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

TL;DR

This paper investigates how small microcanonical systems in an external field exhibit temperature gradients and species-dependent temperatures, contrasting with canonical ensemble predictions, through a detailed mechanical model analysis.

Contribution

It demonstrates that in the microcanonical ensemble, temperature varies with height and differs between species, unlike in the canonical ensemble, using a Lorentz Gas model.

Findings

Kinetic temperature decreases with height in microcanonical ensemble.

Different species can have different temperatures at the same height.

Temperature remains constant with height in canonical ensemble.

Abstract

We consider the statistical mechanics of a small gaseous system subject to a constant external field. As is well known, in the canonical ensemble 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.