190 research outputs found
Free cooling and high-energy tails of granular gases with variable restitution coefficient
We prove the so-called generalized Haff's law yielding the optimal algebraic
cooling rate of the temperature of a granular gas described by the homogeneous
Boltzmann equation for inelastic interactions with non constant restitution
coefficient. Our analysis is carried through a careful study of the infinite
system of moments of the solution to the Boltzmann equation for granular gases
and precise Lp estimates in the selfsimilar variables. In the process, we
generalize several results on the Boltzmann collision operator obtained
recently for homogeneous granular gases with constant restitution coefficient
to a broader class of physical restitution coefficients that depend on the
collision impact velocity. This generalization leads to the so-called
L1-exponential tails theorem. for this model
Fluctuations in granular gases
A driven granular material, e.g. a vibrated box full of sand, is a stationary
system which may be very far from equilibrium. The standard equilibrium
statistical mechanics is therefore inadequate to describe fluctuations in such
a system. Here we present numerical and analytical results concerning energy
and injected power fluctuations. In the first part we explain how the study of
the probability density function (pdf) of the fluctuations of total energy is
related to the characterization of velocity correlations. Two different regimes
are addressed: the gas driven at the boundaries and the homogeneously driven
gas. In a granular gas, due to non-Gaussianity of the velocity pdf or lack of
homogeneity in hydrodynamics profiles, even in the absence of velocity
correlations, the fluctuations of total energy are non-trivial and may lead to
erroneous conclusions about the role of correlations. In the second part of the
chapter we take into consideration the fluctuations of injected power in driven
granular gas models. Recently, real and numerical experiments have been
interpreted as evidence that the fluctuations of power injection seem to
satisfy the Gallavotti-Cohen Fluctuation Relation. We will discuss an
alternative interpretation of such results which invalidates the
Gallavotti-Cohen symmetry. Moreover, starting from the Liouville equation and
using techniques from large deviation theory, the general validity of a
Fluctuation Relation for power injection in driven granular gases is
questioned. Finally a functional is defined using the Lebowitz-Spohn approach
for Markov processes applied to the linear inelastic Boltzmann equation
relevant to describe the motion of a tracer particle. Such a functional results
to be different from injected power and to satisfy a Fluctuation Relation.Comment: 40 pages, 18 figure
Blow up Analysis for Anomalous Granular Gases
We investigate in this article the long-time behaviour of the solutions to
the energy-dependant, spatially-homogeneous, inelastic Boltzmann equation for
hard spheres. This model describes a diluted gas composed of hard spheres under
statistical description, that dissipates energy during collisions. We assume
that the gas is "anomalous", in the sense that energy dissipation increases
when temperature decreases. This allows the gas to cool down in finite time. We
study existence and uniqueness of blow up profiles for this model, together
with the trend to equilibrium and the cooling law associated, generalizing the
classical Haff's Law for granular gases. To this end, we investigate the
asymptotic behaviour of the inelastic Boltzmann equation with and without drift
term by introducing new strongly "nonlinear" self-similar variables.Comment: 20
Hydrodynamics of inelastic Maxwell models
An overview of recent results pertaining to the hydrodynamic description
(both Newtonian and non-Newtonian) of granular gases described by the Boltzmann
equation for inelastic Maxwell models is presented. The use of this
mathematical model allows us to get exact results for different problems.
