Gibbs ensembles in relativistic classical and quantum mechanics

Classical and quantum Gibbs ensembles are constructed for equilibrium statistical mechanics in the framework of an extension to many-body theory of a relativistic mechanics proposed by Stueckelberg. In addition to the usual chemical potential in the grand canonical ensemble, there is a new potential corresponding to the mass degree of freedom of relativistic systems. It is shown that in the nonrelativistic limit the relativistic ensembles we have obtained reduce to the usual ones, and mass fluctuations for the free-particle gas approach the fluctuations in N. The ultrarelativistic limit of the canonical ensemble for the free-particle gas differs from the corresponding limit of the ensemble proposed by Jüttner and Pauli. Due to the mass degree of freedom, the quantum counting of states is different from that of the nonrelativistic theory. If the mass distribution is sufficiently sharp, the thermodynamical effects of this multiplicity will not be large. There may, however, be detectable effects such as a shift in the Fermi level and the critical temperature for Bose-Einstein condensation, and some change in specific heats.     

Equivalence of Gibbs ensembles for quantum systems obeying Maxwell-Boltzmann statistics

A generating functional of the canonical ensemble is introduced for quantum systems of particles obeying Maxwell-Boltzmann statistics. Power estimates for the quantum analogs of the particle distribution functions of the canonical and grand canonical ensembles are obtained for systems of particles with hard cores. It is shown on the basis of the power estimates that limiting generating functionals exist for the systems under consideration and satisfy the same Bogolyubov equation. The ensembles are equivalent in this sense.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–8, July, 1984.The authors thank É. A. Arinshtein for useful discussion.     

The equivalence of ensembles and the Gibbs phase rule for classical lattice systems

We study the relation between the microcanonical, canonical, and grand canonical ensembles in the thermodynamic limit when the system becomes infinite. They are equivalent if there is only one phase in the system. In general it is shown that there is a unique limit of the microcanonical state being a mixture of pure phases if the microcanonical restrictions determine the volume fractions of the phases uniquely, and then the Gibbs phase rule is valid. In this context we show how to define the set of order parameters associated with the state of the system in a natural way.     

Self-similar random processes and infinite-dimensional configuration spaces

We discuss various infinite-dimensional configuration spaces that carry measures quasi-invariant under compactly supported diffeomorphisms of a manifold M corresponding to a physical space. Such measures allow the construction of unitary representations of the diffeomorphism group, which are important to nonrelativistic quantum statistical physics and to the quantum theory of extended objects in M = ?^d. Special attention is given to measurable structure and topology underlying measures on generalized configuration spaces obtained from self-similar random processes (both for d = 1 and d > 1), which describe infinite point configurations having accumulation points.     

Laplace operators on differential forms over configuration spaces

Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given.     

Intermediate filaments in small configuration spaces

Intermediate filaments play a key role in cell mechanics. Apart from their great importance from a biomedical point of view, they also act as a very suitable micrometer-sized model system for semiflexible polymers. We perform a statistical analysis of the thermal fluctuations of individual filaments confined in microchannels. The small channel width and the resulting deflections at the walls give rise to a reduction of the configuration space by about 2 orders of magnitude. This circumstance enables us to precisely measure the intrinsic persistence length of vimentin intermediate filaments and to show that they behave as ideal wormlike chains; we observe that small fluctuations in perpendicular planes decouple. Furthermore, the inclusion of results for confined actin filaments demonstrates that the Odijk confinement regime is valid over at least 1 order of magnitude in persistence length.     

Phase coexistence of cluster crystals: beyond the Gibbs phase rule

We report a study of the phase behavior of multiple-occupancy crystals through simulation. We argue that in order to reproduce the equilibrium behavior of such crystals, it is essential to treat the number of lattice sites as a constraining thermodynamic variable. The resulting free-energy calculations thus differ considerably from schemes used for single-occupancy lattices. Using our approach, we obtain the phase diagram and the bulk modulus for a generalized exponential model that forms cluster crystals at high densities. We compare the simulation results with existing theoretical predictions. We also identify two types of density fluctuations that can lead to two sound modes and evaluate the corresponding elastic constants.     

Orbifolds as configuration spaces of systems with gauge symmetries

In systems like Yang-Mills or gravity theory, which have a symmetry of gauge type, neither phase space nor configuration space is a manifold but rather an orbifold with singular points corresponding to classical states of non-generically higher symmetry. The consequences of these symmetries for quantum theory are investigated. First, a certain orbifold configuration space is identified. Then, the Schrödinger equation on this orbifold is considered. As a typical case, the Schrödinger equation on (double) cones over Riemannian manifolds is discussed in detail as a problem of selfadjoint extensions. A marked tendency towards concentration of the wave function around the singular points in configuration space is observed, which generically even reflects itself in the existence of additional bound states and can be interpreted as a quantum mechanism of symmetry enhancement.     

