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 function for a system of N non-relativistic particles of unit mass obeying Hamiltonian dynamics. We show that this equation is well posed in only one time direction in the sense of the continuous dependence of the solution on the initial data. We interpret the ill-posedness of this equation in the backward time direction as the manifestation of irreversibility of the observable statistical quantities for systems obeying time-reversible dynamical laws.