Nonequilibrium equalities for feedback control

The following theorem provides an equality for feedback control:

The proof is similar to that of Theorem 26. Note that the physical meaning of $P^{†}(X_N^{†},Y_N)$ is the probability obtained from the following two-step experiments:

• Step 1: Carry out the forward experiment and obtain a series of observations Y_N.

• Step 2: Carry out the backward experiment with protocol ${\Lambda }_N^{†}(Y_{N-1})$.

Based on Theorem 29, we also obtain the following theorems:

We can also obtain

$\begin{array}{ll}\langle \sigma \rangle +\langle I_c\rangle \hfill & =\int P(X_N,Y_N)ln\frac{P(X_N,Y_N)}{P^{†}(X_N^{†},Y_N)}d{X}_{N}d{Y}_N\hfill \\ \hfill & =D(P(X_N,Y_N)||P^{†}(X_N^{†},Y_N))\hfill \end{array}$

by taking the expectation of the logarithm of both sides of Eq. (5.307). Note that $D(P(X_N,Y_N)||P^{†}(X_N^{†},Y_N))$ is the Kullback-Leiber distance between the distributions P(X_N, Y_N) and $P^{†}(X_N^{†},Y_N)$. When the reversibility with feedback control holds, i.e.,

$\begin{array}{l}P(X_N,Y_N)=P^{†}(X_N^{†},Y_N),\hfill \end{array}$

we have

$\begin{array}{l}\langle \sigma \rangle +\langle I_c\rangle =0.\hfill \end{array}$

When σ = β(W −ΔF), the following generalized Jarzynski equality holds:

$\begin{array}{l}\langle {\mathrm{e}}^{-\beta (W-\mathrm{\Delta }F)-I_c}\rangle =1,\hfill \end{array}$

which results in

$\begin{array}{l}\langle W\rangle \ge \mathrm{\Delta }F-k_{B}T\langle I_c\rangle .\hfill \end{array}$

This inequality relates the work exerted by external sources, the reduction of free energy of the thermodynamic system, and information obtained from the feedback. Moreover, if ΔF = 0 (namely the free energy of the thermodynamic system does not change), we define W_ext = −W as the work extracted from the thermodynamics system, we have

$\begin{array}{l}\langle W_{\mathrm{ext}}\rangle \le k_{B}T\langle I_c\rangle .\hfill \end{array}$

Hence the work that we can extract from the information provided to the thermodynamic system through the feedback is upper bounded by the mutual information scaled by k_BT.

Take the Szilard engine discussed previously, for example. The information about the location of the particle is binary (0: the left half; 1: the right half). We assume that the measurement is passed through a binary symmetric channel with error probability ϵ; i.e., the controller receives the correct location of the particle with probability 1 − ϵ. Using the argument in Ref. [71], we can obtain

$\begin{array}{l}{W}_{\mathrm{ext}}=k_{B}T(ln2+ϵlnϵ+(1-ϵ)ln(1-ϵ)).\hfill \end{array}$

Nonequilibrium equalities with efficacy

An alternative type of fluctuation theorem can also be derived. Consider a Markovian measurement; i.e.,

$\begin{array}{l}P(y_n|X_n)=P(y_n|x_n).\hfill \end{array}$

A forward experiment is carried out at times t₁, …, t_M with feedback control; a backward experiment is also carried out at times t_N−M, …, t_N−1. The measurements in the backward experiment are denoted by Y_N′ = (y_N−M, …, y_N−1′). Given $X_N^{†}$, we define

$\begin{array}{l}{P}_c(Y_N^{\prime }|X_N†)=\prod _{k=1}^{M}P(y_{N-k}|x_{N-k}^{†}),\hfill \end{array}$

which denotes the probability that Y_N′ is obtained given X_N^†. Given the protocol Λ(Y_N)^†, the probability of observing Y_N′ is given by

$\begin{array}{l}P(Y_N^{\prime }|{\Lambda }_N{(Y_{N-1})}^{†})=\int P_c(Y_N^{\prime }|X_N†)P(X_N^{†}|{\Lambda }_N{(Y_{N-1})}^{†})d{X}_N^{†}.\hfill \end{array}$

The time-reversed sequence of Y_N, denoted by $Y_N^{†}$, is given by

$\begin{array}{l}{Y}_N^{†}=(-y_{N-M},\dots ,-y(N-1)).\hfill \end{array}$

Then the probability that Y_N′ equals $Y_N^{†}$ is given by

$\begin{array}{l}P(Y_N^{†}|{\Lambda }_N{(Y_{N-1})}^{†})=\int P_c(Y_N^{†}|X_N†)P(X_N^{†}|{\Lambda }_N{(Y_{N-1})}^{†})d{X}_N^{†}.\hfill \end{array}$

We further assume time-reversed symmetry; i.e.,

$\begin{array}{l}P(y_n^{*}|x_n^{*})=P(y_n|x_n).\hfill \end{array}$

We then obtain the following theorem, whose detailed proof is similar to that of Theorem 26 and can be found in Ref. [71].

Taking the expectation of both sides of Eq. (5.324), we obtain the following corollary:

For the case σ = β(W −ΔF), we have

$\begin{array}{l}\langle {\mathrm{e}}^{-\beta (W-\mathrm{\Delta }F)}\rangle =\gamma ,\hfill \end{array}$

which is a generalization of the Jarazynski equality.

The values of the efficacy parameter γ are as follows:

• When the feedback control is perfect (namely, the system is expected to return to the initial state with probability 1 in the time-reserved process), γ equals the number of possible observations of Y_N.

• In the Szilard engine, γ = 2.

• When there is no feedback control, γ = 1.