Zusammenfassung
A granular medium is a collection of many macroscopic particles the
interaction between which conserves momentum but dissipates
energy. This gives rise to a novel phenomenology that has attracted
much attention in the literature recently due to the insights it
provides on fundamental questions in the realm of Statistical Physics
Out of Equilibrium.
The homogeneous cooling state (HCS) is characterized by spatial
homogeneity simultaneously with steady decrease of the energy. This
state is unstable against the appearance of vortices of extension greater
than a critical size $L_c$. We have studied the fluctuations
of the total energy $E$ around the HCS of a 2-dimensional system in
a square container of sidelength $L$
as $L := 1-L/L_c 0^+$ for a wide range of values of the particle density and the inelasticity.
We have performed molecular dynamics simulations near the instability
threshold of an ensemble of hard spheres
which collide inelastically, obtaining the following results 1:
(i) The cooling rate, the variance of the
energy fluctuations, and the characteristic decay time of the energy--energy temporal correlation diverge as powers of $L$.
(ii) The probability distribution function (pdf) of the
normalized energy fluctuations, $\varepsilon:= (E-łangle
E\rangle)/\sigma_E$ (with $\sigma_E :=$variance of $E$), is independent
of $L$ and well fitted by Gumbel's distribution of index
$\pi/2$ (see Figure):
$$ G_\pi/2 (\varepsilon) =
A\, exp\,\pi2 łeft
B (\varepsilon-C) - e^B (\varepsilon-C) \right\, ,$$
where $A$, $B$, $C$ are uniquely determined by the
constraints $1 = 1$, $= 0$,
$\varepsilon^2 = 1$. This finding establishes an unexpected
connection with a disparate variety of systems (in
and out of equilibrium) exhibiting critical-like behavior and in which this particular pdf has also been found 3:
$XY$--model, fluid turbulence, models of self--organized
criticality,... A satisfactory explanation for the
''universality'' suggested by this observation is still lacking.
Starting from the equations of fluctuating hydrodynamics we have
developed a mode--coupling theory 2 in which the dynamics of all
the modes is enslaved to the stochastic dynamics of the fundamental
vorticity mode. Both the exponent and the amplitude of the power law
divergences are predicted correctly. The predicted pdf of the energy fluctuations shows,
however, significant departures from $G_\pi/2(\varepsilon)$.\\
1) Brey, Garc\'ıa de Soria, Maynar, Ruiz-Montero, PRL 94 (2005) 098001\\
2) Brey, Dom\'ınguez, Garc\'ıa de Soria, Maynar, PRL 96 (2006) 158002\\
3) Bramwell et al., PRL 84 (2000) 3744
Nutzer