Information Thermodynamics:
Maxwell’s Demon in Nonequilibrium Dynamics
Abstract
We review theory of information thermodynamics which incorporates effects of measurement and feedback into nonequilibrium thermodynamics of a small system, and discuss how the second law of thermodynamics should be extended for such situations. We address the issue of the maximum work that can be extracted from the system in the presence of a feedback controller (Maxwell’s demon) and provide a few illustrative examples. We also review a recent experiment that realized a Maxwell’s demon based on a feedbackcontrolled ratchet.
Contents
1 Introduction
The profound interrelationship between information and thermodynamics was first brought to light by J. C. Maxwell [1] in his gedankenexperiment about a hypothetical being of intelligence which was later christened by William Thomson as Maxwell’s demon. Since then, numerous discussions have been spurred as to whether and how Maxwell’s demon is compatible with the second law of thermodynamics [2, 3, 4].
To understand the roles of Maxwell’s demon, let us consider a situation in which a gas is confined in a box surrounded by adiabatic walls. A barrier is inserted at the center of the box to divide it into two. The temperatures of the gases in the two boxes are assumed to be initially the same. We assume that the demon is present near the barrier and can close or open a small hole in the barrier. If a fasterthanaverage (slowerthanaverage) molecule comes from the right (left) box, the demon opens the hole. Otherwise, the demon keeps it closed. By doing so over and over again, the temperature of the left gas becomes higher than that of the right one, in apparent contradiction with the second law of thermodynamics. This example illustrates the two essential roles of the demon:

The demon observes individual molecules, and obtains the information about their velocities.

The demon opens or closes the hole based on each measurement outcome, which is the feedback control.
In general, feedback control implies that a control protocol depends on the measurement outcome, or equivalently, we control a system based on the obtained information [5, 6]. The crucial point here is that the measurements are performed at the level of thermal fluctuations (i.e., the demon can distinguish the velocities of individual molecules). Therefore, Maxwell’s demon can be characterized as a feedback controller that utilizes information about a thermodynamic system at the level of thermal fluctuations (see also Fig. 1).
In the nineteenth century, it was impossible to observe and control individual atoms and molecules, and therefore it was not necessary to take into account the effect of feedback control in the formulation of thermodynamics. However, due to the recent advances in manipulating microscopic systems, the effect of feedback control on thermodynamic systems has become relevant to real experiments. We can simulate the role of Maxwell’s demon in real experiments and can reduce the entropy of small thermodynamic systems.
In this chapter, we review a general theory of thermodynamics that involves measurements and feedback control [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. We generalize the second law of thermodynamics by including information contents concerning the thermodynamics of feedback control. We note that, by the “demon,” we mean a type of devices that perform feedback control at the level of thermal fluctuations.
This chapter is organized as follows. In Sec. 2, we discuss the Szilard engine which is a prototypical model of Maxwell’s demon and examine the consistency between the demon and the second law. In Sec. 3, we review information contents that are used in the following sections. In Sec. 4, we discuss a generalized second law of thermodynamics with feedback control, which is the main part of this chapter. In Sec. 5, we generalize nonequilibrium equalities such as the fluctuation theorem and the Jarzynski equality to the case with feedback control. In Sec. 6, we discuss the energy cost (work) that is needed for measurement and information erasure. In Sec. 7, we conclude this chapter.
2 Szilard Engine
In 1929, L. Szilard proposed a simple model of Maxwell’s demon that illustrates the quantitative relationship between information and thermodynamics [45]. In this section, we briefly review the model, which is called the Szilard engine, and discuss its physical implications.
The Szilard engine consists of a singleparticle gas that is in contact with a single heat bath at temperature . By a measurement, we obtain one bit of information about the position of the particle and use that information to extract work from the engine via feedback control. While the engine eventually returns to the initial equilibrium, the total amount of the extracted work is positive. The details of the control protocol are as follows (see Fig. 2).
