  We perform numerical studies of a thermally driven, overdamped particle in a
random quenched force field, known as the Sinai model. We compare the unbounded
motion on an infinite 1-dimensional domain to the motion in bounded domains
with reflecting boundaries and show that the unbounded motion is at every time
close to the equilibrium state of a finite system of growing size. This is due
to time scale separation: Inside wells of the random potential, there is
relatively fast equilibration, while the motion across major potential barriers
is ultraslow. Quantities studied by us are the time dependent mean squared
displacement, the time dependent mean energy of an ensemble of particles, and
the time dependent entropy of the probability distribution. Using a very fast
numerical algorithm, we can explore times up top $10^{17}$ steps and thereby
also study finite-time crossover phenomena.
