Stochastic dynamics from maximum entropy in action space

TL;DR

This work recasts stochastic dynamics within an information-theoretic MaxEnt framework where the total action $A$ between spacetime points is the fundamental stochastic variable. By maximizing the entropy of the joint distribution $p(b,A|a)$ subject to normalization and a mean-action constraint, the authors derive a Boltzmann-like form $p(b,A|a) \propto g(A,b)\,e^{-eta A}$ with a density of states $g(A,b)$ that captures path degeneracy; this yields a Gaussian action-space density around the minimal action in the diffusive regime. In the nonrelativistic limit, the framework reproduces the standard Brownian propagator by identifying $\u001beta = 1/(2 m D)$, while in relativistic settings it produces a manifestly Lorentz-covariant diffusion kernel with action bounds set by causality. The paper also develops action-space analogs of fluctuation theorems (Crooks-like and Jarzynski-like relations) and a second-law bound, highlighting how entropic degeneracy and action minimization compete to shape stochastic transitions. Overall, the approach unifies classical and relativistic stochastic processes under a covariant, information-based foundation that avoids functional path integrals and clarifies the role of action-based uncertainties in nonequilibrium dynamics.

Abstract

We develop an information-theoretic formulation of stochastic dynamics in which the fundamental stochastic variable is the total action connecting spacetime points, rather than individual paths. By maximizing Shannon entropy over a joint distribution of actions and endpoints, subject to normalization and a constraint on the mean action, we obtain a Boltzmann-like distribution in action space. This framework reproduces the standard Brownian propagator in the nonrelativistic limit and naturally extends to relativistic regimes, where the Wiener construction fails to preserve Lorentz covariance. The approach bypasses functional integration over paths, makes the role of entropic degeneracy explicit through an action-space density of states, and provides a transparent connection between the principle of least action and statistical inference. We derive the density of states explicitly using large deviation theory, showing that it takes a Gaussian form centered at the minimal action, and rigorously justify the saddle-point approximation in the diffusive regime. The Markovian property of the resulting propagator is verified to hold via the Chapman--Kolmogorov equation, following from the additivity of the minimal action for free-particle dynamics. In the diffusive regime, the resulting dynamics are governed by a competition between extremization of the action and entropic effects, which can be interpreted in terms of an effective action free energy. Our results establish an unified, covariant, and information-based foundation for classical and relativistic stochastic processes.

Figures (1)

• Figure 1: Schematic comparison between the path-based and action-space formulations of stochastic dynamics. (Left) In the conventional path-based approach, stochasticity is described by assigning probabilities to individual trajectories $\gamma$ connecting fixed spacetime endpoints $a=(x_a,t_a)$ and $b=(x_b,t_b)$, with the transition probability obtained from a functional integral over paths weighted by $e^{-\eta A[\gamma]}$. Entropic effects are implicit in the counting of trajectories and require discretization and a Wiener-type measure. (Right) In the action-space formulation proposed here, the total action $A$ is treated as the fundamental stochastic variable. Distinct paths leading to the same action value are grouped into a density of states $g(A,b)$, making path degeneracy explicit. The transition probability is obtained from an ordinary integral over action values, $p(b|a)=\int dA\, g(A,b)\,e^{-\eta A}$, with support restricted to $A\geq A_{\min}(b)$. This reformulation bypasses functional integration, clarifies the role of entropy through $g(A,b)$, and renders Lorentz covariance manifest at the level of the probability distribution.