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Fluctuations in Granular Gases

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Mathematical Models of Granular Matter

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1937))

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.

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Authors and Affiliations

  1. Laboratoire de Physique Théorique (CNRS UMR 8627), Université Paris-Sud, Orsay, France

    A. Barrat

  2. Dipartimento di Fisica, Università di Roma, “La Sapienza” Piazzale Aldo Moro 2, 00185, Roma, Italy

    A. Puglisi

  3. Laboratoire de Physique, Théorique et Modèles Statistiques (CNRS UMR 8626), Université Paris-Sud, Orsay, France

    E. Trizac & P. Visco

  4. Laboratoire Matière et Systèmes Complexes (CNRS UMR 7057), Université Denis Diderot (Paris VII), Paris, France

    F. van Wijland

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Editors and Affiliations

  1. Department of Mathematics, University of Pisa, largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Gianfranco Capriz

  2. Accademia Nazionale dei Lincei, via della Lungara 10, 00165, Roma, Italy

    Gianfranco Capriz

  3. DICeA, University of Florence, via Santa Marta 3, 50139, Firenze, Italy

    Paolo Maria Mariano

  4. Dipartimento di Meccanica e Materiali, Università “Mediterranea” di Reggio Calabria, via Graziella, località Feo di Vito, 89060, Reggio Calabria, Italy

    Pasquale Giovine

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Barrat, A., Puglisi, A., Trizac, E., Visco, P., van Wijland, F. (2008). Fluctuations in Granular Gases. In: Capriz, G., Mariano, P.M., Giovine, P. (eds) Mathematical Models of Granular Matter. Lecture Notes in Mathematics, vol 1937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78277-3_7

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