Nonconvexity of the relative entropy for Markov dynamics: A Fisher information approach

TL;DR

This paper demonstrates that the relative entropy in Markov dynamics can be nonconvex over time, challenging thermodynamic principles, and introduces a Fisher information-based method to identify conditions for convexity or nonconvexity.

Contribution

It provides counterexamples to relative entropy convexity, analyzes conditions for nonconvex solutions, and develops a Fisher information approach to distinguish generator properties.

Findings

Nonconvex solutions are rare and require fine-tuned initial conditions.

Convexity occurs near detailed balance or normality conditions.

The dynamical activity is not a Lyapunov function despite long-term conjectures.

Abstract

We show via counterexamples that relative entropy between the solution of a Markovian master equation and the steady state is not a convex function of time. We thus let down a curtain on a possible formulation of a principle of thermodynamics regarding decrease of the nonadiabatic entropy production. However, we argue that a large separation of typical decay times is necessary for nonconvex solutions to occur, making concave transients extremely short-lived with respect to the main relaxation modes. We describe a general method based on the Fisher information matrix to discriminate between generators that do and don't admit nonconvex solutions. While initial conditions leading to concave transients are shown to be extremely fine-tuned, by our method we are able to select nonconvex initial conditions that are arbitrarily close to the steady state. Convexity does occur when the system is close to satisfy detailed balance, or more generally when certain normality conditions of the decay modes are satisfied. Our results circumscribe the range of validity of a conjecture proposed by Maes et al. [Phys. Rev. Lett. 107, 010601 (2011)] regarding monotonicity of the large deviation rate functional for the occupation probability (dynamical activity), showing that while the conjecture might still hold in the long time limit, the dynamical activity is not a Lyapunov function.