Statistical mechanics of the fluctuating lattice Boltzmann equation

Abstract

We propose a derivation of the fluctuating lattice Boltzmann equation that is consistent with both equilibrium statistical mechanics and fluctuating hydrodynamics. The formalism is based on a generalized lattice-gas model, with each velocity direction occupied by many particles. We show that the most probable state of this model corresponds to the usual equilibrium distribution of the lattice Boltzmann equation. Thermal fluctuations about this equilibrium are controlled by the mean number of particles at a lattice site. Stochastic collision rules are described by a Monte Carlo process satisfying detailed balance. This allows for a straightforward derivation of discrete Langevin equations for the fluctuating modes. It is shown that all nonconserved modes should be thermalized, as first pointed out by Adhikari et al. ͓Europhys. Lett. 71, 473 ͑2005͔͒; any other choice violates the condition of detailed balance. A Chapman-Enskog analysis is used to derive the equations of fluctuating hydrodynamics on large length and time scales; the level of fluctuations is shown to be thermodynamically consistent with the equation of state of an isothermal, ideal gas. We believe this formalism will be useful in developing new algorithms for thermal and multiphase flows.

Multiphase lattice Boltzmann simulations for porous media applications

“…Starting from this general approach, simplifications and extensions have led to the development of, for example, two relaxation time (TRT) models [65,66] as well as models incorporating thermal fluctuations [67][68][69][70].…”

Fluctuating multicomponent lattice Boltzmann model

“…The idea of including noise in LBE is also an active research field, as witnessed by the various publications of the recent years [10][11][12][13][33][34][35]. The basic idea has been pioneered by Ladd [13], who suggested the introduction of noise on the non-conserved hydrodynamic modes, thus reproducing fluctuating viscous stresses in the corresponding hydrodynamic limit (small wavevectors).…”

“…Consequently, the thermalization of the fluid remains incomplete. Adhikari et al [33] were the first to recognize the necessity to include noise on all the non-physical ghost modes, and Dünweg et al [34] reformulated this approach to follow a detailed-balance condition description. In a subsequent work, Kaehler & Wagner [35] also explored the fluctuating LBE for non-vanishing mean velocities.…”

“…The T ai are chosen as linearly independent and orthogonal with respect to a certain norm [33][34][35]. The use of a moment space representation has the advantage that the noise can be constructed such that individual masses and total momentum are explicitly conserved.…”

Modelling thermal fluctuations in non-ideal fluids with the lattice Boltzmann method

“…If such effects are to be simulated within the LB method, the algorithm must be extended to include explicit random forces with magnitudes being determined from an appropriate fluctuation-dissipation theorem (FDT). The theory of thermal fluctuations in the ideal-gas model is now well established [1][2][3], and has been applied to various problems in soft matter physics [4]. Until now, the implementation of thermal fluctuations into non-ideal LB fluids remained unsolved, although there exist many potential applications of such a simulation algorithm in nano-and microfluidics of free surface flows, such as jet instabilities, droplet spreading or dewetting processes [5,6].dynamics of the LB equation (LBE).…”