265 research outputs found
Ergodicity and Slowing Down in Glass-Forming Systems with Soft Potentials: No Finite-Temperature Singularities
The aim of this paper is to discuss some basic notions regarding generic
glass forming systems composed of particles interacting via soft potentials.
Excluding explicitly hard-core interaction we discuss the so called `glass
transition' in which super-cooled amorphous state is formed, accompanied with a
spectacular slowing down of relaxation to equilibrium, when the temperature is
changed over a relatively small interval. Using the classical example of a
50-50 binary liquid of N particles with different interaction length-scales we
show that (i) the system remains ergodic at all temperatures. (ii) the number
of topologically distinct configurations can be computed, is temperature
independent, and is exponential in N. (iii) Any two configurations in phase
space can be connected using elementary moves whose number is polynomially
bounded in N, showing that the graph of configurations has the `small world'
property. (iv) The entropy of the system can be estimated at any temperature
(or energy), and there is no Kauzmann crisis at any positive temperature. (v)
The mechanism for the super-Arrhenius temperature dependence of the relaxation
time is explained, connecting it to an entropic squeeze at the glass
transition. (vi) There is no Vogel-Fulcher crisis at any finite temperature T>0Comment: 10 pages, 9 figures, submitted to PR
Aging in the random energy model
In this letter we announce rigorous results on the phenomenon of aging in the
Glauber dynamics of the random energy model and their relation to Bouchaud's
'REM-like' trap model. We show that, below the critical temperature, if we
consider a time-scale that diverges with the system size in such a way that
equilibrium is almost, but not quite reached on that scale, a suitably defined
autocorrelation function has the same asymptotic behaviour than its analog in
the trap model.Comment: 4pp, P
Slow movement of a random walk on the range of a random walk in the presence of an external field
In this article, a localisation result is proved for the biased random walk
on the range of a simple random walk in high dimensions (d \geq 5). This
demonstrates that, unlike in the supercritical percolation setting, a slowdown
effect occurs as soon a non-trivial bias is introduced. The proof applies a
decomposition of the underlying simple random walk path at its cut-times to
relate the associated biased random walk to a one-dimensional random walk in a
random environment in Sinai's regime
Average shape of fluctuations for subdiffusive walks
We study the average shape of fluctuations for subdiffusive processes, i.e.,
processes with uncorrelated increments but where the waiting time distribution
has a broad power-law tail. This shape is obtained analytically by means of a
fractional diffusion approach. We find that, in contrast with processes where
the waiting time between increments has finite variance, the fluctuation shape
is no longer a semicircle: it tends to adopt a table-like form as the
subdiffusive character of the process increases. The theoretical predictions
are compared with numerical simulation results.Comment: 4 pages, 6 figures. Accepted for publication Phys. Rev. E (Replaced
for the latest version, in press.) Section II rewritte
From interacting particle systems to random matrices
In this contribution we consider stochastic growth models in the
Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large
time distribution and processes and their dependence on the class on initial
condition. This means that the scaling exponents do not uniquely determine the
large time surface statistics, but one has to further divide into subclasses.
Some of the fluctuation laws were first discovered in random matrix models.
Moreover, the limit process for curved limit shape turned out to show up in a
dynamical version of hermitian random matrices, but this analogy does not
extend to the case of symmetric matrices. Therefore the connections between
growth models and random matrices is only partial.Comment: 18 pages, 8 figures; Contribution to StatPhys24 special issue; minor
corrections in scaling of section 2.
Electrical, morphology and structural properties of biodegradable nanocomposite polyvinyl-acetate/ cellulose nanocrystals
In this work, the dielectric properties and the electrical conductivity of polyvinyl acetate (PVAc) polymer doped with cellulose nanocrystals (CNC), extracted from the date palm rachis, are reported. We investigate the filler effect on the molecular mobility of the PVAc polymer chains and the charge transport properties of this material. PVAc/CNC films structure was characterized by powder X-Ray diffraction (XRD), showing the crystalline behavior of the cellulose filler. The dielectric properties were investigated using impedance spectroscopy, in the frequency range of 102–106 Hz and temperatures from 200 to 350 K. A β relaxation, assigned to the motions of the -OCOCH3 side groups, and α relaxation, associated with the glass transition of the PVAc matrix, can be detected.publishe
Fluctuations for the Ginzburg-Landau Interface Model on a Bounded Domain
We study the massless field on , where is a bounded domain with smooth boundary, with Hamiltonian
\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y)). The interaction \CV is assumed
to be symmetric and uniformly convex. This is a general model for a
-dimensional effective interface where represents the height. We
take our boundary conditions to be a continuous perturbation of a macroscopic
tilt: for , , and
continuous. We prove that the fluctuations of linear
functionals of about the tilt converge in the limit to a Gaussian free
field on , the standard Gaussian with respect to the weighted Dirichlet
inner product for some explicit . In a subsequent article,
we will employ the tools developed here to resolve a conjecture of Sheffield
that the zero contour lines of are asymptotically described by , a
conformally invariant random curve.Comment: 58 page
On small-noise equations with degenerate limiting system arising from volatility models
The one-dimensional SDE with non Lipschitz diffusion coefficient is widely
studied in mathematical finance. Several works have proposed asymptotic
analysis of densities and implied volatilities in models involving instances of
this equation, based on a careful implementation of saddle-point methods and
(essentially) the explicit knowledge of Fourier transforms. Recent research on
tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and
S.~Violante. Marginal density expansions for diffusions and stochastic
volatility, part II: Applications. 2013, arxiv:1305.6765] suggests to work with
the rescaled variable : while
allowing to turn a space asymptotic problem into a small- problem
with fixed terminal point, the process satisfies a SDE in
Wentzell--Freidlin form (i.e. with driving noise ). We prove a
pathwise large deviation principle for the process as
. As it will become clear, the limiting ODE governing the
large deviations admits infinitely many solutions, a non-standard situation in
the Wentzell--Freidlin theory. As for applications, the -scaling
allows to derive exact log-asymptotics for path functionals of the process:
while on the one hand the resulting formulae are confirmed by the CIR-CEV
benchmarks, on the other hand the large deviation approach (i) applies to
equations with a more general drift term and (ii) potentially opens the way to
heat kernel analysis for higher-dimensional diffusions involving such an SDE as
a component.Comment: 21 pages, 1 figur
Comparing dynamics: deep neural networks versus glassy systems
We analyze numerically the training dynamics of deep neural networks (DNN) by using methods developed in statistical physics of glassy systems. The two main issues we address are (1) the complexity of the loss landscape and of the dynamics within it, and (2) to what extent DNNs share similarities with glassy systems. Our findings, obtained for different architectures and datasets, suggest that during the training process the dynamics slows down because of an increasingly large number of flat directions. At large times, when the loss is approaching zero, the system diffuses at the bottom of the landscape. Despite some similarities with the dynamics of mean-field glassy systems, in particular, the absence of barrier crossing, we find distinctive dynamical behaviors in the two cases, showing that the statistical properties of the corresponding loss and energy landscapes are different. In contrast, when the network is under-parametrized we observe a typical glassy behavior, thus suggesting the existence of different phases depending on whether the network is under-parametrized or over-parametrized
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