On the Boltzmann–Grad Limit for Smooth Hard-Sphere Systems

Abstract

The problem is posed of the prescription of the so-called Boltzmann–Grad limit operator ( $$\mathcal {L}_{BG}$$ ) for the N-body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. It is proved, that, despite the non-commutative property of the operator $$\mathcal {L}_{BG}$$ , the Boltzmann equation can nevertheless be uniquely determined. In particular, consistent with the claim of Uffink and Valente (Found Phys 45:404, 2015) that there is “no time-asymmetric ingredient” in its derivation, the Boltzmann equation is shown to be time-reversal symmetric. The proof is couched on the “ab initio” axiomatic approach to the classical statistical mechanics recently developed (Tessarotto et al. in Eur Phys J Plus 128:32, 2013). Implications relevant for the physical interpretation of the Boltzmann H-theorem and the phenomenon of decay to kinetic equilibrium are pointed out.