150,884 research outputs found
Functional Integration Approach to Hysteresis
A general formulation of scalar hysteresis is proposed. This formulation is
based on two steps. First, a generating function g(x) is associated with an
individual system, and a hysteresis evolution operator is defined by an
appropriate envelope construction applied to g(x), inspired by the overdamped
dynamics of systems evolving in multistable free energy landscapes. Second, the
average hysteresis response of an ensemble of such systems is expressed as a
functional integral over the space G of all admissible generating functions,
under the assumption that an appropriate measure m has been introduced in G.
The consequences of the formulation are analyzed in detail in the case where
the measure m is generated by a continuous, Markovian stochastic process. The
calculation of the hysteresis properties of the ensemble is reduced to the
solution of the level-crossing problem for the stochastic process. In
particular, it is shown that, when the process is translationally invariant
(homogeneous), the ensuing hysteresis properties can be exactly described by
the Preisach model of hysteresis, and the associated Preisach distribution is
expressed in closed analytic form in terms of the drift and diffusion
parameters of the Markovian process. Possible applications of the formulation
are suggested, concerning the interpretation of magnetic hysteresis due to
domain wall motion in quenched-in disorder, and the interpretation of critical
state models of superconducting hysteresis.Comment: 36 pages, 9 figures, to be published on Phys. Rev.
An inequality for the distance between densities of free convolutions
This paper contributes to the study of the free additive convolution of
probability measures. It shows that under some conditions, if measures
and , are close to each other in terms of the L\'{e}vy metric and
if the free convolution is sufficiently smooth, then
is absolutely continuous, and the densities of measures
and are close to each other. In
particular, convergence in distribution
implies that the density of
is defined for all sufficiently large and
converges to the density of . Some applications are
provided, including: (i) a new proof of the local version of the free central
limit theorem, and (ii) new local limit theorems for sums of free projections,
for sums of -stable random variables and for eigenvalues of a sum of
two -by- random matrices.Comment: Published in at http://dx.doi.org/10.1214/12-AOP756 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Functional Classical Mechanics and Rational Numbers
The notion of microscopic state of the system at a given moment of time as a
point in the phase space as well as a notion of trajectory is widely used in
classical mechanics. However, it does not have an immediate physical meaning,
since arbitrary real numbers are unobservable. This notion leads to the known
paradoxes, such as the irreversibility problem. A "functional" formulation of
classical mechanics is suggested. The physical meaning is attached in this
formulation not to an individual trajectory but only to a "beam" of
trajectories, or the distribution function on phase space. The fundamental
equation of the microscopic dynamics in the functional approach is not the
Newton equation but the Liouville equation for the distribution function of the
single particle. The Newton equation in this approach appears as an approximate
equation describing the dynamics of the average values and there are
corrections to the Newton trajectories. We give a construction of probability
density function starting from the directly observable quantities, i.e., the
results of measurements, which are rational numbers.Comment: 8 page
The large deviation approach to statistical mechanics
The theory of large deviations is concerned with the exponential decay of
probabilities of large fluctuations in random systems. These probabilities are
important in many fields of study, including statistics, finance, and
engineering, as they often yield valuable information about the large
fluctuations of a random system around its most probable state or trajectory.
In the context of equilibrium statistical mechanics, the theory of large
deviations provides exponential-order estimates of probabilities that refine
and generalize Einstein's theory of fluctuations. This review explores this and
other connections between large deviation theory and statistical mechanics, in
an effort to show that the mathematical language of statistical mechanics is
the language of large deviation theory. The first part of the review presents
the basics of large deviation theory, and works out many of its classical
applications related to sums of random variables and Markov processes. The
second part goes through many problems and results of statistical mechanics,
and shows how these can be formulated and derived within the context of large
deviation theory. The problems and results treated cover a wide range of
physical systems, including equilibrium many-particle systems, noise-perturbed
dynamics, nonequilibrium systems, as well as multifractals, disordered systems,
and chaotic systems. This review also covers many fundamental aspects of
statistical mechanics, such as the derivation of variational principles
characterizing equilibrium and nonequilibrium states, the breaking of the
Legendre transform for nonconcave entropies, and the characterization of
nonequilibrium fluctuations through fluctuation relations.Comment: v1: 89 pages, 18 figures, pdflatex. v2: 95 pages, 20 figures, text,
figures and appendices added, many references cut, close to published versio
The mean field theory of spin glasses: the heuristic replica approach and recent rigorous results
The mathematically correct computation of the spin glasses free energy in the
infinite range limit crowns 25 years of mathematic efforts in solving this
model. The exact solution of the model was found many years ago by using a
heuristic approach; the results coming from the heuristic approach were crucial
in deriving the mathematical results. The mathematical tools used in the
rigorous approach are quite different from those of the heuristic approach. In
this note we will review the heuristic approach to spin glasses in the light of
the rigorous results; we will also discuss some conjectures that may be useful
to derive the solution of the model in an alternative way.Comment: 12 pages, 1 figure; lecture at the Flato Colloquia Day, Thursday 27
November, 200
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