Overdamped limit and inverse-friction expansion for Brownian motion in an inhomogeneous medium (original) (raw)
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Journal of Statistical Mechanics: Theory and Experiment, 2009
The non-linear dissipation corresponding to a non-Gaussian thermal bath is introduced together with a multiplicative white noise source in the phenomenological Langevin description for the velocity of a particle moving in some potential landscape. Deriving the closed Kolmogorov's equation for the joint probability distribution of the particle displacement and its velocity by use of functional methods and taking into account the well-known Gibbs form of the thermal equilibrium distribution and the condition of 'detailed balance' symmetry, we obtain the exact master equation: given the white noise statistics, this master equation relates the non-linear friction function to the velocitydependent noise function. In particular, for multiplicative Gaussian white noise this operator equation yields a unique inter-relation between the generally nonlinear friction and the (multiplicative) velocity-dependent noise amplitude. This relation allows us to find, for example, the form of velocity-dependent noise function for the case of non-linear Coulomb friction.
Journal of Statistical Physics, 2012
We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α ∈ R. Interestingly, in addition to the classical Itô (α = 0), Stratonovich (α = 0.5) and anti-Itô (α = 1) integrals, we show that position-dependent α = α(x), and even stochastic integrals with α / ∈ [0, 1] arise. Our findings are supported by numerical simulations.
Brownian motion in a classical ideal gas: A microscopic approach to Langevin’s equation
Pramana, 2004
We present an insightful "derivation" of the Langevin equation and the fluctuation dissipation theorem in the specific context of a heavier particle moving through an ideal gas of much lighter particles. The Newton's Law of motion (mẍ = F ) for the heavy particle reduces to a Langevin equation (valid on a coarser time scale) with the assumption that the lighter gas particles follow a Boltzmann velocity distribution. Starting from the kinematics of the random collisions we show that (1) the average force F ∝ −ẋ and (2) the correlation function of the fluctuating force η = F − F is related to the strength of the average force.
Microscopic theory of brownian motion: Mori friction kernel and langevin-equation derivation
Physica A: Statistical Mechanics and its Applications, 1975
Recei°,,'ed 28 Jammry t975 ~ derivatioa of the plaenome~o]ogical Langevif~ equadon riot d'.e momenu.m~ or a browrxian p. r~icIe from the generalized Langevin equatio~ of Mori is fresemed. This derivatio,q requires a de~aHed examination of the Mori friction kernel {or merr~ory fur~ctio~_~L it is ,demonstra[ed, on :A~e basis of prior work of Ma:zur and Oppenhei_m, that the Mc, ri ker~et doe~ ~':.m admit of a well behaved expar~sion in the ratio of bad>-a~ad brow~ianopar~icie masses. In addition, the Mori kernel is fom~d to decay o~,. t!~e siow time scaie of tI~e bro>miar>par~ide momentum° Both features, which contradict standard assumptio~, are traced to the inH~e~ce c, fcoupli~g to nonlinear powers of the momentum and preclude a Langevi~>equation derh'atier~ solety on the basis of timescale separation arguments. The Langevfi,,-equation is recovered, ho>ever, when the sinai! magnitude of s|owlzy decaying contributions is take,~ it:to accounL
Derivation of a Fokker–Planck equation for generalized Langevin dynamics
Physica A: Statistical Mechanics and its Applications, 2005
A Fokker-Planck equation describing the statistical properties of Brownian particles acted upon by long-range stochastic forces with power-law correlations is derived. In contrast with previous approaches (Wang, Phys. Rev. A 45 (1992) 2), it is shown that the distribution of Brownian particles after release from a point source is broader than Gaussian and described by a Fox function. Transport is shown to be ballistic at short times and either sub-diffusive or super-diffusive at large times. The imposition of occasional trapping events onto the Brownian dynamics can result in confined diffusion (d=dthx 2 i ! 0) at long times when the mean trapping time is divergent. It is suggested that such dynamics describe protein motions in cell membranes.
