Kinetic model for the finite-time thermodynamics of small heat engines

Abstract

We study a molecular engine constituted by a gas of N∼102 molecules enclosed between a massive piston and a thermostat. The force acting on the piston and the temperature of the thermostat are cyclically changed with a finite period τ. In the adiabatic limit τ→∞, even for finite size N, the average work and heat reproduce the thermodynamic values, recovering the Carnot result for the efficiency. The system exhibits a stall time τ* where the net work is zero: for τ<τ* it consumes work instead of producing it, acting as a refrigerator or as a heat sink. At τ>τ* the efficiency at maximum power is close to the Curzorn-Ahlborn limit. The fluctuations of work and heat display approximatively a Gaussian behavior. Based upon kinetic theory, we develop a three-variables Langevin model in which the piston's position and velocity are linearly coupled together with the internal energy of the gas. The model reproduces many of the system's features, such as the inversion of the work's sign, the efficiency at maximum power, and the approximate shape of the fluctuations. A further simplification in the model allows us to compute analytically the average work, explaining its nontrivial dependence on τ.

Attainability of Carnot efficiency with autonomous engines

“…We here adopt a well-used approximation that with given position of the walls and particle number the particles are in equilibrium [28,29]. If the equilibration of particles is much faster than the dynamics of the walls and the exchange of particles with the baths, the above approximation is justified.…”

Thermodynamics of slow solutions to the gas-piston equations

“…For this system there are multiple valid approaches: for example the one particle gas approach [7] and its legacy [8][9][10][11], the explicit-friction formulae approach [12][13][14], and the gas particles-average approach [15][16][17][18]. Among those references [15] is particularly interesting: there, the authors assumed that (i) the gas is perfect and 1-dimensional; (ii) the piston and each gas particle undergoes elastic collisions, so work is the energy exchanged in this way; (iii) the velocity of a gas particle is randomly changed according to the Maxwell-Boltzmann distribution of the reservoir when reservoir-gas particle collisions occurs [19] and heat is the change in energy of the gas; (iv) the gas distribution is always Maxwellian although gas-reservoir and gas-piston collisions change the temperature of the gas over time. Combined in a laborious averaging process, the authors where able to derive a set of dynamical equations for the time evolution of the gas temperature T and piston position x according to any externally prescribed change of the external force Θ and reservoir temperature Ω.…”

“…Combined in a laborious averaging process, the authors where able to derive a set of dynamical equations for the time evolution of the gas temperature T and piston position x according to any externally prescribed change of the external force Θ and reservoir temperature Ω. In [20] we showed with the multiple scales method [21][22][23][24] and some technical assumptions, (discussed in Section II) that the equations derived in [15] allows to find an approximated expression for the heat exchanged with the reservoir in two physically relevant cases: the relaxation to equilibrium and the slow isothermal compression. In the same paper [20] we pointed out the existence of particular solutions, which we called dynamical equilibrium solutions, which describe the asymptotic behavior of the system when externally driven.…”

“…We suggest that similar power series approaches can play a key role in the characterization of thermal cycles where drivings are slow but not necessarily quasi-static. This paper is structured as follows: in Section II we introduce the model from [15] together with the notations and the formalism we will use throughout the paper; in Section III we derived our main results, namely the full recursive relation from which we can derive the coefficients of dynamical equilibrium solution and the exchanged heat; in Section IV we comment the thermodynamical relevance of our result, pointing also out similarities with the virial expansion, and make some conjectures for possible future developments.…”

Fourier’s Law in a Generalized Piston Model

“…The system at its ends interacts with thermal baths at fixed temperatures. It is easy to realize the analogy between such a generalized piston model and the systems of masses and springs: the pistons and the gas compartments play the role of masses and springs, respectively.Our model is an example of partitioning system (as the adiabatic piston), where previous studies showed that the presence of mobile walls can induce interesting behaviours [12][13][14][15][16][17][18][19][20][21]. Basically, in the study of partitioning systems, one can adopt two approaches: in terms of a Boltzmann equation [14] or introducing effective equations (Langevin-like) for suitable observables derivedà la Smoluchowski, i.e., from an analysis of the collisions particles/walls.…”