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

Small object limit of Casimir effect and the sign of the Casimir force

We show a simple way of deriving the Casimir Polder interaction, present some general arguments on the finiteness and sign of mutual Casimir interactions and finally we derive a simple expression for Casimir radiation from small accelerated objects.Comment: 13 pages, late

Does the quark-gluon plasma contain stable hadronic bubbles?

We calculate the thermodynamic potential of bubbles of hadrons embedded in quark-gluon plasma, and of droplets of quark-gluon plasma embedded in hadron phase. This is a generalization of our previous results to the case of non-zero chemical potentials. As in the zero chemical potential case, we find that a quark-gluon plasma in thermodynamic equilibrium may contain stable bubbles of hadrons of radius $R \simeq 1$ fm. The calculations are performed within the MIT Bag model, using an improved multiple reflection expansion. The results are of relevance for neutron star phenomenology and for ultrarelativistic heavy ion collisions.Comment: 12 pages including 8 figures. To appear in Phys. Rev.

Charles Coulston GILLISPIE, Raffaele PISANO, Lazare and Sadi Carnot: A scientific and filial relationship,

Article d'étude sur le livre Charles Coulston GILLISPIE, Raffaele PISANO, Lazare and Sadi Carnot: A scientific and filial relationship,Car-not : A scientific and filial relationship, 2 de édition (Dordrecht : Springer, 2014), XVI-490 p., illustr. n. et bl. et coul., bibliogr., in-dex, « History of mechanism and machine science », vol. 19. Ce livre, « extraordinaire » comme l'écrit Eberhard Knobloch dans sa préface, porte principalement sur un événement unique dans l'histoire, la création d'une science nouvelle. Comme chacun sait, cette création de toutes pièces de la ther-modynamique est marquée par la publication en 1824 des Réflexions sur la puis-sance motrice du feu et sur les machines propres à développer cette puissance de Sadi Carnot (1796-1832). Une bonne partie du présent ouvrage est consacrée à la genèse de ce livre fondateur. Malgré le nombre immense d'études ayant porté sur cette création de la thermodynamique, les auteurs nous présentent une étude d'une grande originalité, dont l'une des lignes directrices est le rôle joué par les idées de Lazare Carnot (1753-1823) sur la pensée de son fils. Le début de l'ouvrage, qui donc concerne l'oeuvre scientifique de Lazare Carnot

The quantum measurement process in an exactly solvable model

An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The $z$-component $\hat s_z$ of a spin-1/2 is measured with an apparatus, which itself consists of magnet coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of $s_z$ of the tested spin. The quantum measurement goes in two steps. On a timescale $1/\sqrt{N}$ the off-diagonal elements of the spin's density matrix vanish due to a unitary evolution of the tested spin and the $N$ apparatus spins; on a larger but still short timescale this is made definite by the bath. Then the system is in a `classical' state, having a diagonal density matrix. The registration of that state is a quantum process which can already be understood from classical statistical mechanics. The von Neumann collapse and the Born rule are derived rather than postulated.Comment: 7 pages revtex, 2 figure

Billiard Systems in Three Dimensions: The Boundary Integral Equation and the Trace Formula

We derive semiclassical contributions of periodic orbits from a boundary integral equation for three-dimensional billiard systems. We use an iterative method that keeps track of the composition of the stability matrix and the Maslov index as an orbit is traversed. Results are given for isolated periodic orbits and rotationally invariant families of periodic orbits in axially symmetric billiard systems. A practical method for determining the stability matrix and the Maslov index is described.Comment: LaTeX, 19 page

New Bardeen-Cooper-Schrieffer-type theory at finite temperature with particle-number conservation

We formulate a new Bardeen-Cooper-Schrieffer (BCS)-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. With its broad applicability, this theory can be a useful tool for investigating the pairing phase transition in finite systems with the particle-number conservation. This theory provides effects of the symmetry-restoring fluctuation (SRF) for the pairing phenomena in finite fermionic systems, distinctively from those of additional quantum fluctuations. It is shown by numerical calculations that the phase transition is compatible with the conservation in this theory, and that the SRF shifts up the critical temperature ($T^\mathrm{cr}$). This shift of $T^\mathrm{cr}$ occurs due to reduction of degrees-of-freedom in canonical ensembles, and decreases only slowly as the particle-number increases (or as the level spacing narrows), in contrast to the conventional BCS theory.Comment: 10 pages including 3 figures, to be published in Phys. Rev.

Mean-field theory of quantum brownian motion

We investigate a mean-field approach to a quantum brownian particle interacting with a quantum thermal bath at temperature $T$, and subjected to a non-linear potential. An exact, partially classical description of quantum brownian motion is proposed, which uses negative probabilities in its intermediate steps. It is shown that properties of the quantum particle can be mapped to those of two classical brownian particles in a common potential, where one of them interacts with the quantum bath, whereas another one interacts with a classical bath at zero temperature. Due to damping the system allows a unique and non-singular classical limit at $\hbar \to 0$. For high $T$ the stationary state becomes explicitly classical. The low-temperature case is studied through an effective Fokker-Planck equation. Non-trivial purely quantum correlation effects between the two particles are found.Comment: 13 pages, 0 figures, revte