Relaxation Processes in Many Particle Systems -- Recurrence Relations Approach

Abstract

The general scheme for the treatment of relaxation processes and temporal autocorrelations of dynamical variables for many particle systems is presented in framework of the recurrence relations approach. The time autocorrelation functions and/or their spectral characteristics, which are measurable experimentally (for example, due to spectroscopy techniques) and accessible from particle dynamics simulations, can be found by means of this approach, the main idea of which is the estimation of the so-called frequency parameters. Model cases with the exact and approximative solutions are given and discussed. I. INTRODUCTIONRelaxation processes, which emerge in many particle systems, are characterized by highly nontrivial features even for the cases of the well-known simplified models [1]. So, for example, the ideal gas dynamics at the drive by external fields exhibit the non-Markovian (memory) effects [2] as well as the manifestations of anomalous transport [3], whilst the chain 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 and intricate 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 complex systems, 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 at the applying the mathematical language of the distributions, the correlation and relaxation functions, as well as the Green functions, that provide a statistical treatment to some extent. Berne and Harp marked the significance of time correlation functions in the consideration of dynamic processes by the phrase [7]: ". . . time correlation functions have done 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 measured quantities due to Kubo's linear response theory [8] as well as by means of nonlinear response approach [9], which obtained recently a rapid development. Importantly, the time correlation functions are associated with the concrete relaxation 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 particle system dynamics [2], the dynamical heterogeneity effects in particle movements [10] and the breakdown of system ergodicity [11].Historically, the formulation of fluctuation-dissipation theorem [12] and the Zwanzig-Mori's projection operator f...

Recent Advances in the Calculation of Dynamical Correlation Functions

“…Also, from RRII one obtains (da 0 (t)/dt)| 0 0, which precludes a pure time exponential as well as other functions that do not have zero derivative at t 0. The method of recurrence relations have since been applied to a variety of problems, such as the electron gas [33][34][35][36], harmonic oscillator chains [37][38][39][40][41][42][43][44][45][46], many-particle systems [47][48][49][50], spin chains [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66], plasmonic Dirac systems [67,68], dynamics of simple liquids [69,70], etc.…”

Recent advances in the calculation of dynamical correlation functions

“…Also, from RRII one obtains (da 0 (t)/dt)| 0 = 0, which precludes a pure time exponential as well as other functions that do not have zero derivative at t = 0. The method of recurrence relations have since been applied to a variety of problems, such as the electron gas [27,28,30,29], harmonic oscillator chains [31,32,33,34,35,36,37,38,39,40], many-particle systems [44,43,41,42], spin chains [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60], plasmonic Dirac systems [61,62], etc.…”

Self-consistent approach to the description of relaxation processes in classical multiparticle systems

“…В случае гауссовой релаксации имеем α ν = 0, в то время как отклонения параметров α ν от нулевых значений будут характеризовать негауссово поведение корреляционной функции φ 0 (t) [65].…”

“…где U (r ij ) -потенциал межчастичного взаимодействия [65], [68]. С другой стороны, учитывая, чтоφ AA 0 (s) -лаплас-образ ВКФ…”