The theory of kinetic processes in anisotropic phonon systems

We present a theoretical investigation of the kinetic properties of strongly anisotropic phonon systems. Such systems can be created in superfluid helium by heat pulses. The general expression for the rates of four-phonon processes are obtained. This expression shows that there is an asymmetry betwe...

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Date:	2001
Main Authors:	Adamenko, I.N., Kitsenko, Y.A., Nemchenko, K.E., Wyatt, A.F.G.
Format:	Article
Language:	English
Published:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2001
Series:	Вопросы атомной науки и техники
Subjects:	Kinetic theory
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/80037
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Cite this:	The theory of kinetic processes in anisotropic phonon systems / I.N. Adamenko, Y.A. Kitsenko, K.E. Nemchenko, A.F.G. Wyatt // Вопросы атомной науки и техники. — 2001. — № 6. — С. 301-305. — Бібліогр.: 8 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-800372025-02-23T20:27:30Z The theory of kinetic processes in anisotropic phonon systems Теория кинетических процессов в анизотропных фононных системах Adamenko, I.N. Kitsenko, Y.A. Nemchenko, K.E. Wyatt, A.F.G. Kinetic theory We present a theoretical investigation of the kinetic properties of strongly anisotropic phonon systems. Such systems can be created in superfluid helium by heat pulses. The general expression for the rates of four-phonon processes are obtained. This expression shows that there is an asymmetry between the creation and decay of the high-energy phonons in the anisotropic phonon systems. Solutions of this expression are then considered. The results presented in this work explain the phenomena which are observed in the anisotropic phonon systems and they will stimulate the conception of new experiments. We express profound gratitude EPSRC (grants GR/N18796 and GR/N20225) UK, and also GFFI Ukraine (grant №02.07/000372) for the support which enables us to carry out our scientific program. 2001 Article The theory of kinetic processes in anisotropic phonon systems / I.N. Adamenko, Y.A. Kitsenko, K.E. Nemchenko, A.F.G. Wyatt // Вопросы атомной науки и техники. — 2001. — № 6. — С. 301-305. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 67.40.Db, 67.40.Fd, 67.90.+z https://nasplib.isofts.kiev.ua/handle/123456789/80037 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Kinetic theory Kinetic theory
spellingShingle	Kinetic theory Kinetic theory Adamenko, I.N. Kitsenko, Y.A. Nemchenko, K.E. Wyatt, A.F.G. The theory of kinetic processes in anisotropic phonon systems Вопросы атомной науки и техники
description	We present a theoretical investigation of the kinetic properties of strongly anisotropic phonon systems. Such systems can be created in superfluid helium by heat pulses. The general expression for the rates of four-phonon processes are obtained. This expression shows that there is an asymmetry between the creation and decay of the high-energy phonons in the anisotropic phonon systems. Solutions of this expression are then considered. The results presented in this work explain the phenomena which are observed in the anisotropic phonon systems and they will stimulate the conception of new experiments.
format	Article
author	Adamenko, I.N. Kitsenko, Y.A. Nemchenko, K.E. Wyatt, A.F.G.
author_facet	Adamenko, I.N. Kitsenko, Y.A. Nemchenko, K.E. Wyatt, A.F.G.
author_sort	Adamenko, I.N.
title	The theory of kinetic processes in anisotropic phonon systems
title_short	The theory of kinetic processes in anisotropic phonon systems
title_full	The theory of kinetic processes in anisotropic phonon systems
title_fullStr	The theory of kinetic processes in anisotropic phonon systems
title_full_unstemmed	The theory of kinetic processes in anisotropic phonon systems
title_sort	theory of kinetic processes in anisotropic phonon systems
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2001
topic_facet	Kinetic theory
url	https://nasplib.isofts.kiev.ua/handle/123456789/80037
citation_txt	The theory of kinetic processes in anisotropic phonon systems / I.N. Adamenko, Y.A. Kitsenko, K.E. Nemchenko, A.F.G. Wyatt // Вопросы атомной науки и техники. — 2001. — № 6. — С. 301-305. — Бібліогр.: 8 назв. — англ.
