74 research outputs found
High-Temperature Expansions of Bures and Fisher Information Priors
For certain infinite and finite-dimensional thermal systems, we obtain ---
incorporating quantum-theoretic considerations into Bayesian thermostatistical
investigations of Lavenda --- high-temperature expansions of priors over
inverse temperature beta induced by volume elements ("quantum Jeffreys' priors)
of Bures metrics. Similarly to Lavenda's results based on volume elements
(Jeffreys' priors) of (classical) Fisher information metrics, we find that in
the limit beta -> 0, the quantum-theoretic priors either conform to Jeffreys'
rule for variables over [0,infinity], by being proportional to 1/beta, or to
the Bayes-Laplace principle of insufficient reason, by being constant. Whether
a system adheres to one rule or to the other appears to depend upon its number
of degrees of freedom.Comment: Six pages, LaTeX. The title has been shortened (and then further
modified), at the suggestion of a colleague. Other minor change
Perturbation and Variational Methods in Nonextensive Tsallis Statistics
A unified presentation of the perturbation and variational methods for the
generalized statistical mechanics based on Tsallis entropy is given here. In
the case of the variational method, the Bogoliubov inequality is generalized in
a very natural way following the Feynman proof for the usual statistical
mechanics. The inequality turns out to be form-invariant with respect to the
entropic index . The method is illustrated with a simple example in
classical mechanics. The formalisms developed here are expected to be useful in
the discussion of nonextensive systems.Comment: revte
Generalised-Lorentzian Thermodynamics
We extend the recently developed non-gaussian thermodynamic formalism
\cite{tre98} of a (presumably strongly turbulent) non-Markovian medium to its
most general form that allows for the formulation of a consistent thermodynamic
theory. All thermodynamic functions, including the definition of the
temperature, are shown to be meaningful. The thermodynamic potential from which
all relevant physical information in equilibrium can be extracted, is defined
consistently. The most important findings are the following two: (1) The
temperature is defined exactly in the same way as in classical statistical
mechanics as the derivative of the energy with respect to the entropy at
constant volume. (2) Observables are defined in the same way as in Boltzmannian
statistics as the linear averages of the new equilibrium distribution function.
This lets us conclude that the new state is a real thermodynamic equilibrium in
systems capable of strong turbulence with the new distribution function
replacing the Boltzmann distribution in such systems. We discuss the ideal gas,
find the equation of state, and derive the specific heat and adiabatic exponent
for such a gas. We also derive the new Gibbsian distribution of states. Finally
we discuss the physical reasons for the development of such states and the
observable properties of the new distribution function.Comment: 13 pages, 1 figur
Boltzmann-Shannon Entropy: Generalization and Application
The paper deals with the generalization of both Boltzmann entropy and
distribution in the light of most-probable interpretation of statistical
equilibrium. The statistical analysis of the generalized entropy and
distribution leads to some new interesting results of significant physical
importance.Comment: 5 pages, Accepted in Mod.Phys.Lett.
Average Entropy of a Subsystem from its Average Tsallis Entropy
In the nonextensive Tsallis scenario, Page's conjecture for the average
entropy of a subsystem[Phys. Rev. Lett. {\bf 71}, 1291(1993)] as well as its
demonstration are generalized, i.e., when a pure quantum system, whose Hilbert
space dimension is , is considered, the average Tsallis entropy of an
-dimensional subsystem is obtained. This demonstration is expected to be
useful to study systems where the usual entropy does not give satisfactory
results.Comment: Revtex, 6 pages, 2 figures. To appear in Phys. Rev.
