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

Local equilibrium properties of ultraslow diffusion in the Sinai model