Generalized Langevin equation and recurrence relations

Lee, M. Howard

doi:10.1103/physreve.62.1769

Cited by 41 publications

(34 citation statements)

References 45 publications

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“…An alternative approach to the dynamics of many-body systems was developed some time ago by Lee [12][13][14]. This method is based on a projection procedure in a Hilbert space which does not require the introduction of memory functions and provides a clearer connection to physical properties via the derivation of particular recurrence relations, though leading to the continued-fraction representation of spectra similar to what is obtained in MZ theory.…”

Section: Introductionmentioning

confidence: 90%

Expansion in Lorentzian functions of spectra of quantum autocorrelations

Barocchi

Bafile

2013

Phys. Rev. E

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We show that in a quantum mechanical many-body system of Boltzmann particles having space inversion symmetry the spectrum of the autocorrelation function of a local observable can always be given, similarly to the classical case [Phys. Rev. E 85, 022102 (2012)], in terms of a series of Lorentzian functions multiplied by the proper quantum detailed balance factor. This is done by transforming the continued fraction representation, which is derived via recurrent relations and without the use of the generalized Langevin equation hierarchy, into a series expansion. In this way characteristic frequencies can be defined, also in quantum mechanics, which refer to the particular autocorrelation. These are the frequencies of the eigenmodes of the relaxation function connected to the observable. We also show that in practical cases of interest in experimental spectroscopy, and particularly in inelastic neutron and x-ray scattering, the use of a finite number of Lorentzian shapes for an approximate description of the data is related to a reduction of the number of the relevant dynamical variables taken into account, equivalent to the lowering of the dimensionality of the orthogonalized space onto which the dynamic of the system is projected. Examples of application are given for the spectrum of the velocity autocorrelation function of liquid parahydrogen, calculated with a quantum simulation algorithm (path-integral centroid molecular dynamics), and for the molecular center-of mass dynamic structure factor in liquid carbon dioxide as computed by means of classical molecular dynamics simulation.

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Section: Introductionmentioning

confidence: 90%

Expansion in Lorentzian functions of spectra of quantum autocorrelations

Barocchi

Bafile

2013

Phys. Rev. E

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“…(ii) F S (t) is the generalized Stokes force with a friction memory kernel γ (t) for modeling viscoelastic properties of a medium [26][27][28] …”

Section: A Generalized Langevin Equationmentioning

confidence: 99%

“…A broad class of such models relies on a linear generalized Langevin equation (GLE) that may include, for instance, inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, eventual optical trapping at long times, as well as active transport by motor proteins [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. This equation can be formally solved by standard Laplace transform techniques.…”

Section: Introductionmentioning

confidence: 99%

Analytical solution of the generalized Langevin equation with hydrodynamic interactions: Subdiffusion of heavy tracers

Grebenkov

Vahabi

2014

Phys. Rev. E

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We consider a generalized Langevin equation that can be used to describe thermal motion of a tracer in a viscoelastic medium by accounting for inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. We derive a Laplace-type integral representation for the linear response function that governs the diffusive dynamics. This representation is particularly well suited for rapid numerical computation and theoretical analysis. In particular, we deduce explicit formulas for the mean and variance of the time averaged (TA) mean square displacement (MSD) and velocity autocorrelation function (VACF). The asymptotic behavior of the TA MSD and TA VACF is investigated at different time scales. Some biophysical and microrheological applications are discussed, with an emphasis on the statistical analysis of optical tweezers' single-particle tracking experiments in polymer networks and living cells.

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“…From the definition (21b) one can see that the physical meaning of the parameters ∆ ν depends on the concrete process and is defined by the corresponding dynamical variables A ν−1 and A ν . Equation (21a) is also known from the recurrence 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 0 , which is associated with the processes experimentally studied.…”

Section: Theoretical Backgroundmentioning

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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Generalized Langevin equation and recurrence relations

Cited by 41 publications

References 45 publications

Expansion in Lorentzian functions of spectra of quantum autocorrelations

Expansion in Lorentzian functions of spectra of quantum autocorrelations

Analytical solution of the generalized Langevin equation with hydrodynamic interactions: Subdiffusion of heavy tracers

Relaxation Processes in Many Particle Systems -- Recurrence Relations Approach

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