Self-Organization of Condensed Matter in Fluctuating Environment
A synergetic conception of plastic deformation is presented allowing for the defects’ interplay resulting in a phase transition. A qualitative-reconstruction pattern of imperfect condensed-matter structure in the presence of fluctuations in both 0-dimensional and distributed (extended) systems is co...
Gespeichert in:
| Datum: | 2005 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут металофізики ім. Г.В. Курдюмова НАН України
2005
|
| Schriftenreihe: | Успехи физики металлов |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/125810 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Self-Organization of Condensed Matter in Fluctuating Environment / D.O. Kharchenko // Успехи физики металлов. — 2005. — Т. 6, № 1. — С. 1-54. — Бібліогр.: 63 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-125810 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1258102025-02-23T17:09:41Z Self-Organization of Condensed Matter in Fluctuating Environment Самоорганизация конденсированной среды во флуктуационном окружении Самоорганізація конденсованого середовища у флуктуаційному оточенні Kharchenko, D.O. A synergetic conception of plastic deformation is presented allowing for the defects’ interplay resulting in a phase transition. A qualitative-reconstruction pattern of imperfect condensed-matter structure in the presence of fluctuations in both 0-dimensional and distributed (extended) systems is considered. As shown, the system undergoes the reversible phase transitions under the increasing intensity of fluctuations, diffusion constituent and deformation-type external parameters. Within the framework of the synergetic approach based on the Lorenz system, a pattern of explosive crystallization is shown to be scripting the same way as the avalanches formation. The relationships are set between the exponent of a size distribution of avalanches, the fractal dimension of a phase space, characteristics of noise, the number of governing equations, the dynamical exponent, and the non-additivity parameter. As shown, the regime of self-organized criticality is associated with an anomalous diffusion processes. Представлена синергетическая концепция пластической деформации, учитывающая взаимодействия между дефектами, в рамках которой исследуются фазовые переходы. Рассмотрена картина качественной перестройки структуры конденсированной среды при наличии флуктуаций в 0-мерных и распределенных системах. Показано, что система способна претерпевать реверсивные фазовые переходы с возрастанием интенсивности флуктуаций, диффузионной составляющей и внешних параметров типа деформации. В рамках синергетического подхода на основе системы Лоренца рассмотрена картина взрывной кристаллизации, протекающей по сценарию процессов лавинообразования. Установлены соотношения между показателями распределения лавин по размерам, фрактальной размерностью фазового пространства, характеристиками флуктуаций, числом управляющих уравнений, динамическим показателем и параметром неаддитивности. Показано, что режим самоорганизованной критичности тесно связан с процессами аномальной диффузии. Представлено синергетичну концепцію пластичної деформації із урахуванням взаємодії між дефектами, в рамках якої досліджуються фазові переходи. Розглянуто картину якісної перебудови структури конденсованого середовища при наявності флюктуацій у 0-вимірних та розподілених системах. Показано, що система здатна зазнавати реверсивні переходи із зростанням інтенсивности флюктуацій, дифузійної складової та зовнішніх параметрів типу деформації. В рамках синергетичного підходу на основі Льоренцової системи розглянуто картину вибухової кристалізації, яка споріднена процесам формування лавин. З'ясовано співвідношення між показником в розподілі за розмірами лавин, фрактальною вимірністю фазового простору, характеристиками флюктуацій, числом керуючих рівнань, динамічним показником, параметром неадитивности. Показано, що режим самоорганізованої критичности тісно пов'язаний із процесами аномальної дифузії. 2005 Article Self-Organization of Condensed Matter in Fluctuating Environment / D.O. Kharchenko // Успехи физики металлов. — 2005. — Т. 6, № 1. — С. 1-54. — Бібліогр.: 63 назв. — англ. 1608-1021 PACS: 05.40.Ca, 05.45.Df, 05.65.+b, 05.70.Fh, 45.70.Ht, 64.60.Ak DOI: https://doi.org/10.15407/ufm.06.01.001 https://nasplib.isofts.kiev.ua/handle/123456789/125810 en Успехи физики металлов application/pdf Інститут металофізики ім. Г.В. Курдюмова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A synergetic conception of plastic deformation is presented allowing for the defects’ interplay resulting in a phase transition. A qualitative-reconstruction pattern of imperfect condensed-matter structure in the presence of fluctuations in both 0-dimensional and distributed (extended) systems is considered. As shown, the system undergoes the reversible phase transitions under the increasing intensity of fluctuations, diffusion constituent and deformation-type external parameters. Within the framework of the synergetic approach based on the Lorenz system, a pattern of explosive crystallization is shown to be scripting the same way as the avalanches formation. The relationships are set between the exponent of a size distribution of avalanches, the fractal dimension of a phase space, characteristics of noise, the number of governing equations, the dynamical exponent, and the non-additivity parameter. As shown, the regime of self-organized criticality is associated with an anomalous diffusion processes. |
| format |
Article |
| author |
Kharchenko, D.O. |
| spellingShingle |
Kharchenko, D.O. Self-Organization of Condensed Matter in Fluctuating Environment Успехи физики металлов |
| author_facet |
Kharchenko, D.O. |
| author_sort |
Kharchenko, D.O. |
| title |
Self-Organization of Condensed Matter in Fluctuating Environment |
| title_short |
Self-Organization of Condensed Matter in Fluctuating Environment |
| title_full |
Self-Organization of Condensed Matter in Fluctuating Environment |
| title_fullStr |
Self-Organization of Condensed Matter in Fluctuating Environment |
| title_full_unstemmed |
Self-Organization of Condensed Matter in Fluctuating Environment |
| title_sort |
self-organization of condensed matter in fluctuating environment |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| publishDate |
2005 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/125810 |
| citation_txt |
Self-Organization of Condensed Matter in Fluctuating Environment / D.O. Kharchenko // Успехи физики металлов. — 2005. — Т. 6, № 1. — С. 1-54. — Бібліогр.: 63 назв. — англ. |
| series |
Успехи физики металлов |
| work_keys_str_mv |
AT kharchenkodo selforganizationofcondensedmatterinfluctuatingenvironment AT kharchenkodo samoorganizaciâkondensirovannojsredyvofluktuacionnomokruženii AT kharchenkodo samoorganízacíâkondensovanogoseredoviŝaufluktuacíjnomuotočenní |
| first_indexed |
2025-11-24T02:28:15Z |
| last_indexed |
2025-11-24T02:28:15Z |
| _version_ |
1849636993298857984 |
| fulltext |
1
PACS numbers: 05.40.Ca, 05.45.Df, 05.65.+b, 05.70.Fh, 45.70.Ht, 64.60.Ak
Self-Organization of Condensed Matter in Fluctuating
Environment
D. O. Kharchenko
Sumy State University,
2 Ryms’ky-Korsakov Str.,
UA-40007 Sumy, Ukraine
A synergetic conception of plastic deformation is presented allowing for the
defects’ interplay resulting in a phase transition. A qualitative-reconstruc-
tion pattern of imperfect condensed-matter structure in the presence of fluc-
tuations in both 0-dimensional and distributed (extended) systems is consid-
ered. As shown, the system undergoes the reversible phase transitions under
the increasing intensity of fluctuations, diffusion constituent and deforma-
tion-type external parameters. Within the framework of the synergetic ap-
proach based on the Lorenz system, a pattern of explosive crystallization is
shown to be scripting the same way as the avalanches formation. The rela-
tionships are set between the exponent of a size distribution of avalanches,
the fractal dimension of a phase space, characteristics of noise, the number of
governing equations, the dynamical exponent, and the non-additivity pa-
rameter. As shown, the regime of self-organized criticality is associated with
an anomalous diffusion processes.
Представлено синергетичну концепцію пластичної деформації із ураху-
ванням взаємодії між дефектами, в рамках якої досліджуються фазові
переходи. Розглянуто картину якісної перебудови структури конденсо-
ваного середовища при наявності флюктуацій у 0-вимірних та розподі-
лених системах. Показано, що система здатна зазнавати реверсивні пе-
реходи із зростанням інтенсивности флюктуацій, дифузійної складової
та зовнішніх параметрів типу деформації. В рамках синергетичного
підходу на основі Льоренцової системи розглянуто картину вибухової
кристалізації, яка споріднена процесам формування лавин. З'ясовано
співвідношення між показником в розподілі за розмірами лавин, фрак-
тальною вимірністю фазового простору, характеристиками флюктуацій,
числом керуючих рівнань, динамічним показником, параметром неади-
тивности. Показано, що режим самоорганізованої критичности тісно
пов'язаний із процесами аномальної дифузії.
Представлена синергетическая концепция пластической деформации,
Успехи физ. мет. / Usp. Fiz. Met. 2005, т. 6, сс. 1—54
Оттиски доступны непосредственно от издателя
Фотокопирование разрешено только
в соответствии с лицензией
2005 ИМФ (Институт металлофизики
им. Г. В. Курдюмова НАН Украины)
Напечатано в Украине.
2 D. O. Kharchenko
учитывающая взаимодействия между дефектами, в рамках которой ис-
следуются фазовые переходы. Рассмотрена картина качественной пере-
стройки структуры конденсированной среды при наличии флуктуаций
в 0-мерных и распределенных системах. Показано, что система способ-
на претерпевать реверсивные фазовые переходы с возрастанием интен-
сивности флуктуаций, диффузионной составляющей и внешних пара-
метров типа деформации. В рамках синергетического подхода на основе
системы Лоренца рассмотрена картина взрывной кристаллизации, про-
текающей по сценарию процессов лавинообразования. Установлены со-
отношения между показателями распределения лавин по размерам,
фрактальной размерностью фазового пространства, характеристиками
флуктуаций, числом управляющих уравнений, динамическим показа-
телем и параметром неаддитивности. Показано, что режим самооргани-
зованной критичности тесно связан с процессами аномальной диффузии.
Key words: noise-induced phase transitions, order parameter, self-organized
criticality, fractal dimension, anomalous diffusion.
(Received April 6, 2004)
1. INTRODUCTION
After years of introduction of the synergetics to describe the complex
behaviour of nonlinear systems [1, 2], the attention of researchers
started to drift toward the nonequilibrium statistical ensembles, which
display the coherent behaviour caused by the collective effects. Along
this line the deterministic approaches to study complex systems were
enriched by the engaging the stochastic methods of analysis [3, 4]. It
allows finding nontrivial peculiarities of nonequilibrium phase transi-
tions in noisy spatially extended systems, which cannot be discovered
by ignoring the fluctuations of environment [5]. As was displayed in a
last decade, among the phase transitions that occur only when a control
parameter is driven above a critical value one has another qualitative
change of the system behaviour associated with the regime of self-
organized criticality [6, 7]. In such a regime, a spontaneous (avalanche-
type) dynamics is observed. A main feature of the systems displaying
self-organized criticality is their self-similarity corresponded to fractal
properties of the system and manifested itself in laws of motion [8, 9].
All mentioned features of the complex behaviour of nonlinear
systems could be found considering a condensed matter where an
atoms' system is moved off the equilibrium state. Such a situation
is realized in the kernel of defect of crystal lattice or in domains of
plastic flow and destruction. It is well known that the plastic flow
of polycrystalline materials is a complex many-stage process, which
occurs on different structure levels being characterized by its own
scale and a type of structure element [10, 11]. In the course of plas-
Self-Organization of Condensed Matter in Fluctuating Environment 3
tic deformation, the nonlinear coupling between the elements of
structure of one type and elements of different structure levels is
realized. It results to the complex cooperative behaviour of the sys-
tem of defects such as appearing regular spatial-temporal structures
on a macroscale level: the chaotic structure of dislocations is trans-
formed to the dislocation tangles and cell boundaries, which can be
transformed to the strip structure. Above-mentioned transforma-
tions of defect structure can be represented on the basis of concep-
tion of nonequilibrium phase transitions. A special attention can be
paid considering the explosive crystallization process [12] relative
to the regime of self-organized criticality. Here, a small number of
embryos of crystallization initiate the crystallization of amorphous
films over whole surface during a small time interval. Such a sys-
tem shows the hierarchical structure, and hence it is characterized
by fractal properties.
Main purpose of this contribution is to present such peculiarities
of complex behaviour of defect structure in the framework of
unique conception for far off-equilibrium systems. It is reached on
the basis of synergetic scheme representing an evolution of hydro-
dynamic modes parametrizing the system. We re aimed to render
the theoretical analysis of the synergetic systems in fluctuation en-
vironment. Section 2 is devoted to development of contemporary
representations of transition from stochastic regime to coherent
one. In Section 2.1, we present most popular models of plastic flow
and reconstruction of defect structure: the ‘activator—inhibitor’
model shown in terms of deformation and tension (Section 2.1.1)
and Lorenz-type scheme to describe a formation of strip of localized
plastic flow (Section 2.1.2). Section 2.2 is devoted to consideration
of the phase transitions induced by fluctuations of linear defects
density and external stress (strain). In Section 2.2.1, we present a
phenomenological model to account the evolution of defects with
different dimensionalities in the field of plastic deformation on the
basis of the assumptions of ‘predator—prey’ model. We show the
correlated fluctuations stimulate the re-entrant noise-induced phase
transitions familiar to the formation of different types of defect
structure (Section 2.2.2). In Section 2.2.3, we consider the effect of
correlations between fluctuations of system parameters. We demon-
strate that this cross correlations can be a reason of the formation
of different types of defect structures. As will be shown, the or-
dered state appears in the course of a chain of phase transitions.
Moreover, we will show that different ordered structures arise as a
result of the first- and second-order phase transitions.
