29 research outputs found
Nonconcave entropies from generalized canonical ensembles
It is well-known that the entropy of the microcanonical ensemble cannot be
calculated as the Legendre transform of the canonical free energy when the
entropy is nonconcave. To circumvent this problem, a generalization of the
canonical ensemble which allows for the calculation of nonconcave entropies was
recently proposed. Here, we study the mean-field Curie-Weiss-Potts spin model
and show, by direct calculations, that the nonconcave entropy of this model can
be obtained by using a specific instance of the generalized canonical ensemble
known as the Gaussian ensemble.Comment: 5 pages, RevTeX4, 3 figures (best viewed in ps
The extended gaussian ensemble and metastabilities in the Blume-Capel model
The Blume-Capel model with infinite-range interactions presents analytical
solutions in both canonical and microcanonical ensembles and therefore, its
phase diagram is known in both ensembles. This model exhibits nonequivalent
solutions and the microcanonical thermodynamical features present peculiar
behaviors like nonconcave entropy, negative specific heat, and a jump in the
thermodynamical temperature. Examples of nonequivalent ensembles are in general
related to systems with long-range interactions that undergo canonical
first-order phase transitions. Recently, the extended gaussian ensemble (EGE)
solution was obtained for this model. The gaussian ensemble and its extended
version can be considered as a regularization of the microcanonical ensemble.
They are known to play the role of an interpolating ensemble between the
microcanonical and the canonical ones. Here, we explicitly show how the
microcanonical energy equilibrium states related to the metastable and unstable
canonical solutions for the Blume-Capel model are recovered from EGE, which
presents a concave "extended" entropy as a function of energy.Comment: 6 pages, 5 eps figures. Presented at the XI Latin American Workshop
on Nonlinear Phenomena, October 05-09 (2009), B\'uzios (RJ), Brazil. To
appear in JPC
Extended gaussian ensemble solution and tricritical points of a system with long-range interactions
The gaussian ensemble and its extended version theoretically play the
important role of interpolating ensembles between the microcanonical and the
canonical ensembles. Here, the thermodynamic properties yielded by the extended
gaussian ensemble (EGE) for the Blume-Capel (BC) model with infinite-range
interactions are analyzed. This model presents different predictions for the
first-order phase transition line according to the microcanonical and canonical
ensembles. From the EGE approach, we explicitly work out the analytical
microcanonical solution. Moreover, the general EGE solution allows one to
illustrate in details how the stable microcanonical states are continuously
recovered as the gaussian parameter is increased. We found out that it
is not necessary to take the theoretically expected limit
to recover the microcanonical states in the region between the canonical and
microcanonical tricritical points of the phase diagram. By analyzing the
entropy as a function of the magnetization we realize the existence of
unaccessible magnetic states as the energy is lowered, leading to a treaking of
ergodicity.Comment: 8 pages, 5 eps figures. Title modified, sections rewritten,
tricritical point calculations added. To appear in EPJ
The generalized canonical ensemble and its universal equivalence with the microcanonical ensemble
This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C2, then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave
Generalized canonical ensembles and ensemble equivalence
This paper is a companion article to our previous paper (J. Stat. Phys. 119,
1283 (2005), cond-mat/0408681), which introduced a generalized canonical
ensemble obtained by multiplying the usual Boltzmann weight factor of the canonical ensemble with an exponential factor involving a continuous
function of the Hamiltonian . We provide here a simplified introduction
to our previous work, focusing now on a number of physical rather than
mathematical aspects of the generalized canonical ensemble. The main result
discussed is that, for suitable choices of , the generalized canonical
ensemble reproduces, in the thermodynamic limit, all the microcanonical
equilibrium properties of the many-body system represented by even if this
system has a nonconcave microcanonical entropy function. This is something that
in general the standard () canonical ensemble cannot achieve. Thus a
virtue of the generalized canonical ensemble is that it can be made equivalent
to the microcanonical ensemble in cases where the canonical ensemble cannot.
