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Brownian-motion approach to statistical mechanics: Langevin equations, fluctuations, and timescales

Sushanta Dattagupta, Aritra Ghosh

Abstract

We briefly review the problem of Brownian motion and describe some intriguing facets. The problem is first treated in its original form as enunciated by Einstein, Langevin, and others. Then, utilizing the problem of Brownian motion as a paradigm and upon using the Langevin equation(s), we present a brief exposition of the modern areas of stochastic thermodynamics and fluctuation theorems in a manner accessible to a non-expert. This is followed by an analysis of non-Markovian Brownian dynamics via generalized Langevin equation(s) in which we particularly shed light onto its derivation, the emergence of the fluctuation-dissipation relation, and the recently-discovered effective-mass framework.

Brownian-motion approach to statistical mechanics: Langevin equations, fluctuations, and timescales

Abstract

We briefly review the problem of Brownian motion and describe some intriguing facets. The problem is first treated in its original form as enunciated by Einstein, Langevin, and others. Then, utilizing the problem of Brownian motion as a paradigm and upon using the Langevin equation(s), we present a brief exposition of the modern areas of stochastic thermodynamics and fluctuation theorems in a manner accessible to a non-expert. This is followed by an analysis of non-Markovian Brownian dynamics via generalized Langevin equation(s) in which we particularly shed light onto its derivation, the emergence of the fluctuation-dissipation relation, and the recently-discovered effective-mass framework.

Paper Structure

This paper contains 15 sections, 85 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Einstein and Langevin
  3. Einstein's treatment of Brownian motion and diffusion
  4. Langevin equation and recovery of equilibrium results
  5. Potentials
  6. Stochastic thermodynamics
  7. Basic ideas
  8. Harmonic oscillator as a classical nano engine
  9. Fluctuation theorems
  10. Jarzynski equality
  11. Beyond Markovian processes
  12. Generalized Langevin equation
  13. System-plus-bath approach
  14. Effective-mass approach to memory
  15. Discussion

Figures (1)

  • Figure 1: The figure shows a large particle of mass $m$ (the one at the centre) interacting with a set of $N \gg 1$ independent harmonic oscillators (represented by the particles at the periphery) with masses $\{m_j\}$ and frequencies $\{\omega_j\};~j=1,2,\dots,N$.