Continuity equation for probability as a requirement of inference over paths

TL;DR

This paper derives the general continuity equation for probability over paths, linking it to fundamental equations in non-equilibrium statistical mechanics, using inference principles like maximum caliber.

Contribution

It introduces the time-slicing equation to connect path probability functionals with time-dependent densities, unifying various statistical mechanics equations.

Findings

Derivation of the general continuity equation from path inference.

Recovery of Liouville and Fokker-Planck equations as special cases.

Application of maximum caliber principle to construct probability functionals.

Abstract

In this work we present the fundamental ideas of inference over paths, and show how this formalism implies the continuity equation, which is central for the derivation of the main partial differential equations that constitute non-equilibrium statistical mechanics. Equations such as the Liouville equation, Fokker-Planck equation, among others can be recovered as particular cases of the continuity equation, under different probability fluxes. We derive the continuity equation in its most general form through what we call the \bf time-slicing equation, which lays down the procedure to go from the representation in terms of a path probability functional $\rho[X()]$ to a time-dependent probability density $\rho(x; t)$. The original probability functional $\rho[X()]$ can in principle be constructed from different methods of inference; in this work we sketch an application using the maximum path entropy or maximum caliber principle.