Optimal Protocols and Optimal Transport in Stochastic Thermodynamics

Abstract

Thermodynamics of small systems has become an important field of statistical physics. Such systems are driven out of equilibrium by a control, and the question is naturally posed how such a control can be optimized. We show that optimization problems in small system thermodynamics are solved by (deterministic) optimal transport, for which very efficient numerical methods have been developed, and of which there are applications in cosmology, fluid mechanics, logistics, and many other fields. We show, in particular, that minimizing expected heat released or work done during a nonequilibrium transition in finite time is solved by the Burgers equation and mass transport by the Burgers velocity field. Our contribution hence considerably extends the range of solvable optimization problems in small system thermodynamics.

Stochastic echo phenomena in nonequilibrium systems

“…The pseudo-heat, evaluated from Eqs. (18) and (19), vanishes while Q = k 2 (λ − Λ) 2 . Therefore the relative entropy S op vanishes and the SEP is perfect.…”

“…Suppose that a system is driven by a time-dependent external parameter λ(t) under the constraint that λ(0) = λ i and λ(τ ) = λ f . One may seek for the optimal control of λ(t) that yields the minimum entropy production [18,19]. Note that this optimal protocol for the minimum entropy production is different from the optimal protocol for the SEP.…”

Applicability of optimal protocols and the Jarzynski equality

“…the smoothest of all, at variance with the results of Refs. [15,16]. In this case, our model provides one simple example in which the predictions for the most efficient protocol obtained using the stochastic approach fail.…”

“…In order to obtain mathematical results, most investigations have focussed on stochastic processes, for which numerous techniques are available. In particular, the works by Schmeidl and Seifert [15] and by Aurell, Mejía-Monasterio and Muratore-Ginanneschi [16] consider Langevin models such as:…”

“…Changing from 9 to 1, these protocols provide a better and better approximation to the discontinuous jumps described by Refs. [15,16]. Each run was repeated 1.5 × 10 5 times for a given simulation length, and the results for the computed free-energy difference ∆F have been averaged over 10 groups of 1.5 × 10 4 samples, and reported in Figure 1b, where the standard error represents their reproducibility, not the error in convergence.…”