Macroscopic equations for the adiabatic piston

A simplified version of a classical problem in thermodynamics—the adiabatic piston—is discussed in the framework of kinetic theory. We consider the limit of gases whose relaxation time is extremely fast so that the gases contained in the left and right chambers of the piston are always in equilibrium (that is, the molecules are uniformly distributed and their velocities obey the Maxwell-Boltzmann distribution) after any collision with the piston. Then by using kinetic theory we derive the collision statistics, from which we obtain a set of ordinary differential equations for the evolution of the macroscopic observables (namely, the piston average velocity and position, the velocity variance, and the temperatures of the two compartments). The dynamics of these equations is compared with simulations of an ideal gas and a microscopic model of a gas devised to verify the assumptions used in the derivation. We show that the equations predict an evolution for the macroscopic variables that catches the basic features of the problem. The results here presented recover those derived, using a different approach, by Gruber, Pache, and Lesne [J. Stat. Phys. 108, 669 (2002); Gruber, Pache, and Lesne,J. Stat. Phys.112, 1177 (2003)].

Figure 1

Sketch of the adiabatic piston. The subscripts $l$ and $r$ indicate the left and right compartments, respectively.

Figure 2

(Color online) (Top) Comparison between the evolution of the piston position $x\left(t\right)$ in a simulation of an ideal gas (red solid line) and of the randomized model (blue open circles). In both models we set $N=1000$, $M=100$, and $L=2000$ corresponding to $R=10$, at time zero $T_l=40$, $T_r=60$, and $x=0.6L$. To have a clean curve we performed an average over about 100 independent realizations. (Bottom) The same as above with $N=1000$, $M=5000$, and $L=2000$, corresponding to $R=0.2$, and the initial state set as $T_l=150$, $T_r=50$, and $x=0.6L$. Here, for the ideal gas, the oscillations last for a much longer time and are damped very slowly before the Brownian-motor-like regime sets in; while the randomized gas is much more efficient in damping the oscillations.

Figure 3

(Color online) Evolution of the system in the temperature volume $\left(T\Omega \right)$ plane. The dotted line indicates the adiabatic phase $T\propto V^{-2}$, and the solid straight line the isobar that characterizes the final phase (Brownian-motor-like regime). The leftmost arrow indicates the initial state of the gas in the left compartment $T_l\left(0\right)=150\mathrm{,}{\Omega }_l\left(0\right)=x\left(0\right)=0.6L$, which evolves (blue dots) toward the final equilibrium in the middle. The rightmost arrow indicates the initial state of the gas in the right compartment $T_r\left(0\right)=50,{\Omega }_r\left(0\right)=L-x\left(0\right)=0.4L$, which evolves (red dots) toward the final equilibrium in the middle. The simulation is done with the ideal gas, for which the initial oscillations are much more evident than for the randomized one.

Figure 4

(Color online) Comparison between the simulations of the noninteracting 1D gas, the randomized one and the prediction given by the macroscopic equations. The parameters are $T_l=40$, $T_r=60$, $L=12000$, $x_0∕L=0.6$, $M={10}^5$, $N={10}^3$ corresponding to $R=0.01$. (Top) Evolution of the piston position. (Bottom) Evolution of the temperatures. Blue open squares refer to the ideal gas and red filled circles to the randomized gas; the solid line is the model (14, 15, 16, 17).

Figure 5

(Color online) The same as the leftmost plot of Fig. 2 (top). The thick black line superimposed on randomized gas data (blue empty circles) is obtained by the numerical integration of (14, 15, 16, 17). The agreement of the model with the randomized gas is perfect also for the temperatures (not shown).

Figure 6

(Color online) Comparison between the molecular dynamics of a noninteracting 1D gas and the model in the Brownian-motor-like regime. Simulations have been performed setting the initial state as $T_l=40$, $T_r=60$, and $x\left(0\right)=0.4L$ with $M=500$ and $L=12N$ with $N={10}^4$: red triangles refer to the randomized gas and blue squares to the ideal gas. The dashed line is the prediction of the macroscopic equation; the solid line (which perfectly superimposes on the randomized gas data) is explained in the text. The simulation data are obtained by performing an average over about ten realizations to reduce the fluctuations.

Figure 7

(Color online) (Top) Evolution of the piston position for parameters as in Fig. 2 (bottom) now plotted in log-linear scale. The (red) symbols refer to the randomized gas simulations, the dashed curve to the evolution predicted by the macroscopic equations, and the solid curve (which perfectly superimposes on the simulation data) is explained in the text. (Bottom) Same as above, but for the evolution of the gas temperatures $T_l$ and $T_r$. The two curves which start from zero show the evolution of the piston temperature $T_p=M{\sigma }_V^2$ as obtained from the macroscopic equations (dashed curve which saturates to $T_p\approx 100$ and as explained in the text the solid one which saturates at $T_p\approx 200$.