Step 1: Initial state. We prepare a singleparticle gas in a box of volume , which is at thermal equilibrium with temperature .
Step 2: Insertion of the barrier. We insert a barrier in the middle of the box, and divide it into two with equal volume .
Step 3: Measurement. We perform an errorfree measurement of the position of the particle to find which box the molecule is in. Because the particle will be found to be in each box with probability , the amount of information gained from this measurement is one bit. We note that one () bit of information in the binary logarithm corresponds to nat of information in the natural logarithm.
Step 4: Feedback. If the particle is in the left (right) box, we quasistatically expand it by moving the barrier to the rightmost (leftmost) position. By this process, we can extract work given by
(1) 
where we used with , , and respectively being the pressure, the volume and the Boltzmann constant. This process corresponds to feedback control, because the direction of the expansion (i.e., left or right) depends on the measurement outcome. After this expansion, the gas returns to the initial thermal equilibrium with volume .
The extracted work is proportional to the obtained information with proportionality constant . This is due to the fact that the entropy of the system is effectively decreased by via feedback control, and the decrease in entropy leads to the increase in the free energy by , which is the resource of the extracted work.
The Szilard engine prima facie seems to contradict the second law of thermodynamics, which dictates that one cannot extract positive work from a single heat bath with a thermodynamic cycle (Kelvin’s principle). In fact, the Szilard engine is consistent with the second law, due to an additional energy cost (work) that is needed for the measurement and information erasure of the measurement device or the demon. This additional cost compensates for the excess work extracted from the engine. In his original paper, Szilard argued that there must be an entropic cost for the measurement process [45]. We stress that the work performed on the demon need not be transferred to the engine; only the information obtained by the measurement should be utilized for the feedback. This is the crucial characteristic of the information heat engine.
By utilizing the obtained information in feedback control, we can extract work from the engine without decreasing its free energy, or we can increase the engine’s free energy without injecting any work to the engine directly. The resource of the work or the free energy is thermal fluctuations of the heat bath; by utilizing information via feedback, we can rectify the thermal energy of the bath and convert it into the work or the free energy. This method allows us to control the energy balance of the engine beyond the conventional thermodynamics. We shall call such a feedbackcontrolled heat engine as an “information heat engine.” The Szilard engine works as the simplest model that illustrates the quintessence of information heat engines. Quantum versions of the Szilard engine have also been studied [12, 31, 38, 43].
3 Information Content in Thermodynamics
In this section, we briefly review the Shannon information and the mutual information [46, 47]. In particular, the mutual information plays a crucial role in thermodynamics of information processing.
3.1 Shannon Information
Let be a probability variable which represents a finite set of possible events. We write as the probability of event being realized. The information content that is associated with event is then defined as , which implies that the rarer an event is, the more information is associated with it. The Shannon information is then given by the average of over all possible events:
(2) 
The Shannon information satisfies , where is the number of the possible events (the size of set ). Here, the lower bound () is achieved if holds for some ; in this case, the event is indeed deterministic, while the upper bound () is achieved if for arbitrary . In general, the Shannon information characterizes the randomness of a probability variable; the more random the variable, the greater the Shannon information. Consider, for example, a simple case in which takes two values: or . We set and with . The Shannon information is then given by , which takes the maximum value for and the minimal value for or .
3.2 Mutual Information
The mutual information characterizes the correlation between two probability variables. Let and be the two probability variables, and be their joint distribution. The marginal distributions are given by , . If the two variables are statistically independent, then . Otherwise, they are correlated. If the two variables are perfectly correlated, the joint distribution satisfies
(3) 
where is the Kronecker’s delta and is a bijection function on . For example, if and with and , the two variables are perfectly correlated with and . If is the identity function that satisfies for any , Eq. (3) reduces to .
The conditional probability of for given is given by . If the two probability variables are statistically independent, the conditional probability reduces to . This implies that we cannot obtain any information about from knowledge of . On the other hand, in the case of the perfect correlation (3), we obtain . This means that we can precisely estimate from by .