series	Вопросы атомной науки и техники
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fulltext	THE THEORY OF KINETIC PROCESSES IN ANISOTROPIC PHONON SYSTEMS I.N. Adamenko1, Y.A. Kitsenko1, K.E. Nemchenko1, A.F.G. Wyatt2 1Kharkov National University, Kharkov, Ukraine e-mail: adamenko@pem.kharkov.ua 2School of Physics, University of Exeter, Exeter, EX4 4QL, UK e-mail: a.f.g.wyatt@exeter.ac.uk We present a theoretical investigation of the kinetic properties of strongly anisotropic phonon systems. Such sys- tems can be created in superfluid helium by heat pulses. The general expression for the rates of four-phonon pro- cesses are obtained. This expression shows that there is an asymmetry between the creation and decay of the high- energy phonons in the anisotropic phonon systems. Solutions of this expression are then considered. The results pre- sented in this work explain the phenomena which are observed in the anisotropic phonon systems and they will stimulate the conception of new experiments. PACS: 67.40.Db, 67.40.Fd, 67.90.+z 1. INTRODUCTION Systems with anisotropic distribution of phonons in momentum space were created in experiments [1-3] with the help of a heater immersed in superfluid helium 4He at such a low temperature that the contribution of thermal excitations can be neglected. The heater is a metal film evaporated onto glass. When current flows through the metal film the surface of the film injects phonons into the superfluid helium within a narrow cone with a solid angle 1< <Ω p and with an axis per- pendicular to the surface of the heater. For a gold film the phonons injected to helium occupied a solid angle 125.0=Ω p sr in momentum space. The dimensions in coordinate space of such strongly anisotropic phonon system are defined by the area of the heater and the du- ration of the thermal pulse. Isotropic phonon systems, for which in momentum space there is no special direction, have been intensively explored theoretically and experimentally during several decades. The first theoretical work, which began sys- tematic theoretical examination of the strongly anisotropic phonon systems in superfluid helium, was published only in 1999 [4]. This work explained the unique characteristics observed in such systems [1-3] and stimulated the design of new experiments. It was shown [4,5,6], that in the strongly anisotropic phonon systems of superfluid helium the kinetic processes differ greatly from those in the isotropic case. The main aim of the present work is to continue the theoretical analy- sis of kinetic processes in the anisotropic phonon sys- tems of superfluid helium begun in 1999. 2. RATES OF PHONON INTERACTIONS IN SUPERFLUID HELIUM The interaction rates in the phonon system of super- fluid 4He are determined by the unusual form of the phonon energy iε and momentum ip relationship, which we write as ( )iii fpc +=ε (1) where ( )ii pff = . The deviation in (1) from a linear dependence is small ( )ii pf < < but nevertheless it de- termines the rates of the kinetic processes amongst the phonons in superfluid helium. At the saturated vapor pressure, for phonons with Kci 10=< εε the function ( ) 0>< ci ppf . This corre- sponds to anomalous dispersion. For such phonons the conservation laws of energy and momentum allow pro- cesses which do not conserve the number of phonons. The fastest of these is the three-phonon process (3pp) where one phonon decays into two or two phonons merge into one. The rate of such process pp3ν in the general case was calculated in [7]. For phonons with ci pp > function ( ) 0<> ci ppf (normal dispersion). For phonons with normal disper- sion the conservation laws of energy and momentum prohibit the three-phonon processes. Then the fastest process is the four-phonon process (4pp). The rate of this process is pp4ν [2], [8] is much smaller that the rate pp3ν . The strong inequality pppp 43 νν > > (2) shows us that phonons of superfluid helium break-up into two subsystems: one subsystem of high-energy phonons (h-phonons) with ci pp > , in which the equi- librium is attained relatively slowly and second subsys- tem of low-energy phonons (l-phonons) with ci pp < , in which the equilibrium occurs relatively quickly. On the time scale of the problem under considera- tion the equilibrium in the subsystem of l-phonons oc- curs instantly and their energy distribution is given by the Bose-Einstein distribution function. The slow establishment of equilibrium in the subsys- tem of h-phonons may be described by a kinetic equa- tion for the distribution function ( ) inpn ≡ , which we write as: db NN dt dn −=1 (3) PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 301-305. 45 where bN and dN are the rates of increasing (b, born) and decreasing (d, decay) number of h1-phonons (the momentum of h1-phonon is 1p ). For four-phonon pro- cesses with the conservation laws of energy 4321 εεεε +=+ (4) and momentum 4321 pppp  +=+ (5) these rates can be written as ( ) ( ) 4 3 3 3 2 3 4321 4321,, , pdpdpdpppp WnN db dbdb  −−+× ×−−+= ∫ Ω δ εεεεδ (6) where ( )4321 ,\|, ppppWW = is defined by the transi- tion probability density; ( )( )2143 11 nnnnnb ++= ( )( )4321 11 nnnnnd ++= ; (7) bΩ and dΩ are the sets of maximum values of solid angles of phonons biΩ (i=3,4) and 2dΩ , taking part in processes of h1-phonon creation and decay, respectively. In the isotropic case π42 =Ω=Ω dbi and in the anisotropic phonon system pdbi Ω=Ω=Ω 2 . In rela- tions (4) - (7) and below it is supposed, that the phonon “1” has the momentum cpp ≥1 while other three phonons may have momentum less than cp or greater than cp . According to (3) the stationary state of the h-phonon subsystem is defined by the equality db NN = . (8) In an isotropic case, when db Ω=Ω , we obtain from the relations (6) - (8) the equation defining a sta- tionary distribution function ( )( ) ( )( )43212143 1111 nnnnnnnn ++=++ . (9) The solution of the equation (9), taking into account (4) and (5), is Bose-Einstein energy distribution function 1 )0( 1 −         −= Tk i B i en ε . (10) We define the rates of creation )(n bν and decay )(n dν for phonons with momentum 1p and an arbitrary distri- bution function ( )ipn  with a help of the relations )()0( 1 n bb nN ν= ; )( 1 n dd nN ν= (11) The rates of creation and decay, calculated using the distribution (10) with a help of equations (6) and (11) we designate accordingly )0( bν and )0( dν . In the isotropic phonon system according to (8)-(11) these rates are equal. However in the anisotropic phonon system, when db Ω≠Ω , these rates are not equal and according to (3) and (11) their difference dt dn ndb 1 )0( 1 )0()0( 1=− νν (12) defines the initial rate of change of the Bose-Einstein distribution function in the phonon system. In Ref. [5], the creation )0( bν and decay )0( dν rates of phonons with momentum 1p directed along the symme- try axis of the anisotropic phonon system with 1< <Ω p were obtained. The results presented in [5] show us the unusual character of the kinetic processes in anisotropic phonon systems and allow us to understand the differ- ence between the stationary h-phonon distribution func- tion and Bose-Einstein distribution (10). However even the complete examination of kinetic processes in the anisotropic phonon systems is possible only after get- ting the rates )0( bν and )0( dν for phonons with arbitrary directions of momenta 1p relative to the symmetry axis of the anisotropic phonon system. Here we obtain a so- lution of this problem in integral form and we discuss the consequences which follows from this general solution. We write the integrals (6) in a spherical coordinate system with the polar axis directed along the symmetry axis z of the phonon system, so that ( )iiii pp ϕθ ,, . In equation (6) we integrate with respect