Stationary properties of maximum entropy random walks

TL;DR

This paper derives the stationary distribution of maximum path entropy Markov processes under state- and path-dependent constraints, revealing differences from Boltzmann distributions and exploring implications for complex systems.

Contribution

It introduces a novel derivation of transition probabilities and stationary distributions for max path entropy Markov processes considering topology and path constraints.

Findings

Stationary distribution differs from Boltzmann distribution.

Path multiplicity influences the inferred probabilities.

Application to particle diffusion on energy landscapes.

Abstract

Maximum entropy (maxEnt) inference of state probabilities using state-dependent constraints is popular in the study of complex systems. In stochastic dynamical systems, the effect of state space topology and path-dependent constraints on the inferred state probabilities is unknown. To that end, we derive the transition probabilities and the stationary distribution of a maximum {\it path} entropy Markov process subject to state- and path-dependent constraints. The stationary distribution reflects a competition between path multiplicity and imposed constraints and is significantly different from the Boltzmann distribution. We illustrate our results with a particle diffusing on an energy landscape. Connections with the path integral approach to diffusion are discussed.