Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics. We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution. The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann. The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures. In terms of these one can prove the Gallavotti–Cohen fluctuation theorem. One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations. At equilibrium one recovers in particular the Onsager reciprocity relations. Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss. The results just mentioned hold so far only for classical systems; they do not involve large size, i.e., they hold without a thermodynamic limit.     

Note on Two Theorems in Nonequilibrium Statistical Mechanics

An attempt is made to clarify the difference between a theorem derived by Evans and Searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state.     

Mathematical Theory of Non-Equilibrium Quantum Statistical Mechanics

We review and further develop a mathematical framework for non-equilibrium quantum statistical mechanics recently proposed in refs. 1–7. In the algebraic formalism of quantum statistical mechanics we introduce notions of non-equilibrium steady states, entropy production and heat fluxes, and study their properties. Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs. We exhibit examples of such systems which have strictly positive entropy production.     

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

Entropy of Quantum States

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.     

Classical Mechanics and Quantum Mechanics

The Newton equation of motion is derived from quantum mechanics.     

Non-Markovian Quantum Kinetics and Conservation Laws

A link between memory effects in quantum kinetic equations and nonequilibrium correlations associated with the energy conservation is investigated. In order that the energy be conserved by an approximate collision integral, the one-particle distribution function and the mean interaction energy are treated as independent nonequilibrium state parameters. The density operator method is used to derive a kinetic equation in second-order non-Markovian Born approximation and an evolution equation for the nonequilibrium quasi-temperature which is thermodynamically conjugated to the mean interaction energy. The kinetic equation contains a correlation contribution which exactly cancels the collision term in thermal equilibrium and ensures the energy conservation in nonequilibrium states. Explicit expressions for the entropy production in the non-Markovian regime and the time-dependent correlation energy are obtained.     

The Casimir Problem of Spherical Dielectrics: A Solution in Terms of Quantum Statistical Mechanics

The Casimir energy for a compact dielectric sphere is considered in a novel way, using the quantum statistical method introduced by Høye and Stell and others. Dilute media are assumed. It turns out that this method is a very powerful one: we are actually able to derive an expression for the Casimir energy that contains also the negative part resulting from the attractive van der Waals forces between the molecules. It is precisely this part of the Casimir energy that has turned out to be so difficult to extract from the formalism when using the conventional field-theoretic methods for a continuous medium. Assuming a frequency cutoff, our results are in agreement with those recently obtained by G. Barton.     

Quantum Mechanics is About Quantum Information

I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles. The notion of quantum information is to be understood as a new physical primitive---just as, following Einsteins special theory of relativity, a field is no longer regarded as the physical manifestation of vibrations in a mechanical medium, but recognized as a new physical entity in its own right.     

Quantum Trajectories, State Diffusion, and Time-Asymmetric Eventum Mechanics

We show that the quantum stochastic Langevin model for continuous in time measurements provides an exact formulation of the von Neumann uncertainty error-disturbance principle. Moreover, as it was shown in the 1980s, this Markov model induces all stochastic linear and nonlinear equations of the phenomenological informational dynamics such as quantum state diffusion and spontaneous localization by a simple quantum filtering method. Here we prove that the quantum Langevin equation is equivalent to a Dirac-type boundary-value problem for the second quantized input offer waves from future in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the trajectories of the output particles. This result supports the wave-particle duality in the form of the thesis of Eventum Mechanics that everything in the future is constituted by quantized waves, everything in the past by trajectories of the recorded particles. We demonstrate how this time arrow can be derived from the principle of quantum causality for nondemolition continuous in time measurements.     

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.     

Einstein Relation for Nonequilibrium Steady States

The Einstein relation, relating the steady state fluctuation properties to the linear response to a perturbation, is considered for steady states of stochastic models with a finite state space. We show how an Einstein relation always holds if the steady state satisfies detailed balance. More generally, we consider nonequilibrium steady states where detailed balance does not hold and show how a generalisation of the Einstein relation may be derived in certain cases. In particular, for the asymmetric simple exclusion process and a driven diffusive dimer model, the external perturbation creates and annihilates particles thus breaking the particle conservation of the unperturbed model.     

Ergodic Properties for a Quantum Nonlinear Dynamics

We present a quantum system composed of infinitely many particles, subject to a nonquadratic Hamiltonian, for which it is possible to investigate the long time behavior of the dynamics and its ergodic properties. We do so both for the KMS states and for a large class of locally normal invariant states, whose very existence is already of some interest.     

Stochastic Isometries in Quantum Mechanics

The class of stochastic maps, that is, linear, trace-preserving, positive maps between the self-adjoint trace class operators of complex separable Hilbert spaces plays an important role in the representation of reversible dynamics and symmetry transformations. Here a characterization of the isometric stochastic maps is given and possible physical applications are indicated.     

Counterfactual Reasoning in Time-Symmetric Quantum Mechanics

A qualification is suggested for the counterfactual reasoning involved in some aspects of time-symmetric quantum theory (which involves ensembles selected by both the initial and final states). The qualification is that the counterfactual reasoning should only apply to times when the quantum system has been subjected to physical interactions which place it in a “measurement-ready condition” for the unperformed experiment on which the counterfactual reasoning is based. The defining characteristic of a “measurement-ready condition” is that a quantum system could be found to have the counterfactually ascribed property without direct physical interaction with the eigenstate corresponding to that property.     

Glafka 2004: Generalizing Quantum Mechanics for Quantum Gravity

Familiar quantum mechanics assumes a fixed spacetime geometry. Quantummechanics must therefore be generalized for quantum gravity where spacetime geometry is not fixed but rather a quantum variable. This extended abstract sketches a fully fourdimensional generalized quantum mechnics of cosmological spacetime geometries that is one such generalization.This contribution to the proceedings of the Glafka Conference is an extended abstract of the author's talk there. More details can be found in the references cited at the end of the abstract expecially (Hartle, 1995).     

Experiments in PT-Symmetric Quantum Mechanics

Extended quantum mechanics using non-Hermitian (pseudo-Hermitian) Hamiltonians H = H is briefly reviewed. A few related mathematical experiments concerning supersymmetric regularizations, solvable simulations and large-N expansion techniques are summarized. We suggest that they could initiate a deeper study of nonlocalized structures (quasi-particles) and/or of their unstable and many-particle generalizations. Using the Klein-Gordon example for illustration, we show how the PT symmetry of its Feshbach-Villars Hamiltonian H ^FV might clarify experimental aspects of relativistic quantum mechanics.