Nonlinear fluctuations in master equation systems. I. Velocity correlation function for the Rayleigh model

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Abstract
The influence of nonlinear velocity fluctuations on the velocity correlation function Π (t) is studied for the Rayleigh model of a massive particle in an ideal gas as an example of a master equation system. It is shown that the Mori kernel K (t), which determines the decay of Π (t), has a slow mass‐dependent decay on the time scale of the decay of Π (t) and has no well‐behaved expansion in the mass ratio. Both features are contrary to standard assumption. The origins of the slow decay are traced to nonlinear fluctuations and the relationship to previous work on requisite conditions for exact exponential decay is discussed. The slow decay of Π (t) is shown to lead to divergent ’’Burnett’’ coefficients in macroscopic friction laws and the resolution of this difficulty is discussed. The relationship of the microscopic ’’bare’’ friction constant to the macroscopic friction constant is considered. Explicit expressions for Π (t) and K (t) for small mass ratio are obtained by mode–mode coupling analysis and perturbation methods. The influence of nonlinear fluctuation effects is found to be numerically negligible despite their long lifetime. The remaining deviation from standard Brownian motion results is examined numerically. The validity of some standard assumptions in mode–mode coupling theory is also examined.