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.

On the derivation of a nonlinear generalized Langevin equation

2021 ◽  
Author(s):  
Loris Di Cairano
Keyword(s):  
Langevin Equation ◽  
Heavy Particle ◽  
Heat Bath ◽  
Mathematical Structure ◽  
Generalized Langevin Equation ◽  
Linear Feature ◽  
Non Linear ◽  
Global Equilibrium ◽  
System Variables ◽  
Instantaneous Configuration

Abstract We recast the Zwanzig's derivation of a non linear generalized Langevin equation (GLE) for a heavy particle interacting with a heat bath in a more general framework showing that it is necessary to readjust the Zwanzig's definitions of the kernel matrix and noise vector in the GLE in order to be able performing consistently the continuum limit. As shown by Zwanzig, the non linear feature of the resulting GLE is due to the non linear dependence of the equilibrium map by the heavy particle variables. Such an equilibrium map represents the global equilibrium configuration of the heat bath particles for a fixed (instantaneous) configuration of the system. Following the same derivation of the GLE, we show that a deeper investigation of the equilibrium map, considered in the Zwanzig's Hamiltonian, is necessary. Moreover, we discuss how to get an equilibrium map given a general interaction potential. Finally, we provide a renormalization procedure which allows to divide the dependence of the equilibrium map by coupling coefficient from the dependence by the system variables yielding a more rigorous mathematical structure of the non linear GLE.

scholarly journals A path integral approach to the Langevin equation

2015 ◽  
Vol 30 (07) ◽  
pp. 1550028 ◽  
Author(s):  
Ashok K. Das ◽  
Sudhakar Panda ◽  
J. R. L. Santos
Keyword(s):  
Brownian Motion ◽  
Path Integral ◽  
Langevin Equation ◽  
Correlation Functions ◽  
Free Particle ◽  
Transition Amplitude ◽  
Planck Equation ◽  
Markovian Process ◽  
Integral Approach ◽  
Generalized Langevin Equation

We study the Langevin equation with both a white noise and a colored noise. We construct the Lagrangian as well as the Hamiltonian for the generalized Langevin equation which leads naturally to a path integral description from first principles. This derivation clarifies the meaning of the additional fields introduced by Martin, Siggia and Rose in their functional formalism. We show that the transition amplitude, in this case, is the generating functional for correlation functions. We work out explicitly the correlation functions for the Markovian process of the Brownian motion of a free particle as well as for that of the non-Markovian process of the Brownian motion of a harmonic oscillator (Uhlenbeck–Ornstein model). The path integral description also leads to a simple derivation of the Fokker–Planck equation for the generalized Langevin equation.

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 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.

COMPLEXITY AND GENERALIZED EXPONENTIAL RELAXATION: MEMORY VERSUS RENEWAL

2008 ◽  
Vol 18 (09) ◽  
pp. 2709-2716 ◽  
Author(s):  
PAOLO GRIGOLINI
Keyword(s):  
Time Series ◽  
Complex Systems ◽  
Langevin Equation ◽  
Survival Probability ◽  
Mathematical Expression ◽  
Real Data ◽  
Generalized Langevin Equation ◽  
Dissipation Process ◽  
Fluctuation Dissipation ◽  
Exponential Relaxation

We illustrate two distinct approaches to the Mittag–Leffler relaxation, as a mathematical expression suitable for the interpretation of real data produced by complex systems, and especially those of physiological interest. The first approach is based on interpreting the fluctuation–dissipation process under study as obtained via Subordination to the Ordinary Fluctuation–Dissipation (SOFD) process. The second approach rests on the Generalized Langevin Equation (GLE). We prove that in the real cases of truncated time series the two theories generate a survival probability in the form of a stretched exponential, and that this property makes it hard to assess if a given time series obeys the GLE or the SOFD prescription. Some conjectures are made on the possibility of distinguishing the GLE from the SOFD predictions through the analysis of a single time series.

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