Statistical Mechanical Estimate of Energy Spectrum for N-Point Vortex Systems

Abstract

A statistical mechanical theory of the energy spectrum for two-dimensional N-point vortex systems is developed. The system in an infinite plane is considered. We focus our attention on the energy spectrum of the system “in equilibrium” and “in a transient state”. For like-sign point vortex systems in an infinite plane, we have succeeded in deriving a scaling law of the energy spectrum E(k)∼k^−α for the intermediate k regime, via the 2-point correlation function R₂(r). However, the applicability of the derived scaling law is limited by the validity of the asymptotic expansion used. By a direct numerical simulation, we obtain various powers α=2.11–3.19 for transient states. Using the Monte Carlo simulation, for equilibrium states we find that the energy spectrum in the intermediate regime does not obey a power law. It is concluded that for point vortex systems in an infinite plane, the scaling law of the energy spectrum in equilibrium and in the transient state depends on the system parameters, and is thus not universal.