From Lagrangian to Brownian motion

We present a Lagrangian describing an idealized liquid interacting with a particle immersed in it. We show that the equation describing the motion of the particle as a functional of the initial conditions of the liquid incorporates noise and friction, which are attributed to specific dynamical processes. The equation is approximated to yield a Langevin equation with parameters depending on the Lagrangian and the temperature of the liquid. The origin of irreversibility and dissipation is discussed.     

On the theory of Brownian motion. I. Interaction between Brownian particles

The molecular theory of the Brownian motion of heavy particles in a homogeneous solvent of light particles is extended to cover the case of interactions between the Brownian particles. This will have physical effects in the concentration dependence of the Brownian particle self-diffusion coefficient. A density expansion for the Brownian particle friction coefficient is derived, and an approximation permitting the first density correction to be calculated is suggested.This work, part of research supported by NSF Grant GP-8497, was done under the tenure of a National Science Foundation Senior Postdoctoral Fellowship, and of a sabbatical leave granted by the University of Oregon.     

Brownian motion near an absorbing sphere

The spherically symmetric solution of the Fokker-Planck equation with absorbing boundary is given in terms of a solution of an equivalent integral equation whose explicit form is found.     

Adiabatic elimination for systems of Brownian particles with nonconstant damping coefficients

We discuss the problem of eliminating the momentum variable in the phase space Langevin equations for a system of Brownian particles in two related situations: (i) position-dependent damping and (ii) existence of hydrodynamic interactions. We discuss the problems associated with the conventional elimination and we develop an alternative elimination procedure, in the Lagevin framework, which leads to the correct Smoluchowski equation. We give a heuristic argument on the basis of stochastic differential equations for the Smoluchowski limit and establish rigorously the limit for the general case of position-dependent friction and diffusion coefficents.     

Brownian motion under a time-dependent periodic potential proportional to the square of the position of a particle and classical fluctuation squeezing

A general stationary case of a Brownian particle with a time-dependent periodic potential proportional to the square of the position of the particle is treated. Even though the vigorous change of the time-dependent proportionality coefficient is applied, there are cases where the fluctuation of the particle decreases in contrary to our intuition, which is called classical fluctuation squeezing. We obtain time-average variances analytically for general cases of an arbitrary change in the coefficient and find conditions favorable for classical fluctuation squeezing. We introduce an asymmetric function behaves like trigonometric cosine one and consider its behavior explicitly.     

On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion

Differential equations governing the time evolution of distribution functions for Brownian motion in the full phase space were first derived independently by Klein and Kramers. From these so-called Fokker-Planck equations one may derive the reduced differential equations in coordinate space known as Smoluchowski equations. Many such derivations have previously been reported, but these either involved unnecessary assumptions or approximations, or were performed incompletely. We employ an iterative reduction scheme, free of assumptions, and calculate formally exact corrections to the Smoluchowski equations for many-particle systems with and without hydrodynamic interaction, and for a single particle in an external field. In the absence of hydrodynamic interaction, the lowest order corrections have been expressed explicitly in terms of the coordinate space distribution function. An additional application of the method is made to the reduction of the stress tensor used in evaluating the intrinsic viscosity of particles in solution. Most of the present work is based on classical Brownian motion theory, but brief consideration is given in an appendix to some recent developments regarding non-Markovian equations for Brownian motion.Supported by the National Science Foundation.     

Brownian motion at short time scales

Brownian motion has played important roles in many different fields of science since its origin was first explained by Albert Einstein in 1905. Einstein's theory of Brownian motion, however, is only applicable at long time scales. At short time scales, Brownian motion of a suspended particle is not completely random, due to the inertia of the particle and the surrounding fluid. Moreover, the thermal force exerted on a particle suspended in a liquid is not a white noise, but is colored. Recent experimental developments in optical trapping and detection have made this new regime of Brownian motion accessible. This review summarizes related theories and recent experiments on Brownian motion at short time scales, with a focus on the measurement of the instantaneous velocity of a Brownian particle in a gas and the observation of the transition from ballistic to diffusive Brownian motion in a liquid.     

布朗运动理论一百年

文章基于作者在2005年纪念爱因斯坦奇迹年的香山会议上的综述报告,扼要叙述了从布朗运动到统计涨落场论的发展历程,特别提及了与中国物理学家有关的贡献.     

布朗运动理论一百年

文章基于作者在2005年纪念爱因斯坦奇迹年的香山会议上的综述报告,扼要叙述了从布朗运动到统计涨落场论的发展历程,特别提及了与中国物理学家有关的贡献.     

Brownian motion in fluctuating periodic potentials

This work deals with the overdamped motion of a particle in a fluctuating one-dimensional periodic potential. If the potential has no inversion symmetry and its fluctuations are asymmetric and correlated in time, a net flow can be generated at finite temperatures. We present results for the stationary current for the case of a piecewise linear potential, especially for potentials being close to the case with inversion symmetry. The aim is to study the stationary current as a function of the potential. Depending on the form of the potential, the current changes sign once or even twice as a function of the correlation time of the potential fluctuations. To explain these current reversals, several mechanisms are proposed. Finally, we discuss to what extent the model is useful to understand the motion of biomolecular motors.     