Section 3 deals with the investigation of the systems where the
avalanche-like dynamics is observed. In condensed matter, such be-
haviour is realized in the process of explosive crystallization. In
4 D. O. Kharchenko
Section 3.1, we present the mechanisms of process of explosive
crystallization in which the formation of avalanche is possible. The
experimental data is discussed in Section 3.1.1. The analytic de-
scription of explosive crystallization in the framework of the Lo-
renz-type model is shown in Section 3.1.2. We treat the transition
to such a regime as the first order phase transition where the veloc-
ity of crystallization sharply increases from zero to the fixed mag-
nitude. The deterministic picture of such a transition assumes the
special form of relaxation process for the velocity of crystallization.
However, such approach does not allow describing the distribution
over the avalanche size or over the velocity. The simplest approach
to account the first-order transition and to obtain above distribu-
tion is achieved considering the fluctuations of main modes of the
model (Section 3.2). To this end, we introduce a unified Lorenz sys-
tem with a fractional feedback and consider the noise-induced tran-
sitions to obtain the phase diagram giving the domain in which the
regime of self-organized criticality is realized. The above generali-
zation puts forward basis of Section 3.2.1 devoted to consideration
of avalanche ensemble. Following famous Edwards paradigm [13,
14], an effective scheme addressed to the nonextensive thermody-
namics [15] is proposed to determine a time-dependent distribution
over energies of equipped sites in the explosive crystallization proc-
ess. Generalization of the Edwards scheme to nonstationary nonex-
tensive systems is reached on the basis of the fractional Lorenz sys-
tem where the avalanche size plays a role of the order parameter,
nonextensive complexity is reduced to the conjugate field and the
nonconserved energy of the equipped sites is the control parameter.
Within the framework of this approach, the phase diagram is calcu-
lated to define the different domains of system behaviour as a func-
tion of noise intensities of the above values. As a result, we arrive
at a natural conclusion that the power-law distribution inherent in
the self-organized criticality regime is caused by noise of the en-
ergy. In Section 3.2.2, we show that this distribution is the solution
of both nonlinear Fokker—Planck equation, which appears in the de-
scription of nonextensive systems [15], and fractional Fokker—
Planck equation inherent in Lévy-type processes [16]. We obtain re-
lations between the main exponents of self-organized criticality re-
gime, dynamical and geometry exponents of the model. Finally,
main results are discussed in conclusion.
Appendix contains the basic properties of fractional integral and
derivative.
2. COOPERATIVE BEHAVIOUR OF DEFECTS’ ENSEMBLE
As well known, the irreversible reconstruction of crystal structure
Self-Organization of Condensed Matter in Fluctuating Environment 5
involves a permanent deformation process. Such a phenomenon is
known as a plastic deformation. At weak deformation, the descrip-
tion of processes of plastic flow can be reached in the framework of
typical scheme of evolution of ensemble of plastic deformation car-
riers in external field [17—23]. In such a case, the density of defects
is a small and they behave themselves independently: in the course
of plastic flow, defects obey the action of external field and interac-
tion forces between them. When the deformation is increased, the
density of carriers becomes so large that the assumption of autono-
mous behaviour of defects is failed [11]. In such an approach, it is
necessary to consider a coherent behaviour of defects whose evolu-
tion should be examined. A transition from an autonomous regime
to a coherent one results in formation of stripe of plastic flow due
to autocatalytic generation of dislocations and vacancies or intersti-
tial atoms. The transition to the coherent regime is a complex proc-
ess where different defect structures are observed. In the course of
plastic deformation, a strong linkage between elements of different
defect structures is realized. It results in appearance of regular spa-
tial-temporal structures on a macroscopic scale. For instance, an
initially homogeneous crystallite is transformed to a weak off-
oriented cellular structure, which transforms into a fragmented
structure at large deformation magnitudes [11, 24] (see Fig. 1).
A mechanism of generation and interaction of defects can be the
following: (i) an interaction between dislocations of different sliding
systems; (ii) formation of generating thresholds, which produce di-
pole configurations of dislocations; (iii) generation of atom defects
as a climb of edge component of elementary thresholds. Point de-
fects themselves have not influence on the plastic behaviour of crys-
tal directly, but their annihilation results in edge dislocations
creation. On the other hand, a coupling point defects with disloca-
tions derives to the plastic deformation contribution with the de-
formation velocity being proportional to production of concentra-
tions of linear and point defects.
To formulate a related scheme we give the most popular syner-
getic models (Section 2.1) to study the plastic deformation and to
treat some special properties of this process. The two-parameter ap-
proach based on the ‘activator—inhibitor’ model is considered in
Section 2.1.1. Here, the role of ‘activator’ is played by the deforma-
tion, whereas a tension is an ‘inhibitor’ to be a conjugate parame-
ter. A collective mode of linear defects in the course of plastic de-
formation is studied in Section 2.1.2. We show that such a model
has a form of the Lorenz-type system, which is the simplest way to
consider a self-organization of defect structure. In Section 2.1.3, we
present the model to account the evolution of defects with different
dimensionalities in the framework of three-parameter synergetic
6 D. O. Kharchenko
conception. The noisy system is considered in Section 2.2. As
shown, the noise induces phase transitions corresponding to the re-
building defect structure in the course of deformation process. The
systems with one and two noises are considered in Sections 2.2.1
and 2.2.2 respectively.
2.1. SYNERGETIC CONCEPTION OF PLASTIC DEFORMATION
The developing the domain of localized deformation results to the
regime where a relaxation processes are realized. It means that the
picture of kinetics of defects ensemble is relevant to the correspond-
ing dynamical problem consideration. Such a situation is realized at
large enough values of density of defects whose coherent evolution
results in collective effects, which cannot be considered in usual
way. To perform the adequate description one needs to pass to the
synergetic scheme to represent time dependences of density of co-
herent ensemble of deformation carriers in the field of deformation.
2.1.1. Activator—Inhibitor Model
The one of the simplest approach to study the process of plastic de-
Fig. 1. Electron microscope views of dislocation structure of ordered Ni3Fe
alloys for differential strains (a—d corresponds to deformation θ = 0.05,
0.05, 0.16, 0.28) [24].
Self-Organization of Condensed Matter in Fluctuating Environment 7
formation is the using the two-component ‘activator—inhibitor’
model where the deformation ε plays a role of an activator and the
shear component τ of the tension is an inhibitor [1]. Such a model
allows one to consider the inhomogeneity of deformation process.
By this, the autocatalytic factor is deformation to be stimulus for
the process where every shear step initiates the corresponding proc-
ess in a neighbour domain with characteristic length , and the de-
formation rate is comparable with the velocity of dislocation motion
dv . On the other hand, at each act of shearing process the elastic
energy is redistributed over the bulk by means of elastic wave of
acoustic emission to produce a tension decrease. The last results in
damping process that has an effective radius comparable with a size
of the system L >> and a rate of order of the velocity of elastic
waves s dv v>> ( sv is the sound velocity). If we assume the envi-
ronment, where the plastic deformation occurs, is of a mosaic of
differently deformed domains, then we can suppose that deforma-
tion is a function of time and space, i.e. ( )tε = ε , r .
To find related field equation we suppose the deformation
e vε = ε + ε is composed in elastic eε and viscous vε components. Ac-
cording the definition of the former, we have v vε = ν∆ε , where ν is
a kinematic-viscosity coefficient. As a result, we derive to the law
of plastic flow as a diffusion process:
( ) ,fε = ε, τ + ν∆ε (2.1a)
where ( )f ε, τ –a generalized force. In a similar manner, we can pos-
tulate the law of motion for a tension field
( )g Dττ = ε, τ + ∆τ. (2.1b)
where ( ),g Dε, τ stand for generalized force and inhomogeneous co-
efficient respectively. It is easy to see the estimations dvν ∼ ,
sD v L∼ . Further, our consideration reduces to definition of explicit
forms for generalized forces– ( )f ε, τ , ( )g ε, τ .
2.1.2. Lorenz-Type Model
The approach given by equations (2.1) does not take into account
the dislocation mechanism of plastic deformation. At large densities
of dislocations ρ the incoherent coupling causes appearing the col-
lective component colρ ≤ ρ , which behaves itself in autonomous re-
gime. Here, the interaction fields of dislocations are became of or-
der of external tension [10] and a long-life hydrodynamic mode
arises with relaxation time 1 1310 sd Dt − −>> ω ≅ ( Dω is the Debye fre-
quency) and amplitude cold b∼ ρ . In autonomous regime, the re-
8 D. O. Kharchenko
laxation of this mode is described by the Debye law dd d t= − / . How-
ever, the coherent coupling between defects causes a positive feed-
back between the tension and the deformation to appear the term
being proportional to τε . Accordingly to the Debye equation for the
tension, ( )e tττ = τ − τ / , which relaxes to the stationary value 0eτ ≠ ,
the negative feedback dε of mode of defects in the field of defor-
mation is added. To construct the complete system of equations we
take into account the Maxwell equation for the viscoelastic matter.
As a result, we arrive at the following Lorenz-type system [25]:
( ) / ,
,
/ .
e
d d
t g d
d d t g
t g d
τ τ
ε ε
τ = τ − τ − ε
= − / + τε
ε = −ε +
(2.2)
For the sake of simplicity, we consider here the homogeneous model
and put 0D Dε τ= = . The scales of the relaxation times tτ , dt , tε are
as follows: for tension relaxation time one has 1
Dt −
τ ∼ ω ; the relaxa-
tion time of collective mode dt is defined by means of the thermoac-
tivated processes of migration, so that 1 exp( / )d Dt Q T−∼ ω where Q is
the energy of activation, T is the temperature; the relaxation time
of deformation is tε = η/µ , where η and µ are a shear viscosity and
an elasticity modulus. It allows treating the deformation ε as the
slowest variable. Within the synergetic scheme, the slowest mode is
known as an order parameter, d plays a role of conjugate field, and
the tension τ corresponds to a control parameter. According to the
relation
,dt t tε τ>> , (2.3)
one can use the adiabatic elimination procedure. It allows reducing
the number of evolution equations and passing from the dynamical
system (2.2) to the one-parameter equation
( )V
tε
∂ ε
ε = − ,
∂ε
(2.4)
where the synergetic potential reads
2 22 1
( ) 1 ln 1 , , .
2 e d m
m m d d
V g g
t t g g
−
ε
τ τ
ε ε ε ε = − θ + θ ≡ τ ε ≡ ε ε
(2.5)
Respectively, the values of τ and d are given by the expressions:
2 2
, .
1 ( / ) 1 ( / )
e
d d e
m m
d t g
τ ε
τ = = τ
+ ε ε + ε ε
(2.6)
Self-Organization of Condensed Matter in Fluctuating Environment 9
It can be shown that the potential (2.5) is addressed to a second-
order phase transition. Indeed, the form of ( )V ε depends on the tensor
of external tension extσ which defines the stationary value ( )exteτ σ of
shear component of internal tensions. At values extσ providing 1θ < ,
the potential (2.5) monotonically increases from the minimum 0 0ε = .
It means the relaxation to the state with 0 0ε = where the collective be-
haviour of defect ensemble has no effect on the value of plastic defor-
mation. Physically, the small values of θ correspond to the relaxation
and strengthening processes. At 1θ > , the potential (2.5) acquires
minimum at nonzero value
1 2
0 ( 1)m
/ε = ε θ − . It means appearing the
plastic deformation caused by the coherent behaviour of ensemble of
dislocations and vacancies. An effective time
1( 1)efft t −
ε= θ − of passage
into the coherent regime increases monotonically at 1θ → .
The main results of above scheme are as follows: (i) autocatalytic
generation of defects causes the coherent coupling, and defects behave
themselves in the collective manner in course of the time efft t∼ under
effect of external tension ( )e extτ σ ; (ii) the deformation ε defined
through the autonomous defects acquires contribution, which in-
creases in course of time efft t>> to the stationary value 0 0ε ≠ . The
proposed scheme can be generalized obviously through the considera-
tion of the stripe of localized plastic deformation that requires ac-
counting the spatial distribution ( )ε r formed at efft t>> .
2.2. NOISE-INDUCED PHASE TRANSITIONS
2.2.1. Evolution of Defects’ Ensemble with Different Dimensionalities
This model generalizes the well-known ‘predator—prey’ model where a
role of ‘predator’ is played by dislocations, and the point defects serve
as ‘prey’. Moreover, we introduce the deformation as a control parame-
ter in addition to above parameters. The main assumption is the linear
form of equations for the density of dislocations ρ and density of the
point defects c. In these equations, we account the Debye relaxation
processes for both the order parameter and conjugate field.
It will be convenient to introduce the order parameter (hydrody-
namic mode amplitude) being the collective mode d. According to the
assumption that dislocation loops are generated through the annihila-
tion of point defects, we set an equation for the relative density of dis-
locations in the form
dt d d c= − + α . (2.7)
Here, dt is the relaxation time, α is the intensity of the generating
the dislocation loops through the transformation of point defects
ensemble. An equation for the point defects is as follows
10 D. O. Kharchenko
ct c c d= − + β , (2.8)
where ct is the relaxation time for point-defects’ concentration, β is a
function of the deformation ε to be measure of the intensity of the
point defects generation due to the climb of the dislocation thresholds
in the field of plastic deformation. An equation for the deformation ε
should contain the relaxation term to define a decrease in the deforma-
tion to the nonzero value 0ε , i.e. 0tεε = ε − ε . Moreover, we should take
into account the term cdε ≅ κ of the deformation velocity given by the
flow of defects produced in the process of plastic deformation. The
proportionality coefficient ( )Tκ = κ Ωτ/ depends on the atom volume
Ω , the temperature T and tension τ . Next, we suppose that the de-
formation occurs in such a way that assumption constκ = is satisfied.
Combining corresponding expressions, we find the equation for the
control parameter in the form:
0t cdεε = ε − ε + κ . (2.9)
Equations (2.7)—(2.9) are complete set to describe the evolution
process of defect structure in the field of plastic deformation. One
remark should be done about the deformation dependence of the co-
efficient β in Eq. (2.8). Indeed, there is the nonlinearity in the
third equation to be a reason for the self-organization in the sys-
tem. Such a positive feedback should be compensated by a negative
feedback in the second equation as a manifestation of the Le Chatel-
ier principle. For this purpose, we expand β in series up to the first
order terms: 0 0 1β ≅ β ε − β ε with 0 1 0β > β > . Then, with accounting
the diffusion character of the evolution of the defects and Eq.
