Stochastic thermodynamics for delayed Langevin systems

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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 <Δstot> ≥ 0 no longer holds true, where Δstot 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 η[χ(t)] which involves the work done by a delay-averaged force F(x,t) along the path χ(t) and equals to the medium entropy change Δsₘ[x(t)] in the absence of delay. We show that the total dissipation functional R = Δs + η, where Δs 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 < Δstot > 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(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.