Stochastic thermodynamics of periodically driven systems: Fluctuation theorem for currents and unification of two classes

Periodic driving is used to operate machines that go from standard macroscopic engines to small nonequilibrium microsized systems. Two classes of such systems are small heat engines driven by periodic temperature variations, and molecular pumps driven by external stimuli. Well-known results that are valid for nonequilibrium steady states of systems driven by fixed thermodynamic forces, instead of an external periodic driving, have been generalized to periodically driven heat engines only recently. These results include a general expression for entropy production in terms of currents and affinities, and symmetry relations for the Onsager coefficients from linear-response theory. For nonequilibrium steady states, the Onsager reciprocity relations can be obtained from the more general fluctuation theorem for the currents. We prove a fluctuation theorem for the currents for periodically driven systems. We show that this fluctuation theorem implies a fluctuation dissipation relation, symmetry relations for Onsager coefficients, and further relations for nonlinear response coefficients. The setup in this paper is more general than previous studies, i.e., our results are valid for both heat engines and molecular pumps. The external protocol is assumed to be stochastic in our framework, which leads to a particularly convenient way to treat periodically driven systems.

Figure 1

Periodically driven system with a stochastic protocol modeled as a bipartite Markov process. For this case, the number of different states of the external protocol is $N=4$. Transition rates that change the state of the system $w_{ij}^n$ depend of the state of the external protocol, whereas a transition rate that changes the state of the protocol ${\gamma }^n$ is independent of the state of the system.

Figure 2

Model for a heat engine. The temperature is cold for $n=0,1$ and hot for $n=2,3$. The transition rate from the state with energy 0 to the states with energy $E^n$ is set to $k{\text{e}}^{-{\beta }^n{E}^n/2}$, while the reversed transition rate is $k{\text{e}}^{{\beta }^n{E}^n/2}$. The transition rate associated with isothermal changes is $\gamma$, whereas the transition rate associated with temperature changes is ${\gamma }^{\prime }$.

Figure 3

Model for a molecular pump. The ellipse in green represents a state with energy ${\mathcal{F}}_e$, and the red circles represent states with energy 0. The transition rates for $n=0$ are $w_{12}^0=k{\mathrm{e}}^{{\mathcal{F}}_e-\mathcal{F}/3}$, $w_{13}^0=k{\mathrm{e}}^{{\mathcal{F}}_e-B}$, $w_{23}^0=k{\mathrm{e}}^{-\mathcal{F}/3}$, $w_{21}^0=k$, $w_{31}^0=k{\mathrm{e}}^{-B-\mathcal{F}/3}$, and $w_{32}^0=k$, where ${\beta }_c=1$. Changing $n$ leads to a rotation in the clockwise direction of the transition rates. For example, $w_{12}^0=w_{23}^0=w_{31}^0$.

Figure 4

Illustration of the comparison between the original bipartite Markov process with rates in Eq. (22) and the one corresponding to reversal of the protocol with rates in Eq. (23).

Figure 5

Scaled cumulant-generating function associated with the entropy current $G_s\left(z\right)$. (a) Symmetric protocol for the model depicted in Fig. 6. Parameters were set to $k={\mathcal{F}}_e=10$, ${\gamma }^0=1$, ${\gamma }^1=2.5$, ${\gamma }^2=4$, and ${\gamma }^3=3$. (b) Nonsymmetric protocol for the molecular pump depicted in Fig. 3. Parameters were set to $k=10$, $\gamma =1$, ${\mathcal{F}}_e=B=10$, and $\mathcal{F}=5$.

Figure 6

Model with a symmetric protocol for $N=4$.

Figure 7

Scaled cumulant-generating functions for a deterministic protocol and for a stochastic protocol. The scaled cumulant-generating function for a stochastic protocol tends to the scaled cumulant-generating function for the deterministic protocol with increasing $N$. The parameters of the model with a symmetric protocol are set to $\gamma =2\pi /N$, $k=1$, and ${\mathcal{F}}_e=2$, where the parameter $\gamma$ is valid only for the stochastic protocol.