Anomalous Diffusion and Growth due to LongRange Correlations.
Abstract
I consider two classes of problems which exhibit the influence of longrange correlations on the behavior of statistical systems. In these cases, the introduction of longrange 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 twodimensional 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 xdirection and it diffuses randomly in the ydirection. 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 newlyadded 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.
 Publication:

Ph.D. Thesis
 Pub Date:
 1992
 Bibcode:
 1992PhDT........60P
 Keywords:

 Physics: General; Statistics