гиперболической моделей. Показано, что закон роста размеров зерен может быть обобщен введением в рассмотрение внутренних флук- туаций, зависимых от поля концентрации. Для исследования стацио- нарной картины (функции распределения, бифуркационных и фазовых диаграмм) развита теория среднего поля, в рамках которой установлено, что соответствующие превращения носят реверсивный характер. Пока- зано, что описание процесса фазового расслоения в стохастических сис- темах с внутренним шумом обеспечивается использованием теории эн- тропийноуправляемых фазовых переходов. Key words: spinodal, diffusion, relaxation, model, liquid, structure factor, stochastic systems. (Received March 6, 2009; in final version March 26, 2009) CONTENTS 1. Introduction 2. Hyperbolic model for spinodal decomposition DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 29 2.1. Hyperbolic transport 2.1.1. Equilibrium fluctuations 2.1.2. Power spectra of density and flux fluctuations 2.2. Hyperbolic spinodal decomposition 2.3. Dispersion relation and speeds for atomic diffusion 2.4. Critical parameters for hyperbolic decomposition 2.4.1. Critical wavelength for decomposition 2.4.2. Amplification rate of decomposition 2.4.3. Critical time for instability 2.4.4. Analysis of a structure function 2.5. Comparison with experimental data 3. Modelling of spinodal decomposition 3.1. 1D modelling 3.2. 3D modelling 4. Stochastic models of spinodal decomposition 4.1. Stochastic parabolic model for spinodal decomposition 4.1.1. An early stage of evolution 4.1.2. A late stage 4.1.3. Stationary case 4.1.4. Influence of external and internal noise sources 4.2. Stochasticity in hyperbolic transport 4.3. Stochastic hyperbolic model for spinodal decomposition 4.3.1. Early stages analysis 4.3.2. The effective Fokker−Planck equation for the hyperbolic model 5. Conclusions Acknowledgments References 1. INTRODUCTION Consider a process of phase separation evolving through spontaneous growth of fluctuations, e.g., through fluctuations of concentration as in liquid−liquid systems or fluctuations of density as in gas—liquid sys- tems. This process is known as a spinodal decomposition, in which, be- cause of spontaneous fluctuations growth, both phases have equivalent symmetry but they differ only in composition. It was observed in many experiments on polymeric mixtures [1], liquid solutions [2, 3], organic systems [4], and metallic systems [5, 6]. This transformation has been widely investigated by using theoretical methods as well [7—10]. Phe- nomenological theory for decomposing phases has been constructed by Ginzburg and Landau [11]. They described magnetic domains in tran- sition from the normal to superconducting phase using non-conserved order parameter. This theory has been successfully advanced by Cahn and Hilliard for using conserved order parameter for description of 30 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV spinodal decomposition in binary liquids and solids [12]. As it has been further derived by Cahn [13, 14], kinetics of decomposition is defined by the growth of unstable fluctuations, and the mean size of a new phase can be given by the most rapidly growing fluctuation. In parallel with detailed analysis [10, 15] and tests against experi- mental data [3, 4], the theory of Cahn and Hilliard has been further ex- plored and developed. In particular, it has been demonstrated in com- putational modelling [16] that the rapidly quenched liquid mixtures under decomposition exhibit unusual non-equilibrium patterns, which do not consistent with predictions of the Cahn and Hilliard’s theory. These inconsistencies might be associated with the phase segregation kinetics induced by hydrodynamic interactions following a rapid quench below spinodal [16]. They also might be attributed to the spi- nodal decomposition upon inhomogeneous quenching [17]. In both cases, there is a boundary for the critical quenching above which the classic Cahn and Hilliard’s approach has to be extended to the case of strongly nonequilibrium decomposition provided by deep supercooling into the spinodal region of a phase diagram. Therefore, earliest stages and periods of decomposition under large supercooling can provide pattern’s dynamics different from those predicted by the Cahn and Hilliard’s theory. A few advancements were made for strongly non-equilibrium phase separation. Binder, Frish, and Jäckle [18] generalized the linearized Cahn—Hilliard’s theory to the case of existence of a slowly relaxing variable. Their calculations showed that the instability of the system is not of the standard diffusive type, but rather it is controlled by the re- laxation of the slow structural variable. Recently, a hyperbolic diffu- sion equation with phase separation was derived in Refs [19, 20] from the formalism of extended irreversible thermodynamics [21]. It has been proposed that the hyperbolic equation is able to describe process of rapidly quenched decomposition for short periods of time, large composition gradients or deep supercoolings within a system. Finally, Grasselli et al. [22] mathematically analyzed extended Cahn—Hilliard’s equation with hyperbolic relaxation of the diffusion flux. Their treat- ments have been devoted to one-, two-, and three-dimensional cases of hyperbolic spinodal decomposition [23—25] to establish existence of the global and exponential attractors for different phase spaces. These in- vestigations [18—20, 22—25] show that evolution of phase separation in deeply supercooled or rapidly quenched systems might be analyzed us- ing predictions of hyperbolic transport equation. It is known that considering the phase separation processes one need to take into account corresponding fluctuations, which lead to memory effects in the system dynamics. Memory effects in generalized trans- port equations play a relevant role at high frequency or high speed of perturbations. The influence of the non-vanishing relaxation time of DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 31 the diffusion flux on the propagation of fast crystallization fronts has been studied [26, 27] in consistency with extended thermodynamics [21]. The memory effects play an important role in the propagation of phase interfaces during fast phase transitions [20]. Fluctuations for slow (i.e. internal energy, solute density etc.) and fast variables (e.g., heat flux and atomic diffusion flux) have been con- sidered frequently. The fluctuations of the heat flux and the viscous pressure were stressed for the first time by Landau and Lifshitz [28], who derived the expressions for their correlation. In Refs [29, 30], the role of rapid fluctuations of the heat flux as a stochastic source has been considered within the extended thermodynamic formalism. In these works, a unified description of slow and fast heat fluctuations has been made [31] for equilibrium and non-equilibrium steady states. The same idea about separation of slow and fast variables to study fluc- tuations in a system of particles with inertia has been realized within the supersymmetric path-integral representation [32]. Besides density fluctuations, we explore the fluctuations of the diffusion flux and in- vestigate their role in two different kinds of descriptions: (i) when the diffusion flux behaves as an independent fluctuating variable; (ii) the fluctuating part of the flux behaves as a stochastic noise in the evolu- tion equation for the density. To study the above-mentioned spatiotemporal phenomena in sto- chastic analysis, several analytical methods can be used. A linear sta- bility analysis allows us to set the stability of a homogeneous state with respect to small perturbations in systems with fluctuating sources [33]. A fundamental study of noise-induced phase transitions can be provided by means of dynamic renormalization group theory [34]. In analytical investigation of noise-induced phenomena, a mean field ap- proach is widely exploited (see Refs [35—38]). Despite the fact that the linear stability analysis can be used for a wide class of systems, the re- normalization group approach cannot be used directly for all models of stochastic dynamics. The mean field theory has several modifications for systems with non-conserved and conserved dynamics. Such ap- proach can be extended to a large number of stochastic systems to give a qualitative prediction of noise induced ordering and disordering phase transitions. The main idea of the present review is to synthesize the previous re- sults on hyperbolic model of spinodal decomposition and to analyze its predictions in comparison with outcomes of the parabolic model of Cahn and Hilliard. Formally, this review can be divided in two parts: the first one is devoted to study the hyperbolic model in the determi- nistic case, where we compare it with parabolic model for phase separa- tion; in the second part, we discuss properties of two above stochastic models. In Section 2, free energy functional leading to hyperbolic gov- erning equation for diffusion and phase separation is analyzed. Using 32 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV dispersion relation, the main propagative peculiarities, such as phase and group speeds, are presented. Critical wavelength and time for in- stability together with amplification rate for hyperbolic decomposition are derived. Peculiarities of evolution of patterns are analyzed in Sec- tion 3 by numerical solution of the hyperbolic and parabolic transport equations. Here, we discuss a method to define the structure factor and results obtained in comparison with the outcomes from the Cahn and Hilliard's theory. Section 4 deals with stochastic approaches related to study the phase separation in parabolic and hyperbolic models. Start- ing from a stochastic parabolic model with a concentration dependent mobility, we introduce internal fluctuations, obeying fluctuation dis- sipation relation with an intensity reduced to the bath temperature. We show that at late stages of the system evolution the domain size growth (Lifshitz—Slyozov) law can be generalized in this model. Study- ing the stationary case in the mean field approximation, we present results of re-entrant behaviour of the effective order parameter, when it takes nontrivial values inside a fixed interval of the system parame- ters, and prove analytical investigations by computer simulations. In order to discuss stochastic hyperbolic model, we start with hyperbolic transport investigations. After, we consider stochastic hyperbolic model for phase separation and compare results obtained for two above stochastic models. Finally, in Section 5, a summary for the results is proposed. 2. HYPERBOLIC MODEL FOR SPINODAL DECOMPOSITION In this Section, we introduce the hyperbolic model for spinodal de- composition. Starting from the hyperbolic transport equation, we analyze equilibrium fluctuations in the system described by two commensurable variables such as solute concentration and diffusion flux and discuss spectral properties of these fluctuations (Subsec- tion 2.1). In Subsection 2.2, we discuss the hyperbolic model for phase separation in the deterministic case. The detailed study of the model is presented in Subsections 2.3, where we obtain the disper- sion relation, group and phase speed, and perform the correspond- ing analysis, in Subsection 2.4 we discuss critical parameters for the hyperbolic model and analyze the structure function behaviour. 2.1. Hyperbolic Transport Let us consider an isothermal and isobaric binary system (both the temperature T and the pressure P are constants) consisting of atoms A and B . Following assumptions of Cahn [13], the system is repre- sented as an isotropic solid solution free from imperfections and with DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 33 the molar volume independent of concentration of A- and B-atoms. The system under study is described by the particle balance equa- tion: = , c t ∂ −∇ ⋅ ∂ J (1) where c is the particle density of a solute in a binary system, J– the diffusion flux, and t–the time. The diffusion flux is assumed to be described by the Maxwell—Cattaneo relaxation equation [20, 21, 26, 27], = ,D D c t ∂τ + − ∇ ∂ J J (2) where Dτ and D are the relaxation time and diffusion constant, re- spectively. The relaxation term is negligible for steady states or low-frequency perturbations. It becomes dominant at high frequen- cies or fast speed of propagation. Density Profiles. Combining Eqs (1) and (2), one gets the following equation of a hyperbolic type .= 2 2 2 cD t c t c D ∇ ∂ ∂+ ∂ ∂τ (3) Equation (3) predicts the propagation of the density profile with a sharp front moving with a finite speed DV inside the undisturbed system. To show this feature of hyperbolic transport, we find an analytical solution of Eq. (3) for the semi-infinite (one-dimensional) space by choosing the initial and boundary conditions in the form ( ,0) fc t c= , 0(0, ) ( , )c x c t x c= → ∞ = , (0, ) 0c x t∂ ∂ = (where x is a spatial coordinate). Under these conditions, the solution is described by the following expressions [39, 40]: behind the diffusion front, 0 < Dx tV≤ , 0 0 0 = / ( , ) = ( ) exp( / ) ( )( / ) ( , )d , t f a f a t x VD c t x c c c x l c c x l f t x t+ − − + − ∫ D D DD t t x V f t x I t x V ⎡ ⎤− τ − = ⎢ ⎥τ− ⎣ ⎦ 2 2 2 1/2 12 2 2 1/2 exp( / 2 ) ( / ) ( , ) 2( / ) ; (4) at the diffusion front, Dx tV= , f a f Dc t x c c c x l c c c t= + − − ≡ + − − τ0 0 0 0( , ) ( ) exp( / ) ( ) exp( / 2 ) ; (5) ahead of the diffusion front, DtV x< < ∞ , 34 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV 0( , ) .c t x c= (6) Here, 1/2( / )D DV D= τ and 1/22( )a Dl D= τ are the diffusion speed and the attenuation distance in the high-frequency limit [41], respectively, and 1I is a modified Bessel function of the first order. The concentration profiles described by Eqs (4)—(6) are shown in Fig. 1. In contrast with the concentration profiles described by the para- bolic differential equation (Fick’s diffusion), the concentration pro- files in the hyperbolic case have a sharp diffusion front which moves with the speed DV (Fig. 1). This diffusion front separates the spatial regions where diffusion occurs ( 0c c> at Dx V t< ; Eq. (4)) and where diffusion is absent ( 0c c= at Dx V t> ; Eq. (6)). Therefore, the position of the diffusion front may be examined as a depth, DtV , of density penetration into a binary system. As it is shown in Fig. 1, the ampli- tude of the diffusive front at Dx V t= decreases with increasing time and spatial coordinate, according to Eq. (5). 2.1.1. Equilibrium Fluctuations Even though solution (4)—(6) describes a smooth profile of density (with sharp diffusion front) fluctuations always exist in a thermody- Fig. 1. Profiles of density c at different moments 1 2 3< <t t t as predicted by solution (4)—(6). Every profile moves with the sharp discontinuity front, which has the diffusion speed DV . The x-coordinate of this discon- tinuity front is given by DtV , and the amplitude of the front is decreasing in time as − τexp[ (2 )]Dt . The density profile at 3 10 Dt t= ≥ τ is matched to those one described by a partial differential equation of a parabolic type (i.e., of the form of Eq. (3) with 0Dτ = ). DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 35 namic system. Indeed, techniques of light scattering or neutron scat- tering allow to explore details of the dynamics of density perturbations in the system, and it has fostered progress in nonequilibrium statisti- cal mechanics [42]. Following Ref. [40], we describe features of the system described by equations (1) and (2) related to c and J fluctua- tions in equilibrium and to stochastic noise. The equilibrium second moments of fluctuations of c and J are ob- tained from the Einstein’s equation for the probability of fluctuations [28, 43], namely 2 exp , 2 B s Pr k T ⎡ ⎤δ∝ ⎢ ⎥ ⎣ ⎦ (7) where entropy ( , )s c J is based on the independent thermodynamic vari- ables c and J. It is known (see, e.g., Ref. [43], Chapter 15) that Ein- stein’s equation (7) considered as an approximate Gaussian distribu- tion function predicts the second moments correctly, but it does not predict third and higher moments accurately. However, since we are only interested in the second moments, we restrict ourselves in this Section to the use of the simple Einstein formula (7). To obtain the second differential 2sδ of entropy in Einstein’s equa- tion (7), one needs to choose the form of the Gibbs equation for en- tropy. The generalized Gibbs equation, which incorporates slow and fast thermodynamic variables, is written as [21] τμ= − − ⋅1 ,Dds du dc d T T TD J J (8) where u is a density of internal energy, D is related to the usual diffu- sion coefficient D through D D c= ∂μ ∂ , and 1 2μ = μ − μ is the relative chemical potential of the solute with respect to the one of the solvent. We focus our attention on the fluctuations of c and J and assume du negligible for the sake of simplicity. Then, from Eq. (8), we get the sec- ond differential of the entropy as 2 2 21 ( ) ( ) .Ds c J T c TD τ∂μδ = − δ − δ ∂ (9) With the definition (7) and taking the second variation of s from Eq. (9), the probability of fluctuations is described by 2 2( , ) exp ( ) ( ) , 2 2 D B B vv Pr c J c J k T c k TD ⎡ ⎤τ∂μ⎛ ⎞δ δ ∝ − δ − δ⎢ ⎥⎜ ⎟∂⎝ ⎠⎢ ⎥⎣ ⎦ (10) where v is a small volume in which the fluctuations cδ and δJ occur. The second moments of fluctuations are given by 36 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV 2 2( ) , ( ) . ( / ) ( / ) B B B T D D T k T k TD k TD c J v c v v c 〈 δ 〉 = 〈 δ 〉 = = ∂μ ∂ τ τ ∂μ ∂ (11) In what follows, we discuss two important points: (i) the power spectra of the fluctuations of c and J, (ii) the description of the stochastic sources in the system (1) and (2). 