Equilibrium Quantum Statistical Mechanics Research Articles

The Entropy-Based Quantum Metric

The von Neumann entropy S( D ^ ) generates in the space of quantum density matrices D ^ the Riemannian metric ds2 = −d2S( D ^ ), which is physically founded and which characterises the amount of quantum information lost by mixing D ^ and D ^ + d D ^ . A rich geometric structure is thereby implemented in quantum mechanics. It includes a canonical mapping between the spaces of states and of observables, which involves the Legendre transform of S( D ^ ). The Kubo scalar product is recovered within the space of observables. Applications are given to equilibrium and non equilibrium quantum statistical mechanics. There the formalism is specialised to the relevant space of observables and to the associated reduced states issued from the maximum entropy criterion, which result from the exact states through an orthogonal projection. Von Neumann’s entropy specialises into a relevant entropy. Comparison is made with other metrics. The Riemannian properties of the metric ds2 = −d2S( D ^ ) are derived. The curvature arises from the non-Abelian nature of quantum mechanics; its general expression and its explicit form for q-bits are given, as well as geodesics.

Generalized Bose-Einstein Condensation into Multiple States in Driven-Dissipative Systems

Bose-Einstein condensation, the macroscopic occupation of a single quantum state, appears in equilibrium quantum statistical mechanics and persists also in the hydrodynamic regime close to equilibrium. Here we show that even when a degenerate Bose gas is driven into a steady state far from equilibrium, where the notion of a single-particle ground state becomes meaningless, Bose-Einstein condensation survives in a generalized form: the unambiguous selection of an odd number of states acquiring large occupations. Within mean-field theory we derive a criterion for when a single state and when multiple states are Bose selected in a noninteracting gas. We study the effect in several driven-dissipative model systems, and propose a quantum switch for heat conductivity based on shifting between one and three selected states.

Generalizing Planck’s distribution by using the Carati–Galgani model of molecular collisions

Classical systems of coupled harmonic oscillators are studied using the Carati–Galgani model. We investigate the consequences for Einstein’s conjecture by considering that the exchange of energy in molecular collisions follows the Lévy type statistics. We develop a generalization of Planck’s distribution admitting that there are analogous relations in the equilibrium quantum statistical mechanics of the relations found using the nonequilibrium classical statistical mechanics approach. The generalization of Planck’s law based on the nonextensive statistical mechanics formalism is compatible with our analysis.

(ħ, k)-Dynamics as some generalization of equilibrium quantum statistical mechanics

We show that the quantum statistical mechanics (QSM) describing quantum and thermal properties of objects has only the sense of a particular semiclassical approximation. We propose a more general (than QSM) micro description of objects in a heat bath taken into account a vacuum as an object environment; we call it (ħ, k)-dynamics (ħkD). We construct a new model of thermostat, namely, a quantum heat bath and study its properties including the cases of a “cold” and “thermal” vacua. We introduce a new generative operator, Schroedingerian, or stochastic action operator, and show its fundamental role in the determination such macro-quantities as internal energy, effective temperature, and effective entropy. We establish that in ħkD the ratio of effective action to effective entropy at zero temperature equals the universal constant ħ/2k. This result corresponds to experimental data taken recently under studying of a new matter state—nearly perfect fluid.

A quantum generalization of equilibrium statistical thermodynamics: Effective macroparameters

We propose a generalization of statistical thermodynamics in which quantum effects are taken into account on the macrolevel without explicitly using the operator formalism while traditional relations between the macroparameters are preserved. In a generalized thermostat model, thermal equilibrium is characterized by an effective temperature bounded from below. We introduce fundamental theoretical macroparameters: the effective entropy and the effective action. Because the effective entropy is nonzero at low temperatures, we can write the third law of thermodynamics in the form postulated by Nernst. The effective action at any temperature coincides with the product of standard deviations of the coordinate and momentum in the Heisenberg uncertainty relation and is therefore bounded from below. We establish that the ratio of the effective action to the effective entropy in the low-temperature limit is determined by a holistic stochastic-action constant depending on the Planck and Boltzmann constants. We show that the same results can be obtained in the framework of a modified version of thermofield dynamics in which the quantum oscillator is described by a temperature-dependent complex macroscopic wave function. We study the discrepancy between the behavior of the action-to-entropy ratio in the low-temperature limit in our proposed theory and that in quantum equilibrium statistical mechanics, which can be verified experimentally.

