We have obtained a microscopic expression for entropy in terms of H function based on nonunitary Λ transformation which leads from the time evolution as a unitary group to a Markovian dynamics and unifies the reversible and irreversible aspects of quantum mechanics. This requires a new representation outside the Hilbert space. In terms of H, we show the entropy production and the entropy flow during the emission and absorption of radiation by an atom. Analyzing the time inversion experiment, we emphasize the importance of pre- and postcollisional correlations, which break the symmetry between incoming and outgoing waves. We consider the angle dependence of the H function in a three-dimensional situation. A model including virtual transitions is discussed in a subsequent paper
Recent developments on the complex spectral representation of the evolution operators have shown that irreversibility is a rigorous dynamical process that takes place outside the Hilbert space for non‐integrable systems, both classical and quantum. Irreversibility is a direct consequence of microscopic dynamical processes associated with resonance singularities, rather than our incomplete knowledge. We discuss the excitation process of a quantum unstable particle by scattering of a wave packet. Using complex eigenstates of the Hamiltonian one may construct a microscopic dynamical entropy (or H function) without any approximation. In order to have scattering, one has to target the wave packet to the particle. This introduces a nonlocal pre‐collisional correlation giving a low dynamical entropy. During the moment the excited particle is created, there appears an entropy flow from the field to the particle. This lowers the particle component of the entropy, even though the total entropy production is always positive. This implies that the excitation of the particle may be considered as the construction of a non‐equilibrium structure due to entropy flow.
The microscopic theory of irreversible processes that we developed is summarized and illustrated, using as a simple example the Friedrichs model. Our approach combines the Poincaré's point of view (dynamical interpretation of irreversibility) with the Gibbs-Einstein ensemble point of view. It essentially consists in a nonunitary transformation theory based on the symmetry properties of the Liouville equation and dealing with continuous spectrum. The second law acquires a microscopic content in terms of a Liapounov function which is a quadratic functional of the density operator. In our new representation of dynamics, which is defined for a restricted set of observables and states, this functional takes a universal form. We obtain, in this way, a semi-group description, the generator of which contains a part directly related to the microscopic entropy production. The Friedrichs model gives us a simple field theoretical example for which the entropy production can be evaluated. The thermodynamical meaning of life-times is explicitly displayed. The transition from pure states to mixtures, as well as the occurrence of long tails in thermodynamic systems, are also briefly discussed.
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 = S max, S′ ≤ S max. 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 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 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, t 0) of the system and finally deduce the infinitesimal unitary operator U (t+τ,t) = 1 + (i/k)τ P by means of very general assumptions. Here P means the generator of the infinitesimal progression of time τ = t − t 0 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 S max. 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.
Within the general formalism of quantum theory irreversibility and the arrow of time and the in the evolution of various physical systems are studied. Irreversible behavior often manifests itself in the guise of “entropy production.” This motivates me to begin this chapter with a brief review of quantum-mechanical entropy, a subject that Elliott Lieb has made outstanding contributions to, followed by an enumeration of examples of irreversible behavior and of an arrow of time analyzed in later sections. Subsequently, a derivation of the laws of thermodynamics from (quantum) statistical mechanics, and, in particular, of the Second Law of thermodynamics, in the forms given to it by Clausius and Carnot, is presented. In a third part, results on diffusive (Brownian) motion of a quantum particle interacting with a quasi-free quantum-mechanical heat bath are reviewed. This is followed by an outline of a theory of friction by emission of Cherenkov radiation of sound waves in a system consisting of a particle moving through a Bose–Einstein condensate and interacting with it. In what may be the most important section of this chapter, the fundamental arrow of time inherent in Quantum Mechanics is discussed.