Efficiency at maximum power output for an engine with a passive piston

Efficiency at maximum power (MP) output for an engine with a passive piston without mechanical controls between two reservoirs is theoretically studied. We enclose a hard core gas partitioned by a massive piston in a temperature-controlled container and analyze the efficiency at MP under a heating and cooling protocol without controlling the pressure acting on the piston from outside. We find the following three results: (i) The efficiency at MP for a dilute gas is close to the Chambadal-Novikov-Curzon-Ahlborn (CNCA) efficiency if we can ignore the side wall friction and the loss of energy between a gas particle and the piston, while (ii) the efficiency for a moderately dense gas becomes smaller than the CNCA efficiency even when the temperature difference of reservoirs is small. (iii) Introducing the Onsager matrix for an engine with a passive piston, we verify that the tight coupling condition for the matrix of the dilute gas is satisfied, while that of the moderately dense gas is not satisfied because of the inevitable heat leak. We confirm the validity of these results using the molecular dynamics simulation and introducing an effective mean-field-like model which we call stochastic mean field model.


I. INTRODUCTION
Equilibrium thermodynamics reveals the relation between work and heat, and the upper bound for the extracted work from heat cycles [1,2]. The milestone of equilibrium thermodynamics is Carnot's theorem, stating that thermodynamic efficiencies for any heat cycles working between two reservoirs characterized by the temperatures T H and T L (T H > T L ) are bounded from above by Carnot's efficiency: η C ≡ 1 − T L /T H , achieved by quasi-static operation [3]. For the practical point of view, Carnot's efficiency itself is useless, because its operation is infinitely slow, i.e. zero power. The extension of thermodynamics toward the finite-time operation, socalled finite-time thermodynamics, has been investigated by many authors [4][5][6][7][8][9][10][11][12]. Chambadal and Novikov independently proposed, and later Curzon and Ahlborn rediscovered that the efficiency for maximum power output is given by the Chambadal-Novikov-Curzon-Ahlborn (CNCA) efficiency: η CA ≡ 1 − T L /T H [4][5][6]. Recently, the validity of the CNCA efficiency has been justified though the linear irreversible thermodynamics in near equilibrium situations [7], molecular kinetics [8] or lowdissipation assumption [9]. Nevertheless, it is believed that the CNCA efficiency is only the efficiency at maximum power near equilibrium situations. Indeed, there are many situations to exceed the CNCA efficiency in power plants or idealized setups [8][9][10].
Recent experimental technique has revealed the relation between work and heat even for small systems, such as single colloidal particle systems, or biological ones. Such small systems are believed to be characterized by the Langevin equation associated with stochastic energetics [15][16][17]. Brownian heat engine, which is a transducer of thermal energy into mechanical work in the Langevin systems [13,14], has been extensively investigated in the last two decades [10,[18][19][20][21][22][23][24][25][26][27][28][29][30]. Carnot's efficiency can be reproduced for a single quasi-static realization for Brownian heat engine, with the aid of a com-plicated protocol, to avoid the irreversible heat [26]. The efficiency at maximum power output for single particle systems agrees with the CNCA efficiency by manipulating the depth of optical tweezers experimentally [30]. A schematic picture of our setup. Finite number N of identical particles (red) are enclosed in an adiabatic container whose cover is an adiabatic piston of the finite mass M at x =X. The density nout and the temperature Tout for the outside gas (blue) are kept to be constants. The temperature T bath of the thermal wall at x = 0 is controlled by an external agent, while thermodynamic quantities, such as the densitynin and the temperatureTin fluctuate in time.
