Relaxation Processes in Many Particle Systems -- Recurrence Relations Approach
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Relaxation Processes in Many Particle Systems – Recurrence Relations Approach
Anatolii V. Mokshin ∗ Kazan Federal University, Kazan, 420008, Russia
The general scheme for the treatment of relaxation processes and temporal autocorrelationsof dynamical variables for many particle systems is presented in framework of the recurrencerelations approach. The time autocorrelation functions and/or their spectral characteristics, whichare measurable experimentally (for example, due to spectroscopy techniques) and accessible fromparticle dynamics simulations, can be found by means of this approach, the main idea of which isthe estimation of the so-called frequency parameters. Model cases with the exact and approximativesolutions are given and discussed.
I. INTRODUCTION
Relaxation processes, which emerge in many particle systems, are characterized by highly nontrivial features even forthe cases of the well-known simplified models [1]. So, for example, the ideal gas dynamics at the drive by external fieldsexhibit the non-Markovian (memory) effects [2] as well as the manifestations of anomalous transport [3], whilst thechain of coupled harmonic oscillators can display the nonlinear dynamics [4] with stochastic resonance peculiarities [5].At the presence of complicated fields of interactions in many particle systems together with structural disorder andintricate spatiotemporal correlations allows one to recognize these as the complex systems. Thus, dense liquids,structural and spin glasses, foams, emulsions and colloidal gels are the typical examples of the physical complexsystems, which combine the complicated dynamics together with the structural inhomogeneity [6].From theoretical standpoint, the description of many particle dynamics reduces oneself into a unified fashion atthe applying the mathematical language of the distributions, the correlation and relaxation functions, as well as theGreen functions, that provide a statistical treatment to some extent. Berne and Harp marked the significance of timecorrelation functions in the consideration of dynamic processes by the phrase [7]: “. . . time correlation functions havedone for the theory of time-dependent processes what the partitions functions have done for the equilibrium theory.The time-dependent problem has became well defined . . . ”. These enthusiastic words become more clear and accepted,if one takes into account that the correlation functions appear to be directly related with the experimentally measuredquantities due to Kubo’s linear response theory [8] as well as by means of nonlinear response approach [9], whichobtained recently a rapid development. Importantly, the time correlation functions are associated with the concreterelaxation processes and, thereby, provide the information about the proper relaxation time scales [2]. Moreover,these functions can be applied to estimate quantitatively and simply the so-called memory effects in many particlesystem dynamics [2], the dynamical heterogeneity effects in particle movements [10] and the breakdown of systemergodicity [11].Historically, the formulation of fluctuation-dissipation theorem [12] and the Zwanzig-Mori’s projection operatorformalism [1] can be distinguished as the important milestones in development of the theoretical description withintime correlation and relaxation functions concept. If the above-mentioned theorem was a forthright indication on therelation between a (system) response function to an external field and the fluctuations of the corresponding degree offreedom, then the projection operator formalism was the first one, which had provided us by a general scheme to definethe time correlation functions at the consideration of many particle system dynamics within unified mathematicalconstrains. The projection operator technique allowed one to establish the exact relations between characteristicsof the different relaxation processes, that had formed later a basis for the construction of such theories like thegeneralized hydrodynamics [13] and the mode-coupling theories [14, 