DOI: 10.22661/AAPPSBL.2019.29.1.10
Stochastic Thermodynamics Study of Microscopic Heat Engine
JAE DONG NOH^{1} ^{1} DEPARTMENT OF PHYSICS, UNIVERSITY OF SEOUL, SEOUL 02504, KOREA (DATED: DECEMBER 13, 2018)
Stochastic thermodynamics is a powerful theoretical framework for studying the thermodynamic properties of microscopic systems driven far from equilibrium in a thermal environment. Microscopic heat engines are an interesting application of stochastic thermodynamics. Recently, we proposed theoretical solvable model systems for a microscopic heat engine consisting of a few Brownian particles. In this article, we review our theoretical results for the efficiency of these heatengine models.
INTRODUCTION
Macroscopic systems in thermal equilibrium have been the subject of statistical mechanics for more than a century. Equilibrium statistical mechanics has been successful in uncovering physical properties of such macroscopic systems. Probably, it is one of the most important achievements of theoretical physics to understand the universality of phase transitions and critical phenomena. Recently, nonequilibrium systems have been attracting growing interest. Given that most physical systems are out of equilibrium, it becomes necessary to have a theoretical framework for nonequilibrium systems. Advances in the technology of fabrication and control of microscopic systems also accelerate the development of nonequilibrium statistical mechanics.
Thermal fluctuations become stronger as a system size becomes smaller. Thus, when a microscopic system is surrounded by a thermal heat bath and driven out of equilibrium, one needs to characterize the system with not only the average values but also with the fluctuations of physical quantities. Stochastic thermodynamics emerges as a powerful and useful statistical mechanics theory for nonequilibrium microscopic systems in thermal contact with a heat bath [1, 2].
Stochastic thermodynamics applies to systems governed by stochastic Markov dynamics including the Langevin equation and the master equation systems. Consider the ensemble of trajectories, generated by stochastic dynamics, in the phase space. Stochastic thermodynamics can be thought as statistical mechanics theory over the trajectory ensemble, which is powerful especially for microscopic systems.
The fluctuation theorem is one of the most important discoveries of stochastic thermodynamics. Consider a thermodynamic system whose states are represented by the phase space coordinate q. It is in thermal contact with the heat bath of temperature T. We are interested in the dynamics during the time interval 0 ≤ t ≤ τ. The initial state q(0) is drawn from a prescribed probability distribution P_{i}(q). It will evolve into the final distribution P_{f} (q) at time t = τ. Let q denote a phase space trajectory during the time interval. Stochastic thermodynamics postulates that the total entropy change caused by the trajectory is given by
 (1)

where the first term is the change in the Shannon entropy of the system and the second term is the entropy change of the heat bath with Q[q] denoting the heat dissipated into the heat bath. Then, the total entropy is shown to satisfy the fluctuation theorem [3]
 (2)

where <·> denotes the average over all trajectories.
The fluctuation theorem (2), being combined with the Jensen's inequality <e^{x}> ≥ e^{<x>}, yields
 (3)

which amounts to the second law of thermodynamics. The fluctuation theorem (2) involves the irreversibility measured in terms of the ratio of probabilities of a trajectory and its timereversed path. Thus, it is not surprising to have the second law from the fluctuation theorem.
The fluctuation theorem (2) has many different representations depending on the initial state. For example, when the system is described by an equilibrium Boltzmann distribution P_{i} (q) = e^{(Hi  Fi)(q)/T} corresponding to a given Hamiltonian H_{i} and free energy F_{i} = T ln ∫ dqe^{Hi(q)/T} and undergoes a thermodynamic process to a final state with Hamiltonian H_{f} and the free energy F_{f}, the external work W[q] done on the system satisfies
 (4)