The aim of this paper is to clarify the thermodynamic structure of a system containing small number of particles confined by a fluctuating piston in a chamber, where the fluctuation of the number density through the fluctuating piston is not negligible. The system treated in this paper can be regarded as an idealized model of small systems bounded or surrounded by a fluctuating membrane or shell, which are important in soft matter physics, e.g. adherent cells or red blood cells [34,35]. Furthermore, our study is relevant for industrial applications to control, e.g. elasticity and osmotic collapse of microcapsules [36][37][38]. Our model would be experimentally realized through two kinds of setups: colloidal suspensions with a semi-permeable membrane or a highly excited di-lute granular gas with a movable piston. Although the hydrodynamic interaction of colloids is important, the osmotic pressure between two dilute solutions separated by a semi-permeable membrane is described by van't Hoff's formula which has an identical form to the state equation for ideal gases. Similarly, inhomogeneity and non-Gaussianity of granular gases can be suppressed, at least, for a specific setup of a highly agitated granular gas [39]. Thus, our model is a simplified and idealized one for such systems. Several authors have studied the similar idealized systems [8,31,33]. It should be stressed that an appropriate stochastic model beyond the simple Langevin equation would be necessary, because the overdamped Langevin equation at a constant temperature cannot reproduce the velocity autocorrelation function for the fluctuating piston [33]. We note that the conventional finite time thermodynamics is not applicable to our system, because the fluctuation of thermodynamic variable is not included in its formulation. Although a molecular kinetic analysis of the heat engine has been studied, it cannot be applied to our aim [8]. Indeed, the fluctuation of the piston has been ignored, because the piston position is perfectly controlled by an external agent. The irreversibility for small systems has been proved to be essential under thermal equilibrium [31,32], while the thermodynamic structure for those under non-equilibrium situations has not been clarified so far.
In this paper, to understand the thermodynamic structure for systems consisting of particles and a fluctuating piston in a chamber, we propose a mean field-like stochastic model of confined particles by a fluctuating piston attached to a thermal wall. We assume that this system can be described by two independent variables: fluctuating densityn in and fluctuating temperatureT in (Fig. 1), where the fluctuation of the molecular density appears through the fluctuating position of the piston. Recently, Hoppenau et al. have investigated the Carnot process of a single particle enclosed by an infinitely-heavy piston [10]. Our model can be regarded as an extended one for their engine, including the fluctuation of the number density and the finite ratio of the piston mass to the molecular mass. Through this paper, variables with "ˆ" denote stochastic variables.
The organization of this paper is as follows. First, in Sec. II, we introduce our setup and its stochastic mean field (SMF) model described by two variablesn in and T in under several assumptions. In Sec. III, implementing our operation protocol for the temperature of the thermal wall T bath , the time evolution of our introduced model is discussed. The results of the SMF are compared with the corresponding even-driven molecular dynamics (MD) simulation. In Sec. IV, we discuss the efficiency at maximum power output, where the semi-analytical expression for the efficiency is derived. The obtained efficiency is shown to be close to the CNCA efficiency, if the piston mass is sufficiently large. In Sec. V, we discuss the relation of our results with previous studies. Lastly, we conclude the paper in Sec. VI. In Appendix A, the effect of the sidewall-friction on the piston is discussed, and in Appendix B, the definition of the work and heat for our system is discussed.

II. SETUP
Ideal gas particles of each mass m are enclosed in a three-dimensional container, where N ideal gas particles are partitioned by an adiabatic piston of mass M and the area A, on the right side of x direction, a diathermal wall on the left side of x direction and four adiabatic walls on the other directions (Fig. 1). We assume that adhesion between particles and the walls of the container can be ignored. The piston is assumed to move in one-dimension without sidewall-friction, and the motion of particles in vertical direction to x is not coupled with the motion of the piston. Post (v ′ , V ′ ) and pre-collision velocities (v, V ) in x direction for gas particles and the piston are related as follows; because the contribution from the horizontal motion of particles to the wall is canceled as a result of statistical average. Here, represents the momentum change of piston due to the collision of a gas particle of velocity v, where e is the restitution coefficient between gas particles and the piston. The reason why we introduce the restitution coefficient is that the wall consists of macroscopic number of particles and the impulses of each collision can be absorbed into the wall as the excitation of internal oscillation.