15]. Despite obvious advances of the Zwanzig-Mori’s projection operator formalism in the description of many particle system dynamics, the universal mathematicalpatterns in these systems became more clear with the appearance of the recurrence relations approach [16, 17]. Inparticular, this approach gave a clear idea that the (dis)similarity of relaxation time scales and the structural recursivenature – structural hierarchy – can be mathematically taken into account by means of recurrence relations.The aim of this paper is to show the general scheme how the relaxation processes in many particle systemscan be treated within the recurrence relations approach. Section presents the some typical relations between theexperimentally measured terms and the correlation functions, which describe relaxation processes in many particle ∗ Electronic address: [email protected] systems. The formulation of the basic general relations is given in section . The simplified model cases with the exactsolutions are presented in sections and , whereas the description of the density fluctuations in simple liquids isconsidered in section . And, finally, we conclude in Section . II. CORRELATION FUNCTIONS VS. OBSERVABLE TERMS
Let us consider the system of N interacted particles, which evolves at the temperature T within the volume V . Thefull dynamics of the system is characterized by a set of the variables A n ( t ) like, for instance, local density, particledisplacement, particle velocity, dipole moment etc. Nevertheless, a concrete problem under study enforces, as a rule, torestrict oneself by a some variable associated with phenomena. The choice of the dynamical variable can be caused bythe experimentally measurable response function, which is related with the corresponding relaxation (or correlation)function of this dynamical variable. Scattering techniques. – The inelastic neutron scattering and the inelastic X-ray scattering techniques allow one tomeasure the dynamic structure factor S ( k, ω ) and the incoherent scattering function S s ( k, ω ) , where k is the wavenumber and ω is the frequency. These terms are related with the autocorrelation functions of the spatial Fouriertransforms for the local density fluctuations ρ ( r , t ) = (1 / √ N ) P i δ [ r − r i ( t )] and for the tagged (single) particledisplacement ρ s ( r , t ) = δ [ r − r i ( t )] , correspondingly, [13] S ( k, ω ) = S ( k )2 πN Z ∞−∞ dt e − iωt X i,j h e − i k · r i (0) e i k · r j ( t ) ih e − i k · [ r i (0) − r j (0)] i (1) S s ( k, ω ) = 12 πN Z ∞−∞ dt e − iωt X i h e − i k · r i (0) e i k · r i ( t ) i , (2)where k is the wave vector, S ( k ) = (1 /N ) P i,j (cid:10) e − i k · [ r i (0) − r j (0)] (cid:11) = R ∞−∞ dω S ( k, ω ) is the static structure factor, k = | k | and ρ ( k , t ) = (1 / √ N ) P i e i k · r i ( t ) is the space Fourier transform of ρ ( r , t ) . Thus, the dynamic structure factor S ( k, ω ) estimates the collective dynamics with frequencies ω over spatial scales ∼ π/k and is related with the densityautocorrelation function (or the so-called intermediate scattering function) φ coh ( k, t ) = 1 N X i,j h e − i k · r i (0) e i k · r j ( t ) ih e − i k · [ r i (0) − r j (0)] i , (3)whereas the incoherent scattering function S s ( k, ω ) performs the same for single particle dynamics and is associatedwith the particle displacement autocorrelation function (the self-intermediate scattering function) φ self ( k, t ) = 1 N X i h e − i k · r i (0) e i k · r i ( t ) i . (4) Dielectric spectroscopy. – The complex dielectric permittivity ε ∗ ( ω ) is measurable due to dielectric spectroscopyexperiments. This term is related with the macroscopic dipole correlation function φ d ( t ) = h M (0) M ( t ) ih M (0) i (5)by the relation [18] − ε ∗ ( ω ) − ε ∞ ε s − ε ∞ = ℑ (cid:20) s Z t dt e − st φ ( t ) (cid:21) , s = iω, (6)where M ( t ) is the macroscopic fluctuating dipole moment of the sample volume unit, which is equal to the vectorsum of all the molecular dipoles, ε s and ε ∞ are the low- and high-frequency limits of the dielectric permittivity,respectively, and ℑ [ . . . ] means imaginary part of [ . . . ] . Transport coefficients. – A feature of resonance techniques