Hyperbolic Equations, (1987)

The Boltzmann equation is widely used in describing the time evolution of a rarefied gas of particles. It looks like: (1) 0, + v • òx)f(x, v,t) = Q (/, / ) (*, v, t) where: f(x, v, t) dxdv = N(A, t) is interpreted as the number of particles in the region A of the one-particle phase space and Q(f, f) is a bilinear operator depending on the interaction among the particles. Since its discovery, due Boltzmann in 1872, such equation stimulated debates among qualified scientists as Poincaré, Zermelo, Loschmidt on one side and Boltzmann himself on the other. The basic point was to conciliate the irreversible character of Eq. 1 (described by the H-Theorem asserting, more or less, the tendency of the system to reach asymptotically the equili-brium) with the Newton law of the dynamics preventing any asymptotically definite behaviour