18,442 research outputs found

### Universal power law tails of time correlation functions

The universal power law tails of single particle and multi-particle time
correlation functions are derived from a unifying point of view, solely using
the hydrodynamic modes of the system. The theory applies to general correlation
functions, and to systems more general than classical fluids. Moreover it is
argued that the collisional transfer part of the stress-stress correlation
function in dense classical fluids has the same long time tail $\sim
t^{-1-d/2}$ as the velocity autocorrelation function in Lorentz gases.Comment: 10 pages, 0 figures, Revised version: old Eqs(7)-(8) are replaced by
new Eqs (7)-(10), based on renormalization of the fluctuating heat conduction
equation for systems with quenched disorder. The new power law tail vanishes
on a periodic lattice, as it shoul

### Microscopic Theory for Long Range Spatial Correlations in Lattice Gas Automata

Lattice gas automata with collision rules that violate the conditions of
semi-detailed-balance exhibit algebraic decay of equal time spatial
correlations between fluctuations of conserved densities. This is shown on the
basis of a systematic microscopic theory. Analytical expressions for the
dominant long range behavior of correlation functions are derived using kinetic
theory. We discuss a model of interacting random walkers with x-y anisotropy
whose pair correlation function decays as 1/r^2, and an isotropic fluid-type
model with momentum correlations decaying as 1/r^2. The pair correlation
function for an interacting random walker model with interactions satisfying
all symmetries of the square lattice is shown to have 1/r^4 density
correlations. Theoretical predictions for the amplitude of the algebraic tails
are compared with the results of computer simulations.Comment: 31 pages, 2 figures, final version as publishe

### Extension of Haff's cooling law in granular flows

The total energy E(t) in a fluid of inelastic particles is dissipated through
inelastic collisions. When such systems are prepared in a homogeneous initial
state and evolve undriven, E(t) decays initially as t^{-2} \aprox exp[ -
2\epsilon \tau] (known as Haff's law), where \tau is the average number of
collisions suffered by a particle within time t, and \epsilon=1-\alpha^2
measures the degree of inelasticity, with \alpha the coefficient of normal
restitution. This decay law is extended for large times to E(t) \aprox
\tau^{-d/2} in d-dimensions, far into the nonlinear clustering regime. The
theoretical predictions are quantitatively confirmed by computer simulations,
and holds for small to moderate inelasticities with 0.6< \alpha< 1.Comment: 7 pages, 4 PostScript figures. To be published in Europhysics Letter

### Scaling Solutions of Inelastic Boltzmann Equations with Over-populated High Energy Tails

This paper deals with solutions of the nonlinear Boltzmann equation for
spatially uniform freely cooling inelastic Maxwell models for large times and
for large velocities, and the nonuniform convergence to these limits. We
demonstrate how the velocity distribution approaches in the scaling limit to a
similarity solution with a power law tail for general classes of initial
conditions and derive a transcendental equation from which the exponents in the
tails can be calculated. Moreover on the basis of the available analytic and
numerical results for inelastic hard spheres and inelastic Maxwell models we
formulate a conjecture on the approach of the velocity distribution function to
a scaling form.Comment: 15 pages, 4 figures. Accepted in J. Statistical Physic

### Asymptotic solutions of the nonlinear Boltzmann equation for dissipative systems

Analytic solutions $F(v,t)$ of the nonlinear Boltzmann equation in
$d$-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 $\nu$, and embedding
inelastic hard spheres ($\nu=1$) and inelastic Maxwell models ($\nu=0$). 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,
$v^d_0(t) F(v,t)$ plotted versus $c=v/v_0(t)$, collapse on a scaling or
similarity solution $f(c)$, where $v_0(t)$ is the r.m.s. velocity. The
dissipative interactions generate overpopulated high energy tails, described
generically by stretched Gaussians, $f(c) \sim \exp[-\beta c^b]$ with $0 < b <
2$, where $b=\nu$ with $\nu>0$ in free cooling, and $b=1+{1/2} \nu$ with $\nu
\geq 0$ when driven by white noise. Power law tails, $f(c) \sim 1/c^{a+d}$, are
only found in marginal cases, where the exponent $a$ 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 $\nu=0$. 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 $\nu>0$.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

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