Statistical mechanics of “negative temperature” states

The dynamics of two-dimensional interacting “line” vortices is identical to that of the two-dimensional electrostatic guiding center plasma. Both are Hamiltonian systems and are therefore susceptible to statistical mechanical treatments. The predictions of the microcanonical ensemble are explored for this sytem. Interest focuses primarily on the regime in which the interaction energy is high enough to be above the Onsager “negative temperature” threshold. Calculations of the probability distribution for a component, by means of the central limit theorem, are carried out in the manner of Khinchin. The probability distribution of a component reduces to the usual Gibbs distribution in the regime of positive temperatures, and is still explicitly calculable for negative temperatures. The negative temperature states are neither quiescent nor spatially uniform. Expressions for the temperature are explicitly provided in terms of the total particle energy and particle number. A BBGKY hierarchy can be derived for both temperature regimes. A Vlasov approximation on the first pair of BBGKY equations leads to the differential equation (two dimensions) ∇2ψ +sinh ψ = 0. The charge (vortex) density has an expectation value proportional to ∇2ψ. Numerical simulations involving solutions of the equations of motion of 4008 particles are presented.