Boltzmann equation and hydrodynamic equations: their equilibrium and non-equilibrium behaviour

Abstract

This short article summarizes the key features of equilibrium and non-equilibrium aspects of Boltzmann and hydrodynamic equations. Under equilibrium, the Boltzmann equation generates uncorrelated random velocity that corresponds to k2 energy spectrum for the Euler equation. The latter spectrum is produced using initial configuration with many Fourier modes of equal amplitudes but with random phases. However, for a large-scale vortex as an initial condition, earlier simulations exhibit a combination of k−5/3 (in the inertial range) and k2 (for large wavenumbers) spectra, with the range of k2 spectrum increasing with time. These simulations demonstrate an approach to equilibrium or thermalization of Euler turbulence. In addition, they also show how initial velocity field plays an important role in determining the behaviour of the Euler equation. In non-equilibrium scenario, both Boltzmann and Navier–Stokes equations produce similar flow behaviour, for example, Kolmogorov’s k−5/3 spectrum in the inertial range.