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Discrete Velocity Models and One Continuous Relation

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Abstract

Some years ago, Cercignani, in studying the energy to mass ratio in a homogeneous equilibrium state, concluded that the associated DVMs (Discrete Velocity Models) ratio would be a drawback for DVMs. Here with planar DVMs, dimension d=2, we try to answer this criticism. First, we study four elementary classes of pth squares planar DVMs with or without rest-particle, where the Hausdorff dimensions of the associated lattices, when p increases up to infinity, are d H =0, 1, 2, 2. The DVM energy to mass ratio leads to a fictitious dimension d p and the problem is to see whether\(d_p \simeq 2\), with p increasing or not. Our result, taking into account the constraints due to the DVM conservation laws, is that this is possible only for the models with d H =2 for the associated lattices with an infinite number of velocities. We also discuss intermediate cases, for instance p finite for the standard DVMs is sufficient in restricted cases. Second, we study two families of intermediate models between the above d H =1,2 with a rest-particle. Only for one family, called α-cross models with d H =2, do we still find that the continuous mass ratio condition for the dimension can be satisfied.

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Authors and Affiliations

  1. Service de Physique théorique, CE Saclay, F-91191, Gif-sur-Yvette, France

    H. Cornille

  2. Unité associée CNRS/SPM/URA 2306, France

    H. Cornille

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Cornille, H. Discrete Velocity Models and One Continuous Relation. Journal of Statistical Physics 114, 977–1014 (2004). https://doi.org/10.1023/B:JOSS.0000012514.21906.4e

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  • DOI: https://doi.org/10.1023/B:JOSS.0000012514.21906.4e

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