2.1.2. Power Spectra of Density and Flux Fluctuations Let us define the correlation functions for the fluctuations of c and J in the following usual form ( , , , ) ( , ) ( , ) , ( , , ) ( , ) ( , ) , c J C t t c t c t C t t t t ′ ′ ′ ′≡ 〈δ δ 〉 ′ ′ ′ ′≡ 〈δ δ 〉 r r r r r, r J r J r (12) where r is the position vector of a point in the system. Since we con- sider equilibrium (homogeneous, time-invariant state), one has ( , , , ) ( , ), ( , , , ) ( , ), c c J J C t t C t t C t t C t t ′ ′ ′ ′= − − ′ ′ ′ ′= − − r r r r r r r r (13) i.e. the correlation functions depend only on relative distances rr ′− and on the difference in time t t′− . We are interested in the Fourier transforms of the quantities in Eq. (13), namely i t i c c i t i J J S e e C t d dt S e e C t d dt ω ω ω = ω = ∫ ∫ kr kr k r r k r r ( , ) ( , ) , ( , ) ( , ) . (14) These expressions represent fluctuation spectra and have special theo- retical and practical interest, as they may be measured by means of light scattering or neutron scattering techniques [42]. To obtain an explicit form of the fluctuation spectra we first write Fourier transform (in space) and Laplace transform (in time) of equa- tions (1) and (2). Using the standard procedure described in Refs [21, 42], we arrive at D S c S i J S c S J S J S i D c S J δ + δ = δ τ δ + δ + δ = δ k k k k k k k k k ( ) ( ) (0), ( ) ( ) ( ) (0), (15) where ( )c Sδ k and ( )J Sδ k are the Fourier−Laplace components of cδ and Jδ , respectively. Then, we have 2 ( ) (0)11 . ( ) (0)(1 ) D D c S cS i D J S Ji SS S Dk δ δ+ τ −⎛ ⎞ ⎛ ⎞⎡ ⎤ =⎜ ⎟ ⎜ ⎟⎢ ⎥δ δ−+ τ + ⎣ ⎦⎝ ⎠ ⎝ ⎠ k k k k k k (16) DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 37 In equilibrium state (where \| \| 0→k ), the crossed second moments, (0) (0)c J〈δ δ 〉k k , vanish because they have opposite time-reversal par- ity. Then, from Eq. (16), we have 2 2 2 2 1 ( ) (0) \| (0)\| , (1 ) ( ) (0) \| (0)\| . (1 ) D D D S c S c c S S Dk S J S J J S S Dk + τ 〈δ δ 〉 = 〈 δ 〉 + τ + 〈δ δ 〉 = 〈 δ 〉 + τ + k k k k k k (17) To obtain the time Fourier transform, one may write c J S k c S i c S k J S i J ω = = ℜ 〈δ = ω δ 〉 ω = = ℜ 〈δ = ω δ 〉 k k k k k k ( , \| \|) 2 [ ( ) (0) ], ( , \| \|) 2 [ ( ) (0) ]. (18) Finally, we obtain c D D J D D Dk S k c D k Dk S k J D k Dk ω = 〈 δ 〉 τ ω + − τ ω + ωω = 〈 δ 〉 τ ω + − τ ω + k k 2 2 2 4 2 2 2 2 2 2 2 4 2 2 2 2 2 ( , ) \| (0) \| , (1 2 ) ( ) 2 ( , ) \| (0) \| . (1 2 ) ( ) The corresponding expressions for 2\| (0)\|c〈 δ 〉k and 2\| (0)\|J〈 δ 〉k in equi- librium obtained from Eq. (11) are described by B B T D T k T k T c J v c v c 〈 δ 〉 = 〈 δ 〉 = ∂μ ∂ τ ∂μ ∂k k 2 2 2 \| (0)\| , \| (0)\| . ( / ) ( / ) (19) Note that, in Eq. (19), the function ( , )cS kω has a maximum at a frequency mω given by 1/22 2(2 1) (2 ) .m D DDk⎡ ⎤ω = τ − τ⎣ ⎦ (20) The fact that the maximum is at 0mω ≠ indicates propagation of density waves with the speed / mk ω , in contrast with the situation when the maximum is at 0mω = , which means purely diffusive transport. It is clear from Eq. (20) that, to observe such a maxi- mum, i.e., the propagation of density wave, it is needed that 1/2(2 )c Dk k D −> ≡ τ . Thus, for ck k< , transport is diffusive, and for ck k> , the den- sity waves may propagate. This analysis is analogous to the analysis of the transverse veloc- ity correlation function in generalized thermodynamics for the Maxwell viscoelastic model [42], which is consistent with the for- malism of extended irreversible thermodynamics [21]. 38 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV 2.2. Hyperbolic Spinodal Decomposition Let us consider a region of phase diagram, in which one phase mix- ture is unstable with respect to decomposition. This is a spinodal region where curvature of free energy is negative: 2 2 0, f c ∂ < ∂ (21) and the spinodal itself is defined by 2 2 0, f c ∂ = ∂ (22) where f is the Helmholtz free energy per unit volume and c is the concentration of B atoms. For a given temperature, the free energy f is based on the follow- ing variables: concentration c, gradient of concentration c∇ , and solute diffusion flux J. Dependence of free energy on concentration is due to existence of a diffuse interface between appearing phases in which high concentration gradients may exist. Dependence of free energy on diffusion flux reflects of the fact that decomposition may proceed with high rates comparable with the speed 1/2( )D DV D= τ of the front of solute diffusion profile, where D is the diffusion coefficient and Dτ the time for relaxation of the sol- ute diffusion flux to its steady-state value. Thus, the selected set of independent variables { , , }c c∇ J consists of slow conserved variable c, fast non-conserved variable J, and gradient variable c∇ . Analo- gous set of variables is generally analyzed within the context of ex- tended thermodynamics [44] and it is used for models of fast phase transformations [20]. Free Energy Density. Expanding the dependence of the free energy density on the concentration gradients and diffusion flux, one gets [45] 2 2 2 2 2 2 ( ) ( , , ) ( ,0,0) ... 2 ( ) ... . 2 c c f c f f c c f c c c c f J f J ∇ = ∇ = = = ⎛ ⎞∂ ∇ ∂⎛ ⎞∇ = + ∇ ⋅ + +⎜ ⎟⎜ ⎟∂∇ ∂ ∇⎝ ⎠ ⎝ ⎠ ⎛ ⎞∂ ∂⎛ ⎞+ ⋅ + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ 0 0 J 0 J 0 J J J (23) The following points regarding Eq. (23) can be accepted. First, we define ( ,0,0) ( )hf c f c= as the free energy density of a homogeneous system with no gradients and fluxes. Second, the term ( )c f c∇ ⋅ ∂ ∂ ∇ must be zero because the free energy of the system does not depend on the sign of the concentration gradient. Third, one can accept DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 39 that the derivative of the free energy with respect to the diffusion flux is linear by the flux: ( 2)Jf∂ ∂ = αJ J as approximation consis- tent with an extended thermodynamics [21]. And fourth, the next terms of expansion (23) can be omitted as a nonlinear terms in flux J. Therefore, Eq. (23) can be rewritten as 2 2( , , ) ( ) ( ) , 2 2 c J h r f c c f c c α ∇ = + ∇ + ⋅J J J (24) where 2 2 2( ( ) )c cr f c ∇ == ∂ ∂ ∇ 0 and coefficient Jα is a characteristic of non-Fickian diffusion which assumed to be [21] D T constTD c = τ ∂μ⎛ ⎞α = ⎜ ⎟∂⎝ ⎠ (25) with a difference μ of the chemical potentials for both chemical components. Within the limits of instant relaxation, i.e., 0Dτ → , the term with fluxes vanishes and Eq. (24) gives the free energy density ( , )f c c∇ of the standard (Ginzburg−Landau or Cahn−Hil- liard) form applicable for local equilibrium system. Interpretation of Free Energy for Local Nonequilibrium States. Going beyond local equilibrium requires re-examination in depth such basic and conceptually relevant concepts as entropy, tempera- ture, pressure or chemical potential under more general circum- stances [21, 46]. Therefore, free energy density (24) has to be inter- preted in terms of a local thermodynamic potential [47, 48]. Equation (24) defines thermodynamic potential with both local equilibrium contribution ( )hf c and purely local nonequilibrium con- tribution ( / 2)α ⋅J J (under spatial inhomogeneity defined by the gradient term). Hence, for the local equilibrium part ( )hf c a local ergodicity (i.e. the system needs to sample the phase space) is true. However, as soon as we postulated diffusion flux with a finite re- laxation time, this means that the local nonequilibrium contribution α ⋅J J reflects the existence of a slow physical process, which is the jump of solute atoms [40]. Considering ergodicity of a phase space for nonequilibrium situation, one may well refer to statistical ef- fects in fast spinodal decomposition due to existence of many parti- cles (atoms and molecules) within local volumes. Since the liquid demixing proceeds very fast, the particles have no time enough to sample all the phase space. Thus, the number of microstates acces- sible to each of them will be lower than in equilibrium. This will imply an increasing in the free energy with respect to the local equilibrium contribution ( )hf c . This is one of the ways to interpret the nonequilibrium contribution ( / 2)α ⋅J J to the free energy (24) that is the simplest conceivable way to express such increasing in the free energy. 40 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV From the phenomenological pint of view, a purely non-equilibrium contribution ( / 2)α ⋅J J to the free energy (or entropy) density for sys- tems under spinodal decomposition controlled by atomic diffusion is explained as a kinetic energy. Thermodynamic interpretation has been recently made [49] for this contribution in the framework of multi- component fluids and of dipolar systems having magnetic moments with non-vanishing inertia. A model, which takes this kinetic contri- bution, is called ‘hyperbolic’ model of spinodal decomposition, because it leads to the constitutive equation of hyperbolic type. In the limit of instantaneous relaxation, i.e. 0Dτ → , the term α ⋅J J vanishes and Eq. (24) gives the free energy density ( , )hf c c∇ of Cahn−Hilliard’s form [12, 13] applicable for local equilibrium system. Free Energy Functional. Taking Eq. (24), the total Helmholtz free en- ergy as a free energy functional is given by 2 2( , , ) ( ) ( ) , 2 2 c J hv r F c c f c c dV ⎡ ⎤α ∇ = + ∇ + ⋅⎢ ⎥ ⎣ ⎦ ∫J J J (26) where V is a sub-volume of the system. Evolution of ( , , )F c c∇ J with time t is described by ex in , dF dF dF dt dt dt ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (27) where ex( )dF dt is the external exchange of the free energy and in( )dF dt is the internal change of the free energy inside the system. The latter is defined as a dissipative function. Using the procedure de- scribed in Refs [19, 20] and applied to Eq. (26), one can obtain ( )2 2 2 ex ( ) ,c n c c n dF c r c f r c J d dt t ′ ∂⎛ ⎞ ⎡ ⎤= ∇ + − + ∇ Ω⎜ ⎟ ⎢ ⎥∂⎝ ⎠ ⎣ ⎦∫ (28) ( )2 2 in ,c c n J dF f r c dV dt t ∂⎛ ⎞ ⎡ ⎤′= ⋅ ∇ − ∇ + α⎜ ⎟ ⎢ ⎥∂⎝ ⎠ ⎣ ⎦∫ J J (29) where Ω is the outer surface of sub-volume V, nJ is the diffusion flux pointed by the normal vector n , and /c hf f c′ = ∂ ∂ . As it follows from Eq. (29), the dissipative function includes the term J tα ∂ ∂J , which has a clear physical meaning: far from equilibrium, the diffusion flux provides additional ordering that is leading to increasing of the dissi- pation. Around a steady state, dissipative function (29) must decrease in time, so that the free energy of the entire system is decreasing. This condition implies a relation between fluxes and forces, which is, in the simplest case, assumed to be linear [21]. For Equation (29), it gives the following evolution equation for the diffusion flux DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 41 ( )2 2 ,c c JM f r c M t ∂′= − ∇ − ∇ − α ∂ J J (30) where M is the atomic mobility. Together with the atomic mass balance , c t ∂ = −∇ ⋅ ∂ J (31) Eq. (30) leads to the following governing equation ( )2 2 2 2 ,D c c c c M f r c t t ∂ ∂ ⎡ ⎤′τ + = ∇ ⋅ ∇ − ∇⎢ ⎥⎣ ⎦∂ ∂ (32) which is the same that it has been previously derived from the entropy functional [19, 20]. Equation (32) is a general partial differential equation of a hyperbolic type with the decomposition delay described by the term 2 2/D c tτ ∂ ∂ . It allows for describe both diffusion mecha- nism and wave propagation of chemical components. A natural boundary condition, originating from external exchange of the free energy (28), is given by ( )2 2 2 0,c n c c n c c f c J t ∂ ′ε ∇ − − ε ∇ = ∂ (33) where nJ and nc∇ are the projections of the diffusion flux and ‘nabla’- operator, respectively, on the normal vector to the boundary of the volume 0υ . Equation (33) represents a dynamical boundary condition, which shows that the product ( ) nc t c∂ ∂ ∇ should be balanced with the product c nJμ on the boundary of the subvolume V. From this, in par- ticular, it follows that if the concentration is fixed, constc = , then the flux is absent, 0nJ = , on the boundary Ω . In the standard parabolic situation described by the Cahn—Hilliard equation ( 0Dτ → ), one has proportionality between the flux and concentration gradient, n nJ c∝ ∇ , and they both can be cancelled from Eq. (33). In this case, equation (33) transforms, with some scaling constant, into the known boundary condition analyzed by Miranville and Zelik (see Eqs (2) and (1.2) from Refs [50, 51], respectively). Hence, Eq. (32), endowed with a dynamic boundary condition (33), is a general partial differential equation of hyperbolic type with the decomposition delay described by the inertial term 2 2/D c tτ ∂ ∂ . Mathematically, the problem of Cahn— Hilliard equation with the term 2 2/D c tτ ∂ ∂ , endowed with proper boundary conditions, has been studied in one-, two-, and three- dimensions [23—25] to establish existence of the global and exponential attractors for different phase spaces. Because we focus on the analysis of the initial stages of decomposi- tion described by Eq. (32) (i.e., when the large concentration gradients exist and short periods of time are important) one may neglect all 42 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV terms not linear in c. This one yields 2 2 2 4 2 ,D cc c c c Mf c Mr c t t ∂ ∂ ′′τ + = ∇ − ∇ ∂ ∂ (34) where 2 2 cc hf f c′′ = ∂ ∂ . As 0Dτ → , Eq. (34) transfers into the classic Cahn—Hilliard equation [12, 13]. In the present form, Eq. (34) can be considered as a modified Cahn—Hilliard equation, which is a linearized partial differential equation of a hyperbolic type. This equation is true for spinodal decomposition with local nonequilibrium diffusion (diffu- sion with relaxation of the solute flux). Such type of decomposition is expected for short periods of time, large characteristic velocities of process, large concentration gradients, or under deep supercoolings. 2.3. Dispersion Relation and Speeds for Atomic Diffusion Main characteristics of diffusion can be found from dispersion analy- sis of the linearized hyperbolic Cahn—Hilliard equation (34). These are the phase speed that characterizes propagation of a single (selected) harmonic, the group speed, which is characteristic of a wave packet, critical wavelength for decomposition, and critical time for instability, which both characterize developing coherent structure in decomposi- tion [52]. We consider the elementary exponential solution of Eq. (34) in the following form 0( , ) exp[ ( ( ) )],kc z t c a i kz k t− = − ω (35) where the dispersion relation ( )kω is given by 1/2 2 2 2 2 ( ) 1 ( ) . 2 4 cc c D D D Mk f r ki k ⎛ ⎞′′ + ⎜ ⎟ω = − ± − ⎜ ⎟τ τ τ⎝ ⎠ (36) The upper and lower signs for ( )kω in Eq. (36) correspond to the branches, which are responsible for the wave propagation in the posi- tive and negative z-directions, respectively. Qualitative behaviour for ( )kω is shown in Fig. 2. It can be seen that the real part of ω begins to exist only from some critical value, 0k k= (Fig. 2, a). This value defines confluence of two branches for imaginary part of ω (Fig. 2, b). In addition, one can de- fine other two critical values for the wave-vector k. The critical value ck k= defines a point from which ω takes positive values of its imagi- nary part (Fig. 2, b). For ck k> , solution (35) exponentially grows in time and decomposition begins to proceed irreversibly. The critical value mk k= gives a maximal positive value for ω (Fig. 2, b). Fre- DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 43 quency ( )mkω defines the mostly unstable mode with which pattern evolves during phase decomposition. Within the local equilibrium limit 0Dτ → , Eq. (36) arrives to the following approximation ( )2 2 2( ) 1 1 2 ( ) . 2 D cc c D i k Mk f r k⎡ ⎤′′ω ≈ − ± − τ +⎢ ⎥⎣ ⎦τ (37) Equation (37) shows that one of the roots is going to −∞ along imaginary axis by the law ( ) Dk iω ∝ τ . This leads to exponential de- cay of the solution (35). The second root of Eq. (37) is finite and it is equivalent to classic Cahn—Hilliard relation 2 2 2( ) ( ).cc ck iMk f r k′′ω ≈ − + (38) Thus, local equilibrium limit for dispersion relation (36) gives two different roots: the first one is diverges and the second one ap- proaches dispersion relation (38) of Cahn and Hilliard. Phase Speed. The values of the wave vector 0k above which relation (36) has the real part, Fig. 2, a, is found from condition 2 2 2 0 2 1 ( ) . 