We next introduce the joint Shannon information and the conditional Shannon information. The Shannon information for the joint probability is given by
(4) 
On the other hand, the Shannon information for the conditional probability , where is the relevant probability variable, is given by
(5) 
By averaging over , we define the conditional Shannon information
(6) 
The conditional Shannon information satisfies the following properties:
(7) 
By definition, and . Hence
(8) 
which implies that the randomness decreases if only one of the two variables is concerned.
The mutual information is defined by
(9) 
or equivalently,
(10) 
As shown below, the mutual information satisfies
(11) 
Here, is achieved if and are statistically independent, i.e., . On the other hand, is achieved if , or equivalently, if for any . This condition is equivalent to the condition that, for any , there exists a single such that , which implies that we can estimate from with certainty. Similarly, is achieved if . In particular, if the correlation between and is perfect such that Eq. (3) holds, . In general, the mutual information describes the correlation between two probability variables; the more strongly and are correlated, the larger is.
The proof of inequalities (11) goes as follows. Because for , where the equality is achieved if and only if , we obtain
(12) 
which implies . On the other hand, Eq. (9) leads to
(13) 
which implies and .
We note that Eqs. (7), (9), and (13) can be illustrated by using a Venn diagram shown in Fig. 3 [46]. This diagram is useful to memorize the relationship among , , and .
The mutual information can be used to characterize the effective information that can be obtained by measurements. Let us consider a situation in which is a phasespace point of a physical system and is an outcome that is obtained from a measurement on the system. In the case of the Szilard engine, specifies the location of the particle (“left” or “right”), and is the measurement outcome. If the measurement is errorfree as we assumed in Sec. 2, is always satisfied and the correlation between the two variables is perfect. In this case, holds, and therefore the obtained information can be characterized by the Shannon information as is the argument in Sec. 2. In general, there exist measurement errors, and the obtained information by the measurement needs to be characterized by the mutual information. The less the amount of the measurement error is, the more the mutual information is.
We next discuss the cases in which the probability variables take continuous values. The probability distributions such as and should then be interpreted as probability densities, where the corresponding probabilities are given by and with and being the integral elements. The Shannon information of can be formally defined as
(14) 
However, Eq. (14) is not invariant under the transformation of the variable. In fact, if we change to such that , Eq. (14) is given by
(15) 
Thus, that the Shannon information is not uniquely defined for the case of continuous variables. Only when we fix some probability variable, we can give the Shannon information a unique meaning. On the other hand, the mutual information is defined as
(16) 
which is invariant under the transformation of the variables. In this sense, the mutual information is uniquely defined for the cases of continuous variables, regardless of the choices of probability variables.
3.3 Examples
We now discuss two typical examples of probability variables: discrete and continuous variables.
Example 1 (Binary channel)
We consider a binary channel with which at most one bit of information is sent from variable to [see Fig. 4 (a)]. Let be the sender’s bit and be the receiver’s bit. We regard this binary channel as a model of a measurement, in which describes the state of the measured system and describes the measurement outcome. We assume that the error in the communication (or the measurement error) is characterized by
(17) 
where and are the error rates for and , respectively. The crucial assumption here is that the error property is only characterized by a pair , which is independent of the probability distribution of . If , this model is called a binary symmetric channel.
Let and be the probability distribution of . The joint distribution of and is then given by , , , , and the distribution of is given by , . By definition, we can show that the mutual information is given by
(18) 
where is the Shannon information for , and we defined for and . From Eq. (18), holds for , , , or . For a binary symmetric channel, Eq. (18) reduces to . Figure 4 (b) shows versus for the case of . The mutual information takes the maximum value if or . In this case, we can precisely estimate from . We note that, if , we can just relabel “” and “” of such that holds.
