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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  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  (in the inertial range) and  (for large wavenumbers) spectra, with the range of  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  spectrum in the inertial range. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

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