2 c cc cc c D r k f f r M ⎛ ⎞ ′′ ′′⎜ ⎟= + − ⎜ ⎟τ⎝ ⎠ (39) a b Fig. 2. Dispersion relations for hyperbolic Cahn—Hilliard equation; Eq. (36). (a) Real part of frequency, ( ( )).kℜ ω (b) Imaginary part of frequency, ( ( )).kℑ ω 44 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV For 2 2 0k k> and real values of ( )kω , one can obtain from Eq. (36) the phase speed: ( )1/2 1 2 2 2( ) / ( ) ( ) (2 )p D D cc ck M f r k k− −′′υ = ℜ ω ℜ = τ τ + − , (40) which may propagate in both positive and negative spatial directions. The speed pυ incorporates a motion for one of separated single har- monics. It can be compared with the predictions of the partial differen- tial equation of a hyperbolic type for solute diffusion without phase separation. Indeed, analysis of dispersion relation for mass transport equation 2 2 2/ /D c t c t D cτ ∂ ∂ + ∂ ∂ = ∇ of a hyperbolic type leads to the following expression [53]: 1/2 2 2 1/2 2 . ( )p D D c D − ⎛ ⎞ υ = ⎜ ⎟τ + τ + ω⎝ ⎠ (41) Taking into account that 0cr = , for the zero spatial atomic correla- tion, ccMf D′′ = is the diffusion coefficient in Eq. (40), we use the rela- tion k ∝ ω for high frequency of disturbances’ propagation. Then, both expressions (40) and (41) lead to the same result p D DD Vυ = τ = ω → ∞1/2( / ) with . (42) In Equation (42), the phase speed pυ is equal to solute diffusion speed DV , which is a maximal speed for propagation of the solute diffusion disturbance (profile). Imaginary part of the phase speed, ( ) (2 )p Di kℑ υ = − τ , specifies the amplification rate for a given harmonic. With 0k k< , harmonics do not move with possible changing of their own amplitudes. For both real and imaginary parts of pυ (with 0k k> ), the harmonics move and change their own amplitudes. The behaviour is shown in Fig. 3 for ( )p kυ . Group Speed. Concentration disturbances propagating by diffusion can be considered as an undistorted wave packet moving with the group speed given by ( ) ( ). k W k k ∂ω = ± ∂ (43) Using Eq. (36), calculation of the group speed W gives 2 2 2 2 2 1/2 2 ( 2 ) ( ) . (4 ( ) 1) cc c D cc c kM f r k W k k M f r k ′′ + = ′′τ + − (44) Dependence ( )W k is shown in Fig. 3. It specifies a speed for concentra- tion profiles envelope. One may see, as for the phase speed pυ , the real values for W given by Eq. (44) exist only at 0k k> . In contrast with the DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 45 behaviour of pυ , the imaginary part of ( )W k may exist only at 0k k< . a b Fig. 3. Phase and group speeds for hyperbolic Cahn—Hilliard equation. (a) Real part ( )pvℜ of phase speed (solid line) and real part ( )Wℜ of group speed (dashed-dotted line). (b) Imaginary part ( )pvℑ of phase speed (solid line), and imaginary part ( )Wℑ of group speed (dashed-dotted line). TABLE 1. Predictions for characteristic speeds of diffusion. Equation Phase speed pυ Group speed W Parabolic diffusion equation ikD− with 2ik Dω = − 2ikD− Hyperbolic diffusion equation 24 1 2 D D i Dk k − ± τ − τ 2 2 4 1D kD Dkτ − Parabolic Cahn— Hilliard equation 2 2( )h ciMk f r k′′− + 2 22 ( 2 )h ciMk f r k′′− + Hyperbolic Cahn— Hilliard equation 2 2 21 1 4 ( ) 2 D h c D i Mk f r k k ⎛ ⎞′′− ± − τ +⎜ ⎟τ ⎝ ⎠ 2 2 2 2 2 2 ( 2 ) 4 ( ) 1 h c D h c kM f r k k M f r k ′′ + ′′τ + − 46 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV Analysis of standard parabolic and hyperbolic diffusion equations [53] as well as parabolic and hyperbolic equations for spinodal de- composition [52], presented in this Subsection for diffusion speeds, leads to comparison of both approximations summarized in Table 1. 2.4. Critical Parameters for Hyperbolic Decomposition 2.4.1. Critical Wavelength for Decomposition Cahn [13] has found a critical wavelength cλ , above which infinitesi- mal sinusoidal fluctuation of concentration is irreversibly grown. Par- ticularly, he confirmed the concept of Hillert [54] that cλ → ∞ with approaching the spinodal, at which one has 2 2/ 0f c∂ ∂ = . To find the critical wavelength for decomposition under local non- equilibrium diffusion, we expand ( )hf c in Eq. (26) about some concen- tration 0c that is 0 0 2 2 0 0 0 2 ( ) ( ) ( ) ( ) ... 2 h h h c c c c c cdf d f f c f c c c dc dc= = ⎛ ⎞−⎛ ⎞= + − + +⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠ . (45) The composition is represented along the z-axis by a series of sinusoi- dal waves with components of the following form 0 cos( ),c zc c a k z− = (46) where ca is the amplitude and zk –the frequency of concentration wave. Substituting Eq. (46) into Eq. (45), we perform integration of the functional (26) over the volume υ . Then, for the difference of the Helmholtz free energy, 0( , , ) ( )hF F c c f c dVΔ = ∇ − ∫J , between a system with concentration (46) and a homogeneous system, respectively, one gets: 2 2 2 . 4 c cc c z aF f r k V Δ ⎡ ⎤′′= + ⎣ ⎦ (47) For the reasonable cases of the positive surface tension, 2 0cr > , one can consider two important points. First, with 0ccf ′′ > the solution is stable against fluctuation of con- centration of any wavelength: the free energy only increases in this case, 0FΔ > . Second, with 0ccf ′′ < the solution is unstable with respect to the critical wavelength for decomposition, which can merely be found by taking the zero value for the square bracket in Eq. (47): DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 47 1/2 22 2 .c c c cc r k f ⎛ ⎞π ⎜ ⎟λ = = π ⎜ ⎟′′−⎝ ⎠ (48) with the critical value for wave vector given by 2 1/2( ) .c cc ck f r′′= − (49) Therefore, with 2 2 0hd f dc < and for cλ > λ , the free energy de- creases, 0FΔ < , and decomposition starts to proceed. Equation (49) clearly shows that as the composition tends to the values lying in the spinodal, 2 2 0f c∂ ∂ = , the critical wavelength approaches to in- finity, cλ → ∞ [13, 54]. 2.4.2. Amplification Rate of Decomposition Consider a real part of the solution (35) in the following form: 0 cos( ) exp( ) cos( ) exp( ).c c a kz t a kz t+ + − −− = ω + ω (50) In this solution, signs ‘plus’ and ‘minus’ correspond to growing or decaying solutions, respectively, in time. Substitution of Eq. (35) into Eq. (34) defines a real part of the frequency as follows ( )( )1/2 1 2 2 2(2 ) 1 1 4 .D D cc ck M f r k− ± ⎡ ⎤′′ω = τ − ± − τ +⎢ ⎥ ⎣ ⎦ (51) After expanding, the square root in Eq. (51) for 24 [D cck M f ′′τ + 2 2] 1cr k+ ≤ one gets in the local equilibrium limit the expression: ( )2 2 2 0 ,lim cc c D k M f r k+ τ → ′′ω = + (52) which is the kinetic amplification rate obtained by Cahn [13] for purely diffusion regime. Therefore, Eq. (51) can be interpreted as the kinetic amplification rate for both dissipative and propagative regimes of atomic transport described by Eq. (34). From the amplification rate +ω of decomposition, the maximum can be obtained by differentiation of Eq. (51) with respect to zk . The extremum condition, / 0zk+∂ω ∂ = , gives maximum frequency ( )1/2 1 2( ) (2 ) 1 1 /m m D D cc ck Mf r− ⎡ ⎤′′ω = τ − + + τ⎢ ⎥⎣ ⎦ (53) at ( )1/2 2/ (2 ) with < 0.m cc c cck f r f′′ ′′= − (54) 48 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV From Eq. (54), there follows that maximum wavelength, 2 /m mkλ = π , is equal to 1/2 22 2 .c m cc r f ⎛ ⎞ ⎜ ⎟λ = π ⎜ ⎟′′−⎝ ⎠ (55) Consequently, maximum amplification rate allows for the wavelength (55) greater in exactly 2 times the critical wavelength (49) of insta- bility against fluctuations of concentration. This result coincides with Cahn and Hilliard’s results for purely diffusion regime. 2.4.3. Critical Time for Instability Let us evaluate the time of transitive period from the beginning of instability (with the beginning of growth of infinitesimal perturba- tion) up to the arriving into the new metastable state. For the fast- est growth of infinitesimal perturbation, the maximal frequency ( )mkω is responsible. Therefore, substitution of Eq. (48) into disper- sion relation (36) leads to [52] ( )1/2 2 2( ) 1 1 ( ) / . 2m m D cc c D i k M f r ⎡ ⎤′′ω = − ± + τ⎢ ⎥τ ⎣ ⎦ (56) Equation (56) adopts both real and imaginary parts for ω . Using maximal frequency (56), solution (35) can be rewritten as 0( , ) exp( ) exp( ),k cc z t c a ikz t t− = (57) where 2 2 1/2 2 (1 ( ) / ) 1 D c D cc c t M f r τ = ′′+ τ − (58) is the time for developing coherent structure. Within the local equilibrium limit, 0Dτ → , we expand square root in Eq. (58) for 2 2( ) / 1D cc cM f r′′τ = . One gets the following approxima- tion 2 2 4 , ( ) c c cc r t M f ≈ ′′ (59) which can be found from the predictions of pure diffusion theory (parabolic transport equation) of Cahn and Hilliard. As a result, com- parative analysis for parabolic and hyperbolic equations in spinodal decomposition is given in Table 2 for dispersion relations, critical wavelengths and times for instability. DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 49 2.4.4. Analysis of a Structure Function Different evolution of spinodally-decomposed systems exhibits differ- ent structures (see Fig. 10). Experimentally, a typical structure after spinodal decomposition is observed as random, interconnected patterns with a characteristic length-scale related to maximal amplification rate of decomposition [4]. Experimental observations using scattering show a broad Bragg-like peaks, from which information about quenched structure during spinodal decomposition and decomposition rate can be read off. For such measurements, a main characteristic for the inten- sity of scattering is a structure factor. Therefore, the structure factor can be taken as a parameter for characterization of analyzed evolutions and for verification of the model predictions with experimental data. Consider the structure factor ( , )S tk , which describes the intensity of quasi-elastic scattering observed at time t after the quenching from the initial temperature iT up to the final temperature fT . The function ( , )S tk can be interpreted as the respective correlation function of the concentration fluctuations, and it is defined as ( , ) ( , ) ( , ) Tf S t c t c t= 〈δ − δ 〉k k k . (60) To obtain expression for the time dependent structure factor (60), TABLE 2. Predictions of parabolic and hyperbolic models. Expression for Parabolic Cahn−Hilliard equation ( 0)Dτ → Hyperbolic Cahn−Hilliard equation Dispersion relation, ( )kω ( )2 2 2 cc ciMk f r k′′− + 1/2 2 2 2 2 ( ) 1 2 4 cc c D D D Mk f r ki ⎛ ⎞′′ + ⎜ ⎟− ± − ⎜ ⎟τ τ τ⎝ ⎠ Critical wave- length, 2 /c ckλ = π ( )1/2 22 ( )c ccr f ′′π − [13] ( )1/2 22 ( )c ccr f ′′π − Amplification rate, +ω ( )2 2 2 cc ck M f r k′′ + [13] ( )( )1/2 2 2 2 2 1 4 1 D D cc ck M f r k τ ′′− τ + − Maximal wave- length, 2 /m mkλ = π ( )1/2 22 2 ( )c ccr f ′′π − [13] ( )1/2 22 2 ( )c ccr f ′′π − Critical time for instability, ct ( )2 24 ( )c ccr M f ′′ [52] ( )1/2 2 2 2 1 ( ) 1 D D cc cM f r τ ′′+ τ − 50 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV we linearize Eq. (34) in terms of concentration inhomogeneity 0( , ) ( , )c t c t cδ = −r r . This one yields 2 2 2 4 2 ( ) ( ) ( ) ( )D cc c c c Mf c Mr c tt ∂ δ ∂ δ ′′τ + = ∇ δ − ∇ δ ∂∂ . (61) Following the approach to the concentration fluctuations for the hyperbolic transport [40], Fourier transforms of ( , )c tδ −k and ( , )c tδ k is expressed as ( , ) exp( ) ( , ), ( , ) exp( ) ( , ). c t d i c t c t d i c t δ − = ⋅ δ − δ = ⋅ δ ∫ ∫ k r k r r k r k r r (62) Then, Eq. (61) for the structure factor (60) is given by ( )2 2 2 2 2 ( , ) ( , ) ( , ).D cc c d S t dS t Mk f r k S t dt dt ′′τ + = − +k k k (63) To solve this equation, we multiply LHS and RHS by exp( )i tω , where ω is a frequency of the concentration inhomogeneity. After some algebra, solution of Eq. (63) is given by 0 2 2 2 2 ( ,0) / (1 ) ( ,0) exp( ) ( , ) ( ) D D D cc c dS dt i S i t S t dt i Mk f r k ∞ τ + + τ ω − ω = ′′ω − τ ω + +∫ k k k . (64) Defining the spectral distribution of fluctuations as 0 ( , ) 2 exp( ) ( , ) ,S i t S t dt ∞ ω = ℜ − ω∫k k (65) one can find from Eq. (64) the spectral distribution for concentra- tion fluctuations. Equations (64) and (65) might describe modelled structure for hyperbolic scenario (with finite Dτ ) and parabolic scenario (for in- stant relaxation with 0Dτ → ) and experimentally observed struc- ture after quenching in spinodal decomposition. Such a description can give information about length scale of concentration fluctua- tions and, as a consequence, about maximal amplification rate of decomposition for given scenario. 2.5. Comparison with Experimental Data The function of the amplification rate predicted by the hyperbolic model has been compared in Refs [48, 55] with experimental data of Andreev et al. on phase-separated glasses [56, 57]. The amplifica- tion rate (51) can be rewritten in the following form DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 51 1/2 1/22 2 2 2 2 2 21 4 (1 ) 1 1 4 (1 ) 1 , 2 2 D c cc D C D D D k r k f l k l k + ⎡ ⎤′′+ τ − − ⎡ ⎤+ − −⎣ ⎦ ⎣ ⎦ω = = τ τ (66) where /C c ccl r f ′′= − is the correlation length, 1/2( )D Dl D= τ is the diffusion length, and ccD Mf ′′= − is the diffusion constant. In addition, one can assume that the free energy of a binary sys- tem can be replaced by 2 4 0 0( , ) ( / 1)( ) ( ) ,h c c cf T c f T T c c B c c⎡ ⎤= − − + −⎣ ⎦ (67) where cT and cc are the critical temperature and concentration, re- spectively, cT T< , and 0 0B > . Equation (67) is often used in analy- sis of kinetics of spinodal decomposition in glasses [58], and the pa- rameters 0f and 0B are treated as phenomenological input parame- ters of the theory, which are fitted to experiment [10]. Figure 4 shows data for the relationship ‘ 2/ k+ω versus 2k ’ ex- tracted from experiments on a binary phase-separated glass [56, 57]. They exhibit non-linear behaviour as predicted by Eqs (66) and (67) for the following material parameters: 142.3 10D −= ⋅ cm2/s, 117.2 10D −τ = ⋅ s, 86.2 10cr −= ⋅ cm⋅ 3/ (mole cm )J ⋅ , / 0.85cT T = , 0 0.15B = , 4 0 1.88 10f = ⋅ J/(mole ⋅ cm3), and 0.8cc c− = mole frac- Fig. 4. Dependence 2/ k+ω upon 2k given by the hyperbolic model (solid line; Eqs (66) and (67)) and scattering data of visible light (points, Refs [56, 57]). Experimental points were obtained on phase-separated SiO2−12 wt.% Na2O glass at T = 803 K. 52 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV tion. In Figure 4, it is shown that good agreement is achieved between theory and experiment. This result is due to the fact that both lengths, correlation length Cl and diffusion length Dl , appear in the theory [48, 55] that is also shown by Eq. (66). The interplay be- tween these two lengths, i.e. the ratio /D Cl l , governs the transition as follows (in contrast with the linear Cahn−Hilliard−Cook model [12, 59] in which only correlation length Cl is important in spinodal decomposition). With the increase of the correlation length Cl in comparison with the diffusion length Cl , spinodal decomposition has the Cahn−Hilliard’s scenario (described by linear or non-linear para- bolic diffusion equation). With D Cl l≈ , one can accept long-range interaction within the system and the Cahn−Hilliard’s scenario takes effect. With D Cl l>> (namely, with 2 2D Cl l≥ [52]), short- range interaction has effect and local nonequilibrium effect (such as relaxation of the diffusion flux to its steady state) plays dominant role in selection of the mode for decomposition. Thus, existence of these two length, existence of these two length, Dl and Cl , makes the theory flexible enough to predict non-linear behaviour for am- plification rate typically observed in experiments and to quantita- tively describe experiments (Fig. 4). 3. MODELLING OF SPINODAL DECOMPOSITION In this Section, we present main numeric procedures to be used in simulations of the spinodal decomposition in deterministic models of the hyperbolic type. Numerical approaches related for 1D and 3D simulations are presented in Subsections 3.1 and 3.2, respectively. Features of hyperbolic spinodal decomposition can be observed in computational dynamics. To model decomposition, Eqs (26), (30), and (31) are taken. From this system, the following dimensionless form of equations is as fol- lows 2 2 ,c c D rF f c c l′ ⎛ ⎞δ = − + ∇⎜ ⎟δ ⎝ ⎠ (68) , F t c ∂ δ⎛ ⎞+ = ∇ ⎜ ⎟∂ δ⎝ ⎠ J J (69) c t ∂ = −∇ ⋅ ∂ J . (70) In these equations, the following scales are introduced: / ccM D f ′′= is the mobility, /D Dl D V= the characteristic spatial length, Dτ the DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 53 time scale as the diffusion relaxation time, and DV the solute diffu- sion speed as the scale for diffusion flux. To complete this system, the free energy density is chosen as a double-well potential 2 2 0( ) (1 ) 4,hf c f c c= − (71) which has minima at 0c = and 1c = , and it has a maximum at 0.5c = . Boundary conditions are established from the previously obtained expression for the external exchange ex( / )dF dt of the free energy: expression in square brackets of Eq. (28) must be zero. This condi- tion implies the following equalities 0nc∇ = and 0nJ = on the boundary of the calculated domain. As a result, we arrive to a set of hyperbolic equations (68)−(71), which describes evolution by the hyperbolic model described in previous Section (in comparison with the parabolic model of Cahn and Hilliard using diffusion equation of a parabolic type). 