On the Brownian motion in a double-well potential in the overdamped limit

The Brownian motion of a particle in a double-well potential is considered and the position correlation function (excluding inertial effects) for the potential V(x)=ax²/2+bx⁴/4 is evaluated first by averaging the governing Langevin equation over its realizations. The exact solution is then obtained via matrix continued fractions by using a representation, which symmetrizes the recurrence relations for the observables generated by the averaging procedure leading to convergence of these recurrence relations unlike the previous approaches to the problem. A reliable approximate solution based on the exponential separation of the time scales of the fast intrawell and low overbarrier relaxation processes associated with the bistable potential is also given. It is shown that a knowledge of the three characteristic relaxation times (the integral, effective and the longest relaxation times) of the position correlation function allows one to accurately predict the relaxation behavior of the system in the overdamped limit for all time scales of interest.     

Microscopic derivation of non-Markovian thermalization of a Brownian particle

In this paper, the first microscopic approach to Brownian motion is developed in the case where the mass density of the suspending bath is of the same order of magnitude as that of the Brownian (B) particle. Starting from an extended Boltzmann equation, which describes correctly the interaction with the fluid, we derive systematically via multiple-time-scale analysis a reduced equation controlling the thermalization of the B particle, i.e., the relaxation toward the Maxwell distribution in velocity space. In contradistinction to the Fokker-Planck equation, the derived new evolution equation is nonlocal both in time and in velocity space, owing to correlated recollision events between the fluid and particle B. In the long-time limit, it describes a non-Markovian generalized Ornstein-Uhlenbeck process. However, in spite of this complex dynamical behavior, the Stokes-Einstein law relating the friction and diffusion coefficients is shown to remain valid. A microscopic expression for the friction coefficient is derived, which acquires the form of the Stokes law in the limit where the meanfree path in the gas is small compared to the radius of particle B.Knowing the interest of Matthieu Ernst in the subtle and fundamental problems of kinetic theory, we have the pleasure to dedicate this study to him.     

Microcanonical variational transition-state theory for reaction rates in dissipative systems

Upper bounds for the classical escape rate of a particle trapped in a metastable well and interacting with a dissipative medium are derived based on the periodic orbits of a reduced two-degree-of-freedom Hamiltonian involving the unstable normal mode and a collective bath mode. It is shown that even in what is usually thought of as the spatial diffusion limit the reactive flux can involve an energy diffusion term due to energy transfer from the dissipative media, in addition to the standard spatial diffusion term.     

Brownian motion in a rotating flow

The Langevin equations for a particle of an arbitrary shape and the correlation functions for the fluctuating forces, torques, or force-torque acting on the particle in a rotating flow are derived from the semimicroscopic level of coarse graining by using fluctuating hydrodynamics. In order to obtain the solution of the Navier-Stokes Langevin equation valid over the entire flow region, use is made of the method of matched asymptotic expansions in ( _f a²/v)^1/2 1. The cases of slow and rapid rotation are analyzed. It is shown that the fluctuation-dissipation theorems hold up to the order of ( _f a²/v)^1/2 in both slow and rapid rotation, and that the diffusivity tensor depends on the angular velocity of the fluid and becomes anisotropic.     

On the theory of Brownian motion. II. Nonuniform systems

An equation of evolution for a heavy particle immersed in a solvent of lighter particles is derived for the case when the system suffers gradients of temperature composition, or velocity. The derivation unifies the theory by applying the same methods which have proved useful in the uniform case. The final equation contains some new terms due to concentration gradients in the solvent, and is applicable to the case when the heavy particles are present at finite concentration and interact with each other.This work, part of research supported by NSF Grant GP-8497, was done under the tenure of a National Science Foundation Senior Postdoctoral Fellowship and a sabbatical leave from the University of Oregon.     

布朗运动理论及其在复杂气候系统研究中的应用

文章将从非平衡态统计物理发展和应用的角度,介绍德国科学家哈塞尔曼荣获2021年诺贝尔物理学奖的研究工作——基于布朗运动理论,建立了描述气象(天气)影响气候长期演化的随机气候学模型,并建立了寻求影响气候主因的最优指纹方法,从而能够分辨出人类活动和自然界局部改变对气候这一复杂系统的影响。哈塞尔曼的工作本质上是理论物理在实际复杂系统领域的成功应用,他采用的基础物理方法——布朗运动理论是我国杰出女物理学家王明贞和其导师乌伦贝克在20世纪40年代基于爱因斯坦的工作发展起来的^[1,2]。文章将介绍布朗运动理论的发展及其相关的非平衡统计物理思想的当代发展,以展示哈塞尔曼如何把相关的物理理论巧妙地用于气候长期预测的实际应用研究：(1)建立了快变的局部“气象”变量涨落通过耗散涨落关系影响缓变的整体气候变量的基础理论;(2)通过最优指纹方法,寻找局部“噪音”和外驱动力影响气候演化的关键要素。