%0 Book Section %1 statphys23_0856 %A Brey, J. %A Dom\'ınguez, A. %A Soria, M.I. Garc\'ıa De %A Maynar, P. %A Montero, M.J. Ruiz %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K criticality granular hydrodynamic instability matter mode--coupling statphys23 topic-7 %T Critical Fluctuations in Granular Fluids %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=856 %X 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

@incollection{statphys23_0856, abstract = {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 $\delta L := 1-L/L_c \to 0^+$ 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 $\delta L$. (ii) The probability distribution function (pdf) of the normalized energy fluctuations, $\varepsilon:= (E-\langle E\rangle)/\sigma_E$ (with $\sigma_E :=$variance of $E$), is independent of $\delta L$ and well fitted by Gumbel's distribution of index $\pi/2$ (see Figure): $$ G_{\pi/2} (\varepsilon) = A\, {\rm exp}\,\frac{\pi}{2} \left[ B (\varepsilon-C) - e^{B (\varepsilon-C)} \right]\, ,$$ where $A$, $B$, $C$ are uniquely determined by the constraints $\langle 1 \rangle = 1$, $\langle \varepsilon \rangle = 0$, $\langle \varepsilon^2 \rangle = 1$. 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{\'\i}a de Soria, Maynar, Ruiz-Montero, PRL {\bf 94} (2005) 098001\\ 2) Brey, Dom{\'\i}nguez, Garc{\'\i}a de Soria, Maynar, PRL {\bf 96} (2006) 158002\\ 3) Bramwell et al., PRL {\bf 84} (2000) 3744}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Brey, J. and Dom{\'\i}nguez, A. and Soria, M.I. Garc{\'\i}a De and Maynar, P. and Montero, M.J. Ruiz}, biburl = {https://www.bibsonomy.org/bibtex/22aa71bf225ba1286637cea0cf7dd74d7/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {39bebc849f62639e7925d095f71a0b7b}, intrahash = {2aa71bf225ba1286637cea0cf7dd74d7}, keywords = {criticality granular hydrodynamic instability matter mode--coupling statphys23 topic-7}, month = {9-13 July}, timestamp = {2007-06-20T10:16:31.000+0200}, title = {Critical Fluctuations in Granular Fluids}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=856}, year = 2007 }