to the variable 4p using δ -function expressing the energy conserva- tion law; also we take into account the relation (11). As a result we obtain ( ) ( ) ( ) 43243232 2 4 2 3 2 2 43214321 43210 )0( , , 1 ζζζϕϕϕ δδ δν dddddddpdpppp pppppppp ppppWn c zzzzyyyy xxxxdb db × ×−−+−−+× ×−−+= ∫ Ω (13) where ( )( ))0( 4 )0( 3 )0( 20 11 nnnn ++= contains the distribu- tion functions (10), φ−−+= 3214 pppp is a function of independent variables 2p and 3p ; 2143 ffff −−+=φ (14) iiix pp ϕcos⊥= , iii pp θsin=⊥ ; iiiy pp ϕsin⊥= ; iiiz pp θcos= , ii θζ cos1−= . Let us integrate the expression (13) with respect to the variables 3ϕ and 4ϕ with the help of the first and the second δ -functions contained in the integrand (13). As a result we have ( ) 43232 2 4 2 3 2 20 44332211 )0( , , 1 ζζζ φζζζζδν ϕ ddddpdppppnI pppp c db db × ×−−−+= ∫ Ω (15) Here ( ) ∫= 2~ ~ 4 ϕ η ϕ ϕ ϕ d R R WI (16) where ( )ϕη R~ - is a step-function equal to unity at 0~ >ϕR and to zero at 0~ <ϕR , and ( ) 22 4 2 3 22 4 2 34~ ⊥⊥⊥∑⊥⊥ −−−= pppppRϕ ; ⊥⊥⊥∑ += 21 ppp  . The function W can be considered as an axially symmetric function if it does not depend on angles, or as in our case of small phonon dispersion, when the mo- mentum of all reacting phonons can be considered as being parallel. Supposing a weak dependence of W on 46 2ϕ we consider ( )2ϕW as a constant W , where 2ϕ is substituted with its effective value. So we can rewrite the integral ϕI in the following way: ( ) ( ) ( )ϕ ϕ ϕ ηαηα RKW R I −= 116 , where ( ) ∫ − = 2 0 22 sin1 π ϕα ϕ α dK is the full elliptic integral of the first kind, ϕ α R pppp ⊥⊥⊥⊥= 43214 is the parameter of ( )αK and ∑∑ = ⊥⊥⊥⊥⊥ => ⊥⊥ −+= 4 1 4 4321 4 1 22 82 i i ki ki pppppppRϕ is a function of the components of transverse momen- tum. Upper bounds of the integration of )( , i dbζ with the variables ( )4,3,2=iiζ in (15) are defined by the param- eter of anisotropy πθζ 2cos1 p pp Ω=−= (17) and are different for creation processes pbpbb ζζζζζ === )4()3()2( ;;2 (18) and decay processes 2;2; )4()3()2( === ddpd ζζζζ (19) The integration limits in (15) for variables 2p and 3p are determined by the conversation laws (4), (5) and values of the moduli of momentum of the phonons par- ticipating in the four-phonon processes. 3. THE ASYMMETRY BETWEEN THE CREATION AND THE DE- CAY OF HIGH-ENERGY PHONONS IN ANISOTROPIC PHONON SYSTEMS Asymmetry between creation and decay of h1- phonons follows from the different limits of integration (18) and (19) the expression (15) for the rates )0( bν and )0( dν . Thus, from conservation laws (4), (5) it could eas- ily be shown that in the anisotropic phonon systems, the restrictions imposed on creation and decay processes of h1-phonons are different. Taking the second power of the left and right parts of equality (5) taking into account the relations (4) and (1) we get 21 43 34 21 21 21 pp pp pp pp ζφζ + + = (20) where ki ki ik pp pp  −= 1ζ . In the anisotropic phonon system, where pi θθ < , the following situation is possible: there is no 2p phonon, which can annihilate the given 1p -phonon within the angle satisfying (20). So, according to (20), when 0>φ the decay of a phonon moving along z-axis is possible only under the following condition φζζ 21 21 min2 pp pp p + => (21) For the creation of the given 1p phonon with 01 =θ , according to (1), (4), (5), we have another restriction φζζ 41 32 min4 pp pp p − => (22) Inequalities (21) and (22) can be extended to the case of the arbitrary pθθ <1 . In the extreme case of an isotropic phonon system, when 2=pζ , inequalities (21) and (22) are always true and the processes of creation and decay are symmetric. In strongly anisotropic phonon systems at 1< <pζ , we can have the situation when one of inequalities (21), (22) is satisfied and other is not. This depends on values ( )4,3,2,1=ip i , which define the signs and magnitudes of the factors contained in the right hand parts of equali- ties (21) and (22). In view of this, it is convenient to ex- amine separately the different types of processes, which describe the interactions of h1-phonon with both l- phonons and with other h-phonons. There are only five possible types of four-phonon in- teractions. 