On the Poisson Structure of the Time-Dependent Mean-Field Equations for Systems of Bosons out of Equilibrium

We analyze the Poisson structure of the time-dependent mean-field equations for bosons and construct the Lie-Poisson bracket associated to these equations. The latter follow from the time-dependent variational principle of Balian and Veneroni when a gaussian Ansatz is chosen for the density operator. We perform a stability analysis of both the full and the linearized equations. We also search for the canonically conjugate variables. In certain cases, the evolution equations can indeed be cast in a Hamiltonian form.Comment: 21 pages. To appear in Annals of Physic

Information in statistical physics

We review with a tutorial scope the information theory foundations of quantum statistical physics. Only a small proportion of the variables that characterize a system at the microscopic scale can be controlled, for both practical and theoretical reasons, and a probabilistic description involving the observers is required. The criterion of maximum von Neumann entropy is then used for making reasonable inferences. It means that no spurious information is introduced besides the known data. Its outcomes can be given a direct justification based on the principle of indifference of Laplace. We introduce the concept of relevant entropy associated with some set of relevant variables; it characterizes the information that is missing at the microscopic level when only these variables are known. For equilibrium problems, the relevant variables are the conserved ones, and the Second Law is recovered as a second step of the inference process. For non-equilibrium problems, the increase of the relevant entropy expresses an irretrievable loss of information from the relevant variables towards the irrelevant ones. Two examples illustrate the flexibility of the choice of relevant variables and the multiplicity of the associated entropies: the thermodynamic entropy (satisfying the Clausius-Duhem inequality) and the Boltzmann entropy (satisfying the H-theorem). The identification of entropy with missing information is also supported by the paradox of Maxwell's demon. Spin-echo experiments show that irreversibility itself is not an absolute concept: use of hidden information may overcome the arrow of time.Comment: latex InfoStatPhys-unix.tex, 3 files, 2 figures, 32 pages http://www-spht.cea.fr/articles/T04/18

Incomplete descriptions and relevant entropies

Statistical mechanics relies on the complete though probabilistic description of a system in terms of all the microscopic variables. Its object is to derive therefrom static and dynamic properties involving some reduced set of variables. The elimination of the irrelevant variables is guided by the maximum entropy criterion, which produces the probability law carrying the least amount of information compatible with the relevant variables. This defines relevant entropies which measure the missing information (the disorder) associated with the sole variables retained in an incomplete description. Relevant entropies depend not only on the state of the system but also on the coarseness of its reduced description. Their use sheds light on questions such as the Second Law, both in equilibrium an in irreversible thermodynamics, the projection method of statistical mechanics, Boltzmann's \textit{H}-theorem or spin-echo experiment.Comment: flatex relevant_entropies.tex, 1 file Submitted to: Am. J. Phy

The Poisson structure of the mean-field equations in the Phi^4 theory

We show that the mean-field time dependent equations in the Phi^4 theory can be put into a classical non-canonical hamiltonian framework with a Poisson structure which is a generalization of the standard Poisson bracket. The Heisenberg invariant appears as a structural invariant of the Poisson tensor. (To be pubished in Annals of Physics)Comment: 12 pages Te

Geometry of the Casimir Effect

When the vacuum is partitioned by material boundaries with arbitrary shape, one can define the zero-point energy and the free energy of the electromagnetic waves in it: this can be done, independently of the nature of the boundaries, in the limit that they become perfect conductors, provided their curvature is finite. The first examples we consider are Casimir's original configuration of parallel plates, and the experimental situation of a sphere in front of a plate. For arbitrary geometries, we give an explicit expression for the zero-point energy and the free energy in terms of an integral kernel acting on the boundaries; it can be expanded in a convergent series interpreted as a succession of an even number of scatterings of a wave. The quantum and thermal fluctuations of vacuum then appear as a purely geometric property. The Casimir effect thus defined exists only owing to the electromagnetic nature of the field. It does not exist for thin foils with sharp folds, but Casimir forces between solid wedges are finite. We work out various applications: low temperature, high temperature where wrinkling constraints appear, stability of a plane foil, transfer of energy from one side of a curved boundary to the other, forces between distant conductors, special shapes (parallel plates, sphere, cylinder, honeycomb).Comment: 44 pages, 8 figures; Proceedings of the 15 th SIGRAV Conference on General Relativity and Gravitational Physics, Villa Mondragone, Monte Porzio Catone, Roma, Italy, September 9-12, 200

Correlation functions from a unified variational principle: trial Lie groups

Time-dependent expectation values and correlation functions for many-body quantum systems are evaluated by means of a unified variational principle. It optimizes a generating functional depending on sources associated with the observables of interest $\ldots$Comment: 42 page

Lattice gauge theory: A retrospective

I discuss some of the historical circumstances that drove us to use the lattice as a non-perturbative regulator. This approach has had immense success, convincingly demonstrating quark confinement and obtaining crucial properties of the strong interactions from first principles. I wrap up with some challenges for the future.Comment: Lattice 2000 (Plenary), 9 pages, 7 figure

Stars and statistical physics: a teaching experience

The physics of stars, their workings and their evolution, is a goldmine of problems in statistical mechanics and thermodynamics. We discuss many examples that illustrate the possibility of deepening student's knowledge of statistical mechanics by an introductory study of stars. The matter constituting the various stellar objects provides examples of equations of state for classical or quantal and relativistic or non-relativistic gases. Maximum entropy can be used to characterize thermodynamic and gravitational equilibrium which determines the structure of stars and predicts their instability above a certain mass. Contraction accompanying radiation induces either heating or cooling, which explains the formation of stars above a minimum mass. The characteristics of the emitted light are understood from black-body radiation and more precisely from the Boltzmann-Lorentz kinetic equation for photons. The luminosity is governed by the transport of heat by photons from the center to the surface. Heat production by thermonuclear fusion is determined by microscopic balance equations. The stability of the steady state of stars is controlled by the interplay of thermodynamics and gravitation.Comment: latex gould_last.tex, 4 files, submitted to Am. J. Phy

Inverse spectral problem for analytic plane domains I: Balian-Bloch trace formula

We give a rigorous version of the classical Balian-Bloch trace formula, a semiclassical expansion around a periodic reflecting ray of the (regularized) resolvent of the Dirichlet Laplacian on a bounded smooth plane domain. It is equivalent to the Poisson relation (or wave trace formula) between spectrum and closed geodesics. We view it primarily as a computational device for explicitly calculating wave trace invariants. Its effectiveness will be illustrated in subsquent articles in the series in which concrete inverse spectral results are proved.Comment: First in a series on the inverse spectral problem for analytic plane domains. 53 pages, 1 figure. Added some reference