Nonextensive Statistical Mechanics

234 7 Thermodynamical and Nonthermodynamical Applications7.1.5.1 Inelastic Maxwell ModelsIn some simple models, such as the so-called inelastic Maxwell models, analyticcalculations can be performed (e.g., in [453]). The velocity distribution that is obtained,from the microscopic dynamics of the system of cooling experiments, fora spatially uniform gas whose temperature is monotonically decreasing with timeis given by (see [450] and references therein) following asymptotic (i.e., t →∞)distributionP(v, t) = 2πv 0 (t)1[ ] 2, (7.5)1 + v2(v 0 (t)) 2with v 0 (t) = v 0 (0) e −λ(r) t , λ(r) being a function of the restitution coefficient r ofthe inelastic collisions (with λ(1) = 0). Equation (7.5) precisely corresponds to aq-Gaussian with q = 3/2.Fig. 7.15 Average vertical velocity profile inside the silo for an aperture 11d (d is the diameter ofthe grains) in eight different stages of its evolution. Time increases from left to right and from topto bottom. In the last one – corresponding to the fully developed flow – a grain on top has fallen adistance equivalent to twice its diameter d. Note the existence of a bounded region in the velocityprofile that travels in the vertical direction (from [451, 452]).