Wave-number dependent current correlation for a harmonic oscillator

Wierling, A.; Sawada, Isao

doi:10.1103/physreve.82.051107

“…In fact, the recurrence relations approach states that if the frequency parameters or the correspondence between of them are known, then the solution for the TACF can be found. Moreover, it imposes the general routine to define these frequency parameters; however, the specific analytical expressions depend on a concrete problem studied [26,[34][35][36].…”

Section: Discussionmentioning

confidence: 99%

Relaxation Processes in Many Particle Systems -- Recurrence Relations Approach

Мокшин¹

2013

DNC

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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...

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“…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.…”

Section: The Methods Of Recurrence Relationsmentioning

confidence: 99%

“…The problem of a mass impurity in the harmonic chain was solved later, and its dynamical correlation functions were found to have the same form as in the quantum electron gas in two dimensions, thus showing that unrelated quantities in these two models displayed the same dynamical behavior, that is, the have dynamic equivalence [76]. It should be mentioned that harmonic oscillator chains have been the subject of a considerable amount of work with the method of recurrence relations [38][39][40][41][42][43][44][45][46].…”

Section: Applications To Interacting Systemsmentioning

confidence: 97%

Recent Advances in the Calculation of Dynamical Correlation Functions

Florencio

¹

,

Bonfim

²

2020

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We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.

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“…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.…”

Section: The Methods Of Recurrence Relationsmentioning

confidence: 99%

Recent advances in the calculation of dynamical correlation functions

Florencio¹,

Bonfim²

2020

Preprint

0

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We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.

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Wave-number dependent current correlation for a harmonic oscillator

Cited by 14 publications

References 45 publications

Relaxation Processes in Many Particle Systems -- Recurrence Relations Approach

Relaxation Processes in Many Particle Systems -- Recurrence Relations Approach

Recent Advances in the Calculation of Dynamical Correlation Functions

Recent advances in the calculation of dynamical correlation functions

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