Example 2 (Gaussian channel)
We next consider a Gaussian channel with continuous variables and . Let be the sender’s variable or the “signal” and be the receiver’s variable or the “outcome.” We can also interpret that describes the phasespace point of a physical system such as the position of a Brownian particle, and that describes the outcome of the measurement on the system. We assume that the error is characterized by a Gaussian noise
(19) 
where is the variance of the noise. For simplicity, we also assume that the probability density of is also Gaussian:
(20) 
where is the variance of . The joint probability density of and is then given by
(21) 
On the other hand, the probability density of is given by
(22) 
which implies that the variance of the outcome is enhanced by factor compared with the original variance of the signal . We can also calculate the conditional probability density as
(23) 
which implies that the variance of the conditional distribution of is suppressed compared with the original variance by a factor of .
We can straightforwardly calculate the mutual information as
(24) 
which is determined by the signaltonoise ratio alone. In the limit of , where the noise dominates the signal, the mutual information vanishes. On the other hand, in the limit of where the noise is negligible compared with the signal, the mutual information diverges, as the variable is continuous.
4 Second Law of Thermodynamics with Feedback Control
In this section, we discuss a universal upper bound of the work that can be extracted from information heat engines such as the Szilard engine. Starting from a general argument for isothermal processes in Sec. 4.1, we discuss two models with which the universal bound is achieved in Sec. 4.2 and 4.3. Moreover, we will discuss an experimental result that demonstrates an information heat engine in Sec. 4.4. We will also discuss the Carnot efficiency with two heat baths in Sec. 4.5.
4.1 General Bound
In the conventional thermodynamics, we extract a work from a heat engine by changing external parameters such as the volume of a gas or the frequency of an optical tweezer. In addition to such parameter changes, we perform measurements and feedback control for the case of information heat engines.
Suppose that we have a thermodynamic engine that is in contact with a single heat bath at temperature . We perform a measurement on a thermodynamic engine, and obtain of mutual information. After that, we extract a positive amount of work by changing external parameters. The crucial point here is that the protocol of changing the external parameters can depend on the measurement outcome via feedback control.
In the case of the Szilard engine, we obtain nat of information, and extract of work. How much work can we extract in principle under the condition that we have of mutual information about the system? The answer of this fundamental question is given by the following inequality [22, 26]:
(25) 
where is the average of the work that is extracted from the engine, and is the freeenergy difference of the engine between the initial and final states. Inequality (25) has been proved for both quantum and classical regimes [22, 26]. However, the mutual information in (25) needs to be replaced by a quantum extension of the mutual information [22, 48, 49] for quantum cases. We will prove inequality (25) for classical cases by invoking the detailed fluctuation theorem in Sec. 5.
Inequality (25) states that we can extract an excess work up to if we utilize of mutual information obtained by the measurement. In the conventional thermodynamics, the upper bound of the extractable work is bounded only by the freeenergy difference , which is determined by the initial and final values of external parameters. For information heat engines, the mutual information is also needed to determine the upper bound of the extractable work. In this sense, inequality (25) is a generalization of the second law of thermodynamics for feedbackcontrolled processes, in which thermodynamic variables ( and ) and the information content () are treated on an equal footing.
The equality in (25) is achieved with the “best” protocol, which means that the process is quasistatic and the postfeedback state is independent of the measurement outcome, i.e., we utilize all the obtained information. This condition is achieved by the Szilard engine, as discussed above. Some models that achieves the equality in (25) have been proposed [23, 34, 35, 44]. Two of them [35, 44] are discussed in Secs. 4.2 and 4.3. The Szilard engine, which gives , , and , achieves the upper bound of the extractable work, and its special role in information heat engines parallels that of the Carnot cycle in conventional thermodynamics.
4.2 Generalized Szilard Engine
We discuss a generalization of the Szilard engine with measurement errors and imperfect feedback [44], with which the equality in (25) is achieved. The control protocol is described as follows (see Fig. 5):
Step 1: Initial state. A singleparticle gas is in thermal equilibrium, which is in contact with a single heat bath at temperature .
