Why the Maxwellian Distribution is the Attractive Fixed Point of the Boltzmann Equation

We know that the velocity distribution of a gas of classical particles in equilibrium is the Maxwellian distribution. This is a very well experimentally confirmed fact. The approach in kinetic theory that gives the time evolution of the velocity distribution of a gas of particles is the Boltzmann equation. Hence, the Boltzmann equation should have the Maxwellian distribution as an attractive fixed point, i.e., when the initial conditions are far from the equilibrium, the distribution function, whose time evolution is given by the Boltzmann equation, should relax to the Maxwellian distribution. This happens when the possibility of binary collisions among the particles is considered. This is called the H-Theorem. Let us remark that this striking result is a non a priori result in the sense that it is obtained as an approximation to the real dynamics after ignoring higher order collisions among the particles of the gas. Thus, the symmetries imposed by the two-body collisions in the asymptotic equilibrium state determines correctly the final distribution, i.e., the Maxwellian distribution. Here we want to offer an alternative view that can be thought as an a priori theoretical argument explaining why the Maxwellian distribution is the velocity distribution found in a gas of classical particles in equilibrium. Usually this fact is shown from the canonical distribution, that is, the gas is considered in thermal equilibrium with a heat reservoir. But we do not need the concept of temperature to reach the Maxwellian distribution nor the supposition of binary collisions in the Boltzmann equation. It can be easily obtained from the microcanonical picture. Let us proceed to show it. We start by supposing a general one-dimensional ideal