1. 4321 lllh +↔+ ; 2. 4321 lhlh +↔+ ; 3. 4321 hhlh +↔+ ; 4. 4321 lhhh +↔+ ; (23) 5. 4321 hhhh +↔+ The arrow to the right indicates the decay of phonon “1” and to the left – creation. The division of processes into five types leads to the division of the integration area, with the variables 2p and 3p (15) into five areas. Accordingly, each integral (15) can be written as the sum of five integrals. As the result, for each of five processes this gives the rates of creation )0( jbν and decay ( )5,4,3,2,1)0( =j jdν . The limits of integration in )0( , jdbν with the variable 2p and 3p are determined by the following: 1. The type of creation or decay process. 2. Conservation laws (4) and (5); and 3. Inequalities (21) and (22) which lead to the appear- ance of the η -functions in the integrands for )0( jdν , when 0min2 >ζ and for )0( jbν , when 0min4 >ζ . According to (15) and (16) the integrals for )0( , jdbν be- come simpler for the case of creation or decay of 1p phonons moving along the z axis. Rates )0( , jdbν calculated for the case 01 =θ we denote jdb,ν where the super- script (0) is understood. When 01 =⊥p the integral ( )αK is equal to 2 π . In (15) it is possible to make the integration numerically and obtain graphs of the dependencies jdb,ν on one of three general parameters of the problem pTp θ,,1 when the other two parameters are fixed. 47 To explain the physical reasons for these dependen- cies for the rates of creation and decay, it is necessary to carry out analytical evaluations alongside the computer calculations. Unfortunately, even at 01 =θ integrals (15) cannot be expressed in terms of elementary func- tions. So the analytical approximation of the integrals was carried out for each jdb,ν by the replacement of the variables of integration, which only cause slow changes in the intergrands, by their typical values. After this, the integration of the remaining expressions can be done. The analytical expressions obtained in this way give nu- merical values which are close to the computer calcula- tions. Creation and decay rates vs the momentum of the h1-phonon for the case of h-phonons interacting with l- phonons. The values of temperature T=1K and the pa- rameter of anisotropy 2102 −⋅=pζ corresponds with those in experiments. The first process: a; the second process: b In the figure the dependencies of creation and decay rates for cases of interactions of h-phonons with l- phonons from 1p are shown. The values T=1K and ζp=2 ⋅10-2 are typical for experiments [3]. The first process is exceptionally important in creat- ing the h-phonon distribution function (Fig. a) where the phonons interchange between h-and l-subsystems. In Fig. a, for comparison, the value of the rate for the isotropic case is shown: ( ) ( )KTKT pdpbisotr 1,21,2 111 ====== ζνζνν . (24) Note in Fig. a the inequality 11 isotrb νν < < , and its growth with increasing 1p is determined by the strong anisotropy of the system ( )1102 2 < <⋅= − pζ and re- striction (22). Unlike the isotropic case (24) at 2102 −⋅=pζ the rate of creation is much greater than the rate of decay. So the different values and the momentum dependence of the rates 1bν and 1dν , shown in Fig. a are caused by the strong anisotropy of the system and the fact that in the first process 21 pp > and 0>φ . As a result at 1< <pζ inequality (21) leads to tighter restrictions than the inequality (22). So according to equality (22), at any 1p there exist 2p and 3p , which satisfy the inequality (22) and rate 1bν differs from zero at any 1p . The situa- tion is different for decay process: according to equality (21), there can be