Step 2: Insertion of the barrier. We insert a barrier to the box and divide it into two with the volume ratio being .
Step 3: Measurement. We perform a measurement to find out which box the particle is in. The possible outcomes are “left” and “right,” which we respectively denote as “” and “.” The measurement can then be modeled by the binary channel with error rates and (see Sec. 3), where () specifies the location of the particle and () shows the measurement outcome.
Step 4: Feedback. We quasistatically move the barrier depending on the measurement outcome. If the outcome is “left” (), we move the barrier, so that the final ratio of the volumes of the two boxes is given by (). If the outcome is “right” (), we move the the barrier, so that the final ratio of the volumes of the two boxes is given by (). We note that the feedback protocol is specified by .
Step 5: Removal of the barrier. We remove the barrier, and the system returns to the initial thermal equilibrium by a free expansion.
We now calculate the amount of the work that is extracted in step 4. By using the equation of states , we find that the extracted work is for , for , for , and for . Therefore the average work is given by
(26) 
The mutual information obtained by the measurement is given by Eq. (18). The upper bound of inequality (25) is not necessarily achieved with a general feedback protocol . We then maximize in terms of and . The optimal feedback protocol with the maximum work is determined by equations and , which lead to and . Therefore, we obtain the maximum work as
(27) 
which achieves the upper bound of the generalized second law (25).
4.3 Overdamped Langevin System
We next discuss a feedback protocol for an overdamped Langevin system, which also achieves the upper bound of inequality (25) as shown in Ref. [35]. We consider a Brownian particle with a harmonic potential, which obeys the following overdamped Langevin equation:
(28) 
where is a friction constant and is a Gaussian white noise satisfying with being the delta function. The harmonic potential can be controlled through two external parameters such that
(29) 
where and respectively describe the spring constant and the center of the potential. We consider the following feedback protocol (see Fig. 6).
Step 1: Initial state. The particle is initially in thermal equilibrium with initial external parameters and .
Step 2: Measurement. We measure the position of the particle and obtain outcome . The measurement error is assumed to be Gaussian that is given by Eq. (19), where is the variance of in the initial equilibrium state (i.e., ), and is the variance of the noise in the measurement [see Eq. (19)]. Immediately after the measurement, the conditional probability is given by Eq. (23). The obtained mutual information is given by Eq. (24).
Step 3: Feedback. Immediately after the measurement, we instantaneously change from to and from to . By this change, the conditional distribution (23) becomes thermally equilibrated with new parameters .
Step 4: Work extraction. We quasistatically expand the potential by changing from to thereby extracting the work. The system then get thermally equilibrated with parameters .
We now calculate the work that can be extracted from this engine. Let be the probability distribution of at time under the condition of . The average of the work for step 3 is given by
(30) 
where is given by in Eq. (23), and is given by Eq. (22). The work for Step 4 is given by
(31) 
where we used the fact that the expansion is quasistatic. Comparing Eqs. (30) and (31) with mutual information (24), we obtain
(32) 
which achieves the upper bound of inequality (25).
4.4 Experimental Demonstration: FeedbackControlled Ratchet
We next discuss a recent experiment that realized an information heat engine by using a realtime feedback control on a colloidal particle in water at room temperature [30].
In the experiment, a colloidal particle with diameter 300 nm is attached to the cover glass, and another particle is attached to the first one [Fig. 7 (a)]. The second particle then moves around the first as a rotating Brownian particle which we observe and control. An AC electric field is applied with four electrodes, and the particle undergoes an effective potential as illustrated in Fig. 7 (b). We note that the potential can take two configurations depending on the phases of the electric field. Each configuration consists of a spatiallyperiodic potential and a constant slope. The slope is created by a constant torque around the circle along which the particle rotates. This potential is like spiral stairs. The depth of the periodic potential is about , and the gradient of the slope per angle is about .
