On the law of increase of entropy for nonequililbrium systems

Under the assumption of a smooth full phase-space distribution function we prove that the nonequilibrium entropy S which is considered as a functional of the distribution vector for an N-body system possesses a lower bound and therefore can not decrease. We also compute the rate of change of S, dS/dt, showing that this is non-negative and having a global minimum at equilibrium. As an aplication we obtain a generalization of the (Bhatnager-Gross-Krook) BGK relaxation model.