Nonequilibrium thermodynamics of feedback control

We establish a general theory of feedback control on classical stochastic thermodynamic systems and generalize nonequilibrium equalities such as the fluctuation theorem and the Jarzynski equality in the presence of feedback control with multiple measurements. Our results are generalizations of the previous relevant works to the situations with general measurements and multiple heat baths. The obtained equalities involve additional terms that characterize the information obtained by measurements or the efficacy of feedback control. A generalized Szilard engine and a feedback-controlled ratchet are shown to satisfy the derived equalities.

Figure 1

Discretization of control protocol $\lambda \left(t\right)$.

Figure 2

Feedback control on nonequilibrium dynamics. The control parameter is denoted as $\lambda$, the point of the phase space of the system as $x$, and the outcome of measurement on the system as $y$. Parameter $\lambda$ depends on $y$ through the real-time feedback control.

Figure 3

(a) Forward outcomes $Y_N$ with forward protocol ${\Lambda }_N\left(Y_N\right)$. (b) Backward outcomes $Y_N^{†}$ with backward protocol [${\Lambda }_N{\left(Y_N\right)}^{†}$].

Figure 4

Generalized Szilard engine with measurement error rate $\varepsilon$, where $x$ denotes the states of the system, and $y$ denote the measurement outcomes. The control protocol is determined by $y$.

Figure 5

Backward processes of the generalized Szilard engine. Corresponding to $y$ that denotes the measurement outcomes in the forward process, we have two control protocols in the backward process, where $y^{\prime }$ denotes the measurement outcomes in the backward process.

Figure 6

Two shapes of potential $V(x,\lambda )$ corresponding to $\lambda =0,1$.

Figure 7

(a) Numerical results of the ensemble average of trajectory $x\left(t\right)$ corresponding to the three control protocols: the trivial control, the flashing ratchet, and the feedback-controlled ratchet. (b) Numerical result of the ensemble average of the work $W\left(t\right)$ corresponding to the flashing ratchet and the feedback-controlled ratchet.

Figure 8

Numerical tests of the Jarzynski equality for the flashing ratchet and a generalized Jarzynski equality (89) for the feedback-controlled ratchet.