Stochastic thermodynamics for delayed Langevin 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 = Δs + η, 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 $<<e^{(}-R)>=1alsoholdstrue.WeapplytheseconceptstoalinearLangevinsystemwithtimedelayandperiodicexternalforce.Numericalresultsdemonstratethatthetotalentropychange$< {Δ{s_{tot}}} >$couldindeedbenegativewhenthedelayfeedbackispositive.Byusinganinversing-mappingapproach,weareabletoobtainthedelay-averagedforce$\bar F({x,t})$fromthestationarydistributionandthencalculatethefunctional$R$aswellasitsdistribution.Thesecondlaw$< R > \ge 0$ and the fluctuation theorem are successfully validated.

16 pages, 5 figures

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