3.1. 1D Modelling For integration of equations (68)−(71), we use implicit Euler method of second order 2( )O τ . Let us introduce the following notations 1 1 2 2 , , n nt t t t c J p s t t= + τ = + τ ∂ ∂= = ∂ ∂ (72) where τ is the time step. Then one can define the system of equa- tions for the time iterations in the following form + + += + τ = + τ ≡ = + τ%( 1) ( ) ( 1/2) ( ) ( 1/2) ( )1 1 , , . 2 2 n n n n n nc c p J J s c c c p (73) Now, Eqs (68)−(71) can be rewritten as ⎧ ∂ ⎛ ⎞+ + τ =⎪ ⎜ ⎟∂ ⎝ ⎠⎪ ⎪ ∂⎛ ⎞⎪ + τ + = −⎜ ⎟⎨ ∂⎝ ⎠⎪ ⎪ ⎛ ⎞ ⎛ ⎞∂ τ ∂′⎪ = − −⎜ ⎟ ⎜ ⎟∂ ∂⎪ ⎝ ⎠ ⎝ ⎠⎩ % % ( ) ( ) 2 22 ( ) 2 2 2 1 0, 2 1 1 ( ) , 2 , 2 n n n c c c D D p J s x s J M c W x r rc p W f l x l x (74) where /c Dr l is the ratio of correlation parameter cr and diffusion length /D Dl D V= . Excluding s from (74), one can obtain the final sys- tem of equations. This one yields 54 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV ⎧ ⎛ ⎞ ⎛ ⎞τ ∂ ∂′⎪ + = −⎜ ⎟ ⎜ ⎟∂ ∂⎪ ⎝ ⎠ ⎝ ⎠⎨ ⎪ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞− + =⎜ ⎟ ⎜ ⎟⎪∂ ∂ τ τ ∂⎝ ⎠ ⎝ ⎠⎩ % % 2 22 2 ( ) 2 2 ( ) , 2 2 2 ( ) 1 . n c c c D D n r rp c W f l x l x J M c W p x x x (75) Equations (75) represent an elliptic set of equations. This system al- lows us to numerically solve Eqs (68)−(71) relatively the functions p and W using the following algorithm. Taking initial data for concen- tration 0 0( ,0) ( )k kc x c x c= = and diffusion flux 0( ,0) 0 kJ x J= = , vari- ables p and W are obtained from the system (75). Then, new data for the concentration and the flux are found from the system (73) for the new time level. These are used for obtaining p and W from Eqs (75). This procedure is iterated in time to compute concentrations and dif- fusion fluxes in spinodal decomposition. Note that the system (75) can be also used for solution of the Cahn and Hilliard’s parabolic equation for spinodal decomposition. This procedure has to assume the steady-state diffusion flux, i.e. the flux needs infinite time for its time changing. Therefore, let τ → ∞ in the second equation of Eqs (75). This excludes from the numerical proce- dure the time dependence of the flux and the second order derivative of concentration with respect to time. In addition, the second equation in Eq. (73) has to be divided on τ with the further taking the same limit τ → ∞ . This leads to equality 0s J t= ∂ ∂ = that allows us to exclude the time dependence of flux from the suggested numeric algorithm. Choosing a finite difference method and using approximation of the second order for coordinate x , elliptic system (75) can be tested against its computational stability. A linearized transfer-matrix ( , )T k n ) is obtained for deviations cδ and Jδ from exact solutions of Eqs (75). This one yields ( 1) ( ) ( 1) ( ) ( , ) , n n n n c c T k n J J + + ⎛ ⎞ ⎛ ⎞δ δ =⎜ ⎟ ⎜ ⎟ δ δ⎝ ⎠ ⎝ ⎠ (76) where ( )( ) 2 2 0 2 2 2 1 1 1 1 1 2 2 2 ( , ) , 1 1 1 1 1 1 3 2 4 ML k i k T k n N iMLk MLk ⎛ ⎞⎛ ⎞ ⎛ ⎞− τ + τ − τ + τ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎜ ⎟= ⎜ ⎟⎛ ⎞⎜ ⎟− τ + τ − τ + + τ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ (77) where k is the wave vector, and factors 0N and L are obtained as ( ) ( )1 12 2 0 1 2 4 1 2 ,N MLk − −= + τ + τ + τ DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 55 and 2( , ) ( / ) .cc c DL n k f r l k′′= +% % The stability condition requires that, for given wave vector k, the eigenvalues for transport matrix (77) have to be not greater than unity. Qualitative dependence of modulus of eigen- values for transfer-matrix ( , )T k n ) from the wave vector k is present in Fig. 5. This dependence is shown at different values of parameters of the time step τ and relation 0 /c Dr lα = between the correlation pa- rameter cr and diffusion length /D Dl D V= . One can see that numeri- cal scheme is conditionally stable. The condition of stability is formed mainly by a largest possible value of wave vector and weakly depends on other parameters, particularly, from the time step τ . Dynamics of spinodal decomposition is presented in Figs. 6 and 7 for the material and computational parameters summarized in Table 3. The dynamics is shown in spatial changing of concentration profiles for a given time step (Figs. 6, 7, a—d). After formation of quasi- sinusoidal profile from an initially random distribution, this distribu- tion becomes unstable in further separation due to up-hill diffusion between decomposing phases (Fig. 6, a—b). This unstable situation evolves much more faster for the system described by Cahn−Hilliard equation than for the local nonequilibrium system described by hyper- bolic equation (Fig. 6, b—c). It occurs, generally, due to propagation of concentration disturbance with infinite diffusion speed in the Cahn−Hilliard’s system. Hyperbolic system has a delay described by the second time derivative in Eqs (32) and (34). As a result, concentra- tion disturbance in the hyperbolic system propagates with the finite speed and the instability realizes with the delay relatively to the Cahn−Hilliard’s system. Fig. 5. Typical dependence of modulus of eigenvalues for transfer-matrix ( , )T k n ) on dimensionless wave vector k with the scale of 2 Dlπ . Region of com- putational stability lies below unity for the eigenvalues of transfer matrix. Dependence is shown at: (a) different values 0 /c Dr lα = that is the ratio of cor- relation parameter cr and diffusion length /D Dl D V= , and (b) different τ . 56 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV Finally, the new stable (or metastable) profile is formed which is the same for both parabolic Cahn−Hilliard’s system and hyperbolic system. Fig. 6. Dynamics of spinodal decomposition of a binary liquid in 1D case with the constant mobility 0 constM M= = : (a) metastable state; (b) beginning of the transition from unstable to metastable state; (c) finishing of the transition from unstable to metastable state; (d) new metastable state. The first metasta- ble state (a) and the following metastable state (d) are equivalent for both clas- sic and modified Cahn−Hilliard equations. The dynamics of transition between two metastable states is much more faster for classic Cahn−Hilliard equation. Fig. 7. Dynamics of spinodal decomposition of a binary liquid in 1D case with the concentration dependence of mobility 0 (1 )M M c c= − : (a) metastable state; (b) beginning of the transition from unstable to metastable state; (c) transient period for the transition from unstable to metastable state; (d) new metastable state. The first metastable state (a) and the following metastable state (d) are equivalent for both classic and modified Cahn−Hilliard equa- tions. The dynamics of transition between two metastable states is much more faster for classic Cahn−Hilliard equation. DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 57 The described dynamical scenario qualitatively does not depend on the atomic mobility of decomposing phases. Comparative dynamics shown in Figs. 6 and 7 for constant mobility and concentration depend- ent mobility, respectively, demonstrates that the both dynamics quali- tatively remain the same. The only difference in dynamics shown in Figs. 6 and 7 is that the transition from the one metastable state to the next metastable state proceeds during various periods of time. Conse- quently, the Cahn and Hilliard model predicts much more faster dy- namics of spinodal decomposition, which can be resulted in much more diffuse boundaries between two separated phases. 3.2. 3D Modelling For 3D numeric solution of Eqs (68)−(71), a cube with linear size N is taken. The cube is approximated by the numerical grid with equal dis- tances xΔ between nodes along Cartesian axes. In such a case, coordi- nates of the nodes are given as x i x= Δ , y i x= Δ , and z i x= Δ , where 1,...,i N= , 1,...,j N= , and 1,...,k N= , respectively. A random distribution around average concentration 0c within the cube is accepted for initial time step 0n = . Then for every time step t n= τ , the following explicit numerical scheme is used: 0 (1 )(1 2 ) 2 n n n n ijk ijk ijk ijk fF c c c c δ⎛ ⎞ = − − − +⎜ ⎟δ⎝ ⎠ TABLE 3. Parameters for a binary system used in numeric computations. Parameter Value and unit for 1D modelling Value and unit for 3D modelling Initial concentration, 0c 0.5 atomic fraction 0.5 atomic fraction Height of the free energy, 0f 0.5 0.5 Time for diffusion relaxation, Dτ 111.5 10−⋅ s 111.5 10−⋅ s Diffusion coefficient, D 95.0 10−⋅ m 2/s 95.0 10−⋅ m 2/s Bulk diffusion speed, 1/2( / )D DV D= τ 18.26 m/s 18.26 m/s Spatial diffusion length, /D Dl D V= 90.27 10−⋅ m 90.27 10−⋅ m Ratio /c Dr l 0.90 0.29 Quantity of computational nodes, N 80 3500 Dimensionless spatial step, xΔ 0.56 0.88 Dimensionless time step, τ 25.13 10−⋅ 35.0 10−⋅ 58 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV 2 1 1 1 1 1 1( 6 ),n n n n n n nc i jk i jk ij k ij k ijk ijk ijk D r c c c c c c c l x + − + − + − ⎛ ⎞ + + + + + + −⎜ ⎟Δ⎝ ⎠ (78) 1 1 1 (1 ) ( 2 n n n n ijk ijk i jk i jk F F J J x c c + + − τ δ δ⎛ ⎞ ⎛ ⎞= − τ + − +⎜ ⎟ ⎜ ⎟Δ δ δ⎝ ⎠ ⎝ ⎠ 1 1 1 1 , n n n n ij k ij k ijk ijk F F F F c c c c+ − + − δ δ δ δ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟δ δ δ δ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (79) 1 1 1 1 1 1 1( ). 2 n n n n n n n n ijk ijk i jk i jk ij k ij k ijk ijkc c J J J J J J x + + − + − + − τ= − − + − + − Δ (80) A set of hyperbolic equations (78)—(80) allows to model decomposi- tion with conserved function of concentration c. It has been resolved numerically with material and computational parameters summarized in Table 3. To test conservation of c, average concentration of a whole Fig. 8. 3D modelling of spinodal decomposition in undercooled binary liquid: (a) 0t = , (b) 150t = τ , (c) 31.5 10t = ⋅ τ , (d) 41.5 10t = ⋅ τ . For every time mo- ment, isoconcentration patterns within the cube with size of 3500 computa- tional nodes is shown. DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 59 computational domain has been compared with the initial value c0. An error of computations 0 0\| \| / 100%c c c− ⋅ was found not higher than 3.8%. That confirms correctness of the present system (78)—(80) to model the spinodal decomposition with conserved concentration c within computational domain. Figure 8 shows evolution of concentration inside the cube. It is seen that initially random parts of distribution with equal concentration (Fig. 8, a) create isoconcentration surface (Fig. 8, b—d] during decom- position. Figure 9 presents snapshots of patterns evolving in local non- equilibrium spinodal decomposition. This sequence exhibits hyperbolic evolution with a sharp boundary between two types of decomposed liq- uid especially at the first moments of decomposition (see Fig. 9, b). The sharp boundary between two liquids follows from the fact that descrip- tion of diffusion in system (78)—(80) is given by hyperbolic type of Fig. 9. Evolution of spinodal decomposition: (a) 0t = , (b) 50t = τ , (c) 310t = τ , (d) 43 10t = ⋅ τ . For every time moment, cross section for a cube with the size of 3500 computational nodes is shown. 60 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV equation (69). Together with conservation law (70), Eq. (69) predicts with a finite speed and, as a consequence, with sharp diffusion fronts between two separating phases. To compare patterns originating in spinodal decomposition, results for both hyperbolic evolution and parabolic evolution were extracted from solution of Eqs (78)—(80). Predictions of complete system (78)— (80) were taken as for hyperbolic evolution. Predictions of the system (78)—(80) without relaxation of the solute diffusion flux, i.e., with the Fick’s diffusion flux , F c δ⎛ ⎞= ∇ ⎜ ⎟δ⎝ ⎠ J (81) and its numerical approximation 1 1 1 1 1 1 1 ( 2 , n n n ijk i jk i jk n n n n ij k ij k ijk ijk F F J x c c F F F F c c c c + − + − + − δ δ⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟Δ δ δ⎝ ⎠ ⎝ ⎠ δ δ δ δ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟δ δ δ δ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (82) instead of Eqs (69) and (79), respectively, were taken as for para- bolic evolution. Results of modelling for patterns in both evolutions are shown in Fig. 10. Evolution of patterns has been spied upon the complete sepa- Fig. 10. Comparison of patterns in spinodal decomposition described by (a—d) hyperbolic equation and (e—h) parabolic equation. Here, (a, e) 10t = τ , (b, f) 40t = τ , (c, g) 22 10t = ⋅ τ , (d, h) 52 10t = ⋅ τ . For every time moment, patterns are shown for a small cube of 350 computational nodes extracted from central part of the cube with the size of 3500 computational nodes. DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 61 ration of liquids (see Fig. 10, d and h). It is seen that the boundaries between two demixing liquid phases are sharper for hyperbolic evolu- tion than for the parabolic evolution. This feature of the hyperbolic evolution is a result of solute diffusion with the finite speed that leads to instability realized with the delay relatively to the Cahn—Hilliard’s system (see comparative dynamics in Figs. 6 and 7). 4. STOCHASTIC MODELS OF SPINODAL DECOMPOSITION In this Section, we consider to above models for spinodal decomposi- tion and illustrate in what a manner concentration fluctuations, con- sidered as internal noise, obeying fluctuation—dissipation relation can affect on the system dynamics and stationary states. It is known that contrary to the naive predictions, assuming that fluctuations lead to disordering effects, phenomena such as noise- induced transitions in zero-dimensional systems [60—64] (when a sto- chastic variable x is a function of the time t only, ( )x x t= ), stochastic resonance [65, 66], noise-induced ordered and disordered phase transi- tions in extended systems [67—70] (when ( , )x x t= r ), noise-induced pat- tern formation processes [71, 72], noise-induced effects in excitable sys- tems [73], and a lot of others are manifestations of the constructive role of fluctuating environment. An increasing interest in the noise-induced phenomena in extended systems results in the discovery of new nonequi- librium universality classes [74, 75] and new types of self-organization processes such as entropy driven phase transitions [76, 77]. In most problems of noise-induced phenomena in extended systems external fluctuating sources are considered; their primary role in self- organizational processes is stated [67, 78]. Recently, a new type of en- tropy driven phase transitions was discovered [76]. Within this type of transitions, it was shown that internal fluctuations with intensity re- lated to a field-dependent kinetic coefficient (mobility) play a principle role in ordering dynamics. Particularly, it was found that the internal multiplicative noise leads to the effective entropy dependence on the stochastic quantity in a functional form but does not change the corre- sponding free energy functional. As a result, noise-induced effects can be understood with a help of the entropy mechanism, which follows from the thermodynamics. Considering parabolic and hyperbolic mod- els for spinodal decomposition within the frame of the linear stability analysis and the mean field theory we compare behaviour of above two models. The aim of this Section is to perform a somewhat detailed study of entropy driven phase transitions mechanisms in phase separation processes. We analyze early and late stages of evolution numerically. For the stationary picture, we extend the mean field approach to the systems with the field dependent mobility and investigate mechanisms 62 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV of entropy driven phase transitions in binary stochastic systems. We generalize the well-known results for early stages of evolution, study late stages and consider re-entrant ordering effects in a stationary case. Our analytical results are compared with computer simulations. The outline of this Section is the following. In Subsection 4.1, we introduce the general stochastic system with conserved dynamics and obtain the corresponding Fokker−Planck equation that can be used in the mean field analysis for both parabolic and hyperbolic models. Con- sidering parabolic model, we discuss effects of the internal multiplica- tive noise influence at early and late stages of spinodal decomposition and consider phase transitions by means of the mean field theory. In Subsection 4.1.4, we discuss effect of a combined effect of both inter- nal and external stochastic sources. In Subsection 4.2, we study sto- chasticity of the hyperbolic transport. Subsection 4.3 is devoted to study of stochastic hyperbolic model for spinodal decomposition. 