Brownian particles in periodic potentials: Coarsegraining versus fine structure
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 coarsegrain, 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 meansquared 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 coarsegrained description and the fine structure of the density, which is given by the BoltzmannGibbs (BG) factor e^{−V (x ) /kBT}, the latter being nonnormalizable. We explain this result and construct a theory of observables using the FokkerPlanck 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 largescale description. Entropy falls into a special category as it has a coarsegrained and a fine structure description. The basic thermodynamic formula F =T S −E is extended to this farfromequilibrium system. The ergodic properties are also studied using tools from infinite ergodic theory.
 Publication:

Physical Review E
 Pub Date:
 February 2023
 DOI:
 10.1103/PhysRevE.107.024122
 arXiv:
 arXiv:2210.10935
 Bibcode:
 2023PhRvE.107b4122D
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 17 pages, 8 figures