scholarly journals Relativistic Langevin Equation derived from a particle-bath Lagrangian

2021 ◽  
Author(s):  
Aleksandr Petrosyan ◽  
Alessio Zaccone
Keyword(s):  
Statistical Mechanics ◽  
Langevin Equation ◽  
Relativistic Correction ◽  
Analytical Approximation ◽  
Space Time ◽  
Langevin Equations ◽  
Generalized Langevin Equation ◽  
Time Translation ◽  
Fluctuation Dissipation ◽  
Wave Forms

Abstract We show how a relativistic Langevin equation can be derived from a Lorentz-covariant version of the Caldeira-Leggett particle-bath Lagrangian. In one of its limits, we identify the obtained equation with the Langevin equation used in contemporary extensions of statistical mechanics to the near-light-speed motion of a tagged particle in non-relativistic dissipative fluids. The proposed framework provides a more rigorous and first-principles form of the weakly-relativistic and partially-relativistic Langevin equations often quoted or postulated as ansatz in previous works. We then refine the aforementioned results to obtain a generalized Langevin equation valid for the case of both fully-relativistic particle and bath, using an analytical approximation obtained from numerics where the Fourier modes of the bath are systematically replaced with covariant plane-wave forms with a length-scale relativistic correction that depends on the space-time trajectory in a parabolic way. We discuss the implications of the apparent breaking of space-time translation and parity invariance, showing that these effects are not necessarily in contradiction with the assumptions of statistical mechanics. The intrinsically non-Markovian character of the fully relativistic generalised Langevin equation derived here, and of the associated fluctuation-dissipation theorem, is also discussed.

scholarly journals Brownian motion of classical spins: Anomalous dissipation and generalized Langevin equation

2017 ◽  
Vol 31 (27) ◽  
pp. 1750189
Author(s):  
Malay Bandyopadhyay ◽  
A. M. Jayannavar
Keyword(s):  
Langevin Equation ◽  
Heat Bath ◽  
Planck Equation ◽  
Generalized Langevin Equation ◽  
Hamiltonian Description ◽  
Lifshitz Equation ◽  
Fluctuation Dissipation ◽  
Anomalous Dissipation ◽  
Dissipative Terms ◽  
Markovian Limit

In this work, we derive the Langevin equation (LE) of a classical spin interacting with a heat bath through momentum variables, starting from the fully dynamical Hamiltonian description. The derived LE with anomalous dissipation is analyzed in detail. The obtained LE is non-Markovian with multiplicative noise terms. The concomitant dissipative terms obey the fluctuation–dissipation theorem. The Markovian limit correctly produces the Kubo and Hashitsume equation. The perturbative treatment of our equations produces the Landau–Lifshitz equation and the Seshadri–Lindenberg equation. Then we derive the Fokker–Planck equation corresponding to LE and the concept of equilibrium probability distribution is analyzed.

scholarly journals Numerical approximation and fast evaluation of the overdamped generalized Langevin equation with fractional noise

2020 ◽  
Vol 54 (2) ◽  
pp. 431-463 ◽  
Author(s):  
Di Fang ◽  
Lei Li
Keyword(s):  
Differential Equation ◽  
Langevin Equation ◽  
Numerical Approximation ◽  
Generalized Langevin Equation ◽  
Fractional Gaussian Noise ◽  
Fast Evaluation ◽  
Fractional Noise ◽  
Fluctuation Dissipation ◽  
Fluctuation Dissipation Theorem

The generalized Langevin equation (GLE) is a stochastic integro-differential equation that has been used to describe the movement of microparticles with sub-diffusion phenomenon. It has been proved that with fractional Gaussian noise (fGn) mostly considered by biologists, the overdamped Generalized Langevin equation satisfying fluctuation dissipation theorem can be written as a fractional stochastic differential equation (FSDE). In this work, we present both a direct and a fast algorithm respectively for this FSDE model in order to numerically study ergodicity. The strong orders of convergence are proven for both schemes, where the role of the memory effects can be clearly observed. We verify the convergence theorems using linear forces, and then verify the convergence to Gibbs measure algebraically for the double well potentials in both 1D and 2D setups. Our work is new in numerical analysis of FSDEs and provides a useful tool for studying ergodicity. The idea can also be used for other stochastic models involving memory.

