132 research outputs found
On Dwork cohomology and algebraic D-modules
After works by Katz, Monsky, and Adolphson-Sperber, a comparison theorem
between relative de Rham cohomology and Dwork cohomology is established in a
paper by Dimca-Maaref-Sabbah-Saito in the framework of algebraic D-modules. We
propose here an alternative proof of this result. The use of Fourier transform
techniques makes our approach more functorial.Comment: latex, 8 page
Coarsening in granular systems
We review a few representative examples of granular experiments or models
where phase separation, accompanied by domain coarsening, is a relevant
phenomenon. We first elucidate the intrinsic non-equilibrium, or athermal,
nature of granular media. Thereafter, dilute systems, the so-called "granular
gases" are discussed: idealized kinetic models, such as the gas of inelastic
hard spheres in the cooling regime, are the optimal playground to study the
slow growth of correlated structures, e.g. shear patterns, vortices and
clusters. In fluidized experiments, liquid-gas or solid-gas separations have
been observed. In the case of monolayers of particles, phase coexistence and
coarsening appear in several different setups, with mechanical or electrostatic
energy input. Phenomenological models describe, even quantitatively, several
experimental measures, both for the coarsening dynamics and for the dynamic
transition between different granular phases. The origin of the underlying
bistability is in general related to negative compressibility from granular
hydrodynamics computations, even if the understanding of the mechanism is far
from complete. A relevant problem, with important industrial applications, is
related to the demixing or segregation of mixtures, for instance in rotating
tumblers or on horizontally vibrated plates. Finally, the problem of compaction
of highly dense granular materials, which has many important applications, is
usually described in terms of coarsening dynamics: there, bubbles of
mis-aligned grains evaporate, allowing the coalescence of optimally arranged
islands and a progressive reduction of total occupied volume.Comment: 12 pages, 10 figures, to appear in "Dynamics of coarsening" Comptes
Rendus Physique special issue,
https://sites.google.com/site/ppoliti/crp-special-issu
Power laws statistics of cliff failures, scaling and percolation
The size of large cliff failures may be described in several ways, for
instance considering the horizontal eroded area at the cliff top and the
maximum local retreat of the coastline. Field studies suggest that, for large
failures, the frequencies of these two quantities decrease as power laws of the
respective magnitudes, defining two different decay exponents. Moreover, the
horizontal area increases as a power law of the maximum local retreat,
identifying a third exponent. Such observation suggests that the geometry of
cliff failures are statistically similar for different magnitudes. Power laws
are familiar in the physics of critical systems. The corresponding exponents
satisfy precise relations and are proven to be universal features, common to
very different systems. Following the approach typical of statistical physics,
we propose a "scaling hypothesis" resulting in a relation between the three
above exponents: there is a precise, mathematical relation between the
distributions of magnitudes of erosion events and their geometry. Beyond its
theoretical value, such relation could be useful for the validation of field
catalogs analysis. Pushing the statistical physics approach further, we develop
a numerical model of marine erosion that reproduces the observed failure
statistics. Despite the minimality of the model, the exponents resulting from
extensive numerical simulations fairly agree with those measured on the field.
These results suggest that the mathematical theory of percolation, which lies
behind our simple model, can possibly be used as a guide to decipher the
physics of rocky coast erosion and could provide precise predictions to the
statistics of cliff collapses.Comment: 20 pages, 13 figures, 1 table. To appear in Earth Surface Processes
and Lanforms (Rocky Coast special issue
Velocity fluctuations in cooling granular gases
We study the formation and the dynamics of correlations in the velocity field
for 1D and 2D cooling granular gases with the assumption of negligible density
fluctuations (``Homogeneous Velocity-correlated Cooling State'', HVCS). It is
shown that the predictions of mean field models fail when velocity fluctuations
become important. The study of correlations is done by means of molecular
dynamics and introducing an Inelastic Lattice Maxwell Models. This lattice
model is able to reproduce all the properties of the Homogeneous Cooling State
and several features of the HVCS. Moreover it allows very precise measurements
of structure functions and other crucial statistical indicators. The study
suggests that both the 1D and the 2D dynamics of the velocity field are
compatible with a diffusive dynamics at large scale with a more complex
behavior at small scale. In 2D the issue of scale separation, which is of
interest in the context of kinetic theories, is addressed.Comment: 24 pages, 16 figures, conference proceedin
Chemical etching of a disordered solid: from experiments to field theory
We present a two-dimensional theoretical model for the slow chemical
corrosion of a thin film of a disordered solid by suitable etching solutions.
This model explain different experimental results showing that the corrosion
stops spontaneously in a situation in which the concentration of the etchant is
still finite while the corrosion surface develops clear fractal features. We
show that these properties are strictly related to the percolation theory, and
in particular to its behavior around the critical point. This task is
accomplished both by a direct analysis in terms of a self-organized version of
the Gradient Percolation model and by field theoretical arguments.Comment: 7 pages, 3 figure
Engineered Swift Equilibration of a Brownian Gyrator
In the context of stochastic thermodynamics, a minimal model for non
equilibrium steady states has been recently proposed: the Brownian Gyrator
(BG). It describes the stochastic overdamped motion of a particle in a two
dimensional harmonic potential, as in the classic Ornstein-Uhlenbeck process,
but considering the simultaneous presence of two independent thermal baths.
When the two baths have different temperatures, the steady BG exhibits a
rotating current, a clear signature of non equilibrium dynamics. Here, we
consider a time-dependent potential, and we apply a reverse-engineering
approach to derive exactly the required protocol to switch from an initial
steady state to a final steady state in a finite time . The protocol can
be built by first choosing an arbitrary quasi-static counterpart - with few
constraints - and then adding a finite-time contribution which only depends
upon the chosen quasi-static form and which is of order . We also get a
condition for transformations which - in finite time - conserve internal
energy, useful for applications such as the design of microscopic thermal
engines. Our study extends finite-time stochastic thermodynamics to
transformations connecting non-equilibrium steady states.Comment: 5 pages, 1 figure plus supplementary material 10 pages, 2 figures. To
appear in PRE Rapid communication
Earthquake dynamics constrained from laboratory experiments: new insights from granular materials
The traction evolution is a fundamental ingredient to model the dynamics of
an earthquake rupture which ultimately controls, during the coseismic phase,
the energy release, the stress redistribution and the consequent excitation of
seismic waves. In the present paper we explore the use of the friction behavior
derived from laboratory shear experiments performed on granular materials at
low normal stress. We find that the rheological properties emerging from these
laboratory experiments can not be described in terms of preexisting governing
models already presented in literature; our results indicate that neither
rate-and state-dependent friction laws nor nonlinear slip-dependent models,
commonly adopted for modeling earthquake ruptures, are able to capture all the
features of the experimental data. Then, by exploiting a novel numerical
approach, we directly incorporate the laboratory data into a code to simulate
the fully dynamic propagation of a 3-D slip failure. We demonstrate that the
rheology of the granular material, imposed as fault boundary condition, is
dynamically consistent. Indeed, it is able to reproduce the basic features of a
crustal earthquake, spontaneously accelerating up to some terminal rupture
speed, both sub- and supershear
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