Jarzyski's equality and Crooks' fluctuation theorem for general Markov chains with application to decision-making systems

TL;DR

This paper generalizes thermodynamic fluctuation theorems, specifically Jarzynski's equality and Crooks' theorem, within the framework of Markov chains, and explores their applications beyond physics, including decision-making systems.

Contribution

It extends fluctuation theorems to general Markov chains and discusses their application to decision-making, addressing discrete-time asymmetries and conditions for validity.

Findings

Derived Jarzynski's equality for Markov chains

Formulated Crooks' fluctuation theorem with conditions for asymmetry

Applied the framework to decision-making systems

Abstract

We define common thermodynamic concepts purely within the framework of general Markov chains and derive Jarzynski's equality and Crooks' fluctuation theorem in this setup. In particular, we regard the discrete time case that leads to an asymmetry in the definition of work that appears in the usual formulation of Crooks' fluctuation theorem. We show how this asymmetry can be avoided with an additional condition regarding the energy protocol. The general formulation in terms of Markov chains allows transferring the results to other application areas outside of physics. Here, we discuss how this framework can be applied in the context of decision-making. This involves the definition of the relevant quantities, the assumptions that need to be made for the different fluctuation theorems to hold, as well as the consideration of discrete trajectories instead of the continuous trajectories, which are relevant in physics.