This is known as the Jarzinsky equality [4]. We also remark that, unlike the work, the heat absorbed by the system does not satisfy the fluctuation theorem [5]. Strong thermal fluctuations of the internal energy prevent the heat from enjoying the fluctuation theorem [6]. The fluctuation theorems have been confirmed in various experimental systems (see e.g., Ref. [7]).
In this article, we review the application of stochastic thermodynamics to microscopic heat engine systems. The experimental technique using optical tool makes it possible to construct a microscopic heat engine consisting of a Brownian particle [8]. It fosters theoretical studies of microscopic heat engines with a statistical mechanical model system. In Sec. II, we review the interesting property of the efficiency of a linear Brownian heatengine model, which was analyzed in Ref. [9]. Section III introduces the information engine model, which was studied theoretically in Ref. [10]. We close this article with the summary in Sec. IV.
LINEAR BROWNIAN HEATENGINE MODEL
Suppose that two Brownian particles of mass m are in thermal contact with heat baths of temperature T_{1} and T_{2} (< T_{1}), respectively, in a onedimensional space. In vector notation x = (x_{1}, x_{2}) and v = (v_{1}, v_{2}) for the position and the velocity of the two particles, the underdamped Langevin equation is given by
 (5)

where γ is the damping coefficient, V(x) = 1/2 Kx^{2} is the harmonic potential confining the particles, f_{nc}(x) = (f_{nc,1}(x), f_{nc,2}(x)) is a nonconservative driving force, and ξ(t) = (ξ_{1}(t), ξ_{2}(t)) is the Gaussian thermal noise satisfying <ξ_{i}(t)> = 0 and <ξ_{i}(t)ξ_{j}(t')> = 2γT_{i} δ_{i,j} δ(tt'). This system is driven out of equilibrium by two ingredients: the temperature difference T_{1} > T_{2} and the nonconservative driving force f_{nc}. The temperature difference generates a heat flow through the system from one bath to the other. In the meanwhile, some of the heat may be converted into the work as the particles work against the driving force f_{nc}. Thus, the system described by equations (5) is a Brownian heat engine. Unlike the conventional Carnot engine, it is an autonomous engine that does not need any timedependent protocol.
In Ref. [9], we investigated the efficiency of the engine with the choice of a linear driving force
 (6)

with control parameters ϵ and δ. With this choice, the Langevin equation is linear and exactly solvable. We can find an analytic solution for an average heat flux q_{1} from the hotter heat bath, q_{2} from the colder bath, and power w generated by the Brownian particles in the steady state. The power denotes the work done against the driving force per unit time, that is, w =<(f_{nc} ov)>_{st} where <·>_{st} denotes the average in the steady state.
Depending on the parameter values, the system has different functionality.
Fig. 1: Function diagram in the (ϵ, δ) plane. The arrows indicate the direction of the energy flow. The figure is taken from Ref. [9].
Fig. 1 is map in the (ϵ, δ) plane showing how the system behaves in each region [9]. There are four distinct regions: (i) The system absorbs heat from the hotter bath (q_{1} > 0) and generates work (w > 0). That is, the system acts as a heat engine. The efficiency is given by
 (7)

(ii) The system consumes external work (w < 0) to direct the heat from the colder bath to the hotter one (q_{1} < 0). This corresponds to a refrigerator or a heat pump. (iii) The system acts as a passive heat conductor consuming external work. (iv) The system dissipates external work as a heat into both heat baths. This study using the solvable model shows that the microscopic nonequilibrium system can have rich behavior.
Small systems suffer from strong thermal fluctuations. Thus, the heat Q_{1}(τ) absorbed from the hot heat bath, the work W(τ) for the time interval τ, and the efficiency η(τ) = W(τ)/Q_{1}(τ) are strongly fluctuating random variables for a Brownian heat engine. It is an interesting task to find the probability distribution of the efficiency [11, 12]. The probability distribution in the large τ limit is well represented by the large deviation function
 (8)

