Decoherent histories and hydrodynamic equations

For a system consisting of a large collection of particles, variables that will generally become effectively classical are the local densities (number, momentum, energy). That is, in the context of the decoherent histories approach to quantum theory, it is expected that histories of these variables will be approximately decoherent and that their probabilities will be strongly peaked about hydrodynamic equations. This possibility is explored for the case of the diffusion of the number density of a dilute concentration of foreign particles in a fluid. This system has the appealing feature that the microscopic dynamics of each individual foreign particle is readily obtained and the approach to local equilibrium may be seen explicitly. It is shown that, for certain physically reasonable initial states, the probabilities for the histories of the number density are strongly peaked about evolution according to the diffusion equation. Decoherence of these histories is also shown for a class of initial states which includes nontrivial superpositions of number density. Histories of phase space densities are also discussed. The case of the histories of number, momentum, and energy density for more general systems, such as a dilute gas, is also discussed in outline. When the initial state is a local equilibrium state, it is shown that the histories are trivally decoherent and that the probabilities for histories are peaked about hydrodynamic equations. An argument for the decoherence of more general initial states is given.