an initial momentum 01 pp = where the inequality (21) is not satisfied. As a result ( ) 0011 => ppdν and the lifetime of such phonons, due to the first process, becomes infinite. It is possible to obtain the analytical expression for 0p from (21) in the following way. We factorize the functions if contained in φ (14) near their typical val- ues of momentum: functions 3,1f - near cp , and 4f - near 2p . We find a minimum value of min2ζ . To do this we must place the maximum value of momentum 2p , which according to (4) is equal 12 2 ppp cup −= for the first process, into the right hand part of equality (21). Then we replace the inequality (21) by equality. The re- sulting equation ( )0122min2 ; pppp upp === ζζ allows us to obtain the analytical expression for 0p ;           ∂ ∂+= − = 2 1 2 2 0 2 1 cppc p c pp c pp εζ (25) For 2102 −⋅=pζ the relation (25) gives a value K k cp B o 11= , which coincides with the results of the computer calculation shown on Fig. a. The infinite lifetime and the finite rate of phonons created for 01 pp > is the cause of the fact that in the anisotropic phonon system the first process cannot cre- ate a dynamic equilibrium between the h-and l-phonon subsystems. However, this equilibrium is ensured by other processes. The second process according to Fig. b operates es- sentially over the whole momentum space 22 db νν > be- cause the inequality (21) in this case is more rigid, than (22). Unlike the first process 2dν differs from zero at all values of 1p because in the second process, the func- tion 0<φ at 31 pp < and there is no restriction (21), because in this case 0min2 <ζ . However the second process cannot compensate for the effect of the first pro- 48 cess at 01 pp > because the second process maintains the total number of h-phonons. Importantly, the dynamic equilibrium between h-and l-phonon subsystems is ensured by the fourth process, for which 44 bd νν > in those area of momentum, where 01 =dν . This result follows from the fact that for all the momenta of phonons taking part in the fourth process, the function 0<φ . As a result for the decay processes 0min2 <ζ and there is no restriction (21) while for cre- ation processes restriction (22) is implemented at 32 pp < , when 0min4 >ζ . The fifth process is similar to second. The function φ (14) may have different signs because of the relative sizes between the momenta of the phonons participating in the fifth process. In the fifth process, as well as in second, the total number of h-phonons is maintained and it also leads to the concentration of h-phonons in momentum space, along z-axis. In the third process 33 db νν > because at all values of momentum function 0>φ and restriction (21) is al- ways implemented. Rates 3bν and 3dν come to zero at cpp →1 , thus the volume of momentum space of a variable 3p , which in the third process satisfies an in- equality, 13 pppc ≤≤ tends to zero. The results of prior computer calculations obtained now with a help of the formula (15) at the arbitrary val- ue pθθ <1 testify that the rates )0( 1bν and )0( 1dν with the increasing of an angle 1θ differ only numerically from values shown in Fig. a. As for the second process there are not only the quantitative, but also qualitative modifi- cations. Beginning with some value eqvθθ =1 in a wide range of 1p values numerical values )0( 2bν become close to the numerical values )0( 2dν . At eqvθθ >1 the rate 2bν is less than the rate 2dν . And this not only quantitative- ly, but also qualitatively differs from the graphs present- ed in Fig. b. As a result the second process will lead to the concentration of h-phonons in momentum space along the z-axis. The effect of h-phonons concentration near z-axis was found in experiments [3] where the h- phonons were confined in a cone with an angle 4°, while l-phonons moved in a cone with an angle 11,4°. CONCLUSION A pulse of phonons