(2.1a), we get finally
2
2
0 0 1
2
0
( )
d d
c c
t d d c d
t c c d c
t cdε
= − + α + ∆ ,
= − + β ε − β ε + ∆ ,
ε = ε − ε + κ + ∆ε,
(2.10)
where d , c are corresponding diffusion lengths and is a charac-
teristic length of deformation field variation. The last term in first
equation reflects processes of interactions of dislocations. It is
physically important that the obtained system (2.10) incorporates
the defect densities with different dimensionalities.
Considering system with hierarchical slaving inhomogeneities for
which ,d c>> and making use of the adiabatic elimination proce-
dure where , c dt t tε << , we derive to the equation for dimensionless
order parameter mx d d= / ( 2
11md = /β κ ) in the form
Self-Organization of Condensed Matter in Fluctuating Environment 11
2 2 2
0 1 0 0 1
( ) 1
( ) ln(1 ) ,
2
( ) .
d d
d V x
t x x V x x x
dt x
∂ = − + ∆ , = − θ + ∂
θ ≡ α β − β ε , β > β
(2.11)
With accuracy of the inhomogeneity term, this equation has the
form of the Lorenz-type equations (2.4), (2.5). It was shown the sta-
tionary magnitude of the order parameter 0x monotonically in-
creases in the course of plastic deformation as 1 2
0 ( 1)x /= θ − . Such a
situation corresponds to the well-known experimental data for the
scalar density of dislocation vs. deformation. However, the model
(2.10) predicts an increasing the dislocation density only and does
not give an information about reconstruction of defect structure. In
reality, there are transitions from chaotic configuration to the tan-
gles and then to a cellular structure. In the cellular structure, we
have dislocation clusters with different directions of the Burgers
vector b. To find the distribution of the clusters we consider spa-
tial-temporal evolution of defect structure and pass to a study of
the statistical system. Next, we are aimed to show correlated fluc-
tuations in the system parameters is one of the reasons of rebuilding
the defect structure in the course of the plastic deformation process.
To account the stochastic effects the model (2.10) should be re-
written in the following form:
2
2
0 0 1
2
0
( )
( ) ( )
( )
d d d d
c c c c
t d d c d r t
t c c d c r t
t cd r tε ε ε
= − + α + ∆ + σ ζ , ,
= − + β ε − β ε + ∆ + σ ζ , ,
ε = ε − ε + κ + ∆ε + σ ζ , .
(2.12)
Here, stochastic terms dζ , cζ , εζ define internal fluctuations of
the main modes with intensities 2
dσ , 2
cσ , 2
εσ . For the sake of sim-
plicity, we assume the stochastic forces are Gaussian distributed
with average ( ) 0tµζ , =r and correlation functions
}( ) ( ) ( ) ( ) { }t t C t t d cµ ν µν′ ′ ′ ′ζ , ζ , = | − | δ − , µ, ν = , , ε .r r r r (2.13)
In other words, we consider the time correlations only neglecting
space correlations. Because the consideration of heterogeneity of
main modes derives to a complex problem, we address the disloca-
tions contribution only. This means, as in Eq. (2.11), the condition
of ,d c>> .
At passage to consideration of stochastic terms in system (2.12), it
should be noted the simplest picture is related to white noise approach.
It means the Fourier components of these terms are frequency inde-
pendent. However, the real physical picture is characterized further by
coloured fluctuations whose spectrum incorporates an assigned fre-
12 D. O. Kharchenko
quency domain. The simplest model to study such fluctuations is the
Ornstein—Uhlenbeck process related to stochastic equation in the form
}( ) { },a t d cµ µ µµτ = −ζ + ξ , µ = , , εζ (2.14)
where
a
iτ is an autocorrelation time for the initial process ( )tµζ ,
whereas ( )tξ is a white noise with following properties: ( ) 0tµξ = ,
( ) ( ) ( )t t t tµ ν µ,ν′ ′ξ ξ = δ δ − . According to Eq. (2.14), the correlation func-
tions of coloured noises ( )tµζ are as follows:
( ) exp
c c
t t
C t t µ ν
µ,ν
µ,ν µ,ν
σ σ ′| − |′, = − , τ τ
(2.15)
where c
µ,ντ is the cross correlation time between the noises µζ and
νζ . At µ = ν , correlation times c
µ,ντ are reduced to the autocorrela-
tion times a
µ,µτ .
Following the assumption that the density of dislocations is a
slow mode, we suppose 0, 0ct c tε≅ ε ≅ , and 2 2 0c c∆ = ∆ε = . Hence,
Eqs. (2.12) yield
( ) ( ) ( )x f x D x g x tµ µ
µ
= + ∆ + ζ∑ (2.16)
where the deterministic force and multiplicative functions are
2 1( )
( ) 1 ( ) (1 ) ( ) ( );x c c
V x
f x g g x x g x xg x
x
−
ε
∂
= − , = , = + , =
∂
(2.17)
the form of the potential V is given by Eq. (2.11)
To explore the picture of noise induced phase transitions, we rep-
resent the equation (2.16) in the regular d-dimension lattice of
mesh size 1l∆ = . Then states of the system will be determined by a
set of scalar variables { }kx , 1 dk … L= , , , defined on the hypercubic
grid of the size L. Making use of this approach the evolution of
{ }kx is defined through the set of equations
( ) ( ) ( )
2
k k ij j k k
j
D
f x D x g x tx
d µ µ,
µ
= + + ζ∑ ∑ (2.18)
where the definition of the Laplacian operator on the grid is used
( )
( )
( 2 )ij nn j kj
j j nn k
D d
∈
= δ − δ∑ ∑ (2.19)
with ( )nn k being a nearest neighbours of the site k.
It is appeared the diffusion can be represented as an effective in-
teraction of nearest neighbours. Indeed, let’s introduce the har-
monic energy of such interactions by the potential
Self-Organization of Condensed Matter in Fluctuating Environment 13
2( )
2int j
j
a
V x x= −∑ (2.20)
where a is the rigidity of effective spring and the sum is carried out over
the locations jx of the nearest neighbours. In the framework of the mean
field approximation jx is replaced by the mean value jxη ≡< > after which
the interaction force int intf V x= −∂ /∂ acquires the form
( )intf D x= − − η (2.21)
where we have introduced the characteristic interaction coefficient
D az= (z is a number of nearest neighbours). Then, we obtain the
equation (space subscripts are dropped)
( ) ( ) ( ) ( )intx f x f x g x tµ µ
µ
= + ;η + ζ∑ (2.22)
whose form follows from Eqs. (2.18), (2.19) within the mean-field
approach. At this stage, Eq. (2.22) should be considered together
with Eq. (2.14) as a complete system.
2.2.2 Effect of Autocorrelations in System with One Noise
Here, we address to a situation where the intensity of one of noises is
much larger than others do. The typical picture of the defect structure
evolution is shown in Fig. 2 where we present the results of computer
simulation of Eqs. (2.18), (2.14) on the grid with 100L = . The analytic
treatment of such a system is achieved on the basis of the unified col-
oured noise approximation. Formally, if we take the time derivative of
Eq. (2.22), replace firstly ζ in terms of ζ and ξ from Eq. (2.14) and
then ζ in terms of x and x from Eq. (2.22), we can obtain the non-
Markovian stochastic differential equation
2 ln 1 ( ) ln ( )int
int int
f f
x x g f f x f f g t
x x g
+∂ ∂ τ − = − − τ + + + + ξ ∂ ∂
(2.23)
with aτ ≡ τ . According to the unified coloured noise approximation
[26, 27] we use the adiabatic elimination that neglects the terms
with x and 2x . Moreover, according to the Ito interpretation of
Eq. (2.14), the white noise ( )tξ (last term in Eq. (2.25)) should be
considered within the Ito calculus. Hence, we come to the equation
( ) ( ) ( ) ( ) ( )intx x f x f x g x tσ ;η = + ;η + ξ (2.24)
with the kinetic coefficient
14 D. O. Kharchenko
( ) ( )
( ) 1 ( ( ) ( )) ln
( )
int
int
f x f x
x f x f x
x g x
+ ;η∂
σ ;η = − τ + ;η . ∂
(2.25)
Since the stochastic equation (2.24) has continuous set of solu-
tions, one should determine their probability density function ( )P x
and then calculate the order parameter xη =< > . The evolution
equation for ( )P x is in the form [28]
( )
2 2
2
2
( ) ( ) ( ) ( ) ( ) ( )
2
P x t x x P x t x P x t
t x x
∂ ∂ σ ∂
, ;η = − φ ;η + ϕ ;η , ;η + ;η , ;η
∂ ∂ ∂
(2.26)
where auxiliary functions are introduced:
( ) ( ( ) ( )) / ( )intx f x f x xφ ;η = + ;η σ ;η , (2.27)
Fig. 2. Time evolution of domains starting in a completely random initial
configuration toward an ordered phase for spatially extended model given
by Eqs.(2.15), (2.19) for the additive noise on a square lattice (L = 100,
θ = σx
2 = 4.0, D = 5.0) at t = 0.0, 2.3, 10.0, 50.4 (a, b, c, d). Dark areas
correspond to a negative values of x, light areas to positive values.
Self-Organization of Condensed Matter in Fluctuating Environment 15
( )
( )
( ; )
g x
x
x
;η = ,
σ η
(2.28)
21( ) ( ) ln ( )
2
x x x
x
∂
ϕ ;η = − ;η σ ;η .
∂
(2.29)
The steady-state solution of Eq. (2.26) has a quasi-Gibbs form
1 2( ) exp ( )efP x N U x−
;η = − ;η /σ , (2.30)
2 2int
2
( ) ( ; )
2 ( ; ) 2 ln ( ) ln ( ; )
( )
x
ef
f x f x
U x dx g x x
g x
′ ′+ η ′ ′= − σ η + σ − σ σ η
′∫ (2.31)
where ( )N N= η takes care of normalization condition:
2( ) exp ( )efN dx U x
∞
−∞
η = − ;η /σ .∫ (2.32)
The order-parameter magnitude is defined through the self-con-
sistency condition:
( ) ( )xP x dx F
∞
−∞
η = ;η ≡ η .∫ (2.33)
The picture of the noise induced phase transitions is achieved by means
of the solutions of the self-consistent Eq. (2.33) and condition
0
( )
1
dF
d η=
η
=
η
(2.34)
to define the phase diagram. For systems where phase transitions
with symmetry breaking occur, coefficient Eq. (2.34) has always a
root 0η = ; nontrivial roots differ only in sign at 0 1dF d η=/ η | > and
define the ordered state.
Consider firstly the additive noise of the dislocation density
where the multiplicative function is 1xg = . Within the linear ap-
proach we get
2( 1 )x D x= θ − − | | .k kk (2.35)
Therefore, the solution 0x =k becomes unstable at kθ > θ where
21 Dθ = + | |k k . It means that the heterogeneity increases the criti-
cal magnitude of the control parameter kθ . On the other hand,
within the linear approximation the stochastic part does not affect
on the stability of the system.
From the phase diagram shown in Fig. 3, it is seen that re-
16 D. O. Kharchenko
entrant phase transition occurs when we change the spectral charac-
teristic τ (the domain of the ordered phase is situated above the
surface). Here at small τ we have the chaotic structure of defects
ensemble where directions of Burgers vector b are equivalently real-
ized. If we pass through the first threshold 1τ , the system becomes
as ordered due to the formation of a dislocation cluster with a fixed
direction of the Burgers vector. A further increase in τ (above 2τ )
allows the system to have clusters with two opposite directions of b
due to formation of a set of dislocation clusters. Varying the pa-
rameter D, we can change the size of fluctuation frequency window
where the ordered state exists and find the critical magnitudes
where one defect structure transforms to another.
Solution of the self-consistency equation (2.33) is shown in
Fig. 4, a. It is seen that an increase in τ narrows the domain of the
deformation θ where the ordered state exists. The solutions of
Eq. (2.33) are in good correspondence with result of computer simu-
lations in 2d grid (see Figs. 4, b, c).
Let us focus now on the affect of the noise of conjugate field be-
ing concentration of point defects with 2 1(1 )cg x −= + . Within the
Fig. 3. Phase diagram for the system with additive coloured noise is shown
at 2 1.0xσ = .
Self-Organization of Condensed Matter in Fluctuating Environment 17
linear stability analysis, we get renormalization of the critical mag-
nitude for the control parameter through the noise intensity 2
cσ .
Following the Novikov theorem [29], we obtain
2 2( 1 2 )cx D x= θ − − σ − | | .k kk (2.36)
Therefore, the coloured multiplicative noise shifts the transition
point to the magnitude of 2 21 2 c Dθ = + σ + | |k k .
a b
c
Fig. 4. Order parameter at the additive noise: a–dependences η(θ, τ) is
shown at D = 1.0, σx
2 = 9.0; b–order parameter and generalized suscepti-
bility vs. noise autocorrelation time τ at θ = 4.0, D = 0.7 (squares); θ = 4.5,
D = 1.0 (circles); θ = 4.0, D = 1.0 (triangles); c–order parameter and gen-
eralized susceptibility vs. control parameter θ (circles and squares corre-
spond to σx
2 = 0.01, D = 10.0, τ = 0.1 and σx
2 = 49.0, D = 10.0, τ = 0.1
(computer simulations on the grid 32×32).