4.1. Stochastic Parabolic Model for Spinodal Decomposition In this subsection, we investigate an influence of the conserved-field (concentration) dependent mobility and the corresponding internal noise on properties of a phase separation scenario in the parabolic model. Formally, in a case of a binary system with concentrations Aρ and Bρ of the components A and B, respectively, the density difference A Bx = ρ − ρ can be introduced. In a phase separation scenario, the quantity x obeys a conservation law ( , )d constx t =∫ r r . In our study, we use the field x to describe the system under consideration and investi- gate a corresponding dynamics and a stationary picture. To define a principle model let us start with a continuity equation for the field ( , )x tr in a d-dimensional space in the form ,x t ∂ = −∇ ⋅ ∂ J (83) where J is the flux. The deterministic part of the flux is of the form det x M x δ= − ∇ δ J [ ]F . (84) Here, M is the mobility, the Helmholtz coarse-grained free energy functional F is of the standard (Ginzburg−Landau or Cahn−Hilliard) form applicable for a local equilibrium system 2[ ] ( ) ( ) , 2 D x dV f x x ⎧ ⎫= + ∇⎨ ⎬ ⎩ ⎭∫F (85) DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 63 where D is the spatial coupling intensity related to the correlation ra- dius in the form 2 cD r= with 2 2 2 0( / ( ) ) \|c xr x ∇ == δ δ ∇F ; ( )f x is the free energy density of the bulk. The mobility we assume in a functional form, i.e. ( )M M x= . To investigate the influence of the flux fluctuations, let us intro- duce fluctuation source ζ into the right hand side of Eq. (84). It yields an expression for the flux in the form as follows: det ( ; , ).x t= + ζJ J r (86) Formally, the stochastic part ζ is assumed to be Gaussian, and gener- ally, it can be a function of the field x. Assuming the x-dependent mo- bility ( )M M x= and using fluctuation dissipation relation, one gets the following definition for the averages 2( ; , ) 0, ( ; , ) ( ; , ) 2 ( ) ( ; ).x t x t x t M x C t t′ ′ ′ ′〈ζ 〉 = 〈ζ ζ 〉 = σ − −r r r r r (87) In the simplest case, we assume the correlation function C in the form of ( ; ) ( ) ( )C t t t t′ ′ ′ ′− − → δ − δ −r r r r , which allows to consider the white noise in space and time. Here, we note that the fluctuation dissipation relation holds that yields an interpretation of the noise intensity 2σ as an effective temperature of the bath. In further study of stochastic dynamics, we consider a general case when control parameters of the system and the noise intensity 2σ are independent quantities. Using conditions corresponding to an equilibrium situation, one gets the flux J as follows [ ] / ( ) ( , )M x x g x t= − ∇δ δ + ξJ rF , where ( ) ( )g x M x= , ( , ) 0t〈ξ 〉 =r , 2( , ) ( , ) 2 ( ) ( )t t t t′ ′ ′ ′〈ξ ξ 〉 = σ δ − δ −r r r r . Sub- stituting the generalized flux into Eq. (83), we get the stochastic con- tinuity equation in the form [ ] ( ) ( ) ( , ). x x M x g x t t x ∂ δ⎛ ⎞= ∇ ⋅ ∇ + ∇ ξ⎜ ⎟∂ δ⎝ ⎠ r F (88) To study statistical properties of the system one needs to find the probability density ([ ], )x t=P P . To this end, we represent the system on a regular d-dimension lattice with a mesh size l . Then, the partial differential equation (88) is reduced to a set of usual differential equa- tions written for an every cell i on a grid in the form ( ) ( ) ( ) ( ),i L ij j R jl L ij j j l dx F M g t dt x ∂= ∇ ∇ + ∇ ξ ∂ (89) where index i labels cells: 1, , di N= K ; the discrete left and right op- erators are introduced ( 2d = ) as follows: 64 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV , 1, 1, , , 1 , , 12 1 1 ( ) ( ), ( ) ( ), 1 ( ) ( ) , ( ) ( ) ( 2 ), L ij i j i j R ij i j i j L ij R ji L ij R jl il i l i l i l − + + − ∇ = δ − δ ∇ = δ − δ ∇ = − ∇ ∇ ∇ = Δ = δ − δ + δ l l l (90) where ijδ is the Kronecker symbol. For stochastic sources, the discrete correlator is of the form 2 2( ) ( ) 2 ( )i j ijt t t t− ′〈ξ ξ 〉 = σ δ δ −l . In the following analysis, we use the Stratonovich interpretation of the Langevin equa- tions (89). Next, to obtain above distribution let us introduce standard defini- tions =1 ([ ], ) ( ( ) ) ( ) , dN i i i x t x t x t= 〈 δ − 〉 ≡ 〈ρ 〉∏P (91) where K and 〈 〉K are averages over initial conditions and noise, re- spectively. To obtain the corresponding Fokker−Planck equation, we use the standard technique and exploit the stochastic Liouville equa- tion ( ).i i i x t x ∂ ∂ρ = − ρ ∂ ∂∑ & (92) Inserting the expression for the time derivative from Eq. (89) and av- eraging over noise, we get ( ) ( ) ( ) ( ) .L ij j R jl L ij j j i iji l i F M g t t x x x ⎛ ⎞∂ ∂ ∂ ∂ 〈ρ〉 = − ∇ ∇ 〈ρ〉 − ∇ 〈 ξ ρ〉⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ∑ ∑ (93) The correlator in the second term can be calculated by means of No- vikov’s theorem that gives [79] δ ρ ′ ′〈 ξ ρ〉 = σ δ δ − 〈 〉 ′δξ∑ ∫2 0 ( ) ( ) 2 ( ) , ( ) t j j j jk k k g t g t dt t t t (94) where for the last multiplier one has ( ) ( ) ( ). ( ) ( ) j l j k l k t t g t x t g t t x t ′= δ ρ δ∂= − ρ ′ ′δξ ∂ δξ (95) In a formal solution of the Langevin equation, the response function takes the form ( ) ( ) ( ) l L lk k k t t x t g t ′= δ = ∇ ′δξ . (96) DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 65 After some algebra, we obtain the Fokker−Planck equation for the total distribution P in the discrete space 2( ) ( ) ( ) ( ) ,L ij j R jl L ij j R ji i i iji l i j F M g g t x x x x ⎛ ⎞∂ ∂ ∂ ∂ ∂= − ∇ ∇ − σ ∇ ∇⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ∑ ∑P P P (97) where relations between left and right gradient operators are used. The one-point probability density is defined as follows: ( ) ([ ], )i mm i P t x t dx ≠ = ∏∫ P . The above equation can be rewritten in the standard form (in our case, it has only formal form) for the functional ({ ( )}, )x t= rP P in the con- tinuum space as follows: 2 2 2 2 ( ) ( ) 2 ( ) ( ) . ( ) M d M t x x x d M x ⎡ ⎤∂ δ δ σ δ⎛ ⎞= − ∇ ∇ − ∇ ∇ −⎢ ⎥⎜ ⎟∂ δ δ δ⎝ ⎠⎣ ⎦ δ−σ ∇ ∇ δ ∫ ∫ r r r r r r F P P P (98) The term proportional to 2σ in the first addendum is the noise-induced drift. The derived Fokker−Planck equation allows us to write down the corresponding Langevin equation 2 ( , ) ( , ), 2 W M x t M g t t x x ∂ δ σ δ⎛ ⎞= ∇ ∇ − ∇ ∇ + ∇ ξ⎜ ⎟∂ δ δ⎝ ⎠ r r F (99) with a process Wξ , which has a strong mathematical definition through the Wiener process ( )W t : ( ) ( ) /W t dW t dtξ = , 2( ) :dW dt [60]. The stationary solution of the Fokker−Planck equation takes the form 2 2 1 [ ] exp [ ] ln ( ) . 2 x x d M x ⎧ ⎫⎛ ⎞σ⎪ ⎪∝ − +⎨ ⎬⎜ ⎟σ⎪ ⎪⎝ ⎠⎩ ⎭ ∫ rP F (100) Exploiting standard thermodynamic definition of the effective inter- nal energy eff ef effT= +U F S and assuming a quasi-Gibbs form for the stationary distribution, we can identify the effective entropy eff (1 / 2) ln ( )d M x= ∫ rS and the effective temperature 2 efT = σ . It is principally important that the stationary distribution is exact and is described not only by the initial functional [ ]xF . Here, the en- tropy contribution modifies the form of the probability density. De- spite, we consider the internal multiplicative noise, its action leads to 66 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV the entropy variability. Thus, controlling the noise intensity or parame- ters in the mobility ( )M x we can govern an ordering process. Therefore, the system studied below is considered as a nonequilibrium one. The same situation is observed in the entropy driven phase transitions in systems with nonconserved dynamics [76, 80−82], where the stationary distribution is obtained exactly and phase transitions picture is con- trolled by the effective entropy variations. Below, we relate the entropy driven phase transitions formalism developed for systems with noncon- served dynamics to the systems with conserved dynamics. The main at- tention will be paid on effects where the entropy variability is principle. Using Langevin or Fokker−Planck equation, one can derive equation for the first statistical moment directly: 2 ( , ) . ( ) 2 ( ) M x t M t x x ∂ δ σ δ 〈 〉 = ∇〈 ∇ 〉 − ∇〈∇ 〉 ∂ δ δ r r r F (101) This equation can be used to analyze the influence of the internal multiplicative noise on the stability of the null phase in the linear ap- proximation. Since the dynamics is conserved, V ( , ) constx t d =∫ r r , V is the system volume), so dynamics of the phase separation can be considered with a help of the Fourier transform of correlation function 2( , ) (1 / ) ( , ) ( , ) ( , )G t V x t x t d x t′ ′ ′= 〈 + 〉 − 〈 〉∫r r r r r r . The corresponding structure function is given as V ( , ) ( , ) iS t G t e d= ∫ krk r r . In practice, it is convenient to use a spherical average of the correla- tion and structure functions, which are as follows: ( , ) ( , ) r g r t G t d Ω = Ω∫ r , ( , ) ( , ) k S k t S t d Ω = Ω∫ k . Here, rΩ and kΩ are spherical shells of radius r and k , respectively. The above values allow us to extract a mean characteristic size of do- mains at time t, ( )R t , using scaling relations: ( , ) ( / ( ))g r t r R t= ϕ , ( , ) ( ) ( ( ))dS k t R t kR t= ϕ . The expected domains growth law is ( ) zR t t∝ , where z is the domain-growth exponent. The Model. In our consideration, we use a model for a binary system, which is described by the free-energy density, ( )f x . Single-phase equi- librium ( )f x has a stable single-well structure. In a two-phase region, ( )f x is of a double-well structure; the corresponding model has the form DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 67 2 4( ) 2 4,f x x x= − ε + (102) where ε is a dimensionless phenomenological constant, playing the role of a control parameter. The mobility is used in the functional form 2 1( ) (1 ) , 0.M x x −= + α α ≥ (103) Variations in the parameter α allow us to consider additive ( 0α = ) or multiplicative ( 0α ≠ ) noises, separately (the difference in such a clas- sification is reduced to the following: if the noise term appears in the evolution equation with a constant multiplier, constg = , then the noise is additive, else, ( )g g x= , it is multiplicative). The model func- tion (103) assumes that fluctuations are large in the case where 0x = , whereas fluctuations are small in cases where 0x ≠ . Formally, assum- ing α to be small, an approximate definition is 2( ) 1M x x≈ − α . As previous studies show, the quantity α can be expressed through the relation between bulk bD and surface sD diffusion constants, i.e. 1 /b sD Dα ≈ − (see Ref. [83]). Further, we are looking for changes in system behaviour when α ranges. In such kind of stochastic models, a possible scenario of phase sepa- ration depends on the initial conditions: at ( ,0) 0x〈 〉 =r , the system evolves by spinodal decomposition scenario (see Fig. 11, a), whereas at ( ,0) 0x〈 〉 ≠r , a nucleation process is realized. (Fig. 11, b). a b Fig. 11. Typical spatial patterns: (a) spinodal decomposition ( ,0) 0,x〈 〉 =r (b) nucleation ( ,0) 0.2x〈 〉 =r . Other parameters are as follows: 120 120,N N× = × 4,D = 1,ε = 0.5,α = 2 0.2.σ = 68 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV 4.1.1. An Early Stage of Evolution At the first, we investigate the internal multiplicative noise influence on instability of the homogeneous phase 0x〈 〉 = . Using Fourier trans- formation for the scalar field ( ) (2 ) ( , )d ix t d x t e− −= π ∫ kr k r r in d-dimensional space, in linear approximation one gets ( )2 2 2 . d x k Dk x dt 〈 〉 = − − ε + ασ 〈 〉k k (104) It is principally important that the noise contribution denoted as 2 xασ 〈 〉k stabilizes the homogeneous state. The same result was ob- served in the case of entropy-driven phase transitions with noncon- served dynamics [76, 81]: as it follows, one can await the similar be- haviour of the stochastic systems with conserved dynamics where the entropy driven phase transition formalism can be generalized. More information of the system behaviour provides the knowledge of the structure function ( , ) ( ) ( )S t x t x t−= 〈 〉k kk . Following the standard approach, a linear evolution equation for the spherically averaged structure function can be derived in the form [84] 2 2 2 2 2 2 2( , ) 1 ( ) ( , ) 2 2 ( , ). (2 )d dS k t k Dk S k t k k d S q t dt = − − ε + ασ + σ − ασ π ∫ q (105) It is seen at 0α = that corresponds to the additive noise case, one ar- rives at the well-known Cahn−Hilliard−Cook equation for the structure function [12, 59]. From exponential solutions of Eqs (104, 105), one can see that only modes with 2( ) /ck k D< = ε − ασ are unstable and grow at early stages of evolution. With an increase in α or 2σ , the size of the unstable modes domain ck k< decreases. Modes with ck k> re- main stable during the linear regime. Note that unstable modes cannot be realized at condition 2ε < ασ . As it follows, the domain growth should be different for additive and multiplicative noise. In Figure 12, we present solutions of the evolution equation (105) at different values of the parameter α . It can be seen that an increase in α leads to a shift of the peak position toward smaller values of k . The peak of ( )S k is less pronounced in the multiplicative noise case than in the case of the additive noise. It follows that, if the multiplicative noise is considered, then the dynamic is slowed. A decrease in the peak height means that an interface is more diffuse in the case of multiplicative noise (see insertions in Fig. 12). We compare analytical results with computer simulations at the same time t on the two-dimensional lattice. In the insertions, a typical patterns and images of spherically averaged DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 69 structure functions are shown. It is seen that in the multiplicative noise case the pattern has more diffuse interface and the resonance ring in S(k)-dependence is less pronounced than at the additive noise. 4.1.2. A Late Stage To get more information about internal multiplicative noise influence, we simulate the system behaviour and study influence of the parameter α at the late stage of the evolution. All simulations were done in two- dimensional lattice with periodic boundary conditions with 120N = . A criterion for the phase separation in the model under consideration is the growth of the averaged second moment of x in the real space. We use the standard definition of the corresponding order parameter 2 2 2 2 =1 ( ) ( ) . N i i M t N x t−= 〈 〉∑ (106) An alternative formula is 2( ) ( )kk M t S t= ∑ . In phase separation scenario during the long time evolution, the quantity 2( )M t grows to the sta- tionary value 2M when the system tends to the nonzero stationary state. Fig. 12. Evolution of the structure function at early stage ( 10t = ) at 4D = , 1ε = , 2 0.3σ = . Different values of the parameter α are used to compare in- fluence of additive 0α = and multiplicative 0.9α = noises (solid and dashed lines, respectively). Insertion shows typical patterns and corresponded images of spherically averaged structure functions at the same time obtained from numerical solution of Eq. (99) at 3 0x = with the initial condition ( ,0) 0x〈 〉 =r . 70 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV Figure 13 displays the evolution of the order parameter 2M at different values of the parameter α and stationary values 2M versus α . It is seen that an increase in α delays slightly the evolution of 2M (Fig. 13, a) at small times and suppresses the stationary values (see Fig. 13, b). Despite the fact that the quantity 2M represents an integral effect, more information about the system behaviour can be found in the structure function ( )S k . A convenient quantity is the spherically av- eraged structure function defined on a circle as follows: a b Fig. 13. Order parameter evolution (a) and its stationary values (b) at dif- ferent values of α and 1.0ε = , 4D = , 2 0.2σ = . DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 71 1 ( , ) ( , ). k k kk S k t S t N ≤ ≤ +Δ = ∑ k k (107) In Figure 14, we display the evolution of the structure function for two values of the parameter α at fixed noise intensity 2σ . Comparing Fig. 14, a, b, one can see that, in the case under consideration, the peaks of the structure function are less pronounced when α increases that cor- responds to the case of the field-dependent mobility case studied in Ref. [85] and relates to the linear stability analysis. As follows from Figs. 14, a, b, the positions of the peaks are the same with an increase in α at equal times. It follows from the linear stability analysis and corresponds to the fact that in the case of multiplicative noise the dy- namics is slowed [85]. This result is different from the deterministic case where at large α peaks are located at higher values of k . The typi- cal behaviour of the Fourier images in Fig. 14, c, shows the diffuse in- terface between two phases and change in the peaks position. As Figure a b c Fig. 14. Evolution of the structure function at ε = 1.0, D = 4.0, σ2 = 0.2: (a) α = 0.01, (b) α = 0.85. The times represented are t = 250, t = 1000, t = 3000. (c) Fourier images of structure function evolution at α = 0.85 and times t = 250, t = 1000, t = 3000. 