Prigogine has shown that in the steady state in which certain macroscopic affinities ${F}_{1}, {F}_{2}, \ensuremath{\cdots} {F}_{k}$ are fixed and other macroscopic affinities ${F}_{k+1}, {F}_{k+2}, \ensuremath{\cdots} {F}_{r}$ are unconstrained, the values assumed by the unconstrained affinities are such as to minimize the rate of production of entropy. We here show that the complete microscopic density matrix of the system is that which minimizes the rate of entropy production subject to the imposed constraints. All magnetic fields are assumed to be zero.