Fluctuations in partitioning systems with few degrees of freedom

We study the behavior of a moving wall in contact with a particle gas and subjected to an external force. We compare the fluctuations of the system observed in the microcanonical and canonical ensembles, by varying the number of particles. Static and dynamic correlations signal significant differences between the two ensembles. Furthermore, velocity-velocity correlations of the moving wall present a complex two-time relaxation that cannot be reproduced by a standard Langevin-like description. Quite remarkably, increasing the number of gas particles in an elongated geometry, we find a typical time scale, related to the interaction between the partitioning wall and the particles, which grows macroscopically.

Figure 1

Sketch of the piston model. A gas of particles is confined by a moving wall that is subjected to a constant external force.

Figure 2

Temperature $k_{B}T$ (in the microcanonical ensemble it is $M\langle V^2\rangle$) plotted as a function of energy $E$ (in the canonical ensemble $E=\langle \mathcal{H}\rangle$) for $N=128$. The dashed line represents the theoretical result for noninteracting particles $k_{B}T=E/(2N+3/2)$, which is expected to hold for high temperatures. The other parameters in the simulations are $L=10$, $F=10$, $m=1$, and $M=128$.

Figure 3

(a) Average position $\langle Y\rangle$ and variance ${\sigma }_Y^2$ plotted as a function of energy $E$ in the microcanonical ensemble with $N=128$. Dashed lines represent the theoretical results for the noninteracting gas $\langle Y\rangle =(N+1)E/F(2N+3/2)$ and ${\sigma }_Y^2=(N+1)(N+1/2)/\left[(2N+5/2){(2N+3/2)}^2\right]{(E/F)}^2$. (b) Same quantities as a function of $k_{B}T$ in the canonical ensemble. The theoretical results for the noninteracting gas are $\langle Y\rangle =(N+1)k_{B}T/F$ and ${\sigma }_Y^2=(N+1){\left(k_{B}T\right)}^2/F^2$. The other parameters in the simulations are $L=10$, $F=10$, $m=1$, and $M=128$.

Figure 4

Variance ${\sigma }_Y^2$ plotted as a function of $k_{B}T$ in the canonical and microcanonical ensembles (in the latter case we consider simulations at constant energy and the temperature is obtained from $k_{B}T=M{\langle V^2\rangle }_E$) for $N=128$. The other parameters in the simulations are $L=10$, $F=10$, $m=1$, and $M=128$.

Figure 5

Velocity autocorrelation functions of the piston in the microcanonical and canonical ensembles for (a) $N=16$, (b) $N=64$, (c) $N=256$, and (d) $N=1024$. The other parameters are $L=30$, $F=150$, $T=10$, $m=1$, and $M=50$.

Figure 6

(a) First relaxation time ${\tau }_0$ of the piston velocity correlation for different values of $N$, in the microcanonical (black circles) and canonical (red squares) ensemble with parameters $M=50$, $F=150$, $L=30$, and $k_{B}T=10$. Notice that the first relaxation saturates for large $N$ and the time ${\tau }_0$ reaches a constant value in both the canonical and microcanonical ensembles. (b) Velocity correlation functions as a function of time rescaled by $N$ in the canonical ensemble with same parameters. In the inset the time $\tau$ shows a linear dependence on $N$ for large $N$.

Figure 7

Autocorrelation of different observables in the case of a piston with noninteracting particles (canonical ensemble) measured in numerical simulations and the Langevin approximation for the piston velocity correlation (black line). It is possible to observe how the oscillation in the autocorrelation of the piston velocity $\langle V\left(t\right)V\left(0\right)\rangle$ (blue circles) are in phase with the one of the center of mass of the gas particles $\langle v_{\mathrm{c}.\mathrm{m}.}\left(t\right)v_{\mathrm{c}.\mathrm{m}.}\left(0\right)\rangle$ (red squares). Green diamonds represent the autocorrelation of a single-particle velocity $\langle v_i\left(t\right)v_i\left(0\right)\rangle$. All the correlations are normalized to one for $t=0$. The parameters are $F=150$, $T=10$, $M=50$, and $N=500$.