72 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV 15 shows, the change in peaks height means that with an increase in α the interface becomes more diffuse. To investigate the domain growth dynamics we use the standard formulas for a relevant length: k k R t k k S k t kdk S k t dk−= 〈 〉 〈 〉 = ∫ ∫max max1 0 0 ( ) , ( , ) ( , ) . (108) The power law behaviour of the function ( ) zR t t∝ is verified at differ- ent values of the parameter α , where the domain growth exponent de- pends on α , i.e. ( )z z= α (see Fig. 16). It is seen that, in the case of ad- ditive noise ( 0α = ), the exponent 1 / 3z ∝ , whereas at 1.0α = , we ob- tain 1 / 4z ∝ . Therefore, with an increase in α a crossover of dynami- cal regimes is observed. Our results are in good correspondence with deterministic and stochastic approaches, which indicate that an in- crease in the parameter α delays the dynamics [86−88]. 4.1.3. Stationary Case To investigate the steady states, we can use an extension of the mean field theory developed for the systems with conserved dynamics [37]. In the framework of this theory, one can use thermodynamic supposi- tions for the deterministic dynamics and after apply it to the stochastic one. Fig. 15. Structure function behaviour at different values of the parameter α at 1.0ε = , 2 0.2σ = , 4.0D = , 3000t = . DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 73 At first, let us define transition and critical points [89]. Considering the deterministic case, we use the model tx M x∂ = ∇ ⋅ ∇ δ δF , where the restriction 0 ( , ) V x d x t= ∫ r r is taken into account, 0x is fixed by the initial conditions. For such a system, the transition point is 0( )T xε : at 0( )T xε < ε the homogeneous state 0x is stable; at 0( )T xε > ε the system separates in bulk phases, 1x a b Fig. 16. Power law for domains size growth: (a) log−log plot of the evolu- tion of ( )R t at different values of the parameter α (insertion shows uni- versal behaviour of the function ( )R t at large times indicated in the rec- tangle); (b) dependence of the power law exponent z versus parameter α . Other parameters are as follows: 1.0ε = , 4.0D = , 2 0.2σ = . 74 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV and 2x , with 0x x〈 〉 = . The transition point coincides with critical one for 0 0x = only, i.e. (0)T cε = ε . The corresponding steady state solutions are given as solutions of the equation 0M x∇ ⋅ ∇ δ δ =F . If no flux condition is applied, then stationary solutions can be obtained from the equation 2 0x∇ δ δ =F , due to the mobility M does not affect on the number and extremes po- sitions of the functional F ; the mobility leads to the change in the dy- namics of the phase transition only. Hence, the bounded solution is x hδ δ =F , where h is a constant effective field of the system, in equi- librium systems h is a chemical potential. In the homogeneous case, the value h does depend on the initial conditions 0x . Above the transi- tion point, the steady state is not globally homogeneous, here the sys- tem separates into two bulk phases with values 1x and 2x . The fraction u of the system can be defined by the lever rule: 1 2 0(1 )ux u x x+ − = . In the case of the symmetric form of the free energy functional where two phases with 1 2x x= − are realized, we get 0h = [37]. Hence, if the field h becomes trivial, then the transition point can be defined. Using the above assumptions, let us move to the stochastic case, fol- lowing prescription [37]. For the one-point probability density, ( ) ([ ], )i mm i P t x t dx ≠ = ∏∫ P , the standard definition of the nearest-neighbours average ( ) ([ ], ) 2 ( )j m ij nn i m i x t x dx d x P t ∈ ≠ = 〈 〉∑ ∏∫ P can be applied. It allows to rewrite Eq. (125) in a more useful form ( ) ( ),i ij j i ji P t M P t t x ∂ ∂= − Δ 〈 〉 ∂ ∂ ∑ % (109) where 2 2 . 2j j j j F M M M M x x x ∂ σ ∂ ∂= − + σ ∂ ∂ ∂ % (110) With no flux condition, the average jM〈 〉% satisfies the equation ( ) 0.ij j s i j M P xΔ 〈 〉 =∑ % (111) Taking i j= , dropping subscripts and using results of the determinis- tic analysis with M h〈 〉 =% , we obtain the mean-field stationary equa- tion 2 2( ) ( ) 2 ( ) ( ), 2s s V M hP x M x dD x x M P x x x x ⎛ ⎞∂ σ ∂ ∂⎡ ⎤= − 〈 〉 − − + σ⎜ ⎟⎢ ⎥∂ ∂ ∂⎣ ⎦⎝ ⎠ (112) DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 75 where we have used mean-field approximation of the Laplacian ( ) ( ) 2 2 ( ),ij j nn i i j nn i x x dx d x x ⎛ ⎞ Δ ≡ − → 〈 〉 −⎜ ⎟ ⎝ ⎠ ∑ ∑ (113) the mean-field value x〈 〉 should be defined self-consistently. A solu- tion of Eq. (112) takes the form 2 2 2 1 ( , , ) exp ( ) ( ) 2 ln ( ) . 2 ( ) s D P x x h N f x x x dx M x h M x ′⎛ ⎡〈 〉 = − + 〈 〉 − +⎜ ⎢σ ⎣⎝ ⎞′ ⎤σ+ − ⎟⎥ ⎟′ ⎦ ⎠ ∫ (114) where 2D dD′ = , next we drop the prime. In order to determine the unknown quantities h and x〈 〉 , we re- call that considered mean field approach is local and expresses sP of a field at a given site of the lattice as a function of the field h and of the mean field 〉〈x in a neighbourhood of the given cell. In the homogeneous case (below the threshold), the mean field is the same everywhere and equals the initial value, i.e. 0x x〈 〉 = . Hence, at the fixed mean-field value, solving the self-consistency equation, sx xP x x h dx〈 〉 = 〈 〉∫ ( , , ) , (115) we obtain the constant effective field h . Above the threshold, the system is separated into two phases with equality 1 2x x〈 〉 = −〈 〉 , and h must be the same for these two phases and must be zero. Hence, above the threshold only x〈 〉 should be defined by solving the self- consistency equation with ( , ,0)sP x x〈 〉 . The values of the constant effective field h are obtained as solu- tions of the self-consistency equation with initial concentration 0 0.2x = and shown in Fig. 17. As seen from Fig. 17, a, the field h decreases monotonically with an increase in the control parameter ε . If h becomes trivial, we get the transition point Tε . After this point, 0h = , and the mean-field value x〈 〉 can be calculated self- consistently. It is clearly seen that the internal multiplicative noise shifts the transition point toward negative values of the control pa- rameter ε . With an increase in D , the same circumstance is observed. The last effect is well defined: an increase in the correlation scale cr or spatial coupling intensity D induces the ordering behaviour in the system. The former is the combined effect of the nonlinearity of the system, multiplicative character of the noise and the spatial coupling. The noise induced effects are well seen in dependence 2( )h σ , shown in Fig. 17, b. Here, at positive values of the control parameter ε (solid 76 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV line), the effective field h increases from the zero value at 2 2 Tσ > σ . Here one has a stable homogeneous phase 0x x〈 〉 ≠ . In the case of nega- tive values of the control parameter (dashed line), the constant effec- tive field h becomes zero inside the domain of the noise intensities 2 2 2 1 2[ , ]T Tσ ∈ σ σ . The value h decreases till the first threshold 2 1Tσ , above the second one 2 2Tσ it increases monotonically. From the formal viewpoint, the corresponding mean field value x〈 〉 obtained as a solu- tion of the self-consistency equation (115) should be nontrivial inside the domain 2 2 2 1 2[ , ]T Tσ ∈ σ σ where the phase separation with 1 2x x〈 〉 = −〈 〉 occurs. Let us discuss the mean field x〈 〉 behaviour. Here, we solve the self- consistency equation, setting 0h = . As Figure 18, a, shows the mean field value changes, its value critically from zero if the parameter ε increases. The critical point cε is defined as a bifurcation point when a b Fig. 17. Constant effective field h versus control parameter ε (a) and noise intensity (b) at fixed initial value 0 0.2x = . Other parameters are shown in legends. DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 77 nontrivial values 1 2x x〈 〉 = −〈 〉 appear. The transition point Tε corre- sponds to the case when 0x x〈 〉 = . With an increase in both the noise intensity 2σ and the parameter α , the critical point cε is shifted to- ward negative values. The dependence of the mean field value versus noise intensity is shown in Fig. 18, b. Here, one can see re-entrant phase transitions at negative values of the control parameter at large spatial coupling intensity. With an increase in ε , the first threshold 2 1cσ is shifted toward small values whereas the second one 2 2cσ becomes larger. Transition points 2 1Tσ and 2 2Tσ are related to the condition 0x x〈 〉 = . With an increase in the noise intensity at 0ε > , the disorder- ing phase transition is observed. The above calculations of the effective field h and the mean field value x〈 〉 allow us to obtain the corresponding phase diagrams. If the initial condition 0 0x ≠ is fixed, then one can obtain the transition lines (dash-dotted lines) which correspond to values of the control pa- a b c d Fig. 18. Mean field value obtained as solution of Eq. (115) at 0h = : (a) dependence x〈 〉 versus ε at different values of the spatial coupling inten- sity, noise intensity and the parameter α ; (b) dependence x〈 〉 versus 2σ at 10D = and 0.8α = and different values of the control parameter ε . 78 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV rameter Tε , the noise intensity 2 Tσ and the spatial coupling TD shown in Figs. 17, 18. The lines of critical points (solid and dashed lines) are obtained under condition when the bifurcations of the mean field x〈 〉 occur at 0h = (see notes in Fig. 18). As seen from Fig. 19, a, an increase in the parameter α , yielding the concentration dependent mobility, leads to a decrease in the values 0cε > at small spatial coupling intensity. If α increases, then the threshold for the noise intensity grows. It results that the bulk states with 1 2x x〈 〉 = −〈 〉 exist at large noise intensities only if the mobility ( )M x decreases more abruptly. An interesting situation can be seen from Fig. 19, b, where the intensity of the spatial coupling is large. Here, at negative values of the control parameter ε , the mean field value should behave in a re-entrant manner with variation in the noise intensity. Indeed, at small and large 2σ the e system is in a homogene- ous state. Inside the bounded domain of the noise intensity 2σ , the sys- a b c d Fig. 19. Mean field diagrams at different values of the parameter α (solid and dashed lines of critical points plotted at 0 0x = , 0h = ) and initial conditions 0x (transition dash-dotted lines are plotted at 0 0.6x = , 0h = ). Figures (a) and (b) correspond to 2.0D = and 10.0D = , respectively. Figures (c) and (d) correspond to 0.2ε = and 0.2ε = − , respectively. DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 79 tem is in inhomogeneous state with 0h = and 0x〈 〉 ≠ . An increase in the parameter α decreases values ε at which the ordered state with 0x〈 〉 ≠ is observed and extends the corresponding domain of the noise intensities. Let us consider the diagram in the plane ( 2,Dσ ) shown in Fig. 19, c. Here, at positive values of the control parameter ε , an in- crease in the noise intensity destroys the state 0x〈 〉 ≠ , as usual. At negative ε , we get the re-entrant behaviour of the mean field x〈 〉 , where with α growth transition values for the spatial coupling inten- sity decrease, and the domain of the noise intensity with the re-entrant behaviour extends. Let us compare our results with computer simulations. To this end, we have computed averaged value 2M〈 〉 , the moment 2 2M〈 〉 , and the value 2 2 2 2 2 M M〈 〉 − 〈 〉χ = σ (116) that can be understood as a generalized susceptibility or variance. Av- eraging was done over 7 experiments in a stationary limit ( ∞→t ) at the time interval of 410t = to 42.3 10t = ⋅ . Obtained results are shown in Fig. 20. From Figure 20, a, one can see a nonmonotonic behaviour of the order parameter 2M〈 〉 versus noise intensity. The variance χ in Fig. 20, b, shows two peaks on observable values of the noise intensity 2σ in the corresponding interval. Peaks in 2( )χ σ dependence are re- lated to two thresholds of the re-entrant phase transition. Finally, to show the entropy-driven phase transitions mechanism in the system under consideration, let us study a topology change of the distribution st ( ; , 0)P x x h〈 〉 = at fixed mean-field values 0x〈 〉 > . In the ordered (in- homogeneous) state, the number of extrema of the above distribution is changed. Indeed, in the domain of point A (see Fig. 21), the stochas- tic distribution has one peak shifted toward positive values of x due to a b Fig. 20. Order parameter 2M (a) and generalized susceptibility χ (b) ver- sus noise intensity 2σ at 0.5ε = − , 10D = , 0.8α = . 80 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV the fact that 0x〈 〉 > . With an increase in the noise intensity 2σ , mov- ing through the dashed line toward the point B, an additional peak in the distribution function appears. The dashed line corresponds to the system parameters when a double degenerated point of the stochastic distribution appears. Therefore, here we get the generalization of noise-induced transitions for extended systems. 4.1.4. Influence of External and Internal Noise Sources Now, let us assume the presence of the nonequilibrium medium, which sets external fluctuations. Since the influence intensity of the medium is determined by the control parameter ε , we may con- sider an assumption about its fluctuations to be suitable for the de- scription of real situations: 0 ( , )tε → ε + ζ r . We endow the Langevin source ( , )tζ r by the Gauss properties 2( , ) 0, ( , ) ( , ) ( ) ( )t t t C t t′ ′ ′ ′〈ζ 〉 = 〈ζ ζ 〉 = σ − δ −r r r r r% (117) with the spatial correlation function ( ) 2 2 \| \| ( ) 2 exp 2 d C − ′⎛ ⎞−′− = λ π −⎜ ⎟λ⎝ ⎠ r r r r . (118) Fig. 21. Phase diagram of phase transitions showing the change of the sto- chastic distribution ( ; 0, 0)stP x x h〈 〉 > = extrema. Insertions display forms of distributions of stochastic field x . Other parameters are as follows: 0.2ε = − , 0.8α = ; the mean-field value is calculated according to the val- ues of D and 2σ for points A and B, respectively. DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 81 Here, λ is the correlation length of the external noise ζ ; 2σ% is the in- tensity. As a result, we arrive at the Langevin equation in the form [ ] ( ) ( , ) ( ) ( , ), x x M x x t g x t t x ⎛ ⎞∂ δ⎡ ⎤= ∇ ⋅ ∇ + ζ + ∇ ξ⎜ ⎟⎢ ⎥∂ δ⎣ ⎦⎝ ⎠ r r F (119) where noises ξ and ζ are assumed to be independent. In what follows, we again pass to a discrete space, representing the continual equation (119) in the form ( ) ( ) ( ) ( ) ( ).i L ij j R jl l l L ij j j l dx F M x t g t dt x ⎡ ⎤∂= ∇ ∇ + ζ + ∇ ξ⎢ ⎥∂⎣ ⎦ (120) Here, left- and right-differences, ( )L ij∇ and ( )R ij∇ , respectively, are consistent with definitions (90). Equation (120) can be rewritten in a more convenient form for a further analysis ( ) ( ) ( ) ( ) ( ) ( ),i L ij j R jl L ij j j ij j j l dx F M g t g t dt x ⎡ ⎤∂= ∇ ∇ + ∇ ξ + Δ ζ⎢ ⎥∂⎣ ⎦ % (121) where j j jg M x=% ; the external noise ( )j tζ obeys Gaussian properties: ( ) 0i t〈ζ 〉 = , 2 \| \|( ) ( ) 2 ( )i j i jt t C t t−′ ′〈ζ ζ 〉 = σ δ −% , where \| \|i jC − is a discrete repre- sentation of the spatial correlation function (\| \|)C ′−r r . Early Stage of the System Evolution with Two Noises. As done in pre- vious sections, we will study the instability of the state ( , ) 0x t =r , tak- ing only into account the linear terms in Eq. (120). In this case, the dy- namical equation for the structure function takes the form 2 2 2 2 2 2 ( , ) ( ) ( , ) 2 2 2 ( , ) ( ) ( , ), (2 ) (2 )d d dS k t k S k t k dt k k d S q t d G q S q t = −ω + σ − ασ σ− + π π∫ ∫q q % (122) where ( )G q is the Fourier transform of the external-noise correlation function ( )C ′−r r . The dispersion relation reads [84] ( )2 2 2 2 2 1 0 1( ) 2 ( ) ,k k D C k d C C⎡ ⎤ω = − σ − ε + σ − σ −⎣ ⎦% % (123) where σ% is the intensity as in Eq. (117). The dispersion relation indi- cates that for ( ) 0kω > , the homogeneous null state is stable. This oc- curs for 2 2 2 0[ (\| \|)] 0rC =−ε + ασ + σ ∇ >r% , so we can define an effective control parameter 2 2 2 ef 0[ (\| \|)] 0,rC =ε = −ε + ασ − σ ∇ >r% (124) 82 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV such that the homogeneous null state is stable for ef 0ε > . Therefore, the onset of stability is now given by 2 2 2 0[ (\| \|)]t rC =ε = ασ − σ ∇ r% , which in a discrete space is 2 2 0 12 ( )t d C Cε = ασ + σ −% . Moreover, the expression for ( )kω has a noise dependent term that can be considered as a modifi- cation of the spatial coupling parameter. Thus, we can define an effec- tive spatial coupling parameter 2 ef 1D D C= − σ% [90]. Mean Field Analysis of the Stationary Case. Let us consider the sta- tionary case. To this end, we can use the standard procedure described above to find the stationary distribution as a solution of the corre- sponding Fokker−Planck equation. The total probability density func- tional obeys the equation [37, 61, 62, 67] 2 2 \| \| , . ij j jr r ij ri j j j j mn nj n m nj n V M D x t x x g g g C g x x − ⎛ ⎡ ⎤∂ ∂ ∂= Δ − + Δ −⎜ ⎢ ⎥⎜∂ ∂ ∂⎢ ⎥⎣ ⎦⎝ ⎞∂ ∂−σ + σ Δ ⎟⎟∂ ∂ ⎠ ∑ ∑ ∑% %% P P (125) To perform mean-field calculations, we exploit the one-point probabil- ity density satisfying the equation ( ) ( ),i ij j i ji P t M P t t x ∂ ∂= Δ 〈 〉 ∂ ∂ ∑ % (126) where 2 2 \| \| , .j j jr r j j j mn nj n r m nj j n V M M D x g g g C g x x x − ⎡ ⎤∂ ∂ ∂= − + Δ − σ + σ Δ⎢ ⎥ ∂ ∂ ∂⎢ ⎥⎣ ⎦ ∑ ∑% % %% (127) Assuming that a stationary distribution ( )s iP x can be obtained under no flux conditions, the quantity jM〈 〉% should obey the equation ( ) 0.ij j s ij M P xΔ 〈 〉 =∑ % (128) Following the standard procedure, one can find that the mean-field distribution function can be computed directly from the stationary equation 2 2 1 0 ( ) ( ) 2 ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ), s s V hP x M x dD x x g x g x x x d g x C g x C g x P x x x ⎛ ∂ ∂⎡ ⎤− = − + 〈 〉 − − σ +⎜ ⎢ ⎥∂ ∂⎣ ⎦⎝ ⎞∂ ∂⎡ ⎤+ σ 〈 〉 − ⎟⎢ ⎥∂ ∂⎣ ⎦ ⎠ % % %% (129) where, according to the mean-field approximation, one can put ( ) ; ( )g x g x〈 〉 〈 〉 [37], and drop the prime for 2D dD′ = . Therefore, for the DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 83 stationary distribution one has ( ; ; ) ( , , ) exp , ( ; )s x x h P x x h N dx x x ′Ω 〈 〉⎛ ⎞′〈 〉 = ⎜ ⎟′Θ 〈 〉⎝ ⎠ ∫ (130) where f x M x g x x x h M x D x x d C h x x x ∂ σ ∂ ∂⎡ ⎤Ω 〈 〉 = − + 〈 〉 − − − σ +⎢ ⎥∂ ∂ ∂⎣ ⎦ 2 2 2 0 ( ) ( ) ( ) ( ; ; ) ( ) ( ) , 2 % % (131) a b Fig. 22. Dependence of the mean field versus the noise intensities (internal and external) at 0.4α = , 0.2ε = − , 10D = (a) and the phase diagram at different values of the correlation radius λ , spatial coupling parameter D and α (b) (curves 1−3 correspond to 0λ = , whereas curves 1′−3′ relate to 1λ = : 1 and 1′– 8.3D = , 0.4α = ; 2 and 2′– 10D = , 0.4α = ,3 and 3′– 8.3D = , 0.6α = ). 