White noise functional approach to the Brownian motion of a free particle
We use white noise functional analysis to derive the velocity-space and configuration-space probability density functions (PDF) for the Brownian motion of a free particle. We then derive the moments, mean square deviations for velocity and position, and the coefficient of diffusion. For long times, we show how the second moment for velocity approaches the equipartition value. We also derive the expression for the distribution law and demonstrate how the Maxwell distribution is achieved.