is that they can measure the transport properties. So, forexample, the nuclear magnetic resonance provides the information about the self-diffusion coefficient D s and viscosity η of study system, which are (as well as other transport coefficients) are related with the autocorrelation functions fromthe corresponding current dynamical variables through the Green-Kubo relations [13]. So, for self-diffusion coefficientone has D = k B Tm Z ∞ dt h v (0) v ( t ) ih v (0) v (0) i , (7)where φ vel = h v (0) v ( t ) ih v (0) v (0) i (8)is the single particle velocity autocorrelation function. Equation, similar to Eq. (7), can be written for rotationaldiffusion coefficient in term of the dynamical variable – angular velocity Ω α , that accounts the molecular reorientationsmeasurable in the depolarization of fluorescence studies.Further, the shear viscosity is η = 1 k B T V Z ∞ dt h P xy (0) P xy ( t ) i , (9)where φ η ( t ) = h P xy (0) P xy ( t ) i (10)is the autocorrelation function of the components of the pressure tensor, which are given by virial formula P αβ = N X i =1 mv iα v iβ + 12 N X i = j F ijα r ijβ , α, β = x, y, z, (11)and F ijα denotes the α -component of the force between particles i and j that are at distance r ij from one another.The thermal conductivity λ can be expressed as λ = 1 k B T V Z ∞ dt h J ez (0) J ez ( t ) i , (12)where φ λ = h J ez (0) J ez ( t ) i (13)is the heat current autocorrelation function with the dynamical variable J ez = N X i =1 v iz m | v i | N X i = j U ( r ij ) − N X i =1 N X i = j v i r ij ∂U ( r ij ) ∂z ij , (14) U ( r ij ) is the potential of particle interaction.The examples given above show clearly that a variety of quantities experimentally observed can be treated interms of the time correlation (or relaxation) functions. Relation of other quantities experimentally measured with thecorresponding correlation functions can be found, for example, in comprehensive review [7]. Thus, the problem of theexplanation of the experimental results in some (not all, but many) cases can be reduced to the problem of the findingeither the proper relaxation function or, at least, the asymptotic behavior of this function. III. THEORETICAL BACKGROUND
Let us assume that we consider the dynamical variable A ( t ) , the time evolution of which is defined by the Heisenbergequation dA ( t ) dt = i[ H, A ( t )] = i ˆ L A ( t ) , A ( t ) | t =0 = A, (15)where H is the Hamiltonian of the system, ˆ L is the Liouville operator, which is taken to be Hermitian and [ . , . ] isthe Poisson bracket. The formal solution of equation (15) can be written as A ( t ) = e i ˆ L t A. (16)On the other hand, the Hamiltonian H defines the averaging operation h A i through the density of phase space ρ ∝ exp [ − β ( H − µN )] , where µ is the chemical potential, β = ( k B T ) − and k B is the Boltzmann constant. Then, inclassical limit one has a simple correspondence between the scalar product of a pair of dynamical variables A and B of the Liouville space and the corresponding correlation function [13] ( A, B ) ≡ h AB ∗ i . (17)The symbol ∗ marks the complex conjugation. In fact, equation (17) provides the identity between the relaxationfunction ( A ( t ) , B ) and the time correlation function h A ( t ) B ∗ i , i.e. ( A ( t ) , B ) ≡ h A ( t ) B ∗ i . Further, we will utilize thetime autocorrelation functions (TACF) in the dimensionless form φ ( t ) = h A (0) ∗ A ( t ) ih| A (0) | i , (18)that ensures the fulfillment of the next conditions φ ( t ) | t =0 = 1 , ≥ φ ( t ) ≥ , dφ ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 . (19)Such a representation of the TACF allows one to focus on the time dependence of φ ( t ) directly, and to do thecomparison of autocorrelations for different dynamical variables.Applying the Gram-Schmidt orthogonalization procedure at the initial condition A ≡ A , we generate the set ofdynamical variables: A = { A , A , A , . . . , A ν , . . . } , (20a) ( A ν , A µ ) = ( A ν , A ν ) δ ν,µ , (20b) ν, µ = 0 , , , , . . . , which are related by the recurrence relation A ν +1 = i ˆ L A ν + ∆ ν A ν − , (21a) ∆ ν = ( A ν , A ν )( A ν − , A ν − ) , (21b) A − = 0 , ∆ = 1 , where ∆ ν are the frequency