We present a theory to obtain the pressure tensor for a class of nonideal multicomponent lattice Boltzmann models, thus extending the theory presented by X. Shan [Phys. Rev. E 77, 066702 (2008)] for single-component fluids. We obtain the correct form of the pressure tensor directly on the lattice and the resulting equilibrium properties are shown to agree very well with those measured from numerical simulations. Results are compared with those of alternative theories.

where ρt = ρ + ρ ′ is the total density, ρ = ρtC the density of the first species, ρ ′ = (1 − C)ρ the density of the second species, P = P (ρt, C) the equation of state and µ = µ(ρt, C) the chemical potential driving diffusion of one species into the other. The capital Greek letters denote stochastic diffusion and momentum fluxes whose variance is fixed by the FDT to be (the superscript T denotes transposition)

The effect of thermal fluctuations near a contact line of a liquid interface partially wetting an impenetrable substrate is studied analytically and numerically. Promoting both the interface profile and the contact line position to random variables, we explore the equilibrium properties of the corresponding fluctuating contact line problem based on an interfacial Hamiltonian involving a "contact" binding potential. To facilitate an analytical treatment we consider the case of a onedimensional interface. The effective boundary condition at the contact line is determined by a dimensionless parameter that encodes the relative importance of thermal energy and substrate energy at the microscopic scale. We find that this parameter controls the transition from a partially wetting to a pseudo-partial wetting state, the latter being characterized by a thin prewetting film of fixed thickness. In the partial wetting regime, instead, the profile typically approaches the substrate via an exponentially thinning prewetting film. We show that, independently of the physics at the microscopic scale, Young's angle is recovered sufficiently far from the substrate. The fluctuations of the interface and of the contact line give rise to an effective disjoining pressure, exponentially decreasing with height. Fluctuations therefore provide a regularization of the singular contact forces occurring in the corresponding deterministic problem.

We prove the hydrodynamic limit globally in time and the exponential relaxation to equilibrium of the solution to the rescaled Boltzmann equation on a torus, for ε small, in L 2 ∩ L ∞ .

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