Nonequilibrium fluctuation theorem for systems under discrete and continuous feedback control

In the time reverse process of a feedback manipulated stochastic system, we allow performing measurements without violating causality. As a result we come across an entropy production due to the measurement process. This entropy production, in addition to the usual system and medium entropy production, constitutes the total entropy production of the combined system of the reservoir, the system, and the feedback controller. We show that this total entropy production of “full” system satisfies an integrated fluctuation theorem as well as a detailed fluctuation theorem as expected. We illustrate and verify this idea through explicit calculation and direct simulation in two examples.

Figure 1

Illustration of the forward trajectory (a) and reverse trajectory (b) with measurements and feedback loops. The pair $(x,m)$ next to each black dot on trajectories (red dashed curves) corresponds to state of the system and measurement outcome, respectively. ${\lambda }_m$ between two black dots represents the protocol between times corresponding to those two dots.

Figure 2

Plot (i) shows that the characteristic function $Z_a$ of ${\Sigma }_a$ satisfies symmetry in Eq. (12) for protocol $\left(\text{a}\right)$. Plot (ii) shows the distribution of ${\Sigma }_b$ during forward process of protocol $\left(\text{b}\right)$ and plot (iii) shows that ${\Sigma }_b$ satisfies the symmetry in Eq. (11). Parameters for this plot are $\tau =2.0,k=1.2,\Delta =0.2,\gamma =1.0,\beta =0.5$, and $\alpha =5.0$.

Figure 3

Here $P\left(\Sigma \right)$ for molecular refrigerator model verifies the symmetry in Eq. (11). Inset compares $P\left(\Sigma \right)$ obtained from direct simulation with the same obtained by numerical inverse Laplace transform of $Z\left(u\right)$ in Eq. (23). Parameters for this plot are $\gamma ={\gamma }^{\prime }=1.0,T=0.6$, and ${\alpha }_0=0.8$.