On the superradiant phase transition for molecules in a quantized radiation field: the dicke maser model

Properties of Quantum Markovian Master Equations

A subalgebraM
0 of a von Neumann algebraM is called weakly sufficient with respect to a pair (φ,ω) of states if the relative entropy of φ and ω coincides with the relative entropy of their restrictions toM
0. The main result says thatM
0 is weakly sufficient for (φ,ω) if and only ifM
0 contains the Radon-Nikodym cocycle [Dφ,Dω]
 t
 . Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.

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Sufficient subalgebras and the relative entropy of states of a von Neumann algebra

Wigner's theorem states that a symmetry operation of a quantum system is induced by a unitary or an anti‐unitary transformation. This note presents a detailed proof which closely follows Wigner's original exposition.

Note on wigner's theorem on symmetry operations

Irreversible Thermodynamics for Quantum Systems Weakly Coupled to Thermal Reservoirs

The formalism of covariant conditional expectations is described as leading to an operational definition of generalized observables in quantum mechanics, wide enough to account for the fuzziness inherent in actual measurement processes, relative to a multidimensional physical continuum. As an application, a position operator for the photon is defined and its intrinsic fuzziness is discussed.

Fuzzy observables in quantum mechanics

A theoretical framework for the nonequilibrium statistical mechanics of open systems is constructed This is concerned with a formulation of a generalized master equation governing the evolution of an arbitrary system S in interaction with a ``large'' reservoir R The dynamics of S are analyzed on the basis of a precise quantum‐mechanical treatment of the microscopic equations of motion for the combined system S + R On proceeding to the thermodynamical limit for R we obtain a generalized master equation for S, subject to specified conditions on the many‐particle structure of R, its initial state, and its coupling to S This master equation corresponds to a self‐contained law of motion for S, in which the R variables appear only in the forms of certain thermal averages, taken over the initial state This dynamical law is a generalization of the quantum‐mechanical Liouville equation to a form appropriate to open systems

Nonequilibrium Statistical Mechanics of Open Systems

We compare the dynamical characterization of pure thermodynamical phases as extremal KMS states and their characterization as extremal time‐ or space‐invariant states. We find that, for a class of Weiss‐Ising models with periodic potentials, the extremal KMS states coincide exactly with the solutions of the self‐consistency equations familiar from molecular field methods. We show that the models considered are not η‐asymptotically Abelian in time. We conclude that the characterization of pure thermodynamical phases as extremal KMS states is the only correct one for these models. We pay special attention (in particular, in the decomposition of an arbitrary KMS state into its extremal KMS components) to the fact that the time evolution is not an automorphism of the C*‐algebra of the quasilocal observables.

Pure Thermodynamical Phases as Extremal KMS States

This paper is intended to provide the axiomatic study of nonequilibrium quantum statistical mechanics with some simple and rigorously solvable models. The models considered here are obtained as generalizations of the Ising model. They illustrate and allow a rational discussion of the following concepts relevant to the theory of irreversible phenomena: coarse‐graining and time‐smoothing, ergodicity, recurrences, impossibility of a Markovian description of the approach to equilibrium for some physical systems, justification of the various random phase assumptions, properties of the interaction responsible for the approach to equilibrium, master equations, etc.

Non‐Markovian Model for the Approach to Equilibrium

A mathematically rigorous discussion of the diffusion equation and of its connection with the Einstein relation linking the diffusion coefficient and the velocity autocorrelation function is presented. Diffusion is then propounded as a typical case for which the logical consistency of a purely mechanistic theory of dissipative phenomena can be established.

Gérard G. Emch

Papers

Fuzzy observables in quantum mechanics

Nonequilibrium Statistical Mechanics of Open Systems

Pure Thermodynamical Phases as Extremal KMS States

Non‐Markovian Model for the Approach to Equilibrium

Diffusion, Einstein formula and mechanics