moving in superfluid helium in one direction is an unusual physical system with unique properties determined by its strong anisotropy. Such system has been studied experimentally for more than ten years [1]-[3]. This paper is the continuation of the theoretical analysis of the anisotropic phonon systems being done by physicists of University of Exeter (UK) and the Kharkov National University (UA) since 1998. As a result of these collaborations it was shown in [5] and [8], that in anisotropic phonon systems, the kinetic processes differ from the usual ones in isotropic phonon systems where the momentum distribution has no unique direction. We have obtained a common expres- sion (15) for the creation and decay rates of high-energy phonons and have examined the consequences follow- ing from this expression. The asymmetry between creation and decay of h- phonons in the anisotropic phonon systems is a conse- quence of different limits of integration (18) and (19) in expression (15). Its physical reasons are explained by the different requirements for decay (21) and creation (22) processes in anisotropic phonon systems. Such asymmetry lead to the interesting predictions that in the anisotropic phonon systems, the energy distribution function of h-phonons should be numerically much greater than that in the Bose-Einstein distribution, and it should have an unusual momentum dependence. However a complete kinetic theory of the anisotrop- ic phonon system in superfluid helium is possible only after an evaluation of all creation and decay rates for h- phonons moving at arbitrary angles pθθ <1 relative to the symmetry axis z of the anisotropic phonon system. Such a problem demands finding all the solutions con- tained in (15). At the moment there are only solutions for all the rates in extreme case 01 =θ , and the previous results of numerical calculations obtained from (15) for the first and the second processes at arbitrary values pθθ <1 . This indicates that it is necessity to carry out further theoretical analysis in this field. This should be accompanied by further experiments, because, although at first, the experiments [1]-[3] stimulated creating the theory [4] and [6], now in some respects the roles are re- versed. We express profound gratitude EPSRC (grants GR/N18796 and GR/N20225) UK, and also GFFI Ukraine (grant №02.07/000372) for the support which enables us to carry out our scientific program. REFERENCES 1. A.F.G. Wyatt. Phonon propogation in liquid 4He: the dependence on injected power and pres- sure // J. Phys: Condens. Matter. 1989, v. 1, p. 8629-8648. 2. M.A.H. Tucker and A.F.G. Wyatt. Four- phonon scattering in superfluid 4He. // J. Phys.: Condens. Matter, 1992, v. 4, p. 7745-7758. 3. M.A.H. Tucker and A.F.G. Wyatt. Phonons in liquid 4He from a heated metal film: II. The an- gular distribution // J. Phys.: Condens. Matter. 1994, v. 6, p. 2825-2834. 4. I.N. Adamenko, K.E. Nemchenko, A.V. Zhukov, M.A.H. Tucker, A.F.G. Wyatt. Cre- ation of high-energy phonons from low-energy phonons in liquid helium // Phys. Rev. Lett. 1999, v. 82, p. 1482-1485. 5. I.N. Adamenko, K.E. Nemchenko, A.F.G. Wyatt. Phonon interactions in anisotropic phonon system leading to a suprathermal distribu- tion in liquid 4He // J. Low Temp. Phys. 2001, v. 125, p. 1-88. 6. A.F.G. Wyatt, M.A.H. Tucker, I.N. Adamenko, K.E. Nemchenko and A.V. Zhukov // Phys. Rev. B. 2000, v. 62, p. 3029- 3032. 49 7. M.A.H. Tucker, A.F.G. Wyatt, I.N. Adamenko, A.V. Zhukov, K.E. Nemchenko. Three-phonon interactions and initial stage of phonon pulse evolution in He II // Low Temp. Phys. 1999, v. 25, p. 488-492. 8. I.N. Adamenko, K.E. Nemchenko, A.V. Zhukov, M.A.H. Tucker, A.F.G. Wyatt. Four- phonon scattering in He II // Physica B. 2000, v. 284-288, p. 31-32. 50

The theory of kinetic processes in anisotropic phonon systems

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