209 research outputs found
On the prevalence of non-Gibbsian states in mathematical physics
Gibbs measures are the main object of study in equilibrium statistical
mechanics, and are used in many other contexts, including dynamical systems and
ergodic theory, and spatial statistics. However, in a large number of natural
instances one encounters measures that are not of Gibbsian form. We present
here a number of examples of such non-Gibbsian measures, and discuss some of
the underlying mathematical and physical issues to which they gave rise
Variational description of Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model
We perform a detailed study of Gibbs-non-Gibbs transitions for the
Curie-Weiss model subject to independent spin-flip dynamics
("infinite-temperature" dynamics). We show that, in this setup, the program
outlined in van Enter, Fern\'andez, den Hollander and Redig can be fully
completed, namely that Gibbs-non-Gibbs transitions are equivalent to
bifurcations in the set of global minima of the large-deviation rate function
for the trajectories of the magnetization conditioned on their endpoint. As a
consequence, we show that the time-evolved model is non-Gibbs if and only if
this set is not a singleton for some value of the final magnetization. A
detailed description of the possible scenarios of bifurcation is given, leading
to a full characterization of passages from Gibbs to non-Gibbs -and vice versa-
with sharp transition times (under the dynamics Gibbsianness can be lost and
can be recovered).
Our analysis expands the work of Ermolaev and Kulske who considered zero
magnetic field and finite-temperature spin-flip dynamics. We consider both zero
and non-zero magnetic field but restricted to infinite-temperature spin-flip
dynamics. Our results reveal an interesting dependence on the interaction
parameters, including the presence of forbidden regions for the optimal
trajectories and the possible occurrence of overshoots and undershoots in the
optimal trajectories. The numerical plots provided are obtained with the help
of MATHEMATICA.Comment: Key words and phrases: Curie-Weiss model, spin-flip dynamics, Gibbs
vs. non-Gibbs, dynamical transition, large deviations, action integral,
bifurcation of rate functio
Gibbs-non-Gibbs transitions via large deviations: computable examples
We give new and explicitly computable examples of Gibbs-non-Gibbs transitions
of mean-field type, using the large deviation approach introduced in [4]. These
examples include Brownian motion with small variance and related diffusion
processes, such as the Ornstein-Uhlenbeck process, as well as birth and death
processes. We show for a large class of initial measures and diffusive dynamics
both short-time conservation of Gibbsianness and dynamical Gibbs-non-Gibbs
transitions
Low-temperature dynamics of the Curie-Weiss Model: Periodic orbits, multiple histories, and loss of Gibbsianness
We consider the Curie-Weiss model at a given initial temperature in vanishing
external field evolving under a Glauber spin-flip dynamics corresponding to a
possibly different temperature. We study the limiting conditional probabilities
and their continuity properties and discuss their set of points of
discontinuity (bad points). We provide a complete analysis of the transition
between Gibbsian and non-Gibbsian behavior as a function of time, extending
earlier work for the case of independent spin-flip dynamics. For initial
temperature bigger than one we prove that the time-evolved measure stays Gibbs
forever, for any (possibly low) temperature of the dynamics. In the regime of
heating to low-temperatures from even lower temperatures, when the initial
temperature is smaller than the temperature of the dynamics, and smaller than
1, we prove that the time-evolved measure is Gibbs initially and becomes
non-Gibbs after a sharp transition time. We find this regime is further divided
into a region where only symmetric bad configurations exist, and a region where
this symmetry is broken. In the regime of further cooling from low-temperatures
there is always symmetry-breaking in the set of bad configurations. These bad
configurations are created by a new mechanism which is related to the
occurrence of periodic orbits for the vector field which describes the dynamics
of Euler-Lagrange equations for the path large deviation functional for the
order parameter. To our knowledge this is the first example of the rigorous
study of non-Gibbsian phenomena related to cooling, albeit in a mean-field
setup.Comment: 31 pages, 24 figure
On the Variational Principle for Generalized Gibbs Measures
We present a novel approach to establishing the variational principle for
Gibbs and generalized (weak and almost) Gibbs states. Limitations of a
thermodynamical formalism for generalized Gibbs states will be discussed. A new
class of intuitively weak Gibbs measures is introduced, and a typical example
is studied. Finally, we present a new example of a non-Gibbsian measure arising
from an industrial application.Comment: To appear in Markov Processes and Related Fields, Proceedings
workshop Gibbs-nonGibb
First-order transitions for very nonlinear sigma models
In this contribution we discuss the occurrence of first-order transitions in
temperature in various short-range lattice models with a rotation symmetry.
Such transitions turn out to be widespread under the condition that the
interaction potentials are sufficiently nonlinear.Comment: Contribution to the Dublin John Lewis memorial meetin
Problems with the definition of renormalized Hamiltonians for momentum-space renormalization transformations
For classical lattice systems with finite (Ising) spins, we show that the
implementation of momentum-space renormalization at the level of Hamiltonians
runs into the same type of difficulties as found for real-space
transformations: Renormalized Hamiltonians are ill-defined in certain regions
of the phase diagram.Comment: 14 pages, late
Clarification of the Bootstrap Percolation Paradox
We study the onset of the bootstrap percolation transition as a model of
generalized dynamical arrest. We develop a new importance-sampling procedure in
simulation, based on rare events around "holes", that enables us to access
bootstrap lengths beyond those previously studied. By framing a new theory in
terms of paths or processes that lead to emptying of the lattice we are able to
develop systematic corrections to the existing theory, and compare them to
simulations. Thereby, for the first time in the literature, it is possible to
obtain credible comparisons between theory and simulation in the accessible
density range.Comment: 4 pages with 3 figure
First-order transitions for n-vector models in two and more dimensions; rigorous proof
We prove that various SO(n)-invariant n-vector models with interactions which
have a deep and narrow enough minimum have a first-order transition in the
temperature. The result holds in dimension two or more, and is independent on
the nature of the low-temperature phase.Comment: late
Discrete approximations to vector spin models
We strengthen a result of two of us on the existence of effective
interactions for discretised continuous-spin models. We also point out that
such an interaction cannot exist at very low temperatures. Moreover, we compare
two ways of discretising continuous-spin models, and show that, except for very
low temperatures, they behave similarly in two dimensions. We also discuss some
possibilities in higher dimensions.Comment: 12 page
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