  We study a classical reparametrization-invariant system, in which ``time'' is
not a priori defined. It consists of a nonrelativistic particle moving in five
dimensions, two of which are compactified to form a torus. There, assuming a
suitable potential, the internal motion is ergodic or more strongly irregular.
We consider quasi-local observables which measure the system's ``change'' in a
coarse-grained way. Based on this, we construct a statistical timelike
parameter, particularly with the help of maximum entropy method and Fisher-Rao
information metric. The emergent reparametrization-invariant ``time'' does not
run smoothly but is simply related to the proper time on the average. For
sufficiently low energy, the external motion is then described by a unitary
quantum mechanical evolution in accordance with the Schr\"odinger equation.
