Temperature gradients in equilibrium: Small microcanonical systems in an external field

We consider the statistical mechanics of a small gaseous system subject to a constant external field. As is well known, in the canonical ensemble, that the system (i) obeys a barometric formula for the density profile, and (ii) the kinetic temperature is independent of height, even when the system is small. We show here that in the microcanonical ensemble the kinetic temperature of the particles affected by the field is not constant with height, but that rather, generally speaking, it decreases with a gradient of order $1/N$. Even more, if we have a mixture of two species, one which is influenced by the field and the other which is not, we find that the two species' kinetic temperatures are generally different, even at the same height. These facts are shown in detail by studying a simple mechanical model: a Lorentz Gas where particles and spinning disks interact and the particles are subjected to a constant external force. In the microcanonical ensemble, the kinetic temperature of the particles is indeed found to vary with height; the disks' kinetic temperature, on the other hand, is height-independent, and thus, differs from that of the particles with which they interact.

Figure 1

The geometric setting of our SLG system: in the closed slab of length $L$ the array of $M$ rotors is set in a (finite horizon) triangular lattice so that $N$ particles (in dots) cannot enter and leave an hexagonal cell surrounding each disk without having at least one scattering collision. Particles are reflected elastically by walls at $x=0$ and $x=L$; the vertical coordinate is periodic. Two cells are the minimum width of the slab in order to avoid consecutive collisions with the same disk. The field strength $\mathcal{E}$ is constant along the slab.

Figure 2

Kinetic temperature profile in the SLG for $N=90$ particles inside the closed slab with length $L=30$ (see Fig. 1), with an external applied field $\mathcal{E}=-0.5$ and system energy of $E=720$. These results were obtained by molecular dynamics simulation (segmented lines). The continuous line (red) is the theoretical prediction, Eq. (22). We observe a slight discrepancy for $T_K\left(x\right)$ along the slab; the reason is presumably related to issues of nonergodicity of the simulation. The short-segmented line (stars) indicates the temperature of the disks. These data are also significantly more noisy than in the Monte Carlo simulations. In the inset, we show a semilog plot of the density of particles $C\left(x\right)$ in the slab, compared to a similar simulation in the usual Lorentz gas (in crosses). All quantities plotted in this figure are dimensionless.

Figure 3

This figure shows results form a Monte Carlo simulation of the SLG model to obtain the kinetic temperature profile for $N=90$ particles inside a closed slab of length $L=30$ with an external applied field $\mathcal{E}=-0.5$ and an initial energy of $E=720$. All quantities plotted here are dimensionless. The continuous line shows the simulation data while the discontinuous line (stars) is the theoretical prediction, Eq. (22). There are thus no approximations involved. The discontinuous line with crosses (green) indicates the temperature of the discs.