Gaussian and Non–Gaussian Distribution–Valued Ornstein–Uhlenbeck Processes

1991 ◽  
Vol 43 (6) ◽  
pp. 1136-1149 ◽  
Author(s):  
Tomasz Bojdecki ◽  
Luis G. Gorostiza
Keyword(s):  
Langevin Equation ◽  
Linear Operators ◽  
Langevin Equations ◽  
Nuclear Space ◽  
Generalized Langevin Equation ◽  
Independent Increments ◽  
Non Gaussian ◽  
Generalized Distribution ◽  
Strong Dual ◽  
Generalized Langevin Equations

Generalized (distribution-valued) Ornstein-Uhlenbeck processes, which by definition are solutions of generalized Langevin equations, arise in many investigations on fluctuation limits of particle systems (eg. Bojdecki and Gorostiza [1], Dawson, Fleischmann and Gorostiza [5], Fernández [7], Gorostiza [8,9], Holley and Stroock [10], Itô [12], Kallianpur and Pérez-Abreu [16], Kallianpur and Wolpert [14], Kotelenez [17], Martin-Löf [19], Mitoma [22], Uchiyama [25]). The state space for such a process is the strong dual Φ′ of a nuclear space Φ. A generalized Langevin equation for a Φ′-valued process X ≡ {Xt} is a stochastic evolution equation of the formwhere {At} is a family of linear operators on Φ and Z ≡ {Zt} is a Φ'-valued semimartingale (in some sense) with independent increments. Equations of the type (1.1) where Z does not have independent increments also arise in applications (eg. [9,14,20]) but here we are interested precisely in the case when Z has independent increments (we restrict the term generalized Langevin equation to this case in accordance with the classical Langevin equation).

Non-Markovian dynamics of dust charge fluctuations in dusty plasmas

2014 ◽  
Vol 80 (3) ◽  
pp. 465-476 ◽  
Author(s):  
H. Asgari ◽  
S. V. Muniandy ◽  
Amir Ghalee
Keyword(s):  
Correlation Function ◽  
Langevin Equation ◽  
Dusty Plasmas ◽  
Langevin Equations ◽  
Transient Time ◽  
Charge Fluctuations ◽  
Dust Charge ◽  
Fluctuation Dissipation ◽  
Dust Charge Fluctuations ◽  
Markovian Systems

Dust charge fluctuates even in steady-state uniform plasma due to the discrete nature of the charge carriers and can be described using standard Langevin equation. In this work, two possible approaches in order to introduce the memory effect in dust charging dynamics are proposed. The first part of the paper provides the generalization form of the fluctuation-dissipation relation for non-Markovian systems based on generalized Langevin equations to determine the amplitudes of the dust charge fluctuations for two different kinds of colored noises under the assumption that the fluctuation-dissipation relation is valid. In the second part of the paper, aiming for dusty plasma system out of equilibrium, the fractionalized Langevin equation is used to derive the temporal two-point correlation function of grain charge fluctuations which is shown to be non-stationary due to the dependence on both times and not the time difference. The correlation function is used to derive the amplitude of fluctuations for early transient time.

scholarly journals Data-driven parameterization of the generalized Langevin equation

2016 ◽  
Vol 113 (50) ◽  
pp. 14183-14188 ◽  
Author(s):  
Huan Lei ◽  
Nathan A. Baker ◽  
Xiantao Li
Keyword(s):  
Explicit Memory ◽  
Random Noise ◽  
Data Driven ◽  
Langevin Equations ◽  
Generalized Langevin Equation ◽  
Memory Kernel ◽  
Laplace Domain ◽  
Fluctuation Dissipation ◽  
Data Driven Approach ◽  
Generalized Langevin Equations

We present a data-driven approach to determine the memory kernel and random noise in generalized Langevin equations. To facilitate practical implementations, we parameterize the kernel function in the Laplace domain by a rational function, with coefficients directly linked to the equilibrium statistics of the coarse-grain variables. We show that such an approximation can be constructed to arbitrarily high order and the resulting generalized Langevin dynamics can be embedded in an extended stochastic model without explicit memory. We demonstrate how to introduce the stochastic noise so that the second fluctuation-dissipation theorem is exactly satisfied. Results from several numerical tests are presented to demonstrate the effectiveness of the proposed method.

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