A Second Law for Open Markov Processes

TL;DR

This paper introduces open Markov processes allowing population flow at boundaries and establishes a second law-like inequality relating relative entropy change to boundary flows, generalizing thermodynamic principles.

Contribution

It defines open Markov processes and proves a second law inequality for their relative entropy dynamics, extending classical thermodynamics to these systems.

Findings

Relative entropy change is bounded by boundary flows.

The second law inequality applies to non-equilibrium distributions.

Provides a thermodynamic interpretation for open Markov processes.

Abstract

In this paper we define the notion of an open Markov process. An open Markov process is a generalization of an ordinary Markov process in which populations are allowed to flow in and out of the system at certain boundary states. We show that the rate of change of relative entropy in an open Markov process is less than or equal to the flow of relative entropy through its boundary states. This can be viewed as a generalization of the Second Law for open Markov processes. In the case of a Markov process whose equilibrium obeys detailed balance, this inequality puts an upper bound on the rate of change of the free energy for any non-equilibrium distribution.