777 research outputs found
Ensemble inequivalence: A formal approach
Ensemble inequivalence has been observed in several systems. In particular it
has been recently shown that negative specific heat can arise in the
microcanonical ensemble in the thermodynamic limit for systems with long-range
interactions. We display a connection between such behaviour and a mean-field
like structure of the partition function. Since short-range models cannot
display this kind of behaviour, this strongly suggests that such systems are
necessarily non-mean field in the sense indicated here. We further show that a
broad class of systems with non-integrable interactions are indeed of
mean-field type in the sense specified, so that they are expected to display
ensemble inequivalence as well as the peculiar behaviour described above in the
microcanonical ensemble.Comment: 4 pages, no figures, given at the NEXT2001 conference on
non-extensive thermodynamic
Accompagner le processus de changement lors d'une fusion : réflexions autour d’une expérience d’intervention psychosociale en organisation
Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems
Classical spin systems with nonadditive long-range interactions are studied
in the microcanonical ensemble. It is expected that the entropy of such a
system is identical to that of the corresponding mean-field model, which is
called "exactness of the mean-field theory". It is found out that this
expectation is not necessarily true if the microcanonical ensemble is not
equivalent to the canonical ensemble in the mean-field model. Moreover,
necessary and sufficient conditions for exactness of the mean-field theory are
obtained. These conditions are investigated for two concrete models, the
\alpha-Potts model with annealed vacancies and the \alpha-Potts model with
invisible states.Comment: 23 pages, to appear in J. Stat. Phy
Ensemble Inequivalence in Mean-field Models of Magnetism
Mean-field models, while they can be cast into an {\it extensive}
thermodynamic formalism, are inherently {\it non additive}. This is the basic
feature which leads to {\it ensemble inequivalence} in these models. In this
paper we study the global phase diagram of the infinite range
Blume-Emery-Griffiths model both in the {\it canonical} and in the {\it
microcanonical} ensembles. The microcanonical solution is obtained both by
direct state counting and by the application of large deviation theory. The
canonical phase diagram has first order and continuous transition lines
separated by a tricritical point. We find that below the tricritical point,
when the canonical transition is first order, the phase diagrams of the two
ensembles disagree. In this region the microcanonical ensemble exhibits energy
ranges with negative specific heat and temperature jumps at transition
energies. These two features are discussed in a general context and the
appropriate Maxwell constructions are introduced. Some preliminary extensions
of these results to weakly decaying nonintegrable interactions are presented.Comment: Chapter of the forthcoming "Lecture Notes in Physics" volume:
``Dynamics and Thermodynamics of Systems with Long Range Interactions'', T.
Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics
Vol. 602, Springer (2002). (see http://link.springer.de/series/lnpp/
An Introduction to the Thermodynamic and Macrostate Levels of Nonequivalent Ensembles
This short paper presents a nontechnical introduction to the problem of
nonequivalent microcanonical and canonical ensembles. Both the thermodynamic
and the macrostate levels of definition of nonequivalent ensembles are
introduced. The many relationships that exist between these two levels are also
explained in simple physical terms.Comment: Revtex4, 5 pages, 1 figur
Breathing mode for systems of interacting particles
We study the breathing mode in systems of trapped interacting particles. Our
approach, based on a dynamical ansatz in the first equation of the
Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy allows us to tackle at
once a wide range of power law interactions and interaction strengths, at
linear and non linear levels. This both puts in a common framework various
results scattered in the literature, and by widely generalizing these,
emphasizes universal characters of this breathing mode. Our findings are
supported by direct numerical simulations.Comment: 4 pages, 4 figure
Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model
We illustrate a novel characterization of nonequivalent statistical
mechanical ensembles using the mean-field Blume-Emery-Griffiths (BEG) model as
a test model. The novel characterization takes effect at the level of the
microcanonical and canonical equilibrium distributions of states. For this
reason it may be viewed as a statistical characterization of nonequivalent
ensembles which extends and complements the common thermodynamic
characterization of nonequivalent ensembles based on nonconcave anomalies of
the microcanonical entropy. By computing numerically both the microcanonical
and canonical sets of equilibrium distributions of states of the BEG model, we
show that for values of the mean energy where the microcanonical entropy is
nonconcave, the microcanonical distributions of states are nowhere realized in
the canonical ensemble. Moreover, we show that for values of the mean energy
where the microcanonical entropy is strictly concave, the equilibrium
microcanonical distributions of states can be put in one-to-one correspondence
with equivalent canonical equilibrium distributions of states. Our numerical
computations illustrate general results relating thermodynamic and statistical
equivalence and nonequivalence of ensembles proved by Ellis, Haven, and
Turkington [J. Stat. Phys. 101, 999 (2000)].Comment: 13 pages, 4 figures, minor typos corrected and one reference adde
Inhomogeneous Quasi-stationary States in a Mean-field Model with Repulsive Cosine Interactions
The system of N particles moving on a circle and interacting via a global
repulsive cosine interaction is well known to display spatially inhomogeneous
structures of extraordinary stability starting from certain low energy initial
conditions. The object of this paper is to show in a detailed manner how these
structures arise and to explain their stability. By a convenient canonical
transformation we rewrite the Hamiltonian in such a way that fast and slow
variables are singled out and the canonical coordinates of a collective mode
are naturally introduced. If, initially, enough energy is put in this mode, its
decay can be extremely slow. However, both analytical arguments and numerical
simulations suggest that these structures eventually decay to the spatially
uniform equilibrium state, although this can happen on impressively long time
scales. Finally, we heuristically introduce a one-particle time dependent
Hamiltonian that well reproduces most of the observed phenomenology.Comment: to be published in J. Phys.
Lyapunov exponents as a dynamical indicator of a phase transition
We study analytically the behavior of the largest Lyapunov exponent
for a one-dimensional chain of coupled nonlinear oscillators, by
combining the transfer integral method and a Riemannian geometry approach. We
apply the results to a simple model, proposed for the DNA denaturation, which
emphasizes a first order-like or second order phase transition depending on the
ratio of two length scales: this is an excellent model to characterize
as a dynamical indicator close to a phase transition.Comment: 8 Pages, 3 Figure
Algebraic damping in the one-dimensional Vlasov equation
We investigate the asymptotic behavior of a perturbation around a spatially
non homogeneous stable stationary state of a one-dimensional Vlasov equation.
Under general hypotheses, after transient exponential Landau damping, a
perturbation evolving according to the linearized Vlasov equation decays
algebraically with the exponent -2 and a well defined frequency. The
theoretical results are successfully tested against numerical -body
simulations, corresponding to the full Vlasov dynamics in the large limit,
in the case of the Hamiltonian mean-field model. For this purpose, we use a
weighted particles code, which allows us to reduce finite size fluctuations and
to observe the asymptotic decay in the -body simulations.Comment: 26 pages, 8 figures; text slightly modified, references added, typos
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