18 D. O. Kharchenko
The corresponding phase diagrams are shown in Fig. 5 where, as in
previous case, both the control parameter θ and noise intensity
2
cσ are
varied (domains of ordered and disordered phases are indicated as ‘O’
and ‘D’). We should stress that in this case of the multiplicative noise
the domain of small values of the noise correlation time τ is essential. It
a
b
Fig. 5. Phase diagrams for the system with multiplicative colored noise:
a–dependences D(θ) are shown at τ = 0.1, σε
2 = 1.18, τ = 0.07, σε
2 = 1.0;
τ = 0.01, σε
2 = 1.0; (curves 1, 2, 3); b–dependences τ(θ) are shown at
D = 0.9, σε
2 = 1.18, D = 0.9, σε
2 = 1.0, D = 1.2, σε
2 = 0 (curves 1, 2, 3).
Self-Organization of Condensed Matter in Fluctuating Environment 19
is seen from Fig. 5, a the ordered phase is realized within bounded do-
main of control parameter and diffusion constant. At extremely small
τ, there is only one transition along D axis (curve 3), whereas a slight
increase in τ shrinks the region of ordered phase (curve 1, 2). Accord-
ing to Fig. 5, b, the system with multiplicative noise is more sensitive
to the variation of the noise autocorrelation time. Here, with an in-
crease in τ the ordered state 0η ≠ is realized in the shrinking domain
of values of the deformation θ. An increase in D expands the domains
of noise correlation values and increases the size of the control parame-
ter values where the ordering occurs. Notice that in the case of multi-
plicative noise only one transition occurs along τ axis, whereas in the
case of additive noise the system undergoes two transitions (re-entrant
ordering transition).
Corresponding behaviour of the order parameter is shown in Fig. 6
where both analytical and numerical results are presented in Figs. 6, a,
b). Here, an increase in the deformation θ results in a rebuilding the
defect structure. Further, an increase in θ suppresses the process of
the dislocation cluster formation with unique direction of the Burgers
vector.
The noise of the control parameter changes crucially the system be-
haviour. Indeed, related amplitude gε becomes trivial at 0x = which
corresponds to formation of the absorbing state [5, 30]. Here the sta-
tionary solution of the Fokker—Planck equation becomes unnormaliz-
able [31] to mean the freezing the configuration points of the phase
space when the domain 1x << is reached [32, 33]. Therefore, in course
of time, the density of dislocations attains the value corresponded to
chaotic structure, and noise does not allow preceding an ordering proc-
ess. Indeed, in the limit 0x → the distribution function is character-
ized by the power law construction
22(1 ) / 3 0DP x x− + −θ σ −∝ , → . (2.37)
Due to 0D > , the logarithm singularity in Eq. (2.37) can be avoided
only at θ > D + 1 + σ2.
2.2.3 System with Two Noises
Considering a stochastic system with more than one noise, we have to
deal with an effect of correlation between them. As shown in Ref. [34],
the system with two noises behaves itself further in unusual manner.
One holds the opinion that cross correlation introduces only weak cor-
rections to results obtained for uncorrelated fluctuations.
Here, we demonstrate the crucial role of cross correlation between
fluctuations and show that they can change the order of phase transi-
tions. Because such a problem can not be solved correctly by using the
20 D. O. Kharchenko
standard methods based on the Novikov theorem [29] or the unified
coloured noise approximation [26, 27], we apply the cumulant expan-
sion method proposed by Van Kampen [35] and developed in Ref. [36].
To construct an equation for the probability density { }( , )kx tρ we
a
b
Fig. 6. Order parameter at the multiplicative noise: a–dependences η(θ, τ)
are shown at D = 1.0, σε
2 = 1.0; b–order parameter and generalized sus-
ceptibility vs. deformation θ (diamonds, circles and squares correspond to
τ = 0.1, D = 2.0, σε
2 = 25; τ = 0.1, D = 2.0, σε
2 = 1.0; τ = 0.1, D = 8.0,
σε
2 = 1.0 (computer simulations on the grid 32×32 are shown at the same
parameters as analytic dependences).
Self-Organization of Condensed Matter in Fluctuating Environment 21
exploit a conventional device and proceed from the continuity equation
{ } { }( , ) ( ( ))k k k
k k
x t x x t
t x
∂ ∂
ρ = − ρ , .
∂ ∂∑ (2.38)
The probability density function is given by the averaging over
noise, i.e. { } { }( , ) ( )k kP x t x t= ρ , . Inserting the time derivative from
Eq. (2.16) into Eq. (2.38), we have
{ } { }( ) ( )k k k k k
k k
x t L g x t
t x
µ µ
µ
∂ ∂
ρ , = − − ξ ρ , ,
∂ ∂∑ ∑ (2.39)
where the evolution operator is denoted as
2k k kj j
jk
D
L f D x
x d
∂
= + .
∂ ∑ (2.40)
According to the well-known cumulant expansion method [35] noise
averaging of Eq. (2.39) derives to the following expression for the
probability density { }( , )kP x t [34]
( ) ( )0 0
0
( ) k kL t L t
k k k
k
P L L C t t e L e d P
t
∞ −
µ µν ν
µ,ν
∂
= − + , − τ τ ∂
∑ ∑ ∫ (2.41)
where we put ({ } )kP P x t≡ , ,
(0) ( ),i k k
k k
L g g x
x xµ µ µ
∂ ∂
≡ =
∂ ∂
(2.42)
and assume the physical time to be much more some correlation
scale , , ,( , )tµ ν µ ν µ µτ >> τ τ . Then, expansion of exponents arrives at
the perturbation expansion
( ) ( )0 ( )
02
nn
k kj j k k
k j nk
D
P f D x L C L P
t x d
∞
µ µν µ = µ,ν
∂ ∂
= − + +
∂ ∂
∑ ∑ ∑ ∑ (2.43)
where moments of the correlation function are determined as
( )
0
1
( )n nC C t t d
n
∞
µν µν= τ , − τ τ.
! ∫ (2.44)
Respectively, operators ( )nL
ν
are defined through the commutator
( 1)( ) nn
i k kLL L −
µ µ = , . (2.45)
So, the first-order approximation yields
22 D. O. Kharchenko
(1)
2 2k k k kj j k kj j k
j jk k k k
D D
L g f D x f D x g
x x d x d x
µ µ µ
∂ ∂ ∂ ∂
= + − + .
∂ ∂ ∂ ∂∑ ∑ (2.46)
According to obtained expressions, the zero-order contribution of cor-
relations gives the following terms for the Kramers—Moyal expansion
[37]
(0) (0)
1
(0) (0)
2
k
k
k
k k
g
D C g
x
D C g g
ν
µν µ
µ,ν
µν µ ν
µ,ν
∂
= ,
∂
= ;
∑
∑
(2.47)
accordingly, a contribution of first-order terms gives
(1) (1)
1
(1) (1)
2
2 2
2 2
kk
k kj j k k kj j
j jk k k
k
k k kj j k k kj j
j jk k
gg D D
D C f D x g f D x
x d x x d
gD D
D C g f D x g f D x
d x x d
µν
µν µ µ,ν
µ
µν ν µ µ,ν
∂∂ ∂
= + + + ,
∂ ∂ ∂
∂ ∂
= + + + .
∂ ∂
∑ ∑ ∑
∑ ∑ ∑
(2.48)
Therefore, the effective Fokker—Planck equation reads
1 2( ) ( )k k
k k k
P D x D x P
t x x
∂ ∂ ∂
= − + , ∂ ∂ ∂
∑ (2.49)
here the drift and the diffusion coefficients are given by
( )
1 1
0
( )
2 2
0
( ) ( )
2
( ) ( )
n
k k kj j k
j n
n
k k
n
D
D x f D x D x
d
D x D x
=
=
= + + ,
= .
∑ ∑
∑
(2.50)
Integrating Eq. (2.49) over all variables, with the exception of
kx , and using the fact that the steady state properties are isotropic
and translationally invariant, one obtains the following stationary
equation for the one-site probability:
1 2( ) ( ) ( ) 0k k k
k
D x D x P x
x
∂
− + = ∂
(2.51)
where conditions ( ) 0, / 0i iP x P x= ∂ ∂ = at ix → ∞ are taken into ac-
count. Within the mean-field theory, the interaction term is in the
form 2 ( )D
kj j kd j
D x D x= η −∑ where the η value is defined from the self-
consistency equation (2.33). Then, equation (2.51) gives the stationary
distribution function
Self-Organization of Condensed Matter in Fluctuating Environment 23
1 1 1
2
20
( )
( ) ( ) ( ) exp
( )
x D x
P x N D x dx
D x
− − ′, η ′, η = η , η . ′, η
∫ (2.53)
Considering the uncorrelated noises, we start with expression
( ) ( )x xC t t t t,ε ε′ ′, = σ σ δ − . In order to suppress correlations defined
through the Novikov theorem [29], we put (0) 0xC ,ε = that derives to
the standard Fokker—Planck equation with independent noises.
Thereby, the terms appearing in Eq. (2.49) can be calculated to be
( )
(0) 2
1
(1) 2
1 int
g
D g
x
g
D g f f
x x
ε
ε ε
ε
ε ε
∂
= σ ,
∂
∂ ∂
= σ τ + ,
∂ ∂
(2.53)
( )
(0) 2 2 2
2
(1) 2
2
x
int
D g
D g g f f
x
ε ε
ε ε ε
= σ + σ ,
∂
= σ τ + .
∂
(2.54)
Inserting 1D , 2D into Eq. (2.52) and the obtained result into Eq.
(2.34), we get an equation to construct the phase diagram. A related
solution is shown in Fig. 7 where letters ‘O’ and ‘D’ denote the domains
of ordered and disordered phases respectively. Note that the self-
consistency equation has two nontrivial solutions, which are differing
in sign. Hence, following the standard scheme we classify this transi-
tion as phase transition with the symmetry breaking. The ordered
Fig. 7. Phase diagram at σx
2 = 1.0, τ = 0.01: curves 1, 2 correspond to
D = 0.9, D = 0.5.
24 D. O. Kharchenko
phase is seen to be realized at large values of the control parameter θ,
so that the noise induces the transition of a second order from the or-
dered phase to disordered one. Another conclusion is that the system is
ordered at large values of the coupling constant D. Thus, we arrive at
the standard synergetic picture of phase transition where the system
becomes ordered one when the control parameter increases. As shown
in Fig. 8, the order parameter η increases monotonically beyond the
critical magnitudes of θ and D.
Most interesting picture of noise induced phase transition can be
found when the noises are correlated. Taking into account terms in
Eqs. (2.47), (2.48) with 0c c
xετ ≡ τ ≠ , we obtain
( )
(0)
1
(1)
1
( )
( )
x
c x int
g
D g
x
g
D g f f
x x
ε
ε ε ε
ε
ε ε ε
∂
= σ σ + σ ,
∂
∂ ∂
= σ τ σ + τσ + ,
∂ ∂
(2.55)
( )
( ) ( )
(0) 2 1 1 2
2
(1) 2 1
2
2x x
c
x int int
D g g
D g g f f g g f f
x x x
− −
ε ε ε ε ε
−
ε ε ε ε ε ε
= σ σ σ σ σ + + ,
∂ ∂ ∂ = σ σ σ τ + + + τ + . ∂ ∂ ∂
(2.56)
In accordance with the self-consistency equation, the order parame-
ter η varies depending on the control parameter θ as is plotted in Fig. 9.
Such a behaviour is determined by the principle contribution into Eq.
(2.33) given by the zero-order term
(0)
xC ,ε . It contains the construction to
break the symmetry of probability-density function (2.53) even in the
Fig. 8. Order parameter η vs. the control parameter θ and the coupling
constant D at σx = σε = 1.0, τ = 0.01.
Self-Organization of Condensed Matter in Fluctuating Environment 25
case of 0D = , and therefore we always have solution of Eq. (2.33) such
as 0η ≠ . Next, we define such an ordered state as a matrix phase.
Let the additive noise intensity be fixed, whereas the intensity of
the multiplicative noise of the control parameter takes on increasing
values. Then, at small εσ (Fig. 9, a) the order parameter takes a non-
trivial positive magnitude at small values of the control parameter
where the ordered phase is stable. A further increase in θ brings the
system to the state with negative solution of the self-consistency equa-
tion. Physically it means a reorientation transition at rθ = θ where the
average Burgers vector of dislocation clusters change the direction.
This solution is unique until the control parameter reaches the critical
value cθ where the bifurcation occurs. Then, two positive solutions of
Fig. 9. Order parameter vs. the control parameter at σx = 1.0, τ = τc = 0.01,
D = 0.9: a–σε
2 = 0.09; b–σε
2 = 1.0; c–σε
2 = 1.69.
26 D. O. Kharchenko
Eq. (2.33) appear which correspond to a metastable state formation. In
such a case, we conclude that the positive solution at small θ and nega-
tive one at large θ means the matrix phase. A new phase formation can
be addressed to branches of the metastable state.
Following the standard theory, we can classify the appearance of new
phase as a biased phase transition. According to Fig. 9, b, the re-
entrance of the new phase formation is observed when we increase the
intensity of the multiplicative noise. Here, the new phase is realized at
, c
c
θ ∈ θ θ . In the domain of small θ, the hysteresis loop is observed,
and the order parameter takes the negative magnitude in discontinuous
manner. Such a metastable state corresponds to a situation where the
Burgers vector of dislocation clusters can have two opposite directions
and some amount of clusters with fixed direction of b can be metastable.
Besides, the new phase at large θ is formed due to the bifurcation. Both
metastable phases appeared in such phase transitions have different
mechanisms of their formation. In such a case, above-mentioned phases
can be associated with different defect structures appearing in the
course of plastic deformation. Finally, a further increase in the noise
Fig. 10. Order parameter vs. the control parameter at σx = σε = 1.0, τ = τc: a–
D = 0.5; b–D = 1.2.
Self-Organization of Condensed Matter in Fluctuating Environment 27
intensity leads to the disappearing hysteresis loop at small θ, and the
matrix phase is characterized by the negative magnitude of η.
