Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium

We consider an analytical solution of the time-dependent elastic Boltzmann transport equation in an infinite uniform isotropic medium with an arbitrar y phase function. We obtain (1) the exact distribution in angle, (2) the ex act first and second spatial cumulants at any angle, and (3) an approximate combined distribution in position and angle and a spatial distribution who se central position and half-width of spread are always exact. The resultin g Gaussian distribution has a center that advances in time, and an ellipsoi dal contour that grows and changes shape providing a clear picture of the t ime evolution of the particle migration from near ballistic, through snakel ike and into the final diffusive regime.