A closer look at the quantum Langevin equation: Fokker-Planck equation and quasiprobabilities
Physics Letters, 1985
The nature of the noise described by the c-number quantum Lange'iin equation is investigated. Subtleties in the stochastic calculus are shown to originate from differences between quantum noise and classical coloured noise. The Fokker-Planck equation for the Wigner distribution and the associated quantum master equation are derived for a linear system. Explicit results for the quasiprobabilities are obtained. Recently there has been a great deal of interest in the influence of dissipation on quantum systems at low temperatures [1,2] where standard weak coupling theories [3,4] fail. In this context it has been suggested that the motion of a quantum mechanical particle of massM and coordinate q can be described by a quantum Langevin equation (QLE) [5-8] M~l'+ M7 4 + OV/Oq = ~(t). (1) Here V(q) is the external potential and-MTq a frictional force caused by environmental coupling. ~(t) is a gaussian random noise with zero mean whose coloured spectrum (~(60) ~(-60))= 2 7~M60 [1-exp(-/3h60)] 1 (2) reflects the quantum nature of the process. From (2) one finds (~(t) ~) =-(TM/2fl)v (sinh lvt)-2 + iTMh d6(t)/dt .
Derivation and solution of a low-friction Fokker-Planck equation for a bound Brownian particle
Zeitschrift f�r Physik B Condensed Matter, 1985
The low-friction region of an anharmonically bound Brownian particle is examined using systematic elimination procedures. We obtain an asymptotic expression for the spectrum of the Fokker-Planck operator. Asymptotic means both small anharmonicities and small friction constants y compared to the oscillatory frequency co. We conclude that Kramers' low-friction equation is generally valid only for 0<7<0.01 co and has to be modified for 7~>0.01 co by including phase-dependent terms. From these the nonlinear part of the force field in connection with a finite temperature is shown to shorten the correlation time of the equilibrium velocity autocorrelation function and to renormalize the frequency of the corresponding spectral density.
Physical Review E, 2020
We present a first-principles thermodynamic approach to provide an alternative to the Langevin equation by identifying the deterministic (no stochastic component) microforce F k,BP acting on a nonequilibrium Brownian particle (BP) in its kth microstate m k. (The prefix micro refers to microstate quantities and carry a suffix k.) The deterministic new equation is easier to solve using basic calculus. Being oblivious to the second law, F k,BP does not always oppose motion but viscous dissipation emerges upon ensemble averaging. The equipartition theorem is always satisfied. We reproduce well-known results of the BP in equilibrium. We explain how the microforce is obtained directly from the mutual potential energy of interaction beween the BP and the medium after we average it over the medium so we only have to consider the particles in the BP. Our approach goes beyond the phenomenological and equilibrium approach of Langevin and unifies nonequilibrium viscous dissipation from mesoscopic to macroscopic scales and provides new insight into Brownian motion beyond Langevin's and Einstein's formulation.
Isothermal Langevin dynamics in systems with power-law spatially dependent friction
Physical Review E, 2016
We study the dynamics of Brownian particles in a heterogeneous one-dimensional medium with a spatially-dependent diffusion coefficient of the form D(x) ∼ |x| c , at constant temperature. The particle's probability distribution function (PDF) is calculated both analytically, by solving Fick's diffusion equation, and from numerical simulations of the underdamped Langevin equation. At large times, the PDFs calculated by both approaches yield identical results, corresponding to subdiffusion for c < 0, and superdiffusion for 0 < c < 1. For c > 1, the diffusion equation predicts that the particles accelerate. Here, we show that this phenomenon, previously considered in several works as an illustration for the possible dramatic effects of spatially-dependent thermal noise, is unphysical. We argue that in an isothermal medium, the motion cannot exceed the ballistic limit (x 2 ∼ t 2). The ballistic limit is reached when the friction coefficient drops sufficiently fast at large distances from the origin, and is correctly captured by Langevin's equation.
Physica A: Statistical Mechanics and its Applications, 1978
The motion of a Brownian particle in an external field can be described on two levels: by a Fokker-Planck equation for the joint probability distribution of position and velocity, and by a Smoluchowski equation for the distribution in position space only. We derive the second description, with corrections, from the first by means of a systematic expansion procedure of the Chapman-Enskog type in terms of the inverse friction coefficient. We also derive equations describing the initial period, when the Smoluchowski description is not yet valid; in particular we find formulae connecting the initial value to be used for the Smoluchowski equation with that of the full Fokker-Planck equation. The special case of an harmonically bound Brownian particle can be solved exactly; the results are used to check and to illustrate our expressions for general potential.