Second quantized reduced Bloch equations and the exact solutions for pairing Hamiltonian

In this article, we present a set of hierarchy Bloch equations for the reduced density operators in either canonical or grand canonical ensembles in the occupation number representation. They provide a convenient tool for studying the equilibrium quantum statistical mechanics for some model systems. As an example of their applications, we solve the equations for the model system with a pairing Hamiltonian. With the aid of its symplectic group symmetry, we obtain the statistical reduced density matrices with different orders. As a special instance for the solutions, we also get the reduced density matrices of the ground state for a superconductor.

Planck's formula and glassy behavior in classical nonequilibrium statistical mechanics

In statistical mechanics it is well known that one has to take into account the role of the relaxation times to equilibrium. We discuss the relevance of this fact for systems of weakly coupled harmonic oscillators in a classical framework, showing how one has a behavior similar to that of glassy systems and how one finds an analogue of Planck's formula. From this point of view, quantum equilibrium statistical mechanics thus appears as a first-order approximation in classical nonequilibrium statistical mechanics very far from equilibrium.

Evaluation of coherent-state path integrals in statistical mechanics by matrix multiplication

The numerical evaluation of coherent-state path-integral representations for partition functions and other quantities in equilibrium quantum statistical mechanics is discussed. Several coherent-state path-integral schemes are introduced, which differ from each other by the order of approximation and by the operator ordering employed in the high-temperature approximation of the density operator. Simple one-dimensional systems are used to test these schemes. For the harmonic oscillator, finite-dimensional approximations to the coherent-state path integral are calculated analytically and compared to each other and to the real-space path integral. For anharmonic systems, integrations must be approximated by quadrature formulas. This leads to a matrix multiplication scheme which is tested for the double-well potential. The results obtained are accurate from zero temperature way up into the high-temperature regime where quantum effects become negligible. This is a significant advantage over traditional real-space path integral schemes which break down at low temperatures.

Quantum statistical mechanics as a construction of an embedding scheme

The aim of the present paper is to show that the formalism of equilibrium quantum statistical mechanics can fully be incorporated into Ludwig's embedding scheme for classical theories in many-body quantum mechanics. A construction procedure based on a recently developed reconstruction procedure for the so-called macro-observable is presented which leads to the explicit determination of the set of classical ensembles compatible with the embedding scheme.

The formalism of equilibrium quantum statistical mechanics revisited

It is shown that the traditional formalism of equilibrium quantum statistical mechanics may fully be incorporated into a general macro-observable approach to quantum statistical mechanics recently proposed by the same author. (1,2) In particular, the partition functions which in the traditional approach are assumed to connect nonnormalized density operators with thermodynamic functions are reinterpreted as functions connecting so-called quantum mechanical effect operators with state parameters. It is argued that these functions although only part of a much richer internal structure of the macro-observable are sufficient to cope with all problems one usually encounters in equilibrium quantum statistical mechanics. p]Denn eigentlich unternehmen wir umsonst, das Wesen eines Dinges auszudrucken. Wirkungen werden wir gewahr, und eine vollstandige Geschichte dieser Wirkungen umfaste wohl allenfalls das Wesen jenes Dinges.

Foundations of the quantum statistics of systems in equilibrium: A measuretheoretic approach

The fundamental equations of equilibrium quantum statistical mechanics are derived in the context of a measure-theoretic approach to the quantum mechanical ergodic problem. The method employed is an extension, to quantum mechanical systems, of the techniques developed by R. M. Lewis for establishing the foundations of classical statistical mechanics. The existence of a complete set of commuting observables is assumed, but no reference is made a priori to probability or statistical ensembles. Expressions for infinite-time averages in the microcanonical, canonical, and grand canonical ensembles are developed which reduce to conventional quantum statistical mechanics for systems in equilibrium when the total energy is the only conserved quantity. No attempt is made to extend the formalism at this time to deal with the difficult problem of the approach to equilibrium.