Lattice Boltzmann Methods for Fluids, Plasmas, and Quantum Systems

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Phenomena in fluid flow are usually described by complicated equations named after Navier and Stokes that treat fluids as continuous, infinitely divisible media, or by computer programs based on these equations. However, real fluids are made up of molecules, each of which behaves according to Newton's laws of motion. Newton's laws are much simpler than the Navier-Stokes equations, but there are so many molecules, perhaps 10^18 per cubic centimetre, that one replaces a handful of complicated equations with vast numbers of simple equations.The lattice Boltzmann approach occupies the middle ground: the equations are each only a little more complicated than Newton's laws, but there are only a few more of them than are already present in the Navier-Stokes equations. This middle ground has proved very fruitful even for simulating phenomena that could be described by the Navier-Stokes equations, and by retaining contact with Newton's laws for molecules one may reach down into smaller lengthscales where the Navier-Stokes equations are no longer adequate. These scales are now routinely reached by ever smaller technological devices, the so-called microelectrical mechanical devices.