The influence of the spatial coupling constant D on the picture of the
system behaviour is shown in Fig. 10. It is seen, that at small D (Fig. 10,
a), we have unique solution of the self-consistency equation which de-
fines the matrix phase. Here the order parameter changes the sign
when we increase the control parameter. At intermediate D, the new
phase appears due to bifurcation and at large D the branches of the new
and matrix phases are merged. It results to formation the hysteresis
loop on which the order parameter changes the sign. The related solu-
tion for the phase diagram arrives at the picture shown in Fig. 11 where
the control parameter θ is built as the function of the multiplicative
noise intensity
2
εσ at different values of the coupling constant D and
the noise cross correlation time cτ . We denote the domain of the matrix
phase as ‘P’ and ‘N’ to indicate the positive and negative magnitudes of
the order parameter; the new phase we denote as ‘N’. The appearance of
the domain of the new phase allows to classify such transitions as first
order phase transitions. The main peculiarities of the system’s behav-
iour are as follows: (i) an increase in the cross correlation time cτ is ac-
companied by extending the domains of a metastable state (cf. Figs. 11,
a, b); (ii) an increase in the spatial coupling constant D is accompanied
by the increase of the domain of a new phase (‘N’) (cf. Figs. 11, c).
The performed analysis shows the crucial role of fluctuations. Con-
sidering the one coloured noise in Section 2.2.2, we address the situa-
tion where reconstruction of defect structure is observed. Accounting
the correlation between noises in Section 2.2.3 allows us to consider
such intermediate structures whose formation is shown as a chain of
phase transitions during the process of plastic deformation.
3. REGIME OF SELF-ORGANIZED CRITICALITY
Among the equilibrium phase transitions or noise-induced ones, a spe-
cial regime of the system behaviour is known as a self-organized criti-
cality (SOC). It explains spontaneous (avalanche-type) dynamics,
unlike the typical phase transitions that occur only when a control pa-
rameter is driven above a critical value. A main feature of the systems
displaying SOC is their self-similarity that leads to a power-law distri-
bution over avalanche sizes. The SOC behaviour appears in a vast vari-
ety of systems, such as real sand pile (ensemble of grains of sand mov-
ing along increasingly tilted surface) [38—41], intermittency in bio-
logical evolution [42], earthquakes and forest-fires, depinning transi-
tions in random medium and explosive crystallization [43].
In this Section, we present an analytical approach, which is able to
describe the behaviour of a whole avalanche ensemble in a phenomenol-
ogical manner considering the process of explosive crystallization.
28 D. O. Kharchenko
In Section 3.1, we use that approach to describe the explosive crystal-
lization process, which manifest the SOC regime. In Section 3.1.1, the
experimental data is presented to show that a course of explosive crys-
tallization of ultradisperse amorphous materials is determined by high
density of crystalline phase embryos. The crystallization phase spread-
Fig. 11. Phase diagram at σx = 1.0, τc = 0.01: a–D = 0.9, τc = 0.05; b–
D = 0.9, τc = 0.01; c–D = 0.95, τc = 0.01.
Self-Organization of Condensed Matter in Fluctuating Environment 29
ing is similar to percolation cluster formation under consideration of
liquid flow in a random medium [44]. Section 3.1.2 is devoted to inves-
tigations of conditions of this picture as a result of self-organization is
carried out. This process is fixed by the velocity of crystallization front
motion, its temperature, and difference of the thermodynamic poten-
tials of amorphous and crystalline states. In Section 3.2, the direct ex-
amination of explosive crystallization is carried out as a self-organized
criticality process caused by fractional feedback and stochastic influ-
ence of the main modes of the system. This assumption allows us to de-
scribe the subcritical regime of the avalanche formation in natural
manner. The above generalization puts forward basis of Section 3.2.1
devoted to consideration of avalanche ensemble. Following famous Ed-
wards paradigm [13, 14], an effective scheme addressed to nonexten-
sive thermodynamics [15] is proposed to determine a time-dependent
distribution over energies of equipped sites in the process of crystalli-
zation. In Section 3.2.2, we show that this distribution is the solution of
both nonlinear Fokker—Planck equation that appears in the description
of nonextensive systems [15], and fractional Fokker—Planck equation
inherent in Lévy-type processes characterized by a dynamical exponent
z [16]. As a result, we obtain relations between the exponent τ of the
distribution inherent to the SOC regime, fractal dimension D of phase
space, characteristic exponent of multiplicative noise, a number of gov-
erning equations needed to present self-consistent behaviour in SOC
regime, dynamical exponent z, and Tsallis nonextensivity parameter q.
3.1. Formation of Avalanche in the Explosive Crystallization Process
The metastable amorphous films are obtained usually by quenching of
a melt or by steam condensation on a cold substrate [45—46]. Experi-
ments show a vast variety and complicated character of the subsequent
transitions into the stable crystalline state. If the film thickness is so
small that the crystallization heat can be absorbed by a thermostat, the
ordinary cold crystallization mechanism works [12]. So, in the crystal-
lized films of semiconductors, the undulating surface is developed un-
der the formation of combs, whose long axes are perpendicular to the
direction of the crystallization front movement [47]. As is known, at
small number of the crystal embryos, this mechanism is realized if
both the nucleation frequency of crystallization centres J and the ve-
locity of crystal growth u are very small.
Sometimes, emerging explosive crystallization can be initiated by the
local heating (for example, by laser or electron impulse). Such a scenario
takes place in the case of instability appearance of the interfacial bound-
ary motion due to both the heat exchange with substrate and the influ-
ence of laser radiation [47]. This instability is ensured with a non-linear
dependence of the crystal growth velocity u as a function of tempera-
30 D. O. Kharchenko
ture. Moreover, the crystallization front instability can be fluctuating
in character that is manifested in the experiments with undulation of
crystallized surface [43]. Such behaviour appears in the partially crys-
tallized film or at the incomplete crystallization of amorphous phase.
Another scenario is observed in the amorphous medium in which as
the crystal growth velocity u, so the nucleation rate J of embryonic
crystals are sufficient large. Indeed, at low temperatures, the quanti-
ties J and u increase with temperature growth, so that the self-heating
stimulates the crystallization. Therefore, the increase of a film thick-
ness can lead to a situation, when the crystallization heat cannot be ab-
sorbed by an environment that causes the heat instability [48]. As a re-
sult, the spontaneous transition to regime of the explosive crystalliza-
tion can be provided by the heating effect. Examples of such amor-
phous mediums are the amorphous ice layers, the some organic matters
[12, 45], and the layers of the germanium amorphous ultradisperse
powder with the admixture of the crystalline phase [45, 46].
Fig. 12. The patterns which arise in the layer of amorphous powder of Ge
at the explosive crystallization: (a) from single center; (b) from several
centers which arise spontaneously.
Self-Organization of Condensed Matter in Fluctuating Environment 31
3.1.1. Experimental
The experimental study of influence of the crystal inclusions distrib-
uted in the amorphous phase volume in the kinetics of the explosive
crystallization is carried out with the germanium [46]. Unfortunately,
there is no information in literature about spontaneous rise of the ex-
plosive crystallization in continuous amorphous thin films of germa-
nium. It is known only that the maximum value of formation fre-
quency of crystalline embryos is
14 1 3
max 10 s cmJ − −∼ in the supercooled
germanium and corresponds to the more high temperatures than real-
ized at the explosive crystallization process [12]. Therefore, in the
germanium amorphous films the natural process of crystals nucleation
has not enough intensity for significant influence in the explosive
crystal growth. Let us point out in this connection that in amorphous
ice layers, where the spontaneous explosive crystallization takes place,
one has
20 1 3
max 10 s cmJ − −∼ .
In order to intensify the mentioned influence of crystal inclusions,
the experiments were carried out with the thin films of amorphous
nanopowders with admixture of large number of smallest crystals hav-
ing not more 3—10% of total mass. The layers of amorphous powder
with the characteristic size of particles 3—10 nm were obtained by
thermal evaporation and following condensation of germanium in the
atmosphere of inert gas at the pressures 10—100 Pa. Changing the
evaporation intensity allows us to regulate the part of the crystalline
particles in amorphous powder. Another peculiarity of our experi-
ments is that the substrates absorb significantly smaller heat due to
porosity of amorphous films.
The spontaneous explosive crystallization has been observed in layers
of nanopowders with the thickness 0.01—0.1 mm at the substrate tem-
perature 300—400 K. In dependence on initial concentration of crystal-
line phase, the movement velocities of crystallization wave have been
changed in the range 0.01—0.1 m/s. Contrary to the transverse undula-
tion being inherent in the usual crystallization mechanism, in our case
the front movement leads almost always to formation of ‘twigs’ along
the movement direction. The characteristic pictures of explosive crys-
tallization in the powder layers are shown in Fig. 12, where the light
background corresponds to non-crystallized domains. It is seen that the
rise of explosive crystallization avalanches occurs from the single cen-
tres, which act as original embryos. The cornerstone of our observation
is that the crystalline phase distribution has a fractal character being
similar to the pattern appeared in diffusion limited aggregation [8].
3.1.2. Conditions to the Explosive Crystallization Process
The experimental data show that two mechanism of amorphous mate-
32 D. O. Kharchenko
rial crystallization, depending on external conditions and presence of
crystalline phase embryos, are possible: the slow growth of a cold crys-
tal and the explosive crystallization that is caused by the phase trans-
formation heating. According to [46] the transition between these re-
gimes is jump-like in character, as a first-order phase transition. We
will show below that such a transition is caused by the system self-
organization due to the positive feedback between the heating and
growth velocity of crystalline phase.
To analyze the problem, the synergetic approach is used where the
time dependences of the crystallization front velocity ( )u t , its tem-
perature ( )T t , and the specific crystallization heat φ (thermodynamic
potential, which can play a role of entropy as a conjugate parameter to
the temperature). The equations define these dependences to take into
account their dissipative character and the positive feedback between
quantities u and φ that is the reason for self-organization. On the other
hand, in order to provide the stability of a system we introduce also the
negative feedback between u and T. In the framework of proposed con-
siderations, the evolution equations for u, T and φ should coincide
formally with the Lorenz system (see Eq. (2.2)) that is the simplest way
to describe the self-organization process [2]. Here instead of ε, d and τ
we use the crystallization front velocity u, temperature T and potential
φ. In the adiabatic limit of the noiseless systems where the relaxation
time for the order parameter (being the velocity u) tu is larger then re-
laxation times Tt , tφ for the conjugate field T and control parameter φ,
the Lorenz system describes the second-order transition. However, the
cold crystallization mode transforms into the explosive one in accor-
dance with the first-order mechanism. To avoid this discrepancy, the
simplest approximation was used [43]
( )20
1 1
1
1ut t u uτ
κ = + ,
+ /
(3.1)
characterized by the positive constants 0, t κ , and uτ . Within the
framework of the adiabatic approximation 0, Tt t tφ << , the obtained
Lorenz system is reduced to the Landau—Khalatnikov equation
0t u V u= −∂ /∂ . (3.2)
Its form is determined by the effective potential
( )
2
22 21
ln 1 ln 1 ,
2 2
V u u u
κα = − θ + + + /α (3.3)
where ( ) 1
0 0/ T u Th h t t A A
−θ ≡ φ ∆ , ∆ ≡ is the scale defined specific crystal-
lization heat, the quantity V is measured in units of
2
m Tu A Aφ= . For
small values of θ, the curve of the V vs. u dependence has a monotoni-
Self-Organization of Condensed Matter in Fluctuating Environment 33
cally increasing shape with its minimum at point 0u = that corre-
sponds to the cold crystallization mechanism. At
0
cθ = θ where
0 2 21 ( 1) 2 1c
θ ≡ + α κ − + α κ − α , (3.4)
a plateau appears, which is transformed (for
0
cθ > θ ) into a minimum
corresponding to the velocity 0eu ≠ and a maximum at point
mu sepa-
rating a minima which meet to the values 0u = and eu u= . When the
parameter θ increases still further, the minimum at point eu u= is
lowered and the height of barrier at
mu u= decreases, vanishing at the
critical value 1cθ = + κ .
The steady-state values of the crystallization-front velocity have the
form
( )
( ) ( )
2
00
1 21 22
00
2 21
00 2
1 1 ( )
1 1
m
e cu
u u
u
//
α
= + θ − θ ,
≡ θ − − + κ α .
∓
(3.5)
If the system’s energy increases slowly, the jump from zero to 002u is
observed at point cθ = θ and then the value eu increases smoothly. If
the parameter θ goes downward quasi-statically, the crystallization
front velocity eu smoothly decreases up to the point, where
0
cθ = θ and
00eu u= , and then jump-like goes down to zero. The hysteresis of such
type takes place only at the presence of energy barrier inherent in a
first-order transition and is manifested itself if only the parameter
mu uτα = / is smaller than unity.
The above analysis shows that the effective potential ( )V u has the
barrier separating the cold and the explosive crystallization modes. As
heat density θ becomes greater than the critical value cθ , this barrier
disappears. Thus, at cθ < θ , the transition to the explosive crystalliza-
tion mechanism requires the penetrating of energy barrier and at op-
posite case it realizes spontaneously. The first of appointed situations
takes place in the case when the explosive crystallization is initiated by
an external beam. With the increase of the coating thickness, the crys-
tallization heat cannot be absorbed by substrate and parameter θ in-
creases. This leads to that the value cθ (at which the function ( )V u
loses barrier) is reached at the critical film thickness and the system
transforms into the explosive crystallization regime spontaneously.
3.2. Self-Organized Criticality of the Explosive Crystallization Process
The presented approach does not incorporate the diffusive peculiari-
ties of the crystallization process indicated in experimental data dis-
cussion. The simplest model to account the diffusive nature of the
34 D. O. Kharchenko
process is the introducing stochastic sources of the Gaussian character
into evolution equations. In this Section, we take into account additive
noises of the main modes of the self-consistent system. By this, an in-
crease in the noise intensities causes avalanche emergence even in non-
driven systems, where the control parameter noise plays a crucial role.