If the periodic potential without the slope was asymmetric and the two potential configurations were periodically switched, the particle would be transported in one direction as a flashing ratchet [50, 51, 52, 53, 54]. In the present setup, however, the periodic potential without the slope is symmetric and is not switched periodically but switched in a manner that depends on the measured position of the particle via feedback control [17, 20, 25, 52]. Such a feedbackcontrolled ratchet has been experimentally realized [20]. In the present experiment, the work and the free energy were measured precisely for quantitatively comparing the experimental results with the theoretical bound (25). It has been pointed out [52, 55] that the feedbackcontrolled ratchet, as well as the flashing ratchet, can be a model of biological molecular motors [56].
The feedback protocol in the experiment [30] was done as follows [see Fig. 7 (c)]. The position of the particle was probed every 40 ms by a microscope, a camera, and an image analyzer. Only if the particle was found in the switching region described by “S” in Fig. 7 (c), the potential configuration was switched after a short delay time . By this switching, when the particle reached the hilltop, the potential is inverted, so that the peak of the potential changed into the bottom of the valley, and therefore the particle is transported to the right direction. Without the switching, the particle would be more likely to go back to the left valley. This positiondependent switching via feedback control induces the reduction of the entropy in a manner analogous to the feedback control in the Szilard engine. By performing this protocol many times, the particle is expected to be transported to the right direction by climbing up the potential slope.
Figure 8 (a) shows typical trajectories of the particle. If the feedback delay is sufficiently shorter than the relaxation time of the particle in each well ( ms), the particle climbs up the potential. If the feedback delay is longer than the relaxation time, the feedback does not work and the particle moves down the potential in agreement with the conventional second law of thermodynamics. Figure 8 (c) shows the averaged velocity of the particle versus , which implies that the shorter the feedback delay is, the faster the average velocity is.
Figure 8 (c) shows the energy balance of this engine. The shaded region is prohibited by the conventional second law of thermodynamics , where is the freeenergy difference corresponding to the height of the potential, is the work performed on the particle during the switching, and represents the ensemble average over all trajectories. By using information via feedback, however, the shaded region is indeed achieved if is sufficiently small. The resource of the excess freeenergy gain is thermal fluctuations of the heat bath, which are rectified by feedback control. This is an experimental realization of an information heat engine.
For the case of ms, . On the other hand, the obtained information is given by , which can be calculated from the histogram of the measurement outcomes by assuming that the measurement is errorfree. By comparing this experimental data with the theoretical bound (25), the efficiency of this information heat engine is determined to be . The reason why the efficiency is less than unity is twofold: (1) the switching is not quasistatic but instantaneous, and (2) the obtained information is not utilized if the particle is found outside of the switching region.
We note that various experiments that are analogous to Maxwell’s demon have been performed with, for example, a granular gas [57, 58], supramolecules [59], and ultracold atoms [60]; however, these examples do not explicitly involve the measurement and feedback, as the controlled system and the demon constitute an autonomous system in those experiments. Such autonomous versions of Maxwell’s demon have also been theoretically studied [61, 62, 63, 64, 65, 66]. In contrast, in the experiment of Ref. [30], the demon (the camera and the computer) is separated from the controlled system (the colloidal particle) as in the case for the Szilard engine. We also note that an information heat engine similar to that in Ref. [30] has been proposed for an electron pump system in Ref. [39].
4.5 The Carnot Efficiency with Two Heat Baths
We next consider the case in which there are two heat baths and the process is a thermodynamic cycle. Without feedback control, the heat efficiency is bounded by the Carnot bound. If we perform measurements and feedback control on this system, the extractable work is bounded from above by [22]
(33) 
where and are the temperatures of the hot and cold heat baths, respectively, and is the heat that is absorbed by the engine from the hot heat bath. The proof of inequality (33) will be given in Sec. 5. The last term on the righthand side (rhs) of (33) describes the effect of feedback. We note that the coefficient of the last term is given by the temperature of the cold bath.