parameters with the dimension of the squared frequency, δ ν,µ is the Kronecker delta.From the definition (21b) one can see that the physical meaning of the parameters ∆ ν depends on the concreteprocess and is defined by the corresponding dynamical variables A ν − and A ν . Equation (21a) is also known from therecurrence relations approach as the first recurrence relation [17, 19, 20]. Usually, it is convenient (but not necessarily)to perform the construction of the set A on the basis of the dynamical variable A , which is associated with theprocesses experimentally studied.Both the projection operators technique [1] as well as the recurrence relations approach [19, 20] yield the chain ofthe integro-differential equations for the dynamical variables A of the form ddt A ν ( t ) = − ∆ ν +1 Z t A ν ( t − τ ) h A ν +1 (0) ∗ A ν +1 ( τ ) ih| A ν +1 (0) | i dτ + A ν +1 ( t ) , (22) here and hereafter ν = 0 , , , , . . . , that is the exact consequence of equation (15). For the TACF’s defined as φ ν ( t ) = h A ν (0) ∗ A ν ( t ) ih| A ν (0) | i , For a case, when the dynamical variable is chosen to be the particle velocity, i.e. A = v , equation (22) represents the known generalizedLangevin equation. the chain (22) takes the following form ddt φ ν ( t ) = − ∆ ν +1 Z t φ ν ( t − τ ) φ ν +1 ( τ ) dτ. (23)Then, applying the Laplace transform operator ˆ L [ f ( τ )] = ˜ f ( s ) = Z ∞ e − sτ f ( τ ) dτ (24)to chain of equations (23), one obtain the recurrence formula ˜ φ ν ( s ) = 1 s + ∆ ν +1 ˜ φ ν +1 ( s ) , (25)which can be transformed to continued fraction representation of frequency spectrum of the TACF: ˜ φ ( s ) = 1 s + ∆ s + ∆ s + ∆ s + . . . . (26)On the one hand, equation (26) indicates that the form of the spectrum ˜ φ ( s ) and, thereby, of the relaxation function φ ( t ) , is completely defined by the frequency parameters ∆ ν as well as by the ratios ∆ ν +1 / ∆ ν . On the other hand, thevalues of ∆ ν ’s are directly associated with the frequency/time range, for which the fraction solution [equation (26)]will be relevant. Thus, the problem of finding φ ( t ) [or ˜ φ ( s ) ] is reduced, mathematically, to the problem of finding afunction, which is representable in the form of continued fraction with the some unique set of values of ∆ ν ’s. Althoughthese frequency parameters are physical characteristics of the concrete relaxation processes [see equation (21b)], onecan possible to consider some different model situations, which can be realized in some general cases. IV. MODELS OF THE FINITE SETS OF VARIABLES
Let us consider the cases with the finite set of dynamical variables ( ν is finite), that corresponds to the finite-dimensional Liouville spaces. Such situations arise at the condition with A ν = 0 and ∆ ν = 0 ; and are relevant tononergodic processes with non-decaying correlation functions expressed by cosine functions. A. A case of ν = 2 At the condition ν = 2 one has A = 0 and ∆ = 0 . Then, continued fraction (26) yields the system of two equations (cid:26) − s ˜ φ ( s ) = ∆ ˜ φ ( s ) ˜ φ ( s ) ,s ˜ φ ( s ) = 1 (27)with simple solutions in the time domain φ ( t ) = cos( p ∆ t ) , (28a) φ ( t ) = 1 . (28b)The relaxation function of the form (28a) reproduces the behavior of undamped harmonic oscillator. This situationis realized, for example, in the case of density fluctuations of homogeneous electron gas at the finite wave numbers k and the temperature T = 0 . Other example, where the case appears, is the dynamics of the chain of classicalharmonic oscillators. Here, the TACF of particle velocity, φ ( t ) = h υ (0) υ ( t ) i / h υ (0) i , is described by equation (28a)(see reference [21]). It is clear that the case of ν = 1 is trivial. Therefore, this case is not considered here. B. A case of ν = 3 One has here that A = 0 and ∆ = 0 . Then, continued fraction (26) in this case transforms into the next system ofequations: − s ˜ φ ( s ) = ∆ ˜ φ ( s ) ˜ φ ( s ) , − s ˜ φ ( s ) = ∆ ˜ φ ( s ) ˜ φ ( s ) ,s ˜ φ ( s ) = 1 , (29)which can be resolved by means of the inverse Laplace transform ˆ L − and yields the solutions φ ( t ) = 1∆ + ∆ h ∆ + ∆ cos( p ∆ + ∆ t ) i , (30a) φ ( t ) = cos( p ∆ t ) . (30b) φ ( t ) = 1 . (30c)Equation (30a) corresponds again to the harmonic behavior of the initial TACF φ ( t ) , where the period is defined bytwo frequency parameters, ∆ and ∆ .From these two cases presented above, one can see that it is possible to find exact analytical solutions for therelaxations functions φ ν ( t ) at any finite dimension ν . V. MODELS OF THE INFINITE SETS OF VARIABLES
