New results in stochastic thermodynamics derived using informational divergences

In prior work we derived simple formulas for how the amount of work dissipated during a time-extended process depends on the initial distribution over states. Here we extend these results to analyze how various rates of change of thermodynamic quantities depend on the current state distribution, thereby deriving novel stochastic thermodynamics results.

Our analysis is grounded in a new theorem that expresses the time-derivative of the KL (Kullback-Leibler) divergence between two distributions as a Bregman divergence between them. Using this theorem, we show that the instantaneous entropy production [EP] of a system with distribution p obeys EP(p) = EP(q) - d/dt D(p||q), where D(. || .) is KL divergence, and q is the distribution that minimizes EP. We also use this theorem and the “Pythagorean theorem for Bregman divergences” to propose a new decomposition of non-adiabatic EP.

In the special case where there is only a single heat bath, the maximal extractable work from a system with distribution p is W_max(p) = D(p||p^eq). We show that in this special case a decomposition similar to the one for EP holds, with EP replaced by -d/dt W_max.

Finally, we derive trajectory-based versions of our results, allowing them to be applied to individual paths as well as averages over all paths.