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Diffusion in a multidimensional energy surface with minima and barriers is aproblem of importance in statistical mechanics and also has wide applications,such as protein folding. To understand it in such a system, we carry out theoryand simulations of a tagged particle moving on a two-dimensional periodicpotential energy surface, both in the presence and absence of noise. Langevindynamics simulations at multiple temperatures are carried out to obtain thediffusion coefficient of a solute particle. Friction is varied from zero tolarge values. Diffusive motion emerges in the limit of long times, even in theabsence of noise, although the trajectory is found to remain correlated over along time. This correlation is manifested in correlated returns to the startingminima following a scattering by surrounding maxima. Noise destroys thiscorrelation, induces chaos, and increases diffusion at small friction.Diffusion thus exhibits a non-monotonic friction dependence at the intermediatevalue of the damping, ultimately converging to our theoretically predictedvalue. The latter is obtained by using the well-established relation betweendiffusion and random walk. An excellent agreement is obtained between theoryand simulations in the high friction limit, but not so in the intermediateregime. The rate of escape from one cell to another is obtained from themultidimensional rate theory of Langer. We find that enhanced dimensionalityplays an important role. In order to quantify the effects of noise on thepotential-imposed coherence on the trajectories, we calculate the Lyapunovexponent. At small friction values, the Lyapunov exponent mimics the frictiondependence of the rate.

Diffusion in the Presence of Correlated Returns in a Two-dimensional  Energy Landscape and non-Monotonic Friction Dependence: Examination of  Simulation Results by a Random Walk Model