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Particle Entropies and Entropy Quanta V. The P–t Uncertainty Relation

Abstract We consider the temporal evolution of an isolated system of N particles from a non-equilibrium state of entropy S = S′ to the equilibrium state of maximum entropy, S = Smax, S′ ≤ Smax. The application of usual density matrix theory to the temporal development 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 Smax. 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 the irreversible 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, 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 of entropy P on one hand and the progression of 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, t0) of the system and finally deduce the infinitesimal unitary operator U(t+τ,t) = 1 + (i/k)τP by means of very general assumptions. Here P means the generator of the infinitesimal progression of time τ = t − t0 and k the Boltzmann constant, representing the atomic entropy unit. Similarly to P we also treat the time t as an observable, defined by t|u〉 = t|u〉. We apply the infinitesimal time-evolution operator U(t+τ,t) to the operator of time t and finally obtain the P–t commutation relation i[P,t] = k, which is independent of τ. It shows that the operators P and t do not commute, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties ΔP and Δt on measuring P and t, which are given by the P–t uncertainty relation ΔPΔt ≥ k/2. It readily allows a discussion of the evolution of the entropy S of the isolated system in time t from S′ to Smax. Now, the irreversible steps are correctly described by the entropy production P = dS/dt > 0, while the thermal equilibrium is given by P = 0, ΔP = 0, and thus the lifetime of the equilibrium state Δt = ∞. According to the P–t uncertainty relation, the Boltzmann constant k is similarly important to the quantum thermodynamics of irreversible processes like Planck′s constant h to usual quantum mechanics.