  We show evidence, based on extensive and carefully performed numerical
experiments, that the system of two elastic hard-point masses in one-dimension
is not ergodic for a generic mass ratio and consequently does not follow the
principle of energy equipartition. This system is equivalent to a right
triangular billiard. Remarkably, following the time-dependent probability
distribution in a suitably chosen velocity direction space, we find evidence of
exponential localization of invariant measure. For non-generic mass ratios
which correspond to billiard angles which are rational, or weak irrational
multiples of pi, the system is ergodic, in consistence with existing rigorous
results.
