We consider the temporal evolution of an isolated system of N particles from a nonequilibrium state of entropy S=S' to the equilibrium state of maximum entropy, S=S-max, S'<=S_max. The application of usual density matrix theory to the temporaldevelopment of S leads us to dS/dt=0: The entropy S does not change in time t. Thereby it is irrelevant, whether we consider a non-equilibrium state or an equilibrium state. Consequently, the system cannot irreversibly change by entropy production dS/dt>0 from S' to S_max. This is a paradoxial result, which contradicts the experience. It can be traced back to the von Neumann equation, which in principle describes reversible processes and hence is unsuitable for calculating theirreversible evolution of the entropy S in time t. Each irreversible process is accompanied by a positive entropy production P=dS/dt>=0 inside the system, which only vanishes in case of the equilibrium state. In order to overcome the above mentioned difficulties, we assign an operator P to the entropy production P=ds/dt>=0 inside the system, which only vanishes in case of the equilibrium state. In order to overcome the above mentioned difficulties, we assign an operator P to the entropy production P, which is defined by the eigenvalue equation P|u>=P|u> with the state vector |u> of the N-particle system. There was an extensive discussion about the relation between the production ofentropy P on one hand and the progressionof time t on the other. Making use of this concept, we combine the operator of entropy production P with the time-development operator U(t,t_0) of the system and finally deduce the indinitesimal unitary operator U(t+tau, t)=1+(i/k)tauP by means of verygeneral assumptions. Here P means the generator of theatomic entropy unit. Similarly to P we also treat the time t as an observable, defined by t|u>=t|u>.We apply the infinites imal time-evolution relation i[P,t]=k, which is independent of tau. It shows that the operators P and t do not commute, and hence P and hence P and t are not sharply defined simultaneously. Instead we have uncertainties DELTAP and DELTAt on measuring P and t, which are given by the P-t uncertainty relation DELTAP DELTAt >=k/2. It readily allows a discussion of the evolution of the entropy S of the isolated system in time t from S' to S_max. Now, the thermal equilibrium is given by P=0, DELTAP=0, and thus the lifetime of the equilibrium state DELTAt=infinity. According to the P-t uncertainty relation, the Boltzmann constant k is similarly important to the quantum thermodynamics of irreversible proccesses like Planck's constant h to usual quantum mechanics.
展开▼
