Statistical mechanics of granular media: An approach A la Boltzmann

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We use an analogy with the statistical mechanics of gas to build the statistical mechanics of granular media. The case of an isotropic disordered packing of equal spheres submitted to an isotropic stress is considered. We use the assumption of a maximal disorder and the method of the Lagrange multipliers to demonstrate that the distribution of the force moduli decreases exponentially and obeys an exponential-decay statistics. This distribution results from the above hypotheses and from the existence of a mean stress, which acts as a constraint. This distribution is confirmed by experiments and simulations. We apply the same method to find the distribution of void defects of a granular assembly. It follows also an exponential-decay statistics.