MONTE CARLO RENORMALIZATION GROUP FOR PERCOLATION OF LINEAR CHAINS

Duygu Balcan and Ayþe Erzan
Department of Physics, Faculty of Sciences and Letters,
Istanbul Technical University, Maslak 80626, Istanbul, Turkey.

ABSTRACT

Steady state fluorescence experiments
performed on a bulk linear polymer reveal a sharp phase transition
characterized by percolation exponents, as a function of the chain length [1].
Although percolation exponents are expected in a gelation phase transition
[2] and indeed have been experimentally observed by the same experimental
method to a very high accuracy [3], there is no a priori reason to expect
such a transition in a system of linear chains [4]. We therefore set out to
compute the critical exponents of a system of growing linear chains, via
Monte Carlo Renormalization Group. We model the polymerization process by
growing nonself intersecting chains from randomly chosen lattice sites.
The Monte Carlo RG [5,6] is implemented by determining the average
fraction of occupied sites at a given time step, in both the original and
coarse grained configurations. The fixed point is determined from
requiring the two densities to be the same, and the correlation function
exponent is determined from the slope of this transformation. Our results
on two dimensional lattices are not distinguishable from the percolation
value. Work is in progress on three dimensional lattices.

REFERENCES 
[1] Y. Yilmaz, O. Pekcan and A. Erzan, "Percolation
exponents at the entanglement transition," condmat/0111339. 
[2] D. Stauffer and A. Aharony, Introduction to Percolation
Theory, (Taylor and Francis, London, 1991). 
[3] Y. Yilmaz, A. Erzan, O. Pekcan, Phys. Rev. E, 58, 7487
(1998). 
[4] A. Coniglio and M. Daoud, J. Phys. A, 12, L259 (1979). 
[5] R. H. Swendsen, Phys. Rev. Lett. 47, 1159 (1981); and in T. W.
Burkhard and J. M. J. van Leeuwen eds., Real Space Renormalization (Springer
Verlag, Heidelberg, 1982). 
[6] P. J. Reynold, H. E. Stanley, W. Klein, Phys.Rev. B, 21, 1223
(1980). 