Operationally Accessible Uncertainty Relations for Thermodynamically Consistent Semi-Markov Processes

TL;DR

This paper derives thermodynamic uncertainty relations for semi-Markov processes, linking entropy production to steady-state current fluctuations, and provides tools to infer underlying Markovian structures from coarse-grained models.

Contribution

It introduces new thermodynamic uncertainty relations specific to semi-Markov processes and their coarse-grained versions, expanding the understanding of thermodynamic consistency in complex stochastic systems.

Findings

Derived a thermodynamic uncertainty relation for semi-Markov processes.

Established a criterion to detect non-Markovian coarse-graining.

Demonstrated the relations with illustrative examples.

Abstract

Semi-Markov processes generalize Markov processes by adding temporal memory effects as expressed by a semi-Markov kernel. We recall the path weight for a semi-Markov trajectory and the fact that thermodynamic consistency in equilibrium imposes a crucial condition called direction-time independence for which we present an alternative derivation. We prove a thermodynamic uncertainty relation that formally resembles the one for a discrete-time Markov process. The result relates the entropy production of the semi-Markov process to mean and variance of steady-state currents. We prove a further thermodynamic uncertainty relation valid for semi-Markov descriptions of coarse-grained Markov processes that emerge by grouping states together. A violation of this inequality can be used as an inference tool to conclude that a given semi-Markov process cannot result from coarse-graining an underlying Markov one. We illustrate these results with representative examples.