Brownian particles in periodic potentials: coarse-graining versus fine structure

TL;DR

This paper investigates the motion of overdamped particles in periodic potentials, comparing coarse-grained and fine-structured descriptions, and develops a theoretical framework to understand their thermodynamics and ergodic properties.

Contribution

It introduces a theory linking coarse-grained diffusion behavior with the fine structure of the density, extending thermodynamic relations to far-from-equilibrium systems.

Findings

Coarse-grained diffusion is slower than free diffusion.

The Boltzmann-Gibbs density is non-normalizable but influences observables.

Entropy has both coarse-grained and fine-structure components.

Abstract

We study the motion of an overdamped particle connected to a thermal heat bath in the presence of an external periodic potential in one dimension. When we coarse-grain, i.e., bin the particle positions using bin sizes that are larger than the periodicity of the potential, the packet of spreading particles, all starting from a common origin, converges to a normal distribution centered at the origin with a mean-squared displacement that grows as $2 D^* t$, with an effective diffusion constant that is smaller than that of a freely diffusing particle. We examine the interplay between this coarse-grained description and the fine structure of the density, which is given by the Boltzmann-Gibbs (BG) factor $e^{-V(x)/k_B T}$, the latter being non-normalizable. We explain this result and construct a theory of observables using the Fokker-Planck equation. These observables are classified as those that are related to the BG fine structure, like the energy or occupation times, while others, like the positional moments, for long times, converge to those of the large-scale description. Entropy falls into a special category as it has a coarse-grained and a fine structure description. The basic thermodynamic formula $F=TS - E$ is extended to this far-from-equilibrium system. The ergodic properties are also studied using tools from infinite ergodic theory.