Operationally accessible uncertainty relations for thermodynamically consistent semi-Markov processes

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-timeMarkov 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.

Automated summary

This article discusses the relationship between entropy production and mean and variance of steady-state currents in semi-Markov processes, as well as the uncertainty relation in coarse-grained Markov processes generated by semi-Markov descriptions. The article introduces a crucial condition in thermodynamic consistency and presents some alternative derivations of results.