The case of quadratic -functions is discussed in detail; it leads to the
so-called Gaussian ensemble.Comment: 8 pages, 4 figures (best viewed in ps), revtex4. Changes in v2: Title
changed, references updated, new paragraph added, minor differences with
published versio
Asymptotics of the mean-field Heisenberg model
We consider the mean-field classical Heisenberg model and obtain detailed
information about the total spin of the system by studying the model on a
complete graph and sending the number of vertices to infinity. In particular,
we obtain Cramer- and Sanov-type large deviations principles for the total spin
and the empirical spin distribution and demonstrate a second-order phase
transition in the Gibbs measures. We also study the asymptotics of the total
spin throughout the phase transition using Stein's method, proving central
limit theorems in the sub- and supercritical phases and a nonnormal limit
theorem at the critical temperature.Comment: 44 page
Nonconcave entropies in multifractals and the thermodynamic formalism
We discuss a subtlety involved in the calculation of multifractal spectra
when these are expressed as Legendre-Fenchel transforms of functions analogous
to free energy functions. We show that the Legendre-Fenchel transform of a free
energy function yields the correct multifractal spectrum only when the latter
is wholly concave. If the spectrum has no definite concavity, then the
transform yields the concave envelope of the spectrum rather than the spectrum
itself. Some mathematical and physical examples are given to illustrate this
result, which lies at the root of the nonequivalence of the microcanonical and
canonical ensembles. On a more positive note, we also show that the
impossibility of expressing nonconcave multifractal spectra through
Legendre-Fenchel transforms of free energies can be circumvented with the help
of a generalized free energy function, which relates to a recently introduced
generalized canonical ensemble. Analogies with the calculation of rate
functions in large deviation theory are finally discussed.Comment: 9 pages, revtex4, 3 figures. Changes in v2: sections added on
applications plus many new references; contains an addendum not contained in
published versio
Mean-field driven first-order phase transitions in systems with long-range interactions
We consider a class of spin systems on with vector valued spins
(\bS_x) that interact via the pair-potentials J_{x,y} \bS_x\cdot\bS_y. The
interactions are generally spread-out in the sense that the 's exhibit
either exponential or power-law fall-off. Under the technical condition of
reflection positivity and for sufficiently spread out interactions, we prove
that the model exhibits a first-order phase transition whenever the associated
mean-field theory signals such a transition. As a consequence, e.g., in
dimensions , we can finally provide examples of the 3-state Potts model
with spread-out, exponentially decaying interactions, which undergoes a
first-order phase transition as the temperature varies. Similar transitions are
established in dimensions for power-law decaying interactions and in
high dimensions for next-nearest neighbor couplings. In addition, we also
investigate the limit of infinitely spread-out interactions. Specifically, we
show that once the mean-field theory is in a unique ``state,'' then in any
sequence of translation-invariant Gibbs states various observables converge to
their mean-field values and the states themselves converge to a product
measure.Comment: 57 pages; uses a (modified) jstatphys class fil
Termodinâmica do modelo de Ising com interações de alcance infinito via ensemble canônico generalizado
Glauber Dynamics for the mean-field Potts Model
We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with
states and show that it undergoes a critical slowdown at an
inverse-temperature strictly lower than the critical
for uniqueness of the thermodynamic limit. The dynamical critical
is the spinodal point marking the onset of metastability.
We prove that when the mixing time is asymptotically
and the dynamics exhibits the cutoff phenomena, a sharp
transition in mixing, with a window of order . At the
dynamics no longer exhibits cutoff and its mixing obeys a power-law of order
. For the mixing time is exponentially large in
. Furthermore, as with , the mixing time
interpolates smoothly from subcritical to critical behavior, with the latter
reached at a scaling window of around . These results
form the first complete analysis of mixing around the critical dynamical
temperature --- including the critical power law --- for a model with a first
order phase transition.Comment: 45 pages, 5 figure