Anomalous Diffusion and Growth due to Long-Range Correlations.

I consider two classes of problems which exhibit the influence of long-range correlations on the behavior of statistical systems. In these cases, the introduction of long-range correlations can be drastic, resulting either in the formation of a steady state, or in a change of the characteristic exponents that describe the dynamic and static properties of the system. First, I examine the two-dimensional motion of a particle in a unidirectional velocity field in which V_{x}(y) is a function of y only. The particle follows the direction of the velocity field in the x-direction and it diffuses randomly in the y-direction. I show that the correlations induced by the environment influence considerably the motion of the particle. The probability distribution deviates from a Gaussian behavior and the mean squared displacement grows with time as t ^{3/2}. In the second class of problems, I examine how correlations alter the shape of an interface grown by deposition of particles. The growth of independent columns (no correlations) produces a surface the width of which increases with the square root of time. The introduction of correlations, in this case interactions between the neighboring columns, results in the formation of a steady state. The roughness of the steady state depends on the detailed form of the correlations. For a restricted solid on solid model in (1 + 1) dimensions, I use exact enumeration techniques to show that the width of the interface scales with the square root of the length of the system in the steady state. I also examine the effects of surface rearrangement on morphology. For a process in which a newly-added particle can move laterally by up to one lattice spacing to maximize the number of saturated bonds, we show that a steady state is achieved. The introduction of a destabilizing factor, in our case antigravity, results in a formation of deep and abrupt valleys. The surface does not stabilize and its width grows linearly with time. Such formations do not appear to be described by a continuous Langevin equation.