Let us introduce our stochastic mean field (SMF) model for the time evolution of the densityn in and the temperatureT in under the following four assumptions (i)-(iv). (i):The enclosed gas is homogeneous, and the density is defined via the position of the pistonX, i.e. n in ≡ N/AX. (ii):The correlations for the gas particles are neglected. (iii):The velocity distribution function (VDF) for the enclosed gas particles is Maxwellian of the temperatureT in at any time: (iv):The equation of state for the enclosed gas follows the equation for ideal gasP in ≡n in k BTin , with the pressure of the gasP in . Thus, the time evolution forn in is governed by the equation of motion for the piston: whereξ v α (α = in, out) denotes the Poissonian noise of the amplitude one whose event probability is given by the collision probability of a gas particle of velocity between v and v + dv against the piston with ε in = −1 and ε out = +1. The symbol " · " in Eq. (6) represents Itô type stochastic product [40,41]. Θ(x) is Heaviside function satisfying Θ(x) = 1 for x ≥ 0 and Θ(x) = 0 for x < 0. The density and temperature for the gas outside are kept to be constants in time, i.e., n out ≡ n out andT out ≡ T out . We note that Eqs. (5) and (6) are an extension of our previous study toward the case that the density and the temperature change in time [42]. Next, we propose the time evolution forT in , considering the energy conservation for the gas. The differential of the internal energy for the gasÛ in ≡ 3N k BTin /2 is given by where dQ wall denotes the heat flow from the thermal wall of the temperature T bath [8], as will be explained in Sec. III B, and dÊ pis denotes the kinetic energy transfer from the piston to the gas. In summary, our model consists of two coupled equations: the equation of motion for the piston (5) and the energy conservation for the enclosed gas (8).

III. OPERATION PROTOCOL AND TIME EVOLUTION
In the following, we consider a heat cycle repeating the heating T bath = T H > T out and cooling T bath = T L = T out protocol (Fig. 2). Initially, the enclosed gas and the gas outside are in a mechanical equilibrium state: P in =n in k BTin = P out andT in = T out = T bath . At t = 0, we attach a heat bath T H on the diathermal wall. For 0 < t < t c , T bath is kept to be T bath = T H (Fig. 2  (a)), and at t = t c , T bath is switched to be T bath = T L FIG. 2: (Color online) The operation protocol. At t = 0, we attach the heat bath TH on the diathermal wall. For 0 < t < tc, T bath is kept to be T bath = TH (a), and at t = tc, T bath is switched to be T bath = TL simultaneously, and is kept to be this state until t = 2tc. Then, we again replace the bath by TH simultaneously (b). After repeating the switching of T bath , the heat cycle reaches the steady cycle.
simultaneously, and is kept to be this state until t = 2t c . Then, we again replace the bath by T H simultaneously ( Fig. 2 (b)). After repeating the switching and attaching of the bath temperature T bath , the heat cycle reaches a steady cycle. It should be noted that the enclosed gas is no longer thermal equilibrium during the cycle, due to dÊ pis . During the operation, we ignore the time for the switching the heat bath. The finite switching time only lowers the amount of the power, but does not affect the efficiency of the cycle and what the maximum-poweroutput process is. Therefore, for simplicity, we ignore the switching time.
A. Details of the simulation for mean field model We numerically solve Eqs. (5) and (8). For our mean field simulation, the unit of length, time and mass are X 0 = X ini ≡ N k B T out /P out A, t 0 ≡ X ini M/k B T out and M , respectively. The number of particle N = 200 is fixed through our simulation. Such a system of small number of particles is strongly influenced by the fluctuating motion of the piston. The numerical integration is performed through Adams-Bashforth method, with dt/t 0 ≡ 0.01ǫ and ǫ ≡ m/M . Calculatingξ v α , the noise is discretized, in terms of v. v and dv are respectively replaced by v i and ∆v, The simulation data is averaged in the steady cycle, which is represented by · · · SC .

B. Details of the Event-Driven simulation
In order to verify the validity of our model, we also perform the event driven molecular dynamics (MD) simulation for our system. We set the diameter of each particle d/ √ A = 0.01, and assume that particles are colliding elastically each other and with adiabatic side walls. The collision rules between the piston and a particle are given by Eqs. (1) and (2). Here, the initial volume fraction of the enclosed gas is estimated to be 10 −6 , i.e., the gas is sufficiently dilute. Collisional force from outside the piston is modeled byF out , through the simulation.
