Stochastic thermodynamics for delayed Langevin systems

We discuss stochastic thermodynamics (ST) for delayed Langevin systems in this paper. 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 way that is similar to that in a system without delay. Because the presence of time delay brings an additional entropy flux into the system, the conventional second law >= 0 no longer holds true, where Delta s(tot) denotes the total entropy change along a stochastic path and <.> stands for the 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 (F) over bar (x,t) along the path chi(t) and equals 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 > >= 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 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 (F) over bar (x,t) from the stationary distribution and then calculate the functional R as well as its distribution. The second law < R > >= 0 and the fluctuation theorem are successfully validated.