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