For thermal wall, we adopt the Maxwell reflection wall, where the post collisional velocity toward the wall x = 0 plane is chosen as a random variables obeying the distribution The energy flow from the thermal wall dQ wall /dt can be calculated as follows. The heat flow outgoingq out wall and incomingq in wall through the wall are, respectively, calculated to bê where we have introduced φ(v) ≡ µ=x,y,z φ(v µ ,T in ). By substituting Eqs. (12) and (13) into dQ wall /dt = q in wall −q out wall , we obtain Eq. (9). Apparently, there exists tc for the maximum power operation, which corresponds to the necessary time for gas to expand toward the mechanical equilibrium. The dotted curve drawn as the guide line proportional to 1/tc. ǫ = 0.1, t c /t 0 = 8.0. The simulation data are averaged from 11th cycle to 20th cycle, where the solid and dashed lines, respectively, represent the data for MD simulation and those for simulation of our SMF model. · · · SC represents the average over the steady cycle. Top figures of Fig. 3 (a) and (b) represent the corresponding averaged time evolution of the volume (piston position), middle ones are for temperature of the gas, and bottom ones are for the piston velocity. Dot-dashed lines represent the operation protocol of T bath . It is remarkable that our SMF model well reproduces the time evolution of MD.
Let us explain the behavior of the system shown in Fig. 3. When the heating process starts, the enclosed gas starts expanding, to find a new mechanical equilibrium densityn in k BTin = P out , because the pressure for the enclosed gas becomes larger than that for the outside after the heating. Similarly, the gas is compressed when the cooling process starts. It should be stressed that the heating (cooling) and expansion (compression) processes take place simultaneously. The time evolutions of the physical quantities can be categorized into two types (a) damped-oscillating type and (b) over-damped type, depending on the mass ratio ǫ ≡ m/M . They can be understood by considering the average force acting on the piston. When the piston is moving near the mechanical equilibrium point, the conditional average of the stochastic force acting on the pistonF gas ≡ α=in,outF α for the given piston position X, velocity V and temperature T in , is represented by where we have introduced the pressure difference ∆P ≡ P in − P out = (N k B T in /AX) − P out , γ gas = γ in + γ out , and γ α ≡ 2(1 + e)P α A M/2πk B T α (α = in, out). When the piston is close to the mechanical equilibrium position, Eq. (14) is equivalent to the force acting on a harmonic oscillator in a viscous medium. If the viscous drag is sufficiently small, i.e. ǫ → 0, the motion of the piston is a damped-oscillating type ( Fig. 3(a)), while the motion turns out to be over-damped type, if ǫ is not small (Fig.  3(b)). Indeed, in Ref. [33], the critical value of ǫ is known to be determined by the approximate condition N ǫ 2 /(1 + ǫ 2 ) = π/2. For a macroscopic piston in the limit ǫ → 0, the onedimensional momentum transfer model Eqs. (1) and (2) only might be too simple to reproduce the realistic motion of the piston. Because, additional degrees of freedom, such as side-wall friction [42,43], the excitation of atoms on the piston surface [44] or tilting of the piston, can be relevant for macroscopic piston motion. In Appendix A, we discuss the effect of side-wall friction on the efficiency for our protocol and show that the side-wall friction lowers the efficiency.

IV. EXISTENCE OF MAXIMUM POWER AND ITS EFFICIENCY
In this section, we discuss the efficiency of the engine at maximum power output. We define the workŴ and the heat spent per a cycleQ H aŝ where and Tµ represent the integral over a single cycle and the integral for T bath = T µ (µ = H or L), respectively. Note that the definition of dQ wall is given in Eq. (9). In Fig. 3(a),T in rapidly relaxes to T bath for ǫ = 0.01, right after T bath is switched. Therefore, the expansion or compression processes after the relaxation ofT in ≃ T bath can be regarded as the adiabatic ones. We also note that the enclosed gas works toward the outside gas, and the validity for the definition of work Eq. (15) is discussed in Appendix B. The efficiency for a single operation protocol [45] can be defined asη We also introduce the conventional efficiency, which is defined asη In this section, we average the data from 11th cycle to 110th cycle. The contact time dependence of the powerp w ≡ W /2t c , for the damped oscillating type ǫ = 0.01 (squares) and the over-damped type ǫ = 0.1 (circle) are shown in Fig. 4, where T H /T L = 5.0 and e = 1.0 are fixed and introduced p 0 ≡ k B T out /t 0 . Apparently, the maximum power output is achieved at a contact time t MP c , which corresponds to the necessary time for gas to expand toward the mechanical equilibrium. We note that the long time heating or cooling ruins the power, because the extracted work would be N k B (T H − T L )ln(T H /T L ), at most. Thus, the power decreases as a function of t c : p w SC ∝ 1/t c as t c → ∞, which is drawn as a dotted line in Fig. 4.
