Stochastic Calculus Approach to Thermodynamics of Jump Processes

Abstract: Stochastic thermodynamics is the field of study relating fluctuations in stochastic systems to thermodynamic quantities. The total entropy production (EP), is central to the thermodynamic classification of systems. Non-equilibrium systems manifestly all have non-zero EP and therefore impose an "arrow of time". Thermodynamic inequalities are lower bounds on the total EP and are especially useful when only parts of systems are operationally accessible. We use a stochastic calculus approach to directly derive and generalise three classes of inequalities for Markov jump processes using correlations of path observables, e.g., currents and densities. Our theoretical predictions are compared with simulations, where a good agreement is observed. The thermodynamic bounds we investigate include the thermodynamic uncertainty relation (TUR), thermodynamic transport bound (TB), and thermodynamic correlation bound (CB). We provide insight into the saturation conditions for these bounds and to what degree saturation can be achieved. Additionally, for the TUR and TB, we show how the bounds are related, which includes identifying a diffusion coefficient for jump dynamics. %An example using a toy model shows how the CB may yield a negative lower bound on the total entropy production, contrary to the non-negative bound that the TUR and TB yield. Comparisons are drawn between the TUR and TB for relaxation and stationary processes in biologically relevant settings. Specifically, calmodulin folding dynamics and secondary active transport, where differences in long-time relaxation and convergence are observed. For a systematic way to construct models, we formulate two methods to drive systems out of equilibrium without changing the stationary probability distribution.