The equality in (33) is achieved in the following example. We consider a singleparticle gas in a box, and quasistatically control it as in the case of the usual Carnot cycle. We then perform the Szilardenginetype operation consisting of measurement and feedback on the system while it is in contact with the cold bath. In this case, we have of excess work, and remains unchanged. Therefore, the equality in (33) is achieved with .
5 Nonequilibrium Equalities with Feedback Control
Since the late 1990’s, a number of universal equalities have been found for nonequilibrium processes [67, 68, 69, 70, 71, 72, 73, 74, 75], and they have been shown to reproduce the second law of thermodynamics and the fluctuationdissipation theorem. The fluctuation theorem and the Jarzynski equality are two prime examples of the nonequilibrium equalities. In this section, we generalize the nonequilibrium equalities to situations in which a thermodynamic system is subject to measurements and feedback control in line with Refs. [26, 29, 44]. As corollaries, we derive inequalities (25) and (33).
5.1 Preliminaries
First of all, we review the nonequilibrium equalities without feedback control. We consider a stochastic thermodynamic system in contact with heat bath(s) with inverse temperatures (). Let be the phasespace point of the system. The system is controlled through external parameters , which describe, for instance, the volume of a gas or the frequency of an optical tweezer. Even when the initial state of the system is in thermal equilibrium, the system can be driven far from equilibrium by changing the external parameters. We consider such a stochastic dynamics of the system from time to . The state of the system stochastically evolves as under a deterministic protocol of the external parameters denoted collectively as .
Let be the trajectory of the phasespace point and be that of the external parameters. The heat that is absorbed by the system from heat bath “” is a trajectorydependent quantity, which we write as . The work that is performed on the system is also trajectorydependent and is denoted as . The first law of thermodynamics is then given by
(34) 
where is the Hamiltonian with external parameters , and the work can be written as
(35) 
Let be the probability density of trajectory with control protocol . It can be decomposed as , where is the probability density under the condition that the initial state is , and is the initial distribution of the forward process.
We next introduce backward processes of the system. Let be the timereversal of . For example, if with being the position and being the momentum, then . Similarly, we denote the timereversal of as . For example, if is the magnetic field, then . Let be the timereversed trajectory of defined as . We also write the timereversal of control protocol as , and write as the probability density of the timereversed trajectory with the timereversed control protocol. We can decompose as , where is the probability density under the condition that the initial state of the backward process is , and the initial distribution of the backward process. We stress that the initial distribution of the backward process can be set independently of the final distribution of the forward process. In experiments, we can initialize the system before we start a backward process so that its initial distribution can be chosen independently of the forward process.
The detailed fluctuation theorem (the transient fluctuation theorem) is given by [68, 69, 70, 71]
(36) 
Defining the entropy production as
(37) 
we obtain
(38) 
By taking the ensemble average of Eq. (38), we have
(39) 
where we used . Therefore, we obtain the integral fluctuation theorem
(40) 
By using the concavity of the exponential function, we find from Eq. (40) that
(41) 
which is an expression of the second law of thermodynamics. In the following, we discuss the physical meanings of entropy production for typical situations. The equality in (41) is achieved if holds for any , which implies the reversibility of the process.
We first consider isothermal processes. In this case, we choose the initial distributions of the forward and backward processes as
(42) 
where is the free energy of the system. We assume that the Hamiltonian has the timereversal symmetry
(43) 
and therefore the canonical distribution satisfies
(44) 
The entropy production then reduces to
(45) 
where . Thus, the integral fluctuation theorem (40) leads to the Jarzynski equality [67]
(46) 
and inequality (41) gives the second law of thermodynamics
(47) 
where is the work that is extracted from the system.
We next consider a case with multiheat baths, and assume that the initial distributions of the forward and the backward processes are given by the canonical distributions as in (42) with a reference inverse temperature . In practice, can be taken as one of ’s, which can be realized if the system is initially attached only to that particular heat bath. We then have
(48) 
where is the difference of the internal energy of the system. Inequality (41) leads to
(49) 
In particular, if the process is a cycle such that and hold, inequality (49) reduces to
(50) 
which is the Clausius inequality. If there are two heat baths with temperatures and , (50) gives the Carnot bound
(51) 
where is the average of the heat that is absorbed by the engine from the hot heat bath.