We present the results for the efficiency at maximum power output (Fig. 5) for (a) ǫ = 0.01 and (b) ǫ = 0.1. The main figures represent the results for elastic piston case e = 1.0, and the insets are those for the corresponding inelastic piston e = 0.9. The open squares η SC and trianglesη are the simulation data for the SMF model, while filled ones are the data for the corresponding MD simulation. Althoughη and η SC are different quantities, they agree with each other. As a comparison with previous studies, we plot the CNCA efficiency η CA (dotted lines).
Our SMF model well reproduces the efficiency at the maximum power operation for MD simulations, at least for ǫ = 0.01, while our model overestimate the efficiency for ǫ = 0.1. We note that the efficiency for our model with e = 1.0 and ǫ = 0.01 are slightly smaller than η CA , which can be understood as follows. The mass ratio ǫ and the inelasticity of the piston correspond to the heat leakage through the fluctuation of the piston position and the internal degree of freedom for the piston, respectively.
Here, we derive the semi-analytical expression onη, in the limit ǫ → 0. In this limit,T in rapidly relaxes to bath temperature, right after T bath is switched. The average of the work Eq. (15) can be calculated to be where we have introduced the volume change of the gas through the cycleX(t c ) ≡ X (t c ) SC / X (0) SC and choose e = 1. Integrating the energy conservation equation (8), we obtain where we have introduced ∆Û = 3N k B (T H − T L )/2 and E (H) pis ≡ TH dÊ pis . Averaging Eq. (20) and expanding in terms of ǫ, we obtain Therefore, the efficiencyη can be calculated as Assuming thatX(t MP c ) depends on the power of T H /T L : we obtain the analytical expression onη for maximum power output: which is shown in Fig. 5 by solid lines. The exponent α can be estimated from the simulation, where α = 3/2 for ǫ = 0.01 and α = 0.79 for ǫ = 0.1 (See Fig. 6). Equation (24) reproduces the MD data for ǫ = 0.01, while the agreement is not good for ǫ = 0.1. The integration of O(ǫ) term would be necessary for the better agreement of ǫ = 0.1. The exponent α = 3/2 for ǫ = 0.01 can be understood as follows. In the limit ǫ → 0, right afterT in relaxes to T bath , the expansion or compression processes can be regarded as adiabatic ones, because the heat flux from the bath is zero ifT in ≃ T bath . Thus, recalling that the expansion or compression ratio is characterized by Poisson's relation for an adiabatic process T 3/2 in X = Const, we expect the exponent α = 3/2, which is verified in Fig. 6. By substituting the obtained α = 3/2 for ǫ = 0.01 into Eq. (24), We note that Eq. (25) is exactly the same as the expansion of η CA up to O(η 2 C ): while the existence of ǫ lowers the efficiency from η CA , even at the leading order O(η C ), because α = 0.79 < 3/2 for ǫ = 0.1.

V. DISCUSSION
In this section, let us discuss the relation of our results with previous works. We showed that the SMF model well reproduces the time evolution of the system (Fig. 3). Note that as in Ref. [33], the overdamped Langevin description is not adequate, at least for the autocorrelation function for the piston velocity.
