Deterministic limit of tagged particle motion: Effect of reflecting boundaries

We study a one-dimensional stochastic motion of a tagged hard point moving among mechanically identical particles. At the initial moment the tagged particle is at rest separating two equal subvolumes L of gases at different in general thermal equilibrium states. Its further evolution is entirely induced by elastic collisions. We determine rigorously the stochastic motion of the tagged particle constructing an explicit form of its distribution function. The scaling x=X/L and τ=t/L of the position and of the time, respectively, preserves the effect of reflecting boundaries in the thermodynamic limit. It is proved that when L→∞, the scaled stochastic tagged particle motion approaches a deterministic trajectory leading to the final equilibrium state. We study the large-L asymptotics of fluctuations around the deterministic trajectory.