A First-Principles Nonequilibrium Deterministic Equation of Motion of a Brownian Particle and Microscopic Viscous Drag

TL;DR

This paper introduces a first-principles deterministic equation of motion for Brownian particles that unifies nonequilibrium viscous dissipation across scales, providing a new perspective beyond Langevin's stochastic framework.

Contribution

It develops a deterministic microforce-based equation of motion for Brownian particles derived from microscopic interactions, bypassing stochastic assumptions of traditional models.

Findings

Reproduces known equilibrium Brownian motion results

Shows viscous dissipation emerges from ensemble averaging microforces

Provides a unified framework for nonequilibrium Brownian dynamics

Abstract

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.