classical mechanics - Where do atoms go after collision?

[I edited the question according to Mark's and Grisha's answers.]

Consider two point-like particles of equal mass colliding centrally in 2D. The final directions of the momenta of the two particles are not determined by conservation of energy and momentum: they can go anywhere.

Under which circumstances the direction of the two particles is determined?

Answer

(Edited to reflect updated question).

The solution has a single parameter, which we can take to be the angle the impacting particle goes after the collision. A particular solution is determined only after this parameter is given. Usually we would talk about the collision of, for example, two spheres, in which case this parameter could be related to whether the spheres are colliding head on or glancing.

For example, in the frame where one particle is initially moving and the other is not, one solution is for the moving particle to come to a halt and for the stationary particle to take up all its motion. Another solution is for both particles to fly off from the impact site at 45 degree angles to the original motion, each with $\sqrt{2}/2$ the initial velocity. Both of these scenarios conserve momentum and kinetic energy.

There are many other solutions. In this frame, the conservation of momentum says

$\vec{p} = \vec{p_1} + \vec{p_2}$

with $\vec{p}$ the momentum before the collision and $\vec{p_1}$ and $\vec{p_2}$ the momenta after. Squaring,

$p^2 = p_1^2 + 2\vec{p_1}\cdot\vec{p_2} + p_2^2$.

Kinetic energy is proportional to $p^2$, so conservation of kinetic energy gives

$p^2 = p_1^2 + p_2^2$.

These last two combine to give

$\vec{p_1}\cdot\vec{p_2} = 0$,

so the momenta point at a 90-degree angle. However, any one particle can choose any angle within 90 degrees of the initial direction of motion to go after the collision. Once this direction is chosen, the direction of motion of the other particle, as well as their momenta, are fixed.