84 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV 2 2 0 1( ; ) ( ) 2 ( )( ( ) ( )).x x M x d g x C g x C g xΘ 〈 〉 = σ + σ − 〈 〉% % %% Unknown field h and x〈 〉 can be obtained form the self-consistency equation described above. Let us consider an influence of spatial correlations λ of external noise on a position of the critical points. Corresponding phase dia- grams are shown in Fig. 22 at different values of D and α (curves 1−3) and at different values of spatial correlation radius λ . As figure shows an increase in D leads to decrease in critical noise intensity magnitudes 2σ% and promotes re-entrance in phase transition picture (see curves 1, 2). The same situation is observed when the pa- rameter α increases (see curves 1, 3). An increase in the correlation radius of the external noise results in increase in its critical values (see curves 1′−3′). It leads to the fact that area of re-entrant behaviour of the order parameter shrinks. Strong Coupling Limit. Considering a strong coupling limit, we as- sume D → ∞ and neglect all possible correlations, i.e. ( ) ; ( )x x〈ϕ 〉 ϕ 〈 〉 . Hence, the stationary distribution functions are given by the mean field approach for each phase and have the form ( , ) ( )sP x x x x〈 〉 = δ − 〈 〉 . Next, to obtain an equation for the effective field h , we integrate Eq. (112) and find 2( ) ( ) ( 2) ( ),h M x V x M x′ ′= 〈 〉 〈 〉 − σ 〈 〉 (132) where prime denotes derivative with respect to the argument. In the homogeneous case, Tε < ε , we have 0x x〈 〉 = , and h becomes a function of the initial conditions. If the value 0x is fixed, then the field h decreases with an increase in ε until it reaches the null value, and increases from the null value with an increase in 2σ . As it follows from the Eq. (132) and mean field analysis, no re-entrance can be found in strong coupling limit. Therefore, a re-entrant behaviour of the mean-field value is realized only at finite magnitudes of the spatial coupling intensity D . In the case of Tε > ε , we have 0h = . Solutions of Eq. (132) give val- ues for the bulk phases ( )1/2 2 2 2 1,2 2 1 (1 ) 4 . 2 x α 〈 〉 = ± − − αε + − αε + α σ α (133) The corresponding transition lines are defined by condition 1 0x x〈 〉 = that leads to 2 2 2 2 0 0 2 0 (1 ) . 1T x x x α σ − + α ε = + α (134) The critical point (for 0 0x = ) is DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 85 2 2,cε = α σ (135) which coincides with linear stability analysis. (One should not be con- fused comparing our results with results in Refs [76, 84]. We consider the internal multiplicative noise influence, whereas in Refs [76, 84] systems with an external multiplicative noise and conserved dynamics were studied. In above papers, it was noted that the critical value cε differs from the obtained in Eq. (135) by the multiplier 2d . Such mul- tiplier arises only if an external noise is introduced. If we follow the procedure described in Ref. [76] and consider external fluctuations, supposing 0 ( , )tε → ε + ση r , where ( , )tη r is the Gaussian noise, then we will recover results with the multiplier 2d .) It follows that the strong coupling limit sets the critical values for both the control pa- rameter and the noise intensity, which correspond to 0ck = . In other words, at 2ε > ασ , unstable modes start to growth. Moreover, we can define values cα < α = ε/σ2 at which phase separation exhibits spatial patterns. Due to [ 1, 1]ε ∈ − and 2 0σ > , 0α ≥ , one gets that cα de- creases with an increase in the noise intensity 2σ . At negative values of the control parameter, we get the phase separation with no patterning. For the system with two stochastic sources, one can find that critical value for the control parameter is renormalized as 2 2 0 12 ( ).c d C Cε = ασ − σ −% (136) From this, it follows that two stochastic sources compete with each other. Here, we get shift of the critical point with the multiplier 2d related to the noise intensity 2 0Cσ% and spatial correlations 1C . 4.2. Stochasticity in Hyperbolic Transport In this Section, we focus on a more conceptual question related to the fluctuations of the flux J or, more concretely, to their conceptual in- terpretation. We shall see that, depending on the value of the relaxa- tion time Dτ and of the observational time scale, such fluctuations can be interpreted in two different ways described in Ref. [40]. To discuss these ideas, we must recall some results concerning hydrodynamic sto- chastic noise. Hyperbolic Transport with Noise. We discuss now the stochastic noise in a system, ,Lτα + α = − α + ζ&& & (137) which may generally represent the hyperbolic transport (3) with noise. Taking into account both independent variables α and α ≡ β& , Eq. (137) may be written as 86 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV α βα = β + ζ β = − α − β + ζ τ τ % & %& 1 , . L (138) The set of equations (138) represents the second-order equation (137) as two first-order evolution equations, in a way that the system be- comes Markovian. In Eq. (138), αζ and βζ are the respective stochastic sources, whose second-order moments have to be obtained. The correla- tor of fluctuating terms is defined as follows 2 2 eq eq eq eq 2 2 eq eq eq eq 0 1 0 ( ) ( ) 1 1 1 L t t L ⎡ ⎤−⎡ ⎤ ⎢ ⎥⎡ ⎤ ⎡ ⎤〈α 〉 〈αβ〉 〈α 〉 〈αβ〉 τ⎢ ⎥′〈ζ ζ 〉 = + ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ 〈βα〉 〈β 〉 〈βα〉 〈β 〉 ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ −⎢ ⎥τ τ⎣ ⎦ ⎢ ⎥τ⎣ ⎦ % % . (139) In Equation (139), we have eq eq 0〈αβ〉 = 〈βα〉 = because α and α& have opposite time-reversal symmetry. Thus, it is found α β α β β α〈ζ 〉 = 〈ζ 〉 = 〈β 〉 〈ζ ζ 〉 = 〈ζ ζ 〉 = 〈α 〉 − 〈β 〉 τ τ % % % % % %2 2 2 2 2 eq eq eq 2 0, , . L (140) To obtain the second moment of equilibrium fluctuations of α and β , we assume, as in Section 2, that α and β are independent variables. Then, including both of these variables into the entropy, one can write 2 2 eq 1 1 ( , ) , 2 2 S S A Bα β = − α − β (141) where only second-order terms have been considered. According to Eq. (7), the probability of fluctuations is described by 2 2( , ) ~ exp . 2 2B B A B Pr k k ⎡ ⎤ α β − α − β⎢ ⎥ ⎣ ⎦ (142) Note that, in Eq. (142), we have identified the fluctuations δα and δβ with α and β , respectively, because their equilibrium average values are zero for both of them due to the form of the evolution equations (138). Then, in equilibrium, we have 2 2 eq eq, .B Bk k A B 〈α 〉 = 〈β 〉 = (143) To evaluate the ratio /A B , we obtain the entropy production corre- sponding to Eq. (141). This one yields [ ] 0.dS dt A B A B A B= − αα − ββ = − αα − αα = −α α + α ≥&& & & && & && (144) From this, and by following the usual methods of nonequilibrium DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 87 thermodynamics [91], it follows the linear relation between thermody- namic flux −α& and its conjugated force A Bα + α&& . This is .A Bα + α = −μα&& & (145) Comparison of Eq. (145) with Eq. (137) yields B = τ , 1μ = , and A L= . Thus, as follows, the second moments of α and β are related by 2 eq 2 eq . B A L 〈α 〉 τ= = 〈β 〉 (146) Introducing this relation into Eq. (140) it is found L α α β α β β〈ζ 〉 = 〈ζ ζ 〉 = 〈ζ ζ 〉 = 〈ζ 〉 = 〈β 〉 = 〈α 〉 τ τ 2 2 2 2 eq eq2 2 2 0, 0, .% % % % % % (147) Since 2 0α〈ζ 〉 =% , the first equation in Eq. (138) may be introduced into the second one. Therefore, we get ,L βτα + α = − α + τζ%&& & (148) with 2 eq(0) ( ) (0) ( ) 2 .t t Lβ β β β〈ζ ζ 〉 = 〈τζ τζ 〉 = 〈α 〉% % (149) Thus, the expression for the noise keeps the same form as in the case with 0τ = . This is in agreement with the ideas of fluctuation- dissipation, which relate the fluctuations to the dissipative part of the equation [the term in L in Eq. (148)]. The transition from noisy hyperbolic transport described by equa- tion (137) to Langevin equation Lα = − α + ζ& might be analyzed by con- sidering the generalized entropy (141) in the following form 2 2 eq( , ) . 2 2 A A S S L τα β = − α − β (150) With 0τ → , the last term in 2β disappears together with the term in α&& in Eq. (137). In this case, the dynamics of α is described by a simple relaxation with a temporal constant given by 1L− . One interesting situation may be found when 1 1L−τ << << . In this case, α decays slowly and β decays fast. Assume, for instance, that 1≈L s −1 and 310−τ ~ s. The typical relaxation of α will be of the order of 1 second and β will decay in a millisecond scale. In this case, βζ% describes the effect of all the variables whose relaxation time is much less than 310− s, in such a way that they may be assumed to decay instantaneously in comparison with β . Fluctuations or Independent Variable. Now, consider a special system 88 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV with two independent variables α and β in such a way that the relaxa- tion rate of β is characterized by its high but finite value. In this case, one may write Lδβ ≡ β + α , i.e. = Lβ − α + δβ , where δβ being the inde- pendent part of β . This part of β is orthogonal to the slow subspace generated by α . Then, we may write α = − α + δβ δβ = − δβ τ && 1 , .L (151) and \| \| 2 \| \|(0) ( ) .t tBLk L t e e A − τ − τ〈δβ δβ 〉 = = 〈α 〉 τ τ (152) The following three cases may be outlined in considering Eqs (151) and (152). (i) If τ is sufficiently short, δβ acts as a ‘noise’ in the equation for α (the first equation of system (151)). (ii) If τ is not completely negligible as compared to 1L− , δβ acts as a coloured noise. (iii) In the limit 0τ → , one has 2 eq(0) ( ) 2 ( ).t L t〈δβ δβ 〉 → 〈α 〉 δ (153) Thus, it is seen that the transition of Eq. (150) and (151) from small τ to vanishing τ is conceptually interesting. It is illustrative of how the variable β (i.e., α& ) goes from an independent variable with its own dynamics to a purely Markovian stochastic noise. In physical terms, the frontier between small τ and vanishing τ is settled by the time scale one is able to measure. For instance, if 1L = s −1 and 810−τ = s, the system will have two independent variables, α and α& , for an ob- server which is able to measure picoseconds ( 1210− s). However, it will have only one independent variable, α , plus a stochastic noise, in a form of α& , for an observer which is only able to measure milliseconds ( 310− s). The latter observer will be able to work in the ‘adiabatic’ ap- proximation with very slow α in comparison with β . The situation described by Eqs (1) and (2) is interesting from this perspective. For negligible values of the diffusion relaxation time Dτ , one should use the Landau−Lifshitz formalism for the fluctuating hy- drodynamics [28] and write 2 ,c c D c t ∂ = ∇ + ζ ∂ (154) where the stochastic noise cζ is interpreted as cζ = −∇ ⋅ δJ (155) DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 89 with δJ being a fluctuating part of the diffusion flux, i.e. .D c= − ∇ + δJ J (156) According to the Landau−Lifshitz approach [28], one has (0) ( ) 2 ( ).Bt k DT t〈δ δ 〉 = δJ J (157) Assume, in contrast, that Dτ is small but still measurable. In this case, we get , ,D J c D c t t ∂ ∂= −∇ ⋅ τ + = − ∇ + ζ ∂ ∂ J J J (158) with c and J being independent variables of the entropy given by the generalized Gibbs equation (8). The second moments of the fluc- tuation of c and J are given by Eq. (11). The noise Jζ would cor- respond to values relaxing in time scales much shorter than Dτ , in such a way that it may be considered as Markovian noise: (0) ( ) 2 ( ).J J Bt k DT t〈ζ ζ 〉 = δ (159) In the limit of vanishing Dτ , the fast part of J becomes a stochas- tic noise, and we get cζ in Eq. (154) described by Eq. (157). Still another form to discuss the interpretation of noise in the context of Eq. (137) is to write Eq. (138) without any added noise, i.e., in the following form , . D D L βα = β β = − α − τ τ && (160) This set of equations may be integrated to give 0 ( ) ( ) ( ) ( ), t t t M t t t dt t′ ′ ′α = − − α + β∫& (161) with the memory function ( ) exp , D D L t t M t t ⎛ ⎞′−′− = −⎜ ⎟τ τ⎝ ⎠ (162) and the exponential relaxation 0 0( ) ( ) exp . D t t t t ⎛ ⎞− β = α −⎜ ⎟τ⎝ ⎠ & (163) In this case, the ‘noise’ is due to the uncertainty in the value of ( )tα& at the initial time 0t . 90 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV Mori’s expression for the fluctuation-dissipation theorem states that 2 eq( ) ( ) ( ) .t t M t t′ ′〈β β 〉 = − 〈α 〉 (164) Indeed, if we use the result (146), we obtain that 2 2 2 0 eq 0 eq eq( ) ( ) . L t t〈α 〉 ≡ 〈β 〉 = 〈α 〉 τ & (165) Combination of Eq. (165) with Eqs (162) and (163) gives Eq. (164). Again, when τ becomes negligible, ( )M t t′− given in Eq. (162) be- comes ( ) ( ),M t t L t t′ ′− = δ − (166) and Eq. (161) becomes ( ) ( ) ,t L t αα = − α + ζ& (167) with 2 22 .Lα〈ζ 〉 = 〈α 〉 (168) Note that, in the limit (166), we consider t t′> , whereas in the limit (168), a factor 2 appears because one considers \| \| 0t t′− > rather than 0t t′− > , i.e. one considers both 0t t′− > and 0t t′ − > . 4.3. Stochastic Hyperbolic Model for Spinodal Decomposition Considering the stochastic hyperbolic model, let us start with a set of equations for the local concentration and the flux in the forms as follow: ,x t ∂ = −∇ ⋅ ∂ J (169) D x M x t t x ∂ δτ = − − ∇ + ζ ∂ δ J J r [ ] ( ; , ) F . (170) Here, Dτ is a relaxation time for the flux J . In further analysis, let us introduce a dimensionless time / xt t′ = τ and operator 1 x x −′∇ = ∇l , where scales xl and xτ are introduced (usually, the time scale for such transition is defined through the Debye frequency Dω , diffusion en- ergy diffE and temperature T as 1 diffexp( / )x D E T−τ = ω ). If the diffu- sion is caused by the vacancy mechanism, then the quantity diffE is en- ergy for migration of vacancies. Next, let us move to dimensionless quantities / DJ J V′ = with /D x xV = τl , 0/M M M′ = , 0/′ ′=F F F , DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 91 DV′ζ = ζ for flux, mobility and free energy, and noise respectively. Inserting such quantities into Eq. (169), dropping the prime, and as- suming 0 0 D x xM V= τlF , we reduce the set of two first-order time equa- tion to the second-order equation in the form of ( ) 2 2 ( ) ( ) ( , ) , x x M x g x t t t ∂ ∂δ + = ∇ ⋅ ∇μ + ζ ∂ ∂ r (171) where /D xδ = τ τ states the ratio of relaxation times for ( ) ( )g x M x= . At 1δ << , one arrives at the parabolic model for the spinodal decomposition discussed previously in Section 4.1. In our consideration, it can be shown that two conjugate variables, as the local concentration and the flux, should be considered as com- mensurable variables as a special case. Moreover, we will explain that even the flux is supposed to be fast variable our results leads to the well-known picture of nonlinear dependence of an amplification rate at early stages, whereas at late stages (stationary case), the hyperbolicity of the model does not affect on the system behaviour essentially. 