Analytical solution and Lie algebra of the relativistic Boltzmann equation

In this work, we present a novel and more efficient approach to constructing the relativistic Bobylev, Krook, and Wu (BKW) solution [A. V. Bobylev, Sov. Phys. Dokl. 20, 820 (1976); M. Krook and T. T. Wu, Phys. Fluids 20, 1589 (1977)] of the relativistic Boltzmann equation. Introducing a new ansatz for the distribution function, we demonstrate that within this specific ansatz space, only the equilibrium and BKW-type forms yield exact solutions to the nonlinear Boltzmann equation. Furthermore, we also derive the Lie algebra of invariant transformations admitted by the relativistic Boltzmann equation and show that the corresponding symmetry group transformations can be systematically constructed from this algebra.