5.2 Measurement and Feedback
We now formulate measurements and feedback on the thermodynamic system [26, 29, 44]. We perform measurements at time () with . Let be the measurement outcome at time . For simplicity, we assume that the measurement is instantaneous; the measurement error of can be characterized only by the conditional probability , which implies that only has the information about . (Note, however, that this assumption can be relaxed [44].) We write the sequence of the measurement outcomes as , and write
(52) 
We then introduce the following quantity:
(53) 
which can be interpreted as a stochastic version of the mutual information. The ensemble average of Eq. (53) gives the mutual information obtained by the measurements as
(54) 
We note that describes the correlation between and that is induced only by measurements, and not by feedback control. The suffix “c” represents this property of . We then identify with in the foregoing arguments. See Ref. [44] for details. We note that has been discussed and referred to as the transfer entropy in Ref. [76]. The following equality also holds by definition:
(55) 
We next consider feedback control by using the obtained outcomes. The control protocol after time can depend on the outcome in the presence of feedback control. We write this dependence as . We can show that the joint probability of is given by
(56) 
which satisfies the normalization condition . Equality (56) is proved in Appendix A (see also Ref. [44]). The probability of obtaining outcome in the forward process is given by the marginal distribution as , and the conditional probability is given by . In the presence of measurement and feedback, the ensemble average is taken over all trajectories and all outcomes; for an arbitrary stochastic quantity , its ensemble average is given by
(57) 
The detailed fluctuation theorem for a given can be written as
(58) 
We note that Eq. (58) is valid in the presence of feedback control, because the detailed fluctuation theorem is satisfied once a control protocol is fixed. Equality (58) provides the basis for the derivations of the formulas in the following section.
5.3 Nonequilibrium Equalities with Mutual Information
In this subsection, we generalize the nonequilibrium equalities by incorporating the mutual information. First of all, from Eq. (53), we have
(59) 
By multiplying the bothhand sides of this equality by those of Eq. (58), we obtain
(60) 
To measure , we follow the backward process corresponding to each forward outcome and count the occurrences of the timereversed trajectories. By taking the ensemble average of the bothhand sides of Eq. (60) with formula (57), we obtain a generalized integral fluctuation theorem with feedback control:
(61) 
Using the concavity of the exponential function, Eq. (61) leads to
(62) 
Inequality (62) is a generalized second law of thermodynamics, which states that the entropy production can be decreased by feedback control, and that the lower bound of the entropy production is given by the mutual information . As shown below, inequalities (25) and (33) in Sec. 4 are special cases of inequality (62). The equality in (62) is achieved if holds for any and , which implies the reversibility with feedback control as discussed in Ref. [34].
The generalized integral fluctuation theorem of the form (61) was first shown in Ref. [26] for a single measurement, and Eq. (60) was obtained in Ref. [29, 44] for multiple measurements. These results has also been generalized to the optimal control process with continuous measurement and the Kalman filter in Ref. [28].
A generalized fluctuation theorem was also obtained in Ref. [19], which is similar to Eq. (61). In Ref. [19], feedback control is performed based on information about the continuouslymonitored velocity of a Langevin system. The result of Ref. [19] includes an quantity that describes the decrease in the entropy by continuous feedback control, instead of the mutual information obtained by the continuous measurement.
We consider isothermal processes with a single heat bath, in which the entropy production is given by Eq. (45). Equality (61) then reduces to a generalized Jarzynski equality
(63) 
and inequality (62) reduces to
(64) 
which implies inequality (25) with identifications , , and .
We next consider the cases in which there are two heat baths and the process is a cycle, in which the entropy production is given by the ensemble average of Eq. (48) with