The decoupling of damped linear systems in configuration and state spaces

It has recently been reported that any viscously damped linear system can be decoupled in the configuration space by a real, nonlinear, time-dependent transformation. The purpose of this rapid communication is to provide a few clarifying remarks about the decoupling operation. It is shown that, for homogeneous systems, the time-dependent configuration-space decoupling transformation is real, linear and time-invariant when cast in state space. In addition, the configuration-space transformation generates a diagonalizing structure-preserving transformation. In non-homogeneous systems, both the configuration and associated state transformations are nonlinear and depend continuously on the excitation. An example is given of a linear system that can be decoupled in configuration but not in state space.     

Generation of GHZ state and cluster state with atomic ensembles via the dipole-blockade mechanism

This paper proposes scalable schemes to generate the Greenberger-Horne-Zeilinger (GHZ) state and the cluster state with atomic ensembles via the dipole blockade mechanism on an atom chip, where the qubit is not carried by a single atom but an atomic ensemble. In the protocols, multiqubit entangled states are determinately prepared. Needlessness for single-photon source further decreases the complexity of the experiment. Based on the present laboratory technique, the schemes may be realized. The achieved results reveal a prospect for large-scale quantum communication and quantum computation.     

Compactness and the maximal Gibbs state for random Gibbs fields on a lattice

We prove for a general class of Gibbsian Random Field on that the set of tempered Gibbs states is compact. This class contains the Euclidean random fields. Moreover if the interaction is attractive, there is a unique minimal and maximal Gibbs state _– and ₊×_± are unique translation invariant ant and have the global Markov property. We also prove that uniqueness of the tempered Gibbs state is equivalent to the magnetizationsm _±=_±(q _x ) being equal which is true if the pressure is differentiable.     

Event-based exploration and comparison on time-varying ensembles

Journal of Visualization - We propose an event-based analysis system for comparison of several ensemble time-varying simulations. In this pipeline, users can customize the selection of events...     

Multifractal Analysis of Weak Gibbs Measures for Intermittent Systems

 In this paper, we establish a multifractal formalism of weak Gibbs measures associated to potentials of weak bounded variation for certain nonhyperbolic systems. We apply our results to Manneville-Pomeau type maps and a piecewise conformal two-dimensional countable Markov map with indifferent periodic points which is related to a complex continued fraction. Received: 6 September 2001 / Accepted: 21 May 2002 Published online: 12 August 2002     

Gibbs entropy and dynamics

Let M be the phase space of a physical system. The dynamics is determined by the map T : M-->M, preserving the measure mu. Let nu be another measure on M, dnu=rho dmu. Gibbs introduced the quantity s(rho)=-integralrho log rho dmu as an analog of the thermodynamical entropy. Attempts to reach a closer analogy between thermodynamical and Gibbs entropy lead to the idea to modify the last one and to replace it by the so-called coarse-grained entropy. The dynamics transforms nu in the following way: nu[mapsto]nu(n), dnu(n)=rho composite functionT(-n)dmu. Hence, we obtain the sequence of densities rho(n)=rho composite functionT(-n) and the corresponding values of the Gibbs and the coarse-grained entropy. We discuss the following question: To what extent the Gibbs and coarse-grained entropy are physical? More precisely: (1) do they grow under the dynamics, generated by T? (2) What properties of the dynamics are responsible for this growth? (3) To what extent can this growth be independent of arbitrariness in the construction of the coarse-grained entropy?     

Dynamical thermostatting and statistical ensembles

Dynamical thermostatting constitutes a procedure for computing thermodynamical mean values of classical dynamical systems that is of interest both from the practical and from the conceptual points of view. Here we extend and unify previous partial results, showing that the dynamical thermostatting approach can be implemented in order to simulate a wide family of statistical ensembles of general dynamical systems with a vanishing divergence and admitting an integral of motion. As a particular illustration, the thermostatting procedure is applied to power law-like maximum entropy ensembles.     

Canonical nonequilibrium ensembles and subdynamics

A method for constructing a canonical nonequilibrium ensemble for systems in which correlations decay exponentially has recently been proposed by Coveney and Penrose. In this paper, we show that the method is equivalent to the subdynamics formalism, developed by Prigogine and others, when the dimension of the subdynamic kinetic subspace is finite. The comparison between the two approaches helps to clarify the nature of the various operators used in the Brussels formalism. We discuss further the relationship between these two approaches, with particular reference to a simple discrete-time dynamical system, based on the baker's transformation, which we call the baker's urn.     

Einstein and ensembles: Response

This article reexamines Einstein's views concerning ensembles and the quantum state function, by way of responding to criticism on this topic. The response calls attention to the range of interpretations found in Einstein's writings, and their function, and emphasizes the nonspecificity of his discussions. It also offers some guidelines for scholarship and criticism in this area.     

Entropic Dynamics on Gibbs Statistical Manifolds

Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles. Here, we develop the entropic dynamics of a system, the state of which is described by a probability distribution. Thus, the dynamics unfolds on a statistical manifold that is automatically endowed by a metric structure provided by information geometry. The curvature of the manifold has a significant influence. We focus our dynamics on the statistical manifold of Gibbs distributions (also known as canonical distributions or the exponential family). The model includes an “entropic” notion of time that is tailored to the system under study; the system is its own clock. As one might expect that entropic time is intrinsically directional; there is a natural arrow of time that is led by entropic considerations. As illustrative examples, we discuss dynamics on a space of Gaussians and the discrete three-state system.