Stochastic thermodynamics for delayed Langevin systems

TL;DR

This paper extends stochastic thermodynamics to systems with delayed Langevin dynamics, defining energy and entropy concepts, and demonstrating a generalized second law and fluctuation theorem through theoretical analysis and numerical validation.

Contribution

It introduces a delay-averaged dissipation functional and generalizes the second law and fluctuation theorem for delayed Langevin systems, which were not previously established.

Findings

The total entropy change can be negative with positive delay feedback.

The total dissipation functional R satisfies < R > ≥ 0, confirming a generalized second law.

The integral fluctuation theorem < <e^(-R)> = 1 holds for delayed systems.

Abstract

Stochastic thermodynamics (ST) for delayed Langevin systems are discussed. By using the general principles of ST, the first-law-like energy balance and trajectory-dependent entropy s(t) can be well-defined in a similar way as that in a system without delay. Since the presence of time delay brings an additional entropy flux into the system, the conventional second law $<{\Delta {s_{tot}}}> \ge 0$ no longer holds true, where $\Delta {s_{tot}}$ denotes the total entropy change along a stochastic path and $<...>$ stands for average over the path ensemble. With the help of a Fokker-Planck description, we introduce a delay-averaged trajectory-dependent dissipation functional $\eta [{\chi(t)}]$ which involves the work done by a delay-averaged force $\bar F({x,t})$ along the path $\chi (t)$ and equals to the medium entropy change $\Delta {s_m}[ {x(t)}]$ in the absence of delay. We show that the total dissipation functional R = \Delta s + \eta, where $\Delta s$ denotes the system entropy change along a path, obeys $< R > \ge 0$, which could be viewed as the second law in the delayed system. In addition, the integral fluctuation theorem $< <e^(-R)>=1 also holds true. We apply these concepts to a linear Langevin system with time delay and periodic external force. Numerical results demonstrate that the total entropy change $< {\Delta {s_{tot}}} >$ could indeed be negative when the delay feedback is positive. By using an inversing-mapping approach, we are able to obtain the delay-averaged force $\bar F({x,t})$ from the stationary distribution and then calculate the functional $R$ as well as its distribution. The second law $< R > \ge 0$ and the fluctuation theorem are successfully validated.