The Boltzmann equation for very hard particles

The nonlinear Boltzmann equation is solved analytically for general initial distributions in a (spatially homogeneous) system of very hard particles (VHP) with two translational degrees of freedom and with a transition probability for binary collisions (vw →v′w′) proportional to δ(v² + w² − v′² −w′²).

The scattering cross-section corresponding to this model increases as the square root of the collision energy (hence the name VHP-model). As the total energy of the system is finite, essentially no highly energetic particles are present to probe the unphysical high-energy behavior of the cross-section.

The VHP-model is extended to a multicomponent mixture of particles, and solved by the same technique, viz. Laplace transformation. An analogous discrete variable model is solved by a generating function method.

Finally the solutions of the nonlinear and linearized Boltzmann equation are compared. Their large-energy behavior at a fixed (large) time is different; their large-time behavior at a fixed energy is the same.