Generalized fluctuation-dissipation relation and statistics for the equilibrium of a system with conformation dependent damping

TL;DR

This paper derives a generalized fluctuation-dissipation relation for systems with conformation-dependent damping, showing that standard equilibrium distributions fail for structured Brownian particles and proposing new distributions consistent with equilibrium.

Contribution

It introduces a new generalized fluctuation-dissipation relation accounting for conformation-dependent damping, extending equilibrium distribution theory beyond classical cases.

Findings

Standard Maxwell and Boltzmann distributions fail for structured Brownian particles.

Derived a generalized fluctuation-dissipation relation ensuring zero current in inhomogeneous space.

Recovered standard distributions as special cases of the generalized theory.

Abstract

Liouville's theorem, based on the Hamiltonian flow (micro-canonical ensemble) for a many particle system, indicates that the (stationary) equilibrium probability distribution is a function of the Hamiltonian. A canonical ensemble corresponds to a micro-canonical one at thermodynamic limit. On the contrary, the dynamics of a single Brownian particle (BP) being explicitly non-Hamiltonian with a force and damping term in it and at the other extreme to thermodynamic limit admits the Maxwell-distribution (MD) for its velocity and Boltmann-distribution (BD) for positions (when in a potential). This is due to the fluctuation-dissipation relation (FDR), as was first introduced by Einstein, which forces the Maxwell distribution to the Brownian particles. For a structureless BP, that, this theory works is an experimentally verified fact over a century now. Considering a structured Brownian particle we will show that the BD and MD fails to ensure equilibrium. We will derive a generalized FDR on the basis of the demand of zero current on inhomogeneous space. Our FDR and resulting generalized equilibrium distributions recover the standard ones at appropriate limits.