Liouville's Theorem from the Principle of Maximum Caliber in Phase Space

TL;DR

This paper derives Liouville's theorem in non-equilibrium statistical mechanics from the principle of maximum caliber, emphasizing inference over continuous paths, applicable to physical and non-physical systems.

Contribution

It provides a novel derivation of Liouville's equation based solely on inference principles, broadening its applicability beyond traditional physical systems.

Findings

Derivation of Liouville's equation from maximum caliber principle

Applicable to systems beyond classical physics, including time series

General framework for phase space probability dynamics

Abstract

One of the cornerstones in non--equilibrium statistical mechanics (NESM) is Liouville's theorem, a differential equation for the phase space probability $\rho(q,p; t)$. This is usually derived considering the flow in or out of a given surface for a physical system (composed of atoms), via more or less heuristic arguments. In this work, we derive the Liouville equation as the partial differential equation governing the dynamics of the time-dependent probability $\rho(q, p; t)$ of finding a "particle" with Lagrangian $L(q, \dot{q}; t)$ in a specific point $(q, p)$ in phase space at time $t$, with $p=\partial L/\partial \dot{q}$. This derivation depends only on considerations of inference over a space of continuous paths. Because of its generality, our result is valid not only for "physical" systems but for any model depending on constrained information about position and velocity, such as time series.