Stochastic dynamics from maximum entropy in action space

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.

21 pages, 1 figures

References in corpus (12)

• The large deviation approach to statistical mechanics
• Irreversible entropy production, from quantum to classical
• Relativistic Brownian Motion
• Diffusive propagation of UHECR and the propagation theorem
• Theory of the Relativistic Brownian Motion. The (1+1)-Dimensional Case
• Diffusion of Cosmic Rays in Expanding Universe. (I)
• Relativistic diffusion processes and random walk models
• Application of the three-dimensional telegraph equation to cosmic-ray transport
• Non quantum uncertainty relations of stochastic dynamics
• Modeling the gamma-ray emission in the Galactic Center with a fading cosmic-ray accelerator
• Survival and extreme statistics of work, heat, and entropy production in steady-state heat engines
• A rigorous solution to the superluminal issue in the diffusion equation