Q:

How can point-like particles in an ideal gas reach thermodynamical equilibrium?

Having learned that the particles of an ideal gas must be point-like (for the gas to be ideal) I wonder how they can reach thermodynamical equilibrium (by "partially" exchanging momentum and energy). First the probability of two point-like particles to collide is literally zero, and second, they can only collide head-on which implies that they can only "swap" their momenta and energies.

How is this puzzle to be solved?

A:

An ideal gas in equilibrium cannot be supposed to have reached its equilibrium from a non-equilibrium state by interaction of its particles, because by definition the particles of an ideal gas do not interact:

Hard spheres of radius $a$ that collide elastically do interact in the time average - even when they interact "almost never". (Among other things they induce a non-vanishing virial coefficient $B_2 = \frac{2}{3}\pi a^3$.) So there is no ideal gas with finite-sized particles.
A system of point-like particles which collide elastically cannot be
distinguished from a system of point-like particles that don't
interact at all.

Thus, an ideal gas is - eventually - just supposed to be in equilibrium state. It cannot have reached it by "internal mechanisms".