We find an analytic expression for the large deviation function for the linear Brownian heatengine model. The result shows that the probability is maximum at the value given in (7). It also shows that the efficiency has a broad distribution. Owing to statistical fluctuation, the efficiency can be larger than the Carnot efficient, although such an event has vanishing probability in the large τ limit. Detailed descriptions are found in Ref. [9].
INFORMATION ENGINE
In 1929, Szilárd proposed a thought experiment for a heat engine where one can extract work from a single gas molecule in contact with a single heat bath by employing the Maxwell's demon [13]. Since it was proposed, a lot of research has been devoted to understanding the thought experiment in the context of the second law of thermodynamics [13]. Without the demon, one cannot extract work from a single heat bath according to the second law of thermodynamics. The role of the demon is to measure the status of the engine and apply feedback control to harvest work. Recently, measurement and feedback control by the demon has been incorporated into thermodynamics successfully under the name of information thermodynamics. In information thermodynamics, the second law of thermodynamics can be extended by including the mutual information between a physical system (heat engine) and a measurement device (demon) [14, 15].
The heat engine driven by the measurement and feedback control is called an information engine. There are several theoretical models for an information engine [16]. However, most models focus on information engines whose engine cycle period is infinite. Here, we introduce our recent work on an information engine whose cycle period is finite [10, 17].
Consider a Brownian particle in a onedimensional thermal environment of temperature T. The particle is confined by a harmonic potential V (X, λ) = 1/2 K(X  λ)^{2} where λ denotes the potential center. For simplicity, we will set T = K = 1. We can extract work from the system by applying the following cyclic protocol.
1. (Measurement) At time t_{n} = nτ, one measures the position of the Brownian particle. The measurement outcome is denoted as X_{n} = X(t = nτ).
2. (Feedback) If (X_{n}  λ) > x_{m}, the potential center is shifted to the new position λ + x_{f}. The work is extracted by the amount of the change in the potential energy of the Brownian particle, W = V (X_{n}, λ)  V (X_{n}, λ + x_{f} ) = x_{f} (X  λ  x_{f} /2). On the other hand, if (X_{n}  λ) ≤ x_{m}, there is no feedback.
3. (Relaxation) The Brownian particle undergoes a relaxation process for the time interval τ until the next engine cycle starts.
By the measurement, one acquires information on the status of the Brownian particle. The information is exploited during the feedback process to extract the work. The Brownian particle then absorbs the heat from the heat bath during the relaxation process After many cycles, the system will reach a steady state where the potential center moves at the constant average velocity and the average work extracted per cycle becomes constant. The steady state property is fully described by the probability distribution function p_{ss} (x;τ) for the relative potion x of the Brownian particle with respect to the potential center. The average work W_{ss} per cycle in the steady state is given by
 (9)

It is challenging to find the steady state probability distribution function p_{ss} (x;τ). It should satisfy the selfconsistent equation
 (10)

where G_{τ} (xz) is the transition probability for the Brownian particle to jump from z to x in time τ. The first (second) term in the righthand side accounts for the events without (with) feedback.
The closed form solution is available in the limiting cases where τ → 0 or ∞. For finite τ, we find that the solution is given by the infinite series
 (11)

where H_{n} denotes the Hermite polynomial of degree n. The coefficients {c_{n}} are given by the solution of the recursive relations, which can be solved easily numerically. Using the solution, we can evaluate the average work W_{ss} or the average power w_{ss} = W_{ss}/τ as a function of x_{m}, x_{f}, and τ [10].
Fig. 2: Density plot for the average work W_{ss} per cycle when τ = ∞ in (a) and τ = log 2 in (b), and the density plot for the average power w_{ss} = W_{ss}/τ when τ → 0 in (c). The x symbols mark the optimal condition where the extracted work is maximum. The figures are taken from Ref. [10].
Fig. 2 presents the density plots for the average work or power in the (x_{m}, x_{f}) plane. One can find that an information engine can be optimized to yield the maximum amount of work.
The theoretical model for the information engine is realized by a recent experiment using a colloidal particle trapped by optical tweezers [17]. The experimental data presented in Ref. [18] match perfectly well with the theoretical results of Ref. [10]. The agreement suggests the feasibility of a microscopic information heat engine.
SUMMARY
In this article, we reviewed the recent progress in our study of microscopic heat engines. Theoretical work suggests a possible mechanism and design for heat engines consisting of Brownian particles. Such engines may operate autonomously or be driven by measurement and feedback. We hope our results trigger further research for realistic microscopic heat engines.
Acknowledgments: The author acknowledges collaborations with Jong Min Park, HyunMyung Chun, Jae Sung Lee, Govind Paneru, Dong Yun Lee, Jin Tae Park, and Hyuk Kyu Pak. This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B2013972).
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Jae Dong Noh is a professor at the Department of Physics, University of Seoul, Korea. After receiving a PhD from the Seoul National University in 1997, he worked at University of Washington, Inha University, University of Saarland, and Chungnam National University before joining University of Seoul in 2006. He serves currently as an editor for the European Physical Journal B and the Journal of Statistical Physics. His research field is statistical physics. 
