Extended Local-Equilibrium Thermodynamics from Chemical Jump Processes

The extended local-equilibrium approach to stochastic thermodynamics assigns thermodynamic state functions to the instantaneous fluctuating state of a system. In this work, we derive the evolution equations used in this approach from a Markov jump process for chemical reactions. We first express the changes of entropy, internal energy and free energy along individual trajectories in terms of the finite changes associated with reaction events. In the weak-noise limit, a midpoint expansion gives the Stratonovich chain rule for these state functions. We then show that the mean and variance of the entropy produced over a mesoscopic time interval obey a thermodynamic uncertainty relation. We also examine the Glansdorff-Prigogine universal evolution criterion along stochastic trajectories. Fluctuations add a contribution with no definite sign to the classical nonpositive term. Within the local Gaussian approximation, we calculate the probability that this contribution reverses the classical sign and obtain a lower bound in terms of entropy production. This bound constrains how reliably the classical sign can be observed over that interval. The results are illustrated with an open linear reaction scheme.