Abstract :
In Part II of this two-part paper we prove that Maxwellʹs demon is unable to accomplish his task of sorting air molecules into swift and slow because in air in a thermodynamic equilibrium state there are no such molecules. The proof is based on the principles of a unified quantum theory of mechanics and thermodynamics.
The key idea of the unified theory is that von Neumannʹs concept of a homogeneous ensemble of identical systems, identically prepared, is valid not only for a density operator ρ equal to a projector (every member of the ensemble is assigned the same projector, ρi=ψi ψi=ρi2, or the same wave function ψi as any other member) but also for a density operator that is not a projector (every member of the ensemble is assigned the same density operator, ρ>ρ2, as any other member). So, the latter ensemble is not a statistical mixture of projectors. The broadening of the validity of the homogeneous ensemble is consistent with the quantum-theoretic postulates about observables, measurement results, and value of any observable.
In the context of the unified theory, among the many novel results is the theorem that each molecule of a system in a thermodynamic equilibrium state has zero value of momentum, that is, each molecule is at a standstill and, therefore, there are no molecules to be sorted as swift and slow. Said differently, if Maxwell were cognizant of quantum theory, he would not have conceived of the idea of the demon.
It is noteworthy that the zero value of momentum is not the result of averaging over different momenta of many molecules. Under the specified conditions, it is the quantum-theoretic value of the momentum of any one molecule, and the same result is valid even if the system consists of only one molecule.