4.3.1. Early Stages Analysis Let us consider an early stage of the system evolution. As done before, we can obtain an evolution equation for the spherically averaged struc- ture function ( , )S k t . In the following, we use the special case of the white noise assumption, C t t t t′ ′ ′ ′− − → σ δ − δ −r r r r2( , ) ( ) ( ) . After some algebra, one gets ( ) 2 2 2 2 2 2 2 2 2 ( , ) ( , ) 1 2 2 ( , ). (2 )d d d S k t k Dk S k t dt dt k k d S q t ⎛ ⎞ δ + = − − ε + ασ +⎜ ⎟ ⎝ ⎠ + σ − ασ π ∫ q (172) This equation is reduced to Eq. (63) in the deterministic case ( 2 0σ = ), whereas in the case of 1δ << and 0α = one arrives at the Cahn−Hilliard−Cook equation [12, 59]. It can be seen that the dis- persion relation now takes the form 2 2 2 2 ( ) 1 ( ) , 2 4 i k k D k − ε + ασω = − ± − δ δ δ (173) which, in the case of the local equilibrium, 1δ << , is as follows: 1 2 2 2 1 ( ) (2 ) 1 2 ( ) .k i k k D− δ= ⎡ ⎤ω = − δ ± δ − ε + ασ⎣ ⎦ (174) At ck k< , the imagine part of the frequency (173) 92 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV − ⎡ ⎤ℑ ω = δ ± − δ − ε + ασ ⎣ ⎦ 1 2 2 2( ( )) (2 ) 1 1 4 ( )k k k D (175) can be used to obtain maximal wave-vector amplitude mk . To obtain a normalized amplification rate, we need to compare dis- persion relations obtained for both parabolic and hyperbolic stochastic models. To that end, we use the quantity CHC m( )kω obtained from the parabolic Cahn−Hilliard−Cook model with multiplicative fluctuations as a normalization factor, 2 m ( ) / 2k D= ε − ασ (see Subsection 4.1.1). Acting in such a manner, the normalized amplification rate takes the form 2 2 hyp hyp CHC m( ) / ( ) / ( ) /q q k k qω = ω ω , where / cq k k= , and disper- sion relation hyp( )kω is taken from Eq. (175) as ω = ℑ ωhyp ( ) ( ( ))k k . Therefore, for the stochastic hyperbolic model one has 2 2 2 2 * 2 2 2 2 ( ) 1 4 (1 ) 1 2 ( ) . ( ) q q D Dq q q ε − ασ+ δ − − ω = δ ε − ασ (176) In such a case, both parabolic and hyperbolic normalized amplification rates take values: 2( ) / 0q qω = at 1q = and 2( ) / 4q qω = at 0q = . At 0δ → (transition to the one slow variable (parabolic) model), one ar- rives at the linear dependence 2( ) /q qω versus 2q . The corresponding dependence from Eq. (176) is shown in Fig. 23. Comparing different curves, one can see that in the nonlinear behav- iour appears only if 2 2( ) 0δ ε − ασ ≠ . It means that in the deterministic case 2 0σ = the nonlinearity is caused by 0δ ≠ , whereas the stochastic contribution leading to renormalization of the control parameter ε is able to promote essential contribution to the above effect. Indeed, at large difference between 2ασ and \| \|ε at fixed α , the nonlinear effect becomes well pronounced. To relate the stochastic approach to the deterministic one, let us re- write the dimensionless dispersion relation in the form 2 2 2( ) 1 ( ) , 2 4 xx ck f r ki k ′′ + ω = − ± − δ δ δ % (177) where the notation 2 xx xxf f′′ ′′= + ασ% is used, 0\|xx xf =′′ = −ε . From this, it fol- lows that the internal noise influence leads to a change in the barrier height for the effective free energy f% due to 2 0\|xx xf =′′ = −ε + ασ% . Moving back to dimensional variables, let us interpret our results for the sto- chastic case. Here, instead of originally exploited correlation and dif- fusion lengths, /C c xxl r f′′= − and 0D xx Dl M f′′= − τ , respectively, it is DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 93 convenient to use the effective lengths, /C c xxl r f′′= − %% and 0D xx Dl M f′′= − τ%% . In such a case, we arrive at formula for the amplifica- tion rate given by Eq. (66). The difference in deterministic and sto- chastic cases lies only in the renormalization of the barrier height for the effective free energy f% . An increase in the intensity 2σ of multi- plicative fluctuations determined by the field dependent mobility de- creases the length 2 Dl ∝ ε − ασ% and increases the scale 21 /Cl ∝ ε − ασ% . The local nonequilibrium effect at small 2σ is en- hanced by the diffusion flux relaxation that is in good correspondence with results obtained for the deterministic case analysis. 4.3.2. The Effective Fokker−Planck Equation for the Hyperbolic Model To describe the system states, we need to know the distribution func- tion of the field x . In order to get it, one should obtain the correspond- ing Fokker−Planck equation. According to the standard procedure, we represent our system in a discrete d-dimensional space with dN cells with the mesh size l . Then, following Eq. (171), the system dynamics will be described by a set of equations for every cell of the space: Fig. 23. Comparison of the function * 2( ) /q qω for hyperbolic (modified Cahn−Hilliard) model for deterministic and stochastic cases at different values of the ratio /D xδ = τ τ and different noise intensities 2σ . Other pa- rameters are as follows: 0.5ε = , 1D = , 0.8α = . 94 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV 2 2 .L Ri i ij j jl j j l d x dx F M g dt dt x ⎛ ⎞∂δ + = ∇ ∇ + ζ⎜ ⎟∂⎝ ⎠ (178) Next, let us introduce a new variable, ip , playing a role of an effective momentum, i ip x= δ & . Then, by definition, the probability density function is given by the averaging of the density function ( , , )x p tρ over noise: ( , , ) ( , , )x p t x p t= 〈ρ 〉P . To construct an equation for the macroscopic density function P , we exploit the conventional device to proceed from the continuity equation: 0i i i i i x p t x p ⎡ ⎤∂ρ ∂ ∂+ + ρ =⎢ ⎥∂ ∂ ∂⎣ ⎦ ∑ & & . (179) Inserting the momentum definition, we obtain ( ) , t ∂ρ = + ζ ρ ∂ L N (180) where the operators ii = ∑L L and ii = ∑N N are defined as follows 1 ,L Ri i ij j jl i i i l p F M p x p x ⎛ ⎞∂ ∂ ∂≡ − − ∇ ∇ −⎜ ⎟δ ∂ ∂ ∂ δ⎝ ⎠ L (181) .L i ij j i g p ∂≡ −∇ ∂ N (182) Within the interaction representation, the density function reads exp( )t℘ = − ρL that allows to rewrite Eq. (180) as , t ∂ ℘ = ℘ ∂ R (183) ( , , ) .t t i i i i i i x p t e e−⎡ ⎤= ≡ ζ ⎣ ⎦∑ ∑ L LR R N (184) The well-known cumulant expansion method serves as standard and effective device to solve such a type stochastic equation [92]. Neglect- ing terms of the order 3( )O R , we get the kinetic equation for the aver- aged quantity ( )t〈℘ 〉 in the form 0 ( ) ( ) ( ) ( ). t t t t dt t t ⎡ ⎤∂ ′ ′〈℘〉 = 〈 〉 〈℘〉⎢ ⎥∂ ⎣ ⎦ ∫ R R (185) Within the original representation, the equation for the probability density reads DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 95 ( ) 0 ( ) ( ) ( ). t P t C e e d P t t τ − τ⎧ ⎫∂ ⎪ ⎪⎡ ⎤= + τ τ⎨ ⎬⎣ ⎦∂ ⎪ ⎪⎩ ⎭ ∫ L LL N N (186) Due to the physical time much larger than a correlation scale ( )t >> τ , one can replace the upper limit of the integration by ∞ . To proceed this one, we use the procedure proposed in Ref. [69, 70] to obtain the effective Fokker−Planck equation for the hyperbolic sto- chastic model. Expanding exponents, we arrive at the equation ( ) ,P Pt ∂ − =∂ L C (187) where the operator (1) (2)≡ +L L L (188) has the components (1) (2) 1 , .L Ri ij j jl i i ii l i i p F M p x x p p ⎛ ⎞∂ ∂ ∂ ∂≡ − + ∇ ∇ ≡⎜ ⎟δ ∂ ∂ ∂ δ ∂⎝ ⎠ ∑ ∑L L (189) The collision operator C is defined as follows: ∞ ∂= = = ≡ − ∇ ∂∑ ∑( ) ( ) ( ) ( ) (0) =0 , ( ), , n n n n L ij j n i i C C g p C M NL L N N , (190) where ( )nL in the collision operator C is defined through the commuta- tor ( ) ( 1)[ , ]n n−=L L L ; moments of the noise correlation function ( )C τ are defined as follows: ( ) 1 0 ( !) ( ) .n nn C d ∞−= τ τ τ∫M (191) Substituting all definitions into commutators, one can calculate the collision operator in the form: ( ) 2 (0) (1) 2 2 2 (1) 2 . L R ij j jl l i i R jl lL R L ij j jl l ij j i i i l i i g g p g g g g p x p x p p ∂= − − ∇ ∇ + ∂ ⎧ ⎫⎛ ⎞∂∇ ⎛ ⎞∂ ∂ ∂⎪ ⎪+ ∇ ∇ − ∇ +⎜ ⎟⎨ ⎬⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭ ∑C M M M (192) Noting that, in further consideration, we are interesting in behav- iour of the distribution ({ }, )ix tP , not the total one, ({ },{ }, )i iP x p t . The reduced distribution can be obtained according to the moments ( ) ({ }, ) ({ },{ }, ) d ,n n i i i i i i P x t x p t p p⎡ ⎤≡ ⎣ ⎦∏∫P (193) 96 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV where the integration is provided over a set { }ip . Then, performing corresponding manipulations with Eq. (187), we arrive at the recursive relations for the moments ( ) ({ }, )n iP x t , where (0) ({ }, )iP P x t≡ , the first moment (1) ({ }, )iP x t can be considered as a flux J of the probability density, i.e. (1)P = J . Indeed, taking the zeroth moment of momentum, p , over Eq. (187), we obtain the expected continuity equation 1 .t i i P x ∂∂ = − δ ∂∑ J (194) The first-moments’ calculation leads to 1 L R t ij j jl i l F M P x ⎡⎛ ⎞∂∂ = − + ∇ ∇ +⎢⎜ ⎟δ ∂⎢⎝ ⎠⎣ ∑J J (2) (1) 1 R jl lL R L ij j jl l ij j i l i gP P g g g P x x x ⎤⎧ ⎫⎛ ⎞∂∇∂ ∂⎪ ⎪ ⎥∇ ∇ + ∇ −⎜ ⎟⎨ ⎬⎜ ⎟∂ ∂ δ ∂ ⎥⎪ ⎪⎝ ⎠⎩ ⎭ ⎦ + M . (195) For the second moment, we obtain (2) (1) (0)1 ( ) .L R ij j jl l i P g g P x ∂ = − ∇ ∇ δ ∂ M M (196) As a result, the evolution equation for the flux J is of the form ⎡⎛ ⎞⎛ ⎞∂∇∂⎢⎜ ⎟∂ = − + ∇ ∇ − ∇ +⎜ ⎟⎜ ⎟⎜ ⎟δ ∂ ∂⎢ ⎝ ⎠⎝ ⎠⎣ ⎤∂+ ∇ ∇ ⎥∂ ⎦ ∑ (1) (0) 1 . R jl lL R L t ij j jl ij j i l l L R ij j jl l i gF M g P x x g g P x J J M M (197) Therefore, we arrive at the set of two differential equations, Eq. (194) and Eq. (197). Eliminating the flux J , we, finally, get the Fok- ker−Planck equation for the hyperbolic stochastic model in the form ⎡ ⎤⎛ ⎞∂∇∂ ∂ ∂ ∂δ + = − ∇ ∇ − ∇ −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂∂ ⎢ ⎥⎝ ⎠⎣ ⎦ ∂− ∇ ∇ ∂ ∑ ∑ 2 (1) 2 2 (0) . R jl lL R L ij j jl ij j i i l j L R ij j jl l ij i j gP P F M g P t x x xt g g P x x M M (198) In the continuum space, the obtained effective Fokker−Planck equa- tion for the probability density functional is as follows: DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 97 ⎡ ⎤δ δ δ⎛ ⎞δ∂ + ∂ = − ∇ ∇ − ∇ ∇ −⎢ ⎥⎜ ⎟δ δ δ⎝ ⎠⎣ ⎦ δ− ∇ ∇ δ ∫ ∫ (1) 2 2 (0) 2 ( ) ( ) 2 ( ) ( ) . ( ) tt t M P P d M P x x x d M P x r r r r r r F M M (199) The stationary probability density can be obtained explicitly under no flux conditions. Indeed, here, we arrive at the result ( )(0) (1) (0) 1 [ ] exp [ ] ( ) ln ( )sP x x d M x ⎧ ⎫∝ − + −⎨ ⎬ ⎩ ⎭∫ rF M M M . (200) This is similar to that presented in Eq. (100). The main difference is that in the distribution (200) we have a contribution related to the first moment of the noise correlation function with respect to the time. It means that the stochastic source appeared in the hyperbolic model should have smeared temporal correlation function. In other words, its frequency spectrum should have some cut-off frequency and the noise should be considered as a coloured noise in time. It looks absolutely natural, because as it was shown in previous subsection the stochastic hyperbolic transport should be characterized with fixed correlation time, ζτ , which may be small comparing to the relaxation time for the concentration field. Taking the temporal correlation function in the exponential form related to the Ornstein−Uhlenbeck process ζ , ( )1( ) exp \| \| /C t t t t− ζ ζ′ ′− = τ − − τ , one can find (0) 2= σM , (1) 2 ζ= τ σM . It is seen that the temporal correlation radius ζτ makes the renormalization of the critical values for the noise intensity, whereas main results ob- tained for the stochastic parabolic model remain the same. Making use the fluctuation dissipation relation one can relate the correlation scale with the relaxation time for the flux, ; Dζτ τ . It follows that, in the white noise limit characterized by 0ζτ → , one gets the parabolic model for the spinodal decomposition, whereas one arrives at the hyperbolic model at fixed but small ζτ . This conclusion is in a good correspondence with the linear stability analysis, where it was shown that in order to get the nonlinear dependence of the amplification rate it is necessary to consider two modes: concentration field and flux. 5. CONCLUSIONS A model for kinetics of fast spinodal decomposition in a binary system free from imperfections and with the molar volume independent of concentration is developed. The model takes into account a finiteness of the diffusion speed DV and assumes that the spinodal decomposition may proceed with the rate of the order of DV . Such an approach leads to the description with independent variables of the concentration and 98 D. O. KHARCHENKO, P. K. GALENKO, and V. G. LEBEDEV atomic diffusion flux having different relaxation times to their own local equilibrium values. It leads to a modified Cahn—Hilliard equa- tion, describing the spinodal decomposition for both diffusion and wave propagation of atoms (components of the binary system). In such a case, the equation describes kinetics of fast separation the rate of which compatible with the diffusion speed DV . To describe fast decomposition, the free energy density e ne( , , )f c c f f∇ = +J includes standard local equilibrium contribution e ( , )f f c c= ∇ and purely non-equilibrium contribution nef proportional to the square of the atomic flux, ⋅J J . The use of ( , , )f c c∇ J in such a form has a statistical basis: there are many particles within every local volume, and it includes a reduction of the available phase space for each particle. It is shown that the fast spinodal decomposition is de- scribed by a hyperbolic type of differential equation. This description is true if the time scale of the process of phase separation has the order of the relaxation time Dτ . It occurs for the case of the fast frequency and short wave-lengths which cannot be neglected in the description of evolution from an unstable state to a new metastable state in spi- nodally-decomposing system and, generally, for fast moving inter- faces. Despite classic Cahn−Hilliard scenario described by parabolic- diffusion equation predicts much more diffuse boundaries, the hyper- bolic scenario exhibits evolution with sharper boundaries between two separating phases. It occurs due to description of hyperbolic evolution by the equation with a finite diffusion speed and description of the parabolic evolution by the equation with an infinite diffusion speed. The provided analysis leads to the obtaining the phase and group atomic speeds. The real values for speeds define the finite propagation of a single harmonic (for the phase speed) and a packet of harmonics (for the group speed). The hyperbolic model is able to give prediction for scenario from very earliest up to latest of spinodal decomposition. We considered the fluctuations of the solute density and the solute dif- fusion flux at the equilibrium steady state. The power spectra of a sol- ute number density and a solute diffusion flux have been reviewed. The latter has a non-vanishing relaxation time leading to a hyperbolic transport equation for the evolution of the density. Several interpreta- tions of the stochastic source related to fast variables eliminated from the description have been examined. Particularly, the phase separation scenario of the system with internal multiplicative noise related to the field dependent mobility for parabolic (Cahn—Hilliard) and hyperbolic models is analyzed. Analysis was performed for early and late stages of the evolution analytically and by computer simulations. The stationary case is considered with the help of the mean field approach. For the system undergoing spinodal decomposition with the field dependent mobility, we have derived the Fokker−Planck equation. It was found that its stationary solution can be written in exact form. DYNAMICS IN SPINODAL DECOMPOSITION OF A BINARY SYSTEM 99 Our theoretical approach shows that we dealt with the entropy driven phase transitions mechanism. As shown, the field-dependent mobility leads to delays in dynamics at early stages and, therefore, leads to de- lays in domain growth law at late stages. Stationary values of the order parameter reduced to the second moment of the stochastic field depend on the parameter that governs the functional dependence of the mobil- ity: with an increase in such parameter, the order is suppressed. Con- sidering stationary states, we extend the mean field approach to the systems with the field-dependent mobility. It was found that an in- crease in the parameter that governs the functional dependence of the mobility, the critical points for the phase transitions are shifted. The system can undergo a re-entrant phase transitions when the mean field becomes nontrivial inside the fixed domain of the noise intensity. The strong coupling regime shows that a position of the critical point de- pends on the constant governing the field-dependent mobility. Our results can be applied to investigations of polymer mixtures where relaxation flows are driven by field-dependent coefficients, phase separation in binary alloys, and microstructure phase transi- tions. ACKNOWLEDGMENTS We thank David Jou and Alexander Olemskoi for fruitful discussions and useful exchanges. Dmitrii Kharchenko acknowledges financial support from the Fundamental Research State Fund of Ukraine (No. GP/F26/0010). 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Deterministic and stochastic dynamics in spinodal decomposition of a binary system

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