Nonlinear fluctuations in transport equations studied for the Rayleigh piston

We treat the master equation in one dimension for the Rayleigh-piston problem of hard rods, i.e., a single heavy tagged particle with finite mass mA moving in an ensemble of light bath particles with mass mB and a time-independent velocity distribution. The backward form of the master equation is used to obtain a transport equation for a conditional average of a time-dependent physical quantity. It is shown how this integro-differential equation can be solved successively by transforming it into a closed set of first-order linear partial differential equations. The solution of the lth linear partial differential equation is completely determined by the solutions of the l-1 linear partial differential equations, meaning that higher approximations do not change lower ones. It is shown how the motion of the tagged particle can be separated into a deterministic (nonfluctuating) part and into fluctuating contributions. It turns out that the l=0 term in the successive approximation scheme satisfies a homogeneous linear partial differential equation and describes the nonfluctuating motion, whereas the higher approximations (l1), which are solutions of nonhomogeneous linear partial differential equations, describe the fluctuating contributions. The calculations for arbitrary time-dependent conditional averages are performed explicitly up to order 2 in the expansion parameter -1=mB(mA+mB). This new method is employed to calculate the conditional averages for the time-dependent mean velocity, the mean-square velocity, the velocity-autocorrelation function, and the self-diffusion coefficient. In addition, the results show that conditional averages calculated via a Gaussian distribution function deviate considerably from the exact results obtained for mass ratios of order 1