A fluctuation regime of this type corresponds to the case, where a dis-
tribution of the order parameter appears in an algebraic form with in-
teger exponent [49]. In order to not being restricted to such a particu-
lar case, we introduce a unified Lorenz system with a fractional feed-
back. This assumption allows us to describe a subcritical regime of the
avalanche formation in natural manner.
We now focus on the affect of additive noises of the velocity compo-
nent u, temperature T, and the thermodynamic potential φ. To this
end, we should add stochastic terms
1 2
uI
/ ξ ,
1 2
TI
/ ξ ,
1 2I /
φ ξ into correspond-
ing equation of motion of main modes. Considering the simplest case,
we define such noises as a white noise.
To get rid off such a restriction, the multiplier u in the nonlinear
terms of Lorenz system is supposed to be replaced by power term
au ,
with an exponent 0 1a≤ ≤ . Taking into account the stochastic addi-
tions, one obtains the basic equations in dimensionless form
0
( )
( )
( ) ( )
u
a
T
a
u u v I t
T T u I t
u T I tφ
= − + + ξ ,
ε = − + φ + ξ ,
δφ = φ − φ − + ξ .
(3.6)
It can be seen that the agreement of the Lorenz self-organization
scheme with SOC conception, related to self-similar systems is
achieved, if one assumes that both positive and negative feedbacks are
fractional. Within such a supposition, the adiabatic approximation
, 1ε δ << leads to the Langevin equation
( ) ( ) ( )a au f u I u t= + ξ , (3.7)
where the force ( )af u and the noise intensity ( )aI u are as follows:
0
12 2 2
( ) ( )
( ) ( ) ( ) 1
a
a a
a a
a u T a a
f u u u d u
I u I I I u d u d u u
−
φ
≡ − + φ ,
≡ + + , ≡ + .
(3.8)
The corresponding distribution
{ }
1
( ) exp ( ) ,
( )a a
a
Z
P u E u
I u
−
= − (3.9)
with a partition function Z, is determined by an effective potential
Self-Organization of Condensed Matter in Fluctuating Environment 35
0
( )
( )
( )
u
a
a
a
f u
E u du
I u
′
′≡ − .
′∫ (3.10)
The extremum points of this distribution are determined by the equation
22 2 1 1 2
02 1 1 2 ( 2 )a a a a a
TaI u u u u u a I I − −
φ φ
+ + φ − + = − , (3.11)
according to which, the boundary of the flux state
2 TI Iφ = , (3.12)
Fig. 13. The φ0-dependence of the steady-state velocity u: (a) at a = 0.75,
IT = 1 (curves 1—4 are related to Is = 1, 2, 3, 5); (b) at IT = 1, Is = 5 (curves
1—4 are related to a = 0.25, 0.5, 0.75, 1.0.
36 D. O. Kharchenko
relates to the condition u = 0. Critical values of state parameters are
fixed by the condition 0/du dφ = ∞ leading to additional equation
( )
( )
22(1 ) 2 1 1 2
1 2 1 1 21
02
1 2 1
1 3 1 2 .
a a a
a a a
u u a a u
u u a a u aI
− − − −
− − − −
φ
+ + + − −
− φ + + + − =
(3.13)
Fig. 14. Three-dimensional phase diagram (the non-avalanche domain is
located under the surface).
Fig. 15. Phase diagram for the system with φ0 = 0 and Is, IT ≠ 0 at a = 0.5,
0.75, 1.0 (dashed, solid and dotted curves, respectively). Diamonds are re-
lated to curves 1—4 in Fig. 18.
Self-Organization of Condensed Matter in Fluctuating Environment 37
Above expressions show that an increase in the noise intensity TI
causes monotonic u-growth, whereas Iφ -increase leads to an effec-
tive barrier formation near the point 0u = , so that the dependence
0( )u φ becomes non-monotonic at magnitudes Iφ above the straight
line (3.12) (see Figs. 13). Here, lower branches of curves correspond
to unstable magnitudes of the order parameter, while the upper
meet the stable ones. According to Fig. 14, the domain, at which
avalanches cannot be created, is located near intermediate magni-
tudes of the state parameters 0φ , TI , Iφ . The phase diagram related
to the avalanche formation reveals the same form as for the sim-
plest case 1a = , but the straight-line shifts abruptly to (3.12) with
escaping the point 1a = (see Fig. 15).
According to Fig. 16, an increase of the noise of the temperature
TI increases the domain of the avalanche formation.
3.2.1. Nonextensive Statistics of Avalanches Ensemble
In contrast to the previous discussion, where the process of a single
avalanche formation has been considered, now we will study ana-
lytically the self-similar size distribution of an avalanche ensemble.
To this end, we will account noises of a complete set of degrees of
freedom, on the one hand, and the fractional feedback type intro-
duced in Section 3.2, on the other one.
Thereby, the Lorenz system unified in the above manner is the
basis of our examination. However, the system under consideration
is now parameterized by a set of pseudo-thermodynamical variables,
which describes the avalanche ensemble in the spirit of the famous
Edwards paradigm [13, 14] generalized to nonstationary system
Fig. 16. Phase diagram in the Iφ—a plane at IT = 2, 3, 4, 5, 6 from bottom
to top (the non-avalanche domain is located inside the curves).
38 D. O. Kharchenko
[49]. With this method, we study time dependences of the avalanche
size (density of equipped sites in the avalanche), nonextensive com-
plexity and nonconserved energy of equipped sites in the process of
crystallization. Within the framework of the usual synergetic ap-
proach, these degrees of freedom play the role of order parameter, con-
jugate field and control parameter, respectively.
It is principally important that the use of the slaving principle of
both synergetics and fractional nature of the system feedback is shown
to stipulate the multiplicative character of noise. It will be shown that
this causes a nonextensivity of the applied thermodynamical scheme,
so that we have to use q-weighted averages instead of usual ones. So,
energy of equipped sites in the process of crystallization is defined by
the expression
q
q i ii
pζ ≡ ζ ,∑ (3.14)
where ip is a probability to enable the site i with energy iζ , 1q ≠ is a
positive parameter that is a measure for the system’s nonextensivity
determined below. Nonextensive complexity of equipping sites is an
analogue to Tsallis entropy [15] that is determined as follows:
1
1
q
i i
q
p
q
−∑Σ ≡ − .
−
(3.15)
The three-parameter set of the standard synergetic scheme [2] is com-
pleted by the avalanche size s.
Following the above-elaborated line, we postulate that a self-
consistent behaviour of the system at issue is presented adequately by
a set of pointed out quantities governed by the Lorenz-type equations
2
0 2
( )
( )
( ) ( )
s s q s
q q q
q q q
t s s a I t
t a s I t
t a s I t
τ/
Σ Σ Σ
τ/
ζ ζ ζ
= − + Σ + ξ ,
Σ = −Σ + ζ + ξ ,
ζ = ζ − ζ − Σ + ξ .
(3.16)
Here st , tΣ , tζ note relaxation times of corresponding values, sa , aΣ ,
aζ are related feedback parameters, sI , IΣ , Iζ are respective noise in-
tensities, τ is a positive exponent and
0ζ is the externally driven en-
ergy of the sand motion. The distinguishing feature of the first of
these equations is that in a noiseless case genuine characteristics s , qΣ
are linearly connected. On the other hand, the two last equations (3.16)
show that the connection of values qζ , qΣ (which are of a thermody-
namic type) with the avalanche size s is nonlinear. Physically, this
means a linear relation between the complexity and the avalanche size
near steady state. Moving away, this leads to negative feedback of the
Self-Organization of Condensed Matter in Fluctuating Environment 39
avalanche size and the complexity on the energy that, in accordance
with Le Chatelier principle, results in the energy decrease. Moreover,
positive feedback appears between the avalanche size and the energy on
the complexity, which causes complexity increase, that is the reason
for the avalanche ensemble’s self-organization.
To analyze the system (3.16), it is convenient to measure the time t
in unit st and introduce the scales for variables s, qΣ , qζ , sI , IΣ , and
Iζ as follows:
1 1
21 1
1 2 1 2
( ) ( ) ( )
( ) ( ) ( ) .
sc sc sc
q s q s
sc sc sc
s s s
s a a a a a a a
I a a I a a a I a a
τ− −− −
Σ ζ Σ ζ Σ
− − − −
Σ ζ Σ Σ ζ ζ Σ
≡ , Σ ≡ , ζ ≡ ;
≡ , ≡ , ≡
(3.17)
Then, the scaled Lorenz system (3.16) takes the simple form
2
0 2
( )
( )
( ) ( )
q s
q q q
q q q
s s I t
s I t
s I t
τ/
Σ
τ/
ζ
= − + Σ + ξ ,
ϑΣ = −Σ + ζ + ξ ,
θζ = ζ − ζ − Σ + ξ ,
(3.18)
with the ratios of relaxation times
s st t t tΣ ζϑ ≡ / , θ ≡ / . (3.19)
It is worth to notice that the system (3.18) is passed to the form of Eqs.
(3.6) if the values s , qΣ , qζ , 2τ/ , ϑ , and θ are replaced by u, T, φ, a, ε,
and δ, respectively.
It is well known that a complete set of SOC systems can be reduced to
one of two families [50]: systems with deterministic dynamics ex-
tremely driven by a random environment (growing interface models,
Bak—Sneppen evolution model, etc.) and the stochastic dynamics fam-
ily (models of earthquakes, forest-fire, etc.). A remarkable peculiarity
of the obtained system (3.18) is the possibility to present both men-
tioned families in a natural manner. The former is related to the noise-
less case, when 0sI I IΣ ζ, , = but the magnitude of the energy relaxation
time is larger than that of the complexity and avalanche size
( st t tζ Σ≥ , ); on the other hand, a parameter of the environment drive
0ζ
has to take a larger value than the critical one 1cζ = [51]. In such a
case, the system (3.18) describes a strange attractor that may repre-
sent the behaviour of SOC systems of the first type. A proper stochas-
tic behaviour is relevant for nonvanishing to nonzero noise intensities
0sI I IΣ ζ, , ≠ that make possible the appearance of the SOC regime even
in the absence of a driven affect (
0 0ζ = ).
We will restrict ourselves to the treatment of the stochastic system,
where the adiabatic conditions , 1ϑ θ << are applicable. Then, the two
last equations of the system (3.18) lead to the following dependences:
40 D. O. Kharchenko
( ) ( ) ( ) ( )q q q q q qt t t tΣ = Σ + Σ ξ , ζ = ζ + ζ ξ , (3.20)
where the deterministic and the fluctuating components are deter-
mined as follows
0 2
0 1
( ) ( )
( ) ( ) ( ) (1 ) .
q q
q q
s d s I I s d s
d s I I s d s d s s
τ/ τ
τ Σ ζ τ
τ τ −
τ ζ Σ τ τ
Σ ≡ ζ , Σ ≡ + ;
ζ ≡ ζ , ζ ≡ + , ≡ +
(3.21)
Due to the slaving principle of synergetics, the initially adiabatic noises
of both complexity and energy are transformed to a multiplicative form.
On the other hand, the relation between the complexity and energy,
0( ),q q q
= ζ −ζ ζ∑ (3.22)
Fig. 17. The energy dependences of the avalanche ensemble temperatures:
(a) nonstationary magnitude T versus ratio 0
0ζ ζ ; (b) stationary tempera-
ture T0 versus ζ0.
Self-Organization of Condensed Matter in Fluctuating Environment 41
deduced from the dependences (3.21), leads to the expression
1
0 0
1 1
2
q q
T
−
ζ ζ= − − − ζ ζ
(3.23)
for the effective temperature qq
T ≡ ∂ /∂ζ Σ . As depicted in Fig. 17, a, T
is a monotonically increasing function of the energy with boundary
values ( 0) 0qT ζ = = and
0( 2)qT ζ = ζ / = ∞ . At the latter point, the
magnitude T changes instantaneously to −∞ and then increases mono-
tonically again to initial value 0T = at
0
qζ = ζ . This means that in the
domain
00 2q≤ ζ < ζ / the avalanche system is dissipative and behaves
in usual manner; on the contrast, in the domain
0 02 qζ / < ζ ≤ ζ a self-
organization process evolves, so that an energy increase leads to a
complexity decrease, in accordance with a negative temperature. At
steady state, where an avalanche has a stationary size
0
0 1s = ζ − , the
temperature takes the stationary value
0
0 0
1
1 2
T
ζ −
= −
− ζ /
, (3.24)
which is negative in the supercritical domain
01 2≤ ζ < . According to
Fig. 17, b, the magnitude 0T decreases monotonically with the driven
energy from value
0
0( 1) 0T ζ = = to
0
0( 2)T ζ → → −∞ .
The presented self-organization regime relates to externally driven
systems, which are relevant for the usual phase transition but not to
the SOC itself. To study the latter within the above consideration, let
us combine Eqs. (3.20), (3.21) with the first of the equations (3.18) in
that way that has been used above for obtaining the Langevin equation.
By analogy with Section 3.2, this leads to stochastic equation (3.7),
where the effective force and noise intensity are given by Eqs. (3.8)
with accuracy to the replacements mentioned after Eqs. (3.19): the
quantities s, qΣ , qζ , 2τ/ have to be taken instead of u, T, φ, a, respec-
tively. Then, all results obtained in Section 3.2 can be used immedi-
ately. Particularly, it is found that the influence of a random scatter-
ing of the avalanche size is non-essential, whereas energy and complex-
ity noises lead to a crucial effect. The related picture is presented by
Fig. 15 taken in plane I Iζ Σ− that is formed by corresponding noise
intensities of the avalanche ensemble. The mixed domain A + N with
respect to the intermittency regime is bounded by the straight line
(3.12) and the bell-shaped curve.
