Statistics of anisotropic random walks

Sang Bub Lee, Purdue University

Abstract

We discuss the three different types of anisotropic random walks based on Monte Carlo simulations and real space renormalization group transformations. Our work for biased self-avoiding walks using Monte Carlo simulation coupled with scaling analyses suggests qualitative differences in the effect of excluded volume on the chain conformation in the stiff limit between two and three dimensions in a manner similar to a suggestion made by Petschek. In the limit of gauche weight p $\to$ 0 and contour length N $\to$ $\infty$, we find scaling for the mean square end-to-end distance $\langle$R$\sp2\rangle>$ with the crossover exponent one as before; however, the scaling function in three dimensions closely matches the random stiff chain results with no excluded volume while that in two dimensions exhibits marked deviations. Our cell renormalization approach also confirms the crossover exponent to be exactly one in any dimensions and for all cell sizes. Our work also shows, by use of renormalization flows, a substantial difference in the crossover between two and three dimensions in the limit of N $\to\infty$ for fixed p. In three dimensions a crossover seems to occur first to random walk limit and then to self-avoiding walk limit, while in two dimensions, it seems to occur directly to self-avoiding walk limit in agreement with our observations based on Monte Carlo simulation. We also study self-avoiding Levy flights in one dimension by Monte Carlo simulation. We find very large corrections to scaling in the node-avoiding Levy flights for a wide range of $\mu$ and also, surprisingly, that the moments of the end-to-end distance of the node-avoiding Levy flights are greater than those of path-avoiding Levy flight when they both exist and are finite. Based on these observations we conclude that the morphology of the node-avoiding Levy flights is far more complex than that of the path-avoiding Levy flights or the random Levy flights, and that the node-avoiding and path-avoiding Levy flights are certainly in different universality classes in one dimension. We also present new results of Monte Carlo simulation for self-avoiding walks on randomly diluted square and simple cubic lattices performed for p very close to the percolation thresholds. The asymptotic behavior obtained is very different from the only previous work of this kind by Kremer for the diamond lattice. While the previous work reported a large increase of Flory exponent compared to the undiluted lattice, our results indicate a behavior rather similar to the ordinary self-avoiding walks.

Degree

Ph.D.

Advisors

Nakanishi, Purdue University.

Subject Area

Condensation

COinS