For the infinite-dimensional Liouville spaces, ν → ∞ , the set of possible scenarios for the TACF φ ( t ) is vast andcontains the decaying functions. A. Gaussian relaxation
Let us now consider the case, where the frequency parameters ∆ ν ’s are related according to the arithmetic progression: ∆ , ∆ = 2∆ , ∆ = 3∆ , . . . , ∆ ν = ν ∆ . (31)Then, the continued fraction (26) takes the following form ˜ φ ( s ) = 1 s + ∆ s + 2∆ s + 3∆ s + . . . , (32)that corresponds in the time domain to the ordinary Gaussian function [22] φ ( t ) = e − ∆ t / . (33)The most well-known examples of the physical realization of such relaxation is the density fluctuations in the perfectgas and one-particle dynamics in liquids (at the limit of high wave numbers k ) [13, 23]. Other case with this relaxationis the dynamics of the one-dimensional XY-model at T → ∞ [24].The exact correspondence between the frequency parameters [given by relations (31)] indicates on the possibility tostudy quantitatively the deviation from the Gaussian relaxation by means of the simple comparison of the parameters ∆ ν ’s: α ν = νν + 1 ∆ ν +1 ∆ ν − . (34)For the Gaussian relaxation one has α ν = 0 , whilst deviations from the zeroth values of α ν will be caused bymanifestations of the non-Gaussian behavior of φ ( t ) . B. Damped relaxation of oscillated correlator
Let us consider the specific case, where the frequency parameters are finite and equal to each other ∆ = ∆ = ∆ = . . . = ∆ ν . (35)that corresponds to the continued fraction (26) of the form ˜ φ ( s ) = 1 s + ∆ s + ∆ s + . . . . (36)As known from the theory of continued fractions, expression (36) is the representation of the next function (over thevariable s ): ˜ φ ( s ) = − s + √ s + 4∆ . (37)Applying the inverse Laplace transform operator ˆ L − to equation (37), one obtains the TACF φ ( t ) = 1 √ ∆ t J (2 p ∆ t ) , (38)where J is the Bessel function of the first order. Such relaxation appears in the processes, which characterized bythe damped harmonic oscillatory behavior. For example, expression (38) is the exact form for the TACF of velocityof the Brownian particle in linear chain of the identical harmonic oscillators [1, 2, 25]. Moreover, the dynamics ofthe two-dimensional electron gas at the temperature T = 0 and at the defined range of the wave number k is othermanifestation of relaxation with the TACF of the form (38) (see reference [26]).Thus, the presented cases demonstrate that at the known correspondence between the frequency parameters F (∆ , ∆ , ∆ , . . . , ∆ ν , . . . ) , one can exactly define the initial TACF φ ( t ) and to estimate the frequency featuresof its spectral image ˜ φ ( s ) . VI. DENSITY FLUCTUATIONS IN SIMPLE LIQUIDSA. Frequency parameters
Let us now consider the liquid system, where the particle interactions are characterized by the spherical symmetryand the potential contains the radial dependence only. Liquid metals and condensate noble gases are the typicalexamples of such systems [13, 27, 28]. Further, we take the space Fourier transform of the local density fluctuations, A ( k ) = ρ ( k , t ) = (1 / √ N ) P i e i k · r i ( t ) , as the initial dynamical variable, the TACF of which, φ coh ( k, t ) , is relatedwith dynamic structure factor S ( k, ω ) [see equation (1)]. Then, the frequency parameters can be found according torelations (21a) and (21b) ( A ( k ) , A ( k )) = S ( k ) , ( A ( k ) , A ( k )) = k B Tm k , ∆ ( k ) = k B Tm k S ( k ) = ( v T k ) S ( k ) , (39a) ∆ ( k ) = 3 k B Tm k + ρm Z ∇ l u ( r )[1 − exp( i kr )] g ( r ) d r − ∆ , (39b) ∆ ( k ) = 1∆ ( k ) Ξ( k ) − [∆ ( k ) + ∆ ( k )] ∆ ( k ) , (39c)where v T is the average thermal velocity of particles, ρ is the number density, g ( r ) is the pair distribution function, u ( r ) is the interparticle potential, the suffix l marks the component parallel to wave vector k . Within the assumptionabout pair-additivity of the interparticle potential u ( r ) , the term Ξ( k ) takes the following form [29]: Ξ( k ) = 15 (cid:18) k B Tm (cid:19) k + k B Tm k ρm Z d r g ( r ) ∇ l u ( r ) (40) + 6 ρ k B Tm k Z d r g ( r ) ∇ l u ( r ) sin( kr )+ 2 ρm Z d r g ( r )[ ∇∇ l u ( r )] [1 − cos( kr )]+ ρ m Z Z d r d r ′ g ( r,r ′ )[1 − cos( kr − kr ′ )] ( ∇∇ l u ( r )) ( ∇ ′ ∇ ′ l u ( r ′ )) , where g ( r , r ′ ) is the three-particle distribution function. The frequency parameters of a higher order willcontain the more complicated integral expressions with the n -particles distribution functions g n (¯ r ) , i.e. ∆ ν ( k ) = F [∆ ( k ) , ∆ ( k ) , . . . , ∆ ν − ( k ); g ( r ) , g (¯ r ) , . . . , g ν (¯ r )] . B. Dynamics at high wave numbers and short time scales
In the ranges of the wave numbers k larger than k m = 2 π/σ [where σ is the effective particle size and k m is the firstmaximum in the static structure factor S ( k ) ] and of the short time scales, which correspond to lengths smaller thanthe mean free path, the interactions between particles in simple liquids can be neglected. Taking into account thatfor this range of k one has S ( k ) → , we obtain directly from equations (39a), (39b) and (39c) ∆ ( k ) = ( v T k ) , ∆ ( k ) = 2( v T k ) , ∆ ( k ) = 3( v T k ) . (41)Comparing (41) and (31) one can see that relations (41) are the first steps of recurrence relation ∆ ν +1 ( k ) = ν + 1 ν ∆ ν ( k ) , (42)which obeys the Gaussian relaxation for the density autocorrelation function φ coh ( k, t ) = e − ( v T kt ) / . (43)Then, for the dynamic structure factor in the frequency domain one has the single Gaussian-like function locatedat the zeroth frequency. This result is completely reasonable, since it means that the short time dynamics in therange of high values of k is defined by free particle movements, and the relaxation occurs over a single time scale τ ∼ p / ∆ ( k ) ∼ ( v T k ) − . C. Microscopic dynamics at wave numbers k ≤ k m At the condition for wave numbers k ≤ k m , the corresponding spatial ranges of a system can be filled by few particlesonly. Within such the ranges, the description of the proper dynamics is relevant if it is performed in terms of two-, three- and four-particle distribution functions, which are contained in the first four frequency parameters, ∆ ν , ν = 1 , , , .Moreover, the treatment of experimental I ( k, ω ) -data of inelastic X-ray scattering [30] as well as the numericalmolecular dynamics simulations results for liquid alkali metals near melting [23, 31, 32] indicate that there is thecorrespondence for this range of wave number: ∆ ( k ) ≃ ∆ ( k ) . (44)The extension of this equality to the frequency parameters of higher order ∆ ν = ∆ , ν > , (45)allows one to define the term ˜ φ ( k, s ) of the chain (25) by analogy with the model case (35) ˜ φ ( k, s ) = − s + p s + 4∆ ( k )2∆ ( k ) . (46)It is necessary to note that extension (45) has a clear sense at consideration of the transition into the regimeof high values of k . Equation (42) indicates on the equality ∆ ν ( k ) = ∆ ν +1 ( k ) at limit of high ν -values, i.e. lim ν →∞ ∆ ν ( k ) / ∆ ν +1 ( k ) = 1 . Thus, for the regime of k ≤ k m the equality of frequency parameters arises at lower levelof chain (23) and smaller values of ν .Then, going down over fraction (26) to the term ˜ φ ( k, s ) one obtains ˜ φ ( k, s ) = 2∆ ( k ) s [2∆ ( k ) − ∆ ( k )] + ∆ ( k ) p s + 4∆ ( k ) . (47)Then, the dynamic structure factor will be as the next S ( k, ω ) = S ( k )2 π ∆ ( k )∆ ( k )∆ ( k ) p ( k ) − ω (cid:8) ∆ ( k )∆ ( k )+ ω (cid:2) ∆ ( k )∆ ( k ) − ( k )∆ ( k ) − ∆ ( k )∆ ( k ) + 2∆ ( k )∆ ( k )∆ ( k ) − ∆ ( k )∆ ( k )∆ ( k )+ ∆ ( k )∆ ( k ) (cid:3) + ω (cid:2) ∆ ( k ) − ( k )∆ ( k ) + 2∆ ( k )∆ ( k ) − ( k )∆ ( k ) + ∆ ( k )∆ ( k ) (cid:3) + ω [∆ ( k ) − ∆ ( k )] (cid:9) − . (48)It is necessary to note that the dynamic structure factor S ( k, ω ) of the form (48) reproduces the three-peak structurein the frequency domain (at the fixed k ), which is observable within the experiments of inelastic neutron scatteringand inelastic x-ray scattering. This spectral form is very similar to the known Rayleigh -Mandelshtam-Brillouin tripletat light scattering corresponded to the hydrodynamic limit ( t → ∞ and k → ). Here, one peak of S ( k, ω ) is locatedat the zeroth frequency ( ω = 0 ), while two other peaks – the so-called doublet – are located at the finite frequencies( ± ω L ). D. Relation with hydrodynamics