According to Fig. 16, where exponent a has to be replaced by 2τ/ ,
the random scattering growth of the complexity extends the SOC do-
main along the axis of the exponent τ.
Now, we will discuss the distribution of the avalanche size on basis
of equations (3.8)—(3.10). For arbitrary noise intensities one has
42 D. O. Kharchenko
{ }
( ) ( )
1
0
0 2
1
2
( )
( ) exp
( )( )
( ) ( )
( ) ( ) ( ) .1
s
s
Z f sP s ds
I sI s
f s s s d s
I s I I I s d s d s s
−
τ/
τ
−τ τ
Σ ζ τ τ
′ ′= ;∫ ′
≡ − + ζ ,
≡ + + , ≡ +
(3.25)
In the SOC regime, the driven energy is vanished, 0 0ζ = , and the
distribution (3.25) behave as depicted in Fig. 18 for different noise
intensities of both energy and complexity. It can be seen that the
power-law dependence inherent in the SOC regime is observed only
in the limits 1s << and , sI I I∑ ζ<< . In this case, the distribution
(3.25) is reduced to the form
2 2
1 1
1
0
( )
( ) exp , ( ) (1 ) .
( )
sd d s
P s I ds d s s
Z s
− −
− τ −τ τ
ζ ττ−
′ ′= − ≡ + ′
∫ (3.26)
It is easy to see that the deviation of this multiplier from a con-
stant value is estimated with term ∼ 2s −τ , which increases with de-
crease of τ and growth of avalanche size to extremely large magni-
tudes 1s ∼ , i.e. with escaping SOC domain. This is confirmed by
Fig. 19, where the deviation δτ of the slope of dependence ( )P s
(Eqs. (3.25)) in the linear domain on the theory parameter τ is de-
picted as a function of the parameter τ itself. In accordance with
the above estimation, it can be seen that the deviation δτ takes a
maximal value 110−δτ < τ at non-essential magnitudes 1τ < or, with
noise intensity growth to enormous values 310sI ∼ .
Fig. 18. Distribution function (3.25) at τ = 1.5 and regimes pointed out by
diamonds in Fig. 15: 1–IT = 0, Iφ = 50 (SOC); 2–IT = 0.5, Iφ = 30 (A + N); 3–
IT = 1, Iφ = 5 (N); 4–IT = Iφ = 0.5 (A).
Self-Organization of Condensed Matter in Fluctuating Environment 43
Computer simulations. In order to perform the computer simula-
tions, one should note that at 1 2a > / the absorbing state takes
Fig. 20. Computer simulation of the explosive crystallization process (back
points correspond to equipped sites). Figures plotted at a = 0.8, D = 1.0,
Θ = 2.0: a, b, c, d correspond to t = 0.0, 0.005, 1.2, 2, 2.3.
Fig. 19. Deviation δτ of the linear slope of curve 1 depicted in Fig. 18 from pa-
rameter τ versus the exponent τ itself (Iφ = 10, 50, 103 from bottom to top).
44 D. O. Kharchenko
place. This state cannot be found considering the most probable
values of stochastic variable s, it is defined through the transfor-
mation of the boundary 0s → of the diffusion process ( )s t . How-
ever, obtained results are in good correspondence with well-known
laws inherent to the SOC regime.
To examine our model on the 2D grid we introduce the Laplacian
into the first equation of the system (3.18). In the adiabatic ap-
proximation, we have evolution pictures of the density of equipped
sites shown in Fig. 20. It is seen that the diffusion in r-space ex-
cites the neighbour sites, which transit the system to the crystal-
lized phase (cf. Fig. 12). A special attention can be paid to consid-
eration of the influence of the spatial coupling. Here, spatial cou-
pling promotes an escape from the absorbing configuration where
system has been trapped.
By proving so, we plot a fraction R of sites in absorbing configu-
ration in Fig. 21. It is seen that at 0D → the system is in absorb-
ing state if 1 2a > / . An increase in D or promotes excitation proc-
esses which transform all inactive sites (being in absorbing state
initially) into active phase (crystallization). Moreover, the process
of crystallization has a short time of transformation of absorbed
(inactive) sites into the active phase: during only 3t ∼ time units, it
corresponds to the explosive behaviour of crystallization. It can be
seen that the fraction R decreases at 3/4a ≅ which corresponds to
the index 3/2τ ≅ where the SOC regime is observed. A further in-
crease in a transforms the system into absorbing state.
Fig. 21. Fraction of sites in absorbing configuration with s = 0: depend-
ences R(a) are shown at D = 0.01, Θ = 0.5; D = 0.01, Θ = 0.1; D = 0.06,
Θ = 1.0 (curves 1, 2, 3).
Self-Organization of Condensed Matter in Fluctuating Environment 45
3.2.2 Linkage between the Anomalous Diffusion Process
and Self-Organized Criticality
A remarkable peculiarity of expression (3.26) is that, within the
limits of 1s << , ,sI I I∑ ζ<< inherent in the SOC regime, it can be
expressed in terms of standard Gamma-function ( )xΓ and fractional
integral 2
sI −τ
− of order 2 − τ (see Appendix):
2
2 2(2 )
( ) exp ( )s
d
P s I d s
IZ
−
−τ −τ
− τ
ζ
Γ − τ= − .
(3.27)
On the other hand, it is well-known [16] that expressions of this
kind appear as a solution of the fractional Fokker—Planck equation
( )
2( ) ( ) ( )t s s
I
D P s t D sP s t D d P s tζω ϖ ϖ
− − τ
, = , + , , Γ ϖ
(3.28)
where the fractional derivative xDω (see (A.2)) is used to be inverted
to the fractional integral (A.1). Multiplying equation (1) by term
2s ϖ and averaging over s according to the definitions,
1
( ) 0s s s s P s t dsα
∞
α α α
−∞
≡ , ≡ , , α > ,∫ (3.29)
one obtains at 2α ≡ ϖ
2
,
z
s t z
ϖ
∼ =
ω
(3.30)
where z is a dynamical exponent. This relation corresponds to the
large time limit, where only the diffusion contribution is essential.
Combining expressions (3.27), (3.30) and (A.1) leads to the relations
2 − 2zτ = ϖ = ω/ , which yield
2
2
zωτ = − . (3.31)
Comparing this equation with the well-known relations for the critical
exponents 1 z Dτ = + / and 2(1 1 )Dτ = − / corresponding to the process
of avalanche formation [50], one obtains
4z
D
ω = . (3.32)
The mean-field magnitudes 1ω = and 4D = are related to the dynami-
cal exponent 1z = that, in accordance with definition (3.30), is related
to the unusual ballistic limit of the SOC regime. On the other hand, the
fractional Fokker—Planck equation (3.28) leads to the usual diffusion
46 D. O. Kharchenko
regime with 2z = only in the artificial case, when the time-derivative
exponent is assumed to be 1 2ω = / .
The obvious reason for such a discrepancy is the non-consistent ap-
plication of the usual field relations for τ to the Lorenz system (3.18).
In this system, the stochastic degrees of freedom s, qΣ , and qζ , whose
number is 3n = , serve as the different space directions. However, the
stochastic process evolves for any of these variables in a plane spanned
by the given variable itself and its conjugated momentum. Moreover,
the multiplicative character of noise, which is determined by the expo-
nent a in expressions (3.8), reduces the fractal dimension of every
plane to the value 2(1 )a− [52]. This result can be achieved considering
the temporal behaviour of main statistical moments of a free particle
system. As was shown in Ref. [53], where the scaling laws of anoma-
lous behaviour of self-similar system were considered, the multiplica-
tive noise with the amplitude given in a power-law form
ax results to
the supper-diffusion processes with
2 2( ) Hx t t∝ ,
1 2(1 )H a− = − . Thus,
the resulting fractal dimension of the phase space, in that the stochas-
Fig. 22. Dependences of exponent τ: (a) on equations number n (a = 0, 1/3,
1/2, 2/3) from top to bottom); (b) on exponent a (n = 2, 3, 4, 6, 8, 10 from bot-
tom to top); (c) phase diagram for mean-field and nonextensivity domains.
Self-Organization of Condensed Matter in Fluctuating Environment 47
tic system evolves, is as follows:
2 (1 ),D n a= − (3.33)
where 3n = for the used Lorenz system. Inserting this dimension into
expression (3.32) leads to 2zω = , which, in contrast to the relation
1zω = obtained above, is correct in the simplest case 1ω = , 2z = [the
latter is relevant to a single stochastic degree of freedom ( 1n = ) with
additive noise ( 0a = )]. In the general case, equations (3.31)—(3.33)
yield the final result
12 1
2 (1 )n a
τ = − . −
(3.34)
The respective dependences are depicted in Figs. 22, a, b to show
that the exponent τ increases monotonically from its minimum magni-
tude 1τ = at the critical number
1(1 )a −− to upper value 2τ = in the
limit n → ∞ ; thereby, an a-growth shifts the dependence ( )nτ to large
magnitudes n, i.e. decreases the exponent τ.
It is easy to see that relation (3.34) reproduces known results of dif-
ferent approaches for the dimension D (see Ref. [54]). In the case re-
lated to mean-field theory, one has 3 2τ = / and equation (3.34) ex-
presses the number of self-consistent stochastic equations needed for
treating the SOC behaviour as a function of the exponent of the corre-
sponding multiplicative noise:
2
1
n
a
=
−
. (3.35)
In accordance with Fig. 22, c, a self-consistent mean-field treatment is
possible, if the number of relevant equations is larger than the minimum
magnitude 2cn = . Approaches [39—41, 50, 55] represent examples of
such considerations, where noise is supposed to have additive character
( 0a = ). Switching the multiplicative noise leads to an a-growth and
non-contradicting representation of the SOC demands an increase in the
number of self-consistent equations: for example, within the field
scheme [56] related to directed percolation ( 1 2a = / ), the mean-field ap-
proximation is applicable for dimensions larger than the critical magni-
tude 4cd = ; here the Lorenz scheme ( 3n = ) with multiplicative noise is
characterized by the exponent 1 3a = / (see below).
Let us now focus on the relation of the above exponents to the nonex-
tensivity parameter q related to Tsallis definitions (3.14), (3.15) [15].
The relevant kinetic behaviour could be described by the nonlinear
Fokker—Planck equation
2( ) ( )q
t sD P s t D P s tω
−, = , , (3.36)
48 D. O. Kharchenko
where tDω
is the fractional derivative and the measure units are chosen
in such a way that the effective diffusion coefficient disappears
( 0ω > , 0q > are the relevant exponents [57—59]). Supposing a nor-
malized distribution function in a self-similar form
1( , ) ( ) ( ) c c c cP s t s P x s s t x s s−= ; ≡ , ≡ / , (3.37)
we obtain
1 1 2, , 0q q
cs t P x x+ ω −∼ ∼ → . (3.38)
On the other hand, we could use the fractional Fokker—Planck equation
of the type of Eq. (3.28):
2( ) ( )t sD P s t D P s tω ϖ
−, = , . (3.39)
Inserting the solution (3.37), one finds the dependences
2 2 1, ,cs t P xϖ ω ϖ−∼ ∼ (3.40)
whose comparison with Eqs. (3.38) yields
2 1q = ϖ − . (3.41)
Because the average s in Eq. (3.29) is reduced to the scale cs in the
case of self-similar systems, the relevant dependences (3.30), (3.38)
and (3.40) give
1.q z= ω − (3.42)
Combining this equality with Eqs. (3.32), (3.33) leads to the resulting ex-
pression for the nonextensivity parameter of the considered system:
2 1
(1 )
q
n a
= − .
−
(3.43)
The maximum magnitude 2 1q n= / − is related to systems with addi-
tive noise ( 0a = ), which is relevant to the mean-field picture at a
number of governing equations 2n < ( 1n = ). Switching the multipli-
cative noise with increasing exponent 0a > leads to a q-growth and the
self-organizing system gets a nonextensive character ( 1q ≥ ) in the
limit 1 (1 )q a≤ / − of the mean-field applicability domain. In accor-
dance with the above estimation, the fractional Lorenz system is non-
extensive, essentially if the exponent 2 3a > / .
It is worth to remember that above we have assumed the superdiffu-
sion process only to be related to Lévy flights at discrete time instant
Self-Organization of Condensed Matter in Fluctuating Environment 49
with arbitrary displacements, including infinite ones [60]. Related to
the Fokker—Planck equation (3.28), such processes are characterized
by exponents 1ω = and 1ϖ < , the first of which is constant, whereas
the second one characterizes the fractal time-sequence of the Lévy
flights and leads to the dynamical exponent 2 2z ≡ ϖ/ω < (see the last
of equations (3.30)). The probability distribution of the displacement
x , that is dependent on microscopic conditions, reads
( )( ) Dp x− + γ∼x (3.44)
and is characterized by the fractal dimension D and the microscopic
step exponent γ. It is obvious that in the case of rare events, when
2γ < , the dynamical exponent z is reduced to the microscopic step ex-
ponent ( 2z = γ < ), whereas at 2γ ≥ one has 2z = [61].
In the opposite case of subdiffusion processes, a microscopical in-
gredient is the random Lévy walks instead of the discrete Lévy flights.
This process evolves continuously in course of the time over discrete
placed traps, so that the exponent 1ω < describes fractal properties of
this space that depend on microscopic conditions. These properties
generate the transformation of the usual Boltzmann—Gibbs statistics
in a nonextensive manner [15]. The subdiffusion process is presented
by the Tsallis-type distribution [62]
1
12( ) 1 ( 1) const 0qp q x −−
∝ + β − , β = > ,x (3.45)
where the deviation 1q − of the nonextensive parameter is caused by
the fractal nature of the system phase space that is connected to the
step exponent γ as follows:
.
