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

We study a molecular engine constituted by a gas of $N\sim {10}^2$ 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 $\tau$. In the adiabatic limit $\tau \to \infty$, 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 ${\tau }^{*}$ where the net work is zero: for $\tau <{\tau }^{*}$ it consumes work instead of producing it, acting as a refrigerator or as a heat sink. At $\tau >{\tau }^{*}$ 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 $\tau$.

Figure 1

Sketch of the piston model. A gas of particles is confined by a fixed wall (the thermal bath) and a moving wall (the piston) that is subject to a constant external force.

Figure 2

Graph of $F$ and $T_0$ as a function of time over a cycle period $\tau$.

Figure 3

Average values per cycle of work $W$ and heat $Q_1,Q_2$ as a function of the cycle typical time $\tau$. Dashed horizontal lines represent the adiabatic value of such quantities. Inset: schematic of the cycle protocol in the space of parameters $F,T_o$.

Figure 4

Average power vs efficiency for MD (red curve) and for the reduced “$3V$” model, Eq. (11) (blue curve). The Curzon-Ahlborn estimate and the Carnot efficiency are also indicated.

Figure 5

Study of fluctuations. (a) and (b) PDF of the work in a cycle for two different values of $\tau$ (“$E$” and “$D$” regimes), from the MD and from the reduced “$3V$” model [Eq. (11)]. (c) PDF of the fluctuating efficiency in a cycle at $\tau =500$. (d) rescaled st.dev. (see the text) of $P\left(W\right)$ and of $P\left(Q_2\right)$ as a function of $\tau$. The statistics in this figure is obtained from 2000 cycles.

Figure 6

Average work and heat per cycle. Comparison between the MD, the three-variables model, Eq. (11), and the analytical solution in Eq. (17) rescaled by their asymptotic values.