Prigogine has shown that in the steady state in which certain macroscopic affinities $F_1, F_2, \cdots F_k$ are fixed and other macroscopic affinities $F_{k+1\mathrm{}}, F_{k+2\mathrm{}}, \cdots F_r$ 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.

It is shown that the kinetic coefficients connecting Casimer's $\alpha$-type and $\beta$-type variables alwaýs vanish. The validity of the minimum entropy production theorem in the absence of a magnetic field depends on this fact. The limitations on the validity of the minimum entropy production theorem in the presence of a magnetic field are briefly discussed.

Calculations on particular models by Klein and Meijer and by Klein corroborate the theorem here proved. An analysis of magnetic resonance by Wangsness suggests certain modifications necessary in this case of a nonzero, nonstationary, magnetic field.

• Received 27 August 1956

DOI:https://doi.org/10.1103/PhysRev.105.360