Generalized Boltzmann Distribution for Systems out of Equilibrium

TL;DR

This paper introduces a generalized Boltzmann distribution that models all stochastic systems, including those far from equilibrium, offering a unified framework inspired by electronic circuit analogies to understand complex non-equilibrium behaviors.

Contribution

It extends the Boltzmann distribution to a universal form applicable to non-equilibrium systems, providing new analytical solutions and insights into their fundamental limits.

Findings

Generalized distribution models all stochastic systems.

Experimental data show living systems operate at fundamental limits.

New closed-form solutions for strongly-driven systems.

Abstract

The Boltzmann distribution predicts the collective behavior of systems at thermodynamic equilibrium as a function of their constituent parts. Yet most systems in nature are not at equilibrium, and a unified theory of their behavior does not currently exist. Here, I show that the Boltzmann distribution is a special case of a general distribution that governs all stochastic systems, even if far from equilibrium. The generalized Boltzmann distribution is explained as an analog of the voltage equation in electronics, where resistors, batteries, node voltages, and path currents correspond to equilibrium rate constants, driven rate constants, probabilities, and probability flows, respectively. The general distribution recapitulates known properties of weakly driven systems and enables new closed-form solutions for strongly-driven systems. These solutions provide insight into fundamental limits on system performance, and experimental data show that living systems can operate at those limits. The formal mapping between non-equilibrium systems and electronic circuits may provide a unified framework to simplify, understand, and ultimately control the behavior of complex non-equilibrium systems.