First, the Navier--Stokes constitutive equations with explicit expressions for
the corresponding transport coefficients are derived by applying the
Chapman--Enskog method to inelastic gases. Second, the non-Newtonian
rheological properties in the uniform shear flow (USF) are obtained in the
steady state as well as in the transient unsteady regime. Next, an exact
solution for a special class of Couette flows characterized by a uniform heat
flux is worked out. This solution shares the same rheological properties as the
USF and, additionally, two generalized transport coefficients associated with
the heat flux vector can be identified. Finally, the problem of small spatial
perturbations of the USF is analyzed with a Chapman--Enskog-like method and
generalized (tensorial) transport coefficients are obtained.Comment: 40 pages, 10 figures; v2: final version published in a special issue
devoted to "Granular hydrodynamics
Asymptotic solutions of the nonlinear Boltzmann equation for dissipative systems
Analytic solutions of the nonlinear Boltzmann equation in
-dimensions are studied for a new class of dissipative models, called
inelastic repulsive scatterers, interacting through pseudo-power law
repulsions, characterized by a strength parameter , and embedding
inelastic hard spheres () and inelastic Maxwell models (). The
systems are either freely cooling without energy input or driven by
thermostats, e.g. white noise, and approach stable nonequilibrium steady
states, or marginally stable homogeneous cooling states, where the data,
plotted versus , collapse on a scaling or
similarity solution , where is the r.m.s. velocity. The
dissipative interactions generate overpopulated high energy tails, described
generically by stretched Gaussians, with , where with in free cooling, and with when driven by white noise. Power law tails, , are
only found in marginal cases, where the exponent is the root of a
transcendental equation. The stability threshold depend on the type of
thermostat, and is for the case of free cooling located at . Moreover we
analyze an inelastic BGK-type kinetic equation with an energy dependent
collision frequency coupled to a thermostat, that captures all qualitative
properties of the velocity distribution function in Maxwell models, as
predicted by the full nonlinear Boltzmann equation, but fails for harder
interactions with .Comment: Submitted to: "Granular Gas Dynamics", T. Poeschel, N. Brilliantov
(eds.), Lecture Notes in Physics, Vol. LNP 624, Springer-Verlag,
Berlin-Heidelberg-New York, 200
Velocity fluctuations in a one dimensional Inelastic Maxwell model
We consider the velocity fluctuations of a system of particles described by
the Inelastic Maxwell Model. The present work extends the methods, previously
employed to obtain the one-particle velocity distribution function, to the
study of the two particle correlations. Results regarding both the homogeneous
cooling process and the steady state driven regime are presented. In particular
we obtain the form of the pair correlation function in the scaling region of
the homogeneous cooling process and show that some of its moments diverge. This
fact has repercussions on the behavior of the energy fluctuations of the model.Comment: 16 pages, 1 figure, to be published on Journal of Statistical
Mechanics: Theory and Experiment
Which is the temperature of granular systems? A mean field model of free cooling inelastic mixtures
We consider a mean field model describing the free cooling process of a two
component granular mixture, a generalization of so called Maxwell model. The
cooling is viewed as an ordering process and the scaling behavior is attributed
to the presence of an attractive fixed point at for the dynamics. By
means of asymptotic analysis of the Boltzmann equation and of numerical
simulations we get the following results: 1)we establish the existence of two
different partial granular temperatures, one for each component, which violates
the Zeroth Law of Thermodynamics; 2) we obtain the scaling form of the two
distribution functions; 3) we prove the existence of a continuous spectrum of
exponents characterizing the inverse-power law decay of the tails of the
velocity, which generalizes the previously reported value 4 for the pure model;
4) we find that the exponents depend on the composition, masses and restitution
coefficients of the mixture; 5) we also remark that the reported distributions
represent a dynamical realization of those predicted by the Non Extensive
Statistical Mechanics, in spite of the fact that ours stem from a purely
dynamical approach.Comment: 23 pages, 9 figures. submitted for publicatio
What is the temperature of a granular medium?
In this paper we discuss whether thermodynamical concepts and in particular
the notion of temperature could be relevant for the dynamics of granular
systems. We briefly review how a temperature-like quantity can be defined and
measured in granular media in very different regimes, namely the glassy-like,
the liquid-like and the granular gas. The common denominator will be given by
the Fluctuation-Dissipation Theorem, whose validity is explored by means of
both numerical and experimental techniques. It turns out that, although a
definition of a temperature is possible in all cases, its interpretation is far
from being obvious. We discuss the possible perspectives both from the
theoretical and, more importantly, from the experimental point of view
Driven granular gases with gravity
We study fluidized granular gases in a stationary state determined by the
balance between an external driving and the bulk dissipation. The two
considered situations are inspired by recent experiments, where the gravity
plays a major role as a driving mechanism: in the first case gravity acts only
in one direction and the bottom wall is vibrated, in the second case gravity
acts in both directions and no vibrating walls are present. Simulations
performed under the molecular chaos assumption show averaged profiles of
density, velocity and granular temperature which are in good agreement with the
experiments. Moreover we measure the velocity distributions which show strong
non-Gaussian behavior, as experiments pointed out, but also density
correlations accounting for clustering, at odds with the experimental results.
The hydrodynamics of the first model is discussed and an exact solution is
found for the density and granular temperature as functions of the distance
from the vibrating wall. The limitations of such a solution, in particular in a
broad layer near the wall injecting energy, are discussed.Comment: Revised version accepted for publication. New results added and
discussions considering tangential forces. 27 pages (19 figures included), to
appear in Phys.Rev.
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