A derivation of the master equation from path entropy maximization

TL;DR

This paper derives the master equation and Markov processes from the principle of maximum entropy, providing a rigorous, systematic foundation for these models as solutions to an inverse problem under constraints.

Contribution

It offers a novel derivation of the master equation as the maximum entropy solution to an inverse problem, generalizing stochastic models beyond traditional assumptions.

Findings

Markov processes are uniquely derived as maximum entropy solutions.

The approach provides a systematic way to generalize stochastic models.

It offers a rigorous alternative to traditional justifications of the master equation.

Abstract

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive n-th order Markov processes and the master equation as unique solutions to an inverse problem. In particular, we find that when the constraints are not enough to uniquely determine the stochastic model, the n-th order Markov process emerges as the unique maximum entropy solution to this otherwise under-determined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from first principles.