On the well-posedness of equations for smoothed phase space distribution functions and irreversibility in classical statistical mechanics

We obtain an exact equation for a smoothed phase space distribution functio n for a system of N non-relativistic particles of unit mass obeying Hamilto nian dynamics. We show that this equation is well posed in only one time di rection in the sense of the continuous dependence of the solution on the in itial data. We interpret the ill-posedness of this equation in the backward time direction as the manifestation of irreversibility of the observable s tatistical quantities for systems obeying time-reversible dynamical laws.