2
1q
D
= +
+ γ
(3.46)
A formal advantage of this distribution (3.45) is that the corre-
sponding q-weighted average
2 2 ( ) ,q D
q p d x≡ ∫x x x (3.47)
where the integrand varies as (1 )x− + γ , converges for arbitrary step
exponents 0γ > . As a result, the equation of motion of the random
Lévy walker is given by
2 ,
q
tω∼x
2
1, at 2,
1 ( 1) , at 2 1.D
q q
q q
− γ, <
ω = − − γ ≥ , >
(3.48)
In contrast to Eq. (3.40), where the exponents 1ϖ < , 1ω = are
relevant for the superdiffusion, one here has inverted relations:
50 D. O. Kharchenko
1ϖ = , 1ω < . Thus, in accordance with the subdiffusion nature, the
last equality (3.30) yields the dynamical exponent 2z > .
In the general case of 1ϖ, ω ≠ , inserting Eqs. (3.48) into the relation
(3.42) leads to the result
2
1
1 ( 1)
1
1
at 1 ,
at 2,
D
q
Dq
q
Dq
q q
z
q q
+
− −
+
−
< ≤=
≤ ≤
(3.49)
where the boundary value of the nonextensivity parameter is intro-
duced:
4 .
2D
Dq
D
+≡
+
(3.50)
To avoid a mistake, let us focus on Eqs. (3.31), (3.32), which could be
related to the real phase space and the configurational one (the latter is
spanned by variables of governing equations), the above-obtained rela-
tions (3.48)—(3.50) are relevant for the real phase space only. This is
reflected by addressing the fractal dimension D to the only real coordi-
nate space in the former case, whereas in the latter it is reduced to the
effective value (3.33). According to our treatment, relation (3.33)
plays a central role since, in analogy with a renormalization group, we
have considered the properties of the configurational space but not
real diffusion process.
4. CONCLUSIONS
This review is concerned the study of synergetic systems in noisy envi-
ronment. We addressed our approach to the consideration of self-
organization processes during the plastic flow. It was shown that ran-
dom fluctuation of the media can be a reason to the qualitative recon-
struction of condensed matter structure in the course of plastic defor-
mation. We discussed the representation of the explosive crystalliza-
tion processes, which are familiar to the self-organ—ized criticality re-
gime. At last, the evolution of general type of models to describe the
condensed matter is investigated in detail.
Considering the synergetic models with different dimensionalities,
we found that the correlation between fluctuations of the freedom de-
grees of the system plays a crucial role and makes the system to pass
from one structure to another through re-entrant phase transitions. De-
veloping the well-known cumulant expansion method, we investigated
the character of diffusion of linear and point defects separately within
the framework of the mean-field approximation, which gives the quali-
tative results, and with using computer simulation procedure. Our re-
sults are in a good correspondence with a well-known conclusion that the
Self-Organization of Condensed Matter in Fluctuating Environment 51
reconstruction of macroscopic picture of plastic flow is a result of proc-
esses, which occur in microscales where the density of dislocations var-
ies discontinuously. As shown, the discontinuous behaviour of the den-
sity of linear defects is explained as a result of cross correlations be-
tween fluctuations of different freedom degrees.
The description of the explosive crystallization is given on the basis
of the Lorenz-type model. The conditions to the explosive crystalliza-
tion processes are determined. We showed that the transition to the ex-
plosive crystallization has properties inherent to the regime of self-
organized criticality. The avalanche-like dynamics is represented for
both single avalanche system and an avalanche ensemble. Considering
the noise induced transitions we have shown that the absorbing state
appeared at large noise intensities can be destroyed introducing the
spatial coupling in the system. It was shown that the main laws of self-
organized criticality are applicable to the explosive crystallization
process. We found main exponents and set relations between dynamical
and geometrical characteristics of the system phase space. It was shown
that the explosive crystallization is related to the anomalous diffusion.
We are looking forward that the described results, models and as-
sumptions presented in this review could give a stimulus to develop the
theory of self-organization of defect structure during the plastic flow.
One can expect that obtained results will be of interest for the re-
searches in the field of plastic deformation to predict the resources of
details and constructions. The representations of reconstruction of de-
fect structure can be useful to interpret the experimental data in which
different defects are observed with increasing deformation or diffu-
sion constant.
Perspectives of the work can be as follows: (i) generalization of the
theory of fluctuation reconstruction of phase transition; from mathe-
matical point of view picture of noise induced phase transition need to
be established in a more rigorous way to account spatial correlations;
(ii) the theory of escaping from absorbing configuration should be de-
veloped to investigate the ergodicity breaking effect more precisely;
(iii) theory of SOC should be examined introducing the fractional de-
rivatives into motion equation to reflect the anomalous diffusion re-
gime.
APPENDIX
Here, the basic properties of fractional integral and derivative are pre-
sented. The integral of fractional order ϖ is defined by equality [9, 63]
( ) 1
0
( )1( ) 0
( )
x
x
f x
I f x dx
x x
ϖ
−ϖ
′ ′≡ , ϖ > ,
′Γ ϖ −∫ (A.1)
52 D. O. Kharchenko
where ( )f x is an arbitrary function, ( )xΓ is the standard Gamma-
function. To be inverted to the fractional integral, the relevant de-
rivative x xD Iϖ −ϖ≡ of order 0ϖ > is determined as follows:
( ) 1
0
( )1( )
( )
x
x
f x
D f x dx
x x
ϖ
+ϖ
′ ′≡ .
′Γ −ϖ −∫ (A.2)
In the region 0 < 1ϖ < , it is convenient to use the expression
( ) 1
0
( ) ( )
( )
1 ( )
x
x
f x f x
D f x dx
x x
ϖ
+ϖ
′−ϖ ′≡ ,
′Γ − ϖ −∫ (A.3)
where we take into account the known equality ( ) ( 1)x x xΓ = Γ + for
x ≡ −ϖ .
REFERENCES
1. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems
(New York, London, Sydney, Toronto: A Wiley-Interscience Publ., John
Wiley & Sons: 1977).
2. H. Haken, Synergetics–An Introduction (Berlin, Heidelberg, New York:
Springer-Verlag: 1983).
3. C. W. Gardiner, Handbook of Stochastic Methods (Berlin: Springer-Verlag:
1983).
4. W. Horsthemke and R. Lefever, Noise-Induced Transitions (Berlin:
Springer-Verlag: 1984).
5. J. Garcia-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems
(New York: Springer-Verlag: 1999).
6. P. Bak, How Nature Works: the Science of Self-Organized Criticality (Ox-
ford: Oxford University Press: 1997).
7. H. J. Jensen, Self-Organized Criticality. Emergent Complex Behaviour in
Physical and Biological Systems. Cambridge Lecture Notes in Physics (Cam-
bridge: Cambridge University Press: 1998).
8. J. Feder, Fractals (New York: Plenum: 1989).
9. Applications of Fractional Calculus in Physics (Ed. R. Hilfer) (Singapore:
World Scientific: 2000).
10. V. V. Rubin, Bol’shie Plasticheskie Deformatsii i Razrushenie Metallov
(Large Plastic Deformations and Failure of Metals) (Moscow: Metallurgiya:
1985) (in Russian).
11. V. E. Panin, V. A. Likhachyov, and Yu. V. Grinyayev, Strukturnyye Urovni
Deformatsii Tvyordykh Tel (Structural Levels of Deformation of Condensed
Matter) (Novosibirsk: Nauka: 1985) (in Russian).
12. V. P. Skripov and V. P. Koverda, Spontannaya Kristallizatsiya
Pereokhlazhdyonnykh Zhydkostej (Spontaneous Crystallization of Super-
cooled Liquids) (Moscow: Nauka: 1984) (in Russian).
13. S. F. Edwards and R. B. S. Oakeshott, Physica A, 157: 1080 (1989).
14. S. F. Edwards, Granular Matter: An Interdisciplinary Approach (Ed. A.
Metha) (New York: Springer-Verlag: 1994).
Self-Organization of Condensed Matter in Fluctuating Environment 53
15. C. Tsallis, Lecture Notes in Physics (Eds. S. Abe and Y. Okamoto)
(Heidelberg: Springer-Verlag: 2001).
16. G. M. Zaslavsky, Chaos, 4: 25 (1994); idem, Physica D, 76: 110 (1994); A. I.
Saichev and G. M. Zaslavsky, Chaos, 7: 753 (1997).
17. A. Damask and J. Diens, Point Defects in Metals (New York: Gordon and
Breach: 1963).
18. J. Friedel, Dislocations (Oxford, London, Edinburg, NewYork, Paris, Frank-
furt: Pergamon Press: 1964).
19. J. D. Vishnyakov, Defekty Upakovki v Kristallicheskoj Strukture (Defects of
Packing in Crystal Structure) (Moscow: Metallurgiya: 1970) (in Russian).
20. J. P. Hirth and J. Lothe, Theory of Dislocations (New York: McGraw-Hill:
1968).
21. B. I. Smirnov, Dislokatsionnaya Struktura i Uprochnenie Kristallov (Disloca-
tion Structure and Strengthening) (Leningrad: Nauka: 1981) (in Russian).
22. V. I. Vladimirov and A. E. Romanov, Disklinatsii v Kristallakh (Disclina-
tions in Crystals) (Leningrad: Nauka: 1986) (in Russian).
23. R. de Witt, Continual Theory of Disclinations. (Moscow: Mir Publishers:
1977) (Russian translation).
24. A. I. Olemskoi, Physics Reviews (Ed. I. M. Khalatnikov) (London: Gordon &
Breach: 1996), vol. 18, Part 1, p. 1.
25. A. I. Olemskoi and A. V. Khomenko, Usp. Fiz. Met., 2: 189 (2001).
26. S. E. Mangioni, R. R. Deza, R. Toral, and H. S. Wio, Phys. Rev. E, 61: 223,
(2000).
27. P. Jung and P. Hänggi, Phys. Rev. A, 35: 4467 (1987); F. Castro, H. S. Wio,
and G. Abramson, Phys. Rev. E, 52: 159 (1995).
28. D. O. Kharchenko and I. A. Knyaz’, Mettallofiz. Noveishie Tekhnol., 24, No.
3: 389 (2002).
29. E. A. Novikov, JETP, 47: 1919 (1964).
30. M. A. Munoz, Phys. Rev. E, 57: 1377 (1998).
31. A. I. Olemskoi and D. O. Kharchenko, Met. Phys. Adv. Tech., 16: 841 (1996).
32. A. I. Olemskoi and D. O. Kharchenko, Physics of Solid State, 42: 520 (2000).
33. D. O. Kharchenko, Physica A, 308: 101 (2002).
34. D. O. Kharchenko and I. A. Knyaz’, Eur. Phys. J. B, 32: 375 (2003).
35. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Amster-
dam: North Holland: 1992).
36. V. E. Shapiro, Phys. Rev. E, 48: 109 (1993).
37. H. Risken, The Fokker—Planck Equation (Berlin: Springer Verlag: 1989).
38. S. F. Edwards and D. R. Wilkinson, Proc. Roy. Soc. A, 381: 17 (1982).
39. A. Mehta and G. C. Barker, Rep. Prog. Phys., 57: 383 (1994).
40. J. -P. Bouchaud, M. E. Cates, J. R. Prakash, and S. F. Edwards, J. Phys. I
(France), 4: 1383 (1994).
41. K. P. Hadeler and C. Kuttler, Granular Matter, 2: 9 (1999).
42. P. Bak and K. Sneppen, Phys. Rev. Lett., 71: 4083 (1993).
43. A. I. Olemskoi, A. V. Khomenko, and V. P. Koverda, Physica A, 284: 79
(2002).
44. L. Pietronero, A. Vespignani, and S. Zapperi, Phys. Rev. Lett., 72: 1690
(1994); idem, Phys. Rev. E, 51: 1711 (1995).
45. V. P. Koverda and V. P. Skripov, Rasplavy, 1, No. 4: 3 (1987) (in Russian).
46. N. M. Bogdanov, V. P. Koverda, V. N. Skokov, V. P. Skripov, and
A. A. Dik, Kristallografiya, 33, No. 5: 1251 (1988) (in Russian).
54 D. O. Kharchenko
47. D. A. Kurtze, Phys. Rev. B, 34: 1770 (1986).
48. V. P. Koverda, Zh. Tekhn. Fiz., 64, No. 3: 62 (1994) (in Russian).
49. A. I. Olemskoi, A. V. Khomenko, and D. O. Kharchenko, Physica A, 323:
263 (2003).
50. A. Vespignani and S. Zapperi, Phys. Rev. Lett., 78: 4793 (1997); idem, Phys.
Rev. E, 57: 6345 (1998).
51. A. I. Olemskoi and A. V. Khomenko, JETP, 83: 1180 (1996).
52. A. I. Olemskoi, Physics–Uspekhi, 41: 269 (1998).
53. D. O. Kharchenko, Fluct. and Noise Lett., 2, No. 4: L287 (2002).
54. A. Chessa, E. Marinari, A. Vespignani, and S. Zapperi, Phys. Rev. E, 57:
R6241 (1998).
55. L. Gil and D. Sornette, Phys. Rev. Lett., 76: 3991 (1996).
56. A. Vespignani, R. Dickman, M. A. Muсoz, and S. Zapperi, Phys. Rev. Lett.,
81: 5676 (1998); idem, Phys. Rev. E, 62: 4564 (2000).
57. A. I. Olemskoi, JETP Let., 71: 285 (2000).
58. A. I. Olemskoi and D. O. Kharchenko, Physica A, 293: 178 (2001).
59. D. O. Kharchenko and S. V. Kohan, Eur. Phys. J. B, 29: 97 (2002).
60. J. -P. Bouchaud and A. Georges, Phys. Rep., 195: 127 (1991).
61. H. C. Fogedby, Phys. Rev. E, 58: 1690 (1998).
62. D. H. Zanette and P. A. Alemany, Phys. Rev. Lett., 75: 366 (1995).
63. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and
Derivatives–Theory and Applications (New York: Gordon and Breach:
1993).
|