Steady shear flow thermodynamics based on a canonical distribution approach
A non-equilibrium steady state thermodynamics to describe shear flows is
developed using a canonical distribution approach. We construct a canonical
distribution for shear flow based on the energy in the moving frame using the
Lagrangian formalism of the classical mechanics. From this distribution we
derive the Evans-Hanley shear flow thermodynamics, which is characterized by
the first law of thermodynamics relating infinitesimal
changes in energy , entropy and shear rate with kinetic
temperature . Our central result is that the coefficient is given by
Helfand's moment for viscosity. This approach leads to thermodynamic stability
conditions for shear flow, one of which is equivalent to the positivity of the
correlation function of . We emphasize the role of the external work
required to sustain the steady shear flow in this approach, and show
theoretically that the ensemble average of its power must be
non-negative. A non-equilibrium entropy, increasing in time, is introduced, so
that the amount of heat based on this entropy is equal to the average of
. Numerical results from non-equilibrium molecular dynamics simulation
of two-dimensional many-particle systems with soft-core interactions are
presented which support our interpretation.Comment: 23 pages, 7 figure
Derivation of the Blackbody Radiation Spectrum from a Natural Maximum-Entropy Principle Involving Casimir Energies and Zero-Point Radiation
By numerical calculation, the Planck spectrum with zero-point radiation is
shown to satisfy a natural maximum-entropy principle whereas alternative
choices of spectra do not. Specifically, if we consider a set of
conducting-walled boxes, each with a partition placed at a different location
in the box, so that across the collection of boxes the partitions are uniformly
spaced across the volume, then the Planck spectrum correspond to that spectrum
of random radiation (having constant energy kT per normal mode at low
frequencies and zero-point energy (1/2)hw per normal mode at high frequencies)
which gives maximum uniformity across the collection of boxes for the radiation
energy per box. The analysis involves Casimir energies and zero-point radiation
which do not usually appear in thermodynamic analyses. For simplicity, the
analysis is presented for waves in one space dimension.Comment: 11 page
Notes on the Third Law of Thermodynamics.I
We analyze some aspects of the third law of thermodynamics. We first review
both the entropic version (N) and the unattainability version (U) and the
relation occurring between them. Then, we heuristically interpret (N) as a
continuity boundary condition for thermodynamics at the boundary T=0 of the
thermodynamic domain. On a rigorous mathematical footing, we discuss the third
law both in Carath\'eodory's approach and in Gibbs' one. Carath\'eodory's
approach is fundamental in order to understand the nature of the surface T=0.
In fact, in this approach, under suitable mathematical conditions, T=0 appears
as a leaf of the foliation of the thermodynamic manifold associated with the
non-singular integrable Pfaffian form . Being a leaf, it cannot
intersect any other leaf const. of the foliation. We show that (N) is
equivalent to the requirement that T=0 is a leaf. In Gibbs' approach, the
peculiar nature of T=0 appears to be less evident because the existence of the
entropy is a postulate; nevertheless, it is still possible to conclude that the
lowest value of the entropy has to belong to the boundary of the convex set
where the function is defined.Comment: 29 pages, 2 figures; RevTex fil
The imprints of superstatistics in multiparticle production processes
We provide an update of the overview of imprints of Tsallis nonextensive
statistics seen in a multiparticle production processes. They reveal an
ubiquitous presence of power law distributions of different variables
characterized by the nonextensivity parameter q > 1. In nuclear collisions one
additionally observes a q-dependence of the multiplicity fluctuations
reflecting the finiteness of the hadronizing source. We present sum rules
connecting parameters q obtained from an analysis of different observables,
which allows us to combine different kinds of fluctuations seen in the data and
analyze an ensemble in which the energy (E), temperature (T) and multiplicity
(N) can all fluctuate. This results in a generalization of the so called
Lindhard's thermodynamic uncertainty relation. Finally, based on the example of
nucleus-nucleus collisions (treated as a quasi-superposition of nucleon-nucleon
collisions) we demonstrate that, for the standard Tsallis entropy with degree
of nonextensivity q < 1, the corresponding standard Tsallis distribution is
described by q' = 2 - q > 1.Comment: 12 pages, 3 figures. Based on invited talk given by Z.Wlodarczyk at
SigmaPhi2011 conference, Larnaka, Cyprus, 11-15 July 2011. To be published in
Cent. Eur. J. Phys. (2011
Gibbs' Paradox according to Gibbs and slightly beyond
The so-called Gibbs paradox is a paradigmatic narrative illustrating the necessity to account for
the N! ways of permuting N identical particles when summing over microstates. Yet, there exist
some mixing scenarios for which the expected thermodynamic outcome depends on the viewpoint
one chooses to justify this combinatorial term. After a brief summary on Gibbs' paradox and what is
the standard rationale used to justify its resolution, we will allow ourself to question from a historical
standpoint whether the Gibbs paradox has actually anything to do with Gibbs' work. In so doing,
we also aim at shedding a new light with regards to some of the theoretical claims surrounding its
resolution. We will then turn to the statistical thermodynamics of discrete and continuous mixtures
and introduce the notion of composition entropy to characterise these systems. This will enable us to
address, in a certain sense, a "curiosity" pointed out by Gibbs in a paper published in 1876. Finally,
we will �nish by proposing a connexion between the results we propose and a recent extension of
the Landauer bound regarding the minimum amount of heat to be dissipated to reset one bit of
memory
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