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Separation of dense colloidal suspensions in narrow channels: A stochastic model

The flow of a colloidal suspension in a narrow channel of periodically varying width is described by the one-dimensional generalized asymmetric exclusion process. Each site admits multiple particle occupancy. We consider particles of two different sizes. The sites available to particles form a comblike geometry: entropic traps due to variation of channel width are modeled by dead ends, or pockets, attached individually to each site of a one-dimensional chain. This geometry, combined with periodically alternating external driving, leads to a ratchet effect which is very sensitive to particle size, thus enabling particle sorting. A typical behavior is reversal of the current orientation when we change the density of small and big particles. In an optimal situation, the two types of particles move in opposite directions, and particle separation is in principle perfect. We show that in the simplest situation with one type of particles only, this model is exactly soluble. In the general case we use enhanced mean-field approximation as well as direct numerical simulations.

Variational calculation of transport coefficients in diffusive lattice gases

A diffusive lattice gas is characterized by the diffusion coefficient depending only on the density. The Green-Kubo formula for diffusivity can be represented as a variational formula, but even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient can be computed only in the exceptional situation when the lattice gas is gradient. In the general case, minimization over an infinite-dimensional space is required. We propose an approximation scheme based on minimizing over finite-dimensional subspaces of functions. The procedure is demonstrated for one-dimensional generalized exclusion processes in which each site can accommodate at most two particles. Our analytical predictions provide upper bounds for the diffusivity that are very close to simulation results throughout the entire density range. We also analyze nonequilibrium density profiles for finite chains coupled to reservoirs. The predictions for the profiles are in excellent agreement with simulations.