Theory of collisional invariants for the Master kinetic equation

• Statistical treatment of the Boltzmann–Sinai classical dynamical system.
• Investigation of integral conservation properties of the Master kinetic equation.
• Proof of existence of the Boltzmann collisional invariants for the Master equation.
• Proof of existence of new collisional invariants holding for finite-size hard spheres.
• Connection with the Boltzmann equation and physical interpretation are pointed out.

The paper investigates the integral conservation properties of the Master kinetic equation, which provides an exact kinetic statistical description for the Boltzmann–Sinai classical dynamical system. It is proved that, besides the customary Boltzmann collisional invariants, this equation admits also a class of generalized collisional invariants (GCI). The result applies only when the number N and the diameter σ of hard-spheres are finite. This includes the case of dilute gases for which suitable asymptotic ordering conditions hold. However, when the Boltzmann–Grad limit is performed on the Master kinetic equation, it is shown that the existence of GCI is not permitted anymore.