The following theorem provides an equality for feedback control:
The proof is similar to that of Theorem 26. Note that the physical meaning of is the probability obtained from the following two-step experiments:
• Step 1: Carry out the forward experiment and obtain a series of observations YN.
• Step 2: Carry out the backward experiment with protocol .
Based on Theorem 29, we also obtain the following theorems:
We can also obtain
by taking the expectation of the logarithm of both sides of Eq. (5.307). Note that is the Kullback-Leiber distance between the distributions P(XN, YN) and . When the reversibility with feedback control holds, i.e.,
we have
When σ = β(W −ΔF), the following generalized Jarzynski equality holds:
which results in
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 Wext = −W as the work extracted from the thermodynamics system, we have
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 kBT.
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
An alternative type of fluctuation theorem can also be derived. Consider a Markovian measurement; i.e.,
A forward experiment is carried out at times t1, …, tM with feedback control; a backward experiment is also carried out at times tN−M, …, tN−1. The measurements in the backward experiment are denoted by YN′ = (yN−M, …, yN−1′). Given , we define
which denotes the probability that YN′ is obtained given XN†. Given the protocol Λ(YN)†, the probability of observing YN′ is given by
The time-reversed sequence of YN, denoted by , is given by
Then the probability that YN′ equals is given by
We further assume time-reversed symmetry; i.e.,
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
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 YN.
• In the Szilard engine, γ = 2.
• When there is no feedback control, γ = 1.
We have discussed various approaches of evaluating the communication requirements for controlling CPSs in different situations. As we have seen, several approaches are closely related to the entropy, either deterministic or stochastic, of the physical dynamics. Hence it is a common and generic approach to analyze the increase of the entropy of the physical dynamics and then study how the control can alleviate the entropy increase. We are still awaiting an elegant framework to unify these approaches.