Here, we compare the velocity autocorrelation function C(t) ≡ V (t)V (0) / V 2 (0) for MD with our model, by choosing T bath = T out . As can be seen in Fig. 7, due to the implementation of the fluctuation ofT in , our SMF model well reproduces the behavior of C(t) for ǫ = 0.01, which is better than that obtained by an underdamped Langevin equation at a constant temperature. The operation protocol for our engine is equivalent to that for Stirling engine [46], containing heating, cooling, expansion and compression processes, while the expansion (compression) and heating (cooling) processes occur simultaneously in our case. Although the gas is attached to the thermal wall, because the processes can be regarded as adiabatic ones, onceT in relaxes to T bath , our engine is also equivalent to finite-time Carnot's engine, effectively.

VI. CONCLUDING REMARKS
In this paper, we proposed a stochastic mean field (SMF) model for a piston motion confining not too many number of particles. The validity of the stochastic model is verified through the event-driven MD simulation. The introduced model consists of two types of stochastic equations, the equation of motion for the piston Eq. (5), and the energy conservation Eq. (8), which correspond to the time evolution of the density and the temperature the enclosed gas, respectively. We considered the heat cycle, consisting of heating and cooling processes (Fig.  2). Average time evolution of the volume, temperature of the gas and the piston velocity for MD simulation is well reproduced by our SMF model (Fig. 3). Under the protocol repeating heating and cooling processes, we found the existence of the maximum power (Fig. 4), and obtained the semi-analytical expression for the efficiency Eq. (22) and (24). We revealed that the obtained efficiency at the maximum power operation is close to the CNCA efficiency, if the elastic piston mass is sufficiently large (Fig. 5). We also showed that our stochastic model can reproduce MD results for the velocity autocorrelation C(t) (Fig. 7). type, even if the piston is heavy. Because the friction is a kind of zero temperature "bath," we define the efficiency under friction [43] by introducing the frictional heat: The simulated data for the efficiency for maximum power output with γ/γ out = 1.0, e = 1.0 are plotted in Fig. 8. The asymptotic behavior of η SC and η fri SC in ǫ → 0 limit for T H /T L = 5.0 are shown in Fig. 8 (a). In Fig. 8 (b), we plot the temperature dependence of η SC and η fri SC for maximum power output with ǫ = 0.001, where the efficiencies are lower than η CA (see Fig. 5 (a)). Thus, as expected, the friction on the sidewall lowers the efficiency.

Appendix B: On the Definition of Work
In the text, we define the work as "Pressure times Volume change," which is not trivial. In this appendix, we justify the definition, i.e. we decompose the kinetic energy change of piston into heat and work by considering the path probability of (X(t),V (t)) underT in (t) = T in . The discussion in the following is the extension of Ref. [47], toward the case that the volume of the enclosed gas fluctuates in time. Let us consider the path probability for the forward evolution P([X,V |τ ) of (X,V ) during the interval τ from (X(0),V (0)) to (X(τ ),V (τ )) and the backward one P([X,V ] † |τ ) from (X(τ ), −V (τ )) to (X(0), −V (0)), where n collisions between the piston and gas particles take place at time {t i } n i=1 with 0 = t 0 < t 1 < · · · < t n = τ . The jump rate for the piston velocity from V i−1 ≡V (t i−1 ) to V i ≡V (t i ) at the piston position X i−1 ≡X(t i−1 ) caused by the collision from gas particles inside and outside the container are respectively written as The escape rate per a unit time κ(V i−1 |X i−1 ) for (X i−1 , V i−1 ) is represented as Thus, P([X, V ]|τ ) and P([X, V ] † |τ ) are represented as Here, the position of the piston at time t i < s i < t i+1 is given by X Xi n in (X)k B T in AdX + β out P out AV j (t i+1 − t i ), Here we have introduced the inverse temperature β α ≡ 1/k B T α and the energy change of α side gas ∆E α through the piston fluctuation (α = in, out). Using Eqs. (B8) and (B9), we obtain the following expression on the definition of the work: where we have introduced the abbreviation X 0 ≡ X(0), V 0 ≡V (0), X τ ≡X(τ ) and V τ ≡V (τ ). From Eq.
(B11), the change of the internal energy for the enclosed gas ∆E in is apparently decomposed into the change of work and heat. Thus, we adopt the definition of work Eq. (15) in the text.