Fluctuation theorems and inequalities generalizing the second law of thermodynamics out of equilibrium

We present a general framework for systems which are prepared in a nonstationary nonequilibrium state in the absence of any perturbation and which are then further driven through the application of a time-dependent perturbation. By assumption, the evolution of the system must be described by Markovian dynamics. We distinguish two different situations depending on the way the nonequilibrium state is prepared; either it is created by some driving or it results from a relaxation following some initial nonstationary conditions. Our approach is based on a recent generalization of the Hatano-Sasa relation for nonstationary probability distributions. We also investigate whether a form of the second law holds for separate parts of the entropy production and for any nonstationary reference process, a question motivated by the work of M. Esposito et al. [Phys. Rev. Lett. 104, 090601 (2010)]. We find that although the special structure of the theorems derived in this reference is not recovered in the general case, detailed fluctuation theorems still hold separately for parts of the entropy production. These detailed fluctuation theorems contain interesting generalizations of the second law of thermodynamics for nonequilibrium systems.

Figure 1

Illustration of the detailed fluctuation relation obeyed by the quantity $\mathcal{Y}$. Orange squares correspond to a long protocol with fast relaxation of the ${\pi }_t$ function towards equilibrium (orange dashed line of the insets), whereas black dots are for a short protocol with slow relaxation towards equilibrium (black solid lines of the insets). Inset (i) shows the half sinusoidal protocols and inset (ii) shows the relaxation, as a function of time $t$, of the distribution ${\pi }_t(b,h)$ towards equilibrium distribution for a given $h$ value.

Figure 2

Transition between nonstationary oscillating states for different times of driving: (from left to right) $t_d=50$ and $t_d=7.91$. (Top) Protocol imposed (orange dashed lines) and the total force applied on the system (black solid line). (Bottom) Exact (black) and accompanying probability distribution (orange) of state $b$ as a function of time. For the shortest protocol, we see that the accompanying distribution differs from the exact solution while the protocol is changing.

Figure 3

Illustration of the modified second law for nonstationary systems created by periodic driving. Symbols represent the driving entropy production $\langle \mathcal{Y}\rangle$ (filled squares), the nonadiabatic entropy production $\langle \Delta S_{\mathrm{na}}\rangle$ (filled triangles), the difference of traffic $\langle \Delta \mathcal{T}\rangle$ (bullets), and $〈\Delta S_b〉$ (stars) as a function of the duration of the driving $t_d$. Note that $\langle \mathcal{Y}\rangle$, $\langle \Delta S_{\mathrm{na}}\rangle$, and $\langle \Delta \mathcal{T}\rangle$ are evaluated between the times $t_{\mathrm{di}}$ and the final time $T$, while $〈\Delta S_b〉$ is evaluated at time $t_{\mathrm{df}}$. The solid line is simply $1/t_d$ and shows that $\langle \mathcal{Y}\rangle$ and $\langle \Delta S_{\mathrm{na}}\rangle$ behave as the inverse of the duration of the driving $t_d$ in the limit of large $t_d$.

Figure 4

(Inset) Transitions between two nonstationary states corresponding to the forces $h_{t_{\mathrm{di}}}=1.3$ and $h_{t_{\mathrm{df}}}=3$. The spring constant is oscillating with a period of $0.5$ around the value 1 and with an amplitude of $0.5$. The inverse temperature is taken at $\beta =0.2$; the friction is $\gamma =1$.