The features of the high-frequency (inelastic) peaks of the dynamic structure factor S ( k, ω ) will be defined by thesolution of the next equation s + ∆ ( k ) s + ∆ ( k ) ˜ φ ( k, s ) = 0 , (49)which is general within the recurrence relation approach [23]. In the considered case, the function ˜ φ ( k, s ) for thepresented scheme has the form (47), while equation (49) will have complex solutions s = ℜ [ s ( k )] + i ℑ [ s ( k )] . Here, theimaginary part ℑ [ s ( k )] defines the high-frequency peak positions, while the real part ℜ [ s ( k )] is associated with thewidths of the peaks.To analyze equation (49) we introduce two dimensionless quantities (at the fixed k ): ξ ( k ) = s ∆ ( k ) , (50a) ς ( k ) = 2∆ ( k )∆ ( k ) − . (50b)At the transition to the hydrodynamical limit the next condition should be satisfied: | ξ ( k ) | ≪ , s + 2 p ∆ ( k ) ς ( k ) s + (cid:20) ∆ ( k ) + ∆ ( k ) + ∆ ( k ) ς ( k ) (cid:21) s + 2∆ ( k ) p ∆ ( k ) ς ( k ) = 0 . (51)The solution of this cubic equation can be found within the convergent scheme for approximating solutions appliedby Mountain (see reference [33]) s , ( k ) = ± s ∆ ( k ) (cid:26) ( k )[1 + ς ( k )]∆ ( k ) ς ( k ) (cid:27) − p ∆ ( k ) ς ( k ) (cid:18) − ∆ ( k ) ς ( k )∆ ( k ) ς ( k ) + ∆ ( k ) ς ( k ) + ∆ ( k ) (cid:19) , (52a) s ( k ) = − ( k ) ς ( k ) ∆ ( k ) ς ( k )∆ ( k ) ς ( k ) + ∆ ( k ) ς ( k ) + ∆ ( k ) . (52b)These approximated solutions corresponds to the results of the hydrodynamical Landau-Placzek theory with thefollowing parameters: the adiabatic sound velocity c s is ( c s k ) = lim k → ( v T k ) = lim k → ∆ ( k ) , the sound damping parameter is Γ = γ − γ p ∆ ( k ) ς ( k ) , where the ratio of the specific heat at constant pressure to the specific heat at constant volume γ = c p /c v is γ = 1 + ∆ ( k ) + ∆ ( k ) ς ( k )∆ ( k ) ς ( k ) . Thus, the expression for the dynamic structure factor S ( k, ω ) of the form (48) satisfies completely the transition intohydrodynamic regime; and the parameters of hydrodynamic theory are expressed through the frequency parameters ∆ ( k ) , ∆ ( k ) , ∆ ( k ) and ∆ ( k ) . VII. CONCLUSIONS
In this paper, we have presented the approach, by means of which the relaxation processes in many particle systemscan be studied. This approach reformulates the equation of motion for the considered dynamical variable in termsof the recurrence relations, where the set of frequency parameters appear. In fact, the recurrence relations approachstates that if the frequency parameters or the correspondence between of them are known, then the solution for theTACF can be found. Moreover, it imposes the general routine to define these frequency parameters; however, thespecific analytical expressions depend on a concrete problem studied [26, 34–36].Here, we have shown some simple model cases with the exact solutions, which are realized, nevertheless, in thephysical problems. As the other nontrivial case, we have considered the problem of the description of local densityfluctuations in simple liquids. As it was shown, the solution for the spectral image of the local density fluctuation TACFcan be obtained within the recurrence relations, and this solution is correctly consistent with the known asymptoticscenarios.The important advantage of the approach is the absence of the abstract parameters at the construction of theoreticaldescription. All the quantities are expressed only through the frequency parameters, which contain the details ofinterparticle interactions and static structure correlations. Therefore, such the approach, in our opinion, is the perfectone within of which it is possible to construct the explanation how the many particle system structure defines thesystem dynamics.1
VIII. ACKNOWLEDGEMENTS
We would like to thank M. Howard Lee (University of Georgia, USA) for very useful discussions. [1] Zwanzig, R. (2001)
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