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 k 2 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 k 2 (for large wavenumbers) spectra, with the range of k 2 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. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

Variable energy flux in turbulence

“…For example, Cichowlas et al [58] showed that Taylor-Green vortex as an initial condition yields a mixture of k −5/3 and k 2 spectra. On the contrary, for white noise as initial condition, the system exhibits k 2 spectrum for the whole range of wavenumbers [59,60]. The former system exhibits variable energy flux, but the latter system (equilibrium configuration) has no energy flux.…”

The Galerkin-truncated Burgers equation: crossover from inviscid-thermalized to Kardar–Parisi–Zhang scaling

“…Later, Kraichnan [4] proposed a different approach for these absolute equilibrium states by considering that the complex amplitudes of the Fourier modes followed a canonical distribution that is controlled by the mean values of the invariants of the system. The Galerkin-truncated hydrodynamical system that has been investigated most extensively is the time-reversible Euler equation for a classical, ideal fluid [6–8], which can be studied efficiently, in a spatially periodic domain, by the Fourier pseudospectral method [9,10]. Absolute-equilibrium solutions have also been examined in a variety of hydrodynamical systems including compressible flows [11], the Gross-Pitaevskii equation in both three and two dimensions [12,13], and the Euler equation and ideal magnetohydrodynamics (MHD) in two dimensions [14].…”

Equilibrium states of Burgers and KdV equations

“…Hydrodynamic and Burgers turbulence, crack propagation, and earthquakes are some of the prominent examples of nonequilibrium systems. Interestingly, dissipation-less hydrodynamic and Burgers turbulence exhibit both nonequilibrium and equilibrium behaviour depending on the initial condition [2][3][4]. This is the subject of this paper.…”

“…Clearly, the equilibrium configurations of the KdV and dissipation-less Burgers equations discussed above are very similar to that of Euler turbulence. Verma [4] and Verma et al [27] simulated Euler turbulence using delta-correlated random initial condition and obtained the aforementioned equilibrium state. Note that a smooth large-scale velocity field as an initial condition, as in Cichowlas et al [2], yields nonzero energy flux as transients.…”

“…Note that zero energy flux implies detailed balance of energy transfer among the Fourier modes. The delta-correlated random solutions of dissipation-less Burgers, KdV, and Euler equations are under equilibrium and they yield zero energy flux [4,12,13]. However, the dissipative Burgers and hydrodynamic turbulence with large-scale forcing exhibit constant nonzero energy flux with power-law (k −2 and k −5/3 respectively) energy spectra [7,8,10,[16][17][18].…”