Microcanonical simulations of adsorbing self-avoiding walks

Abstract

Abstract. Linear polymers adsorbing on a wall can be modelled by half-space selfavoiding walks adsorbing on a line in the square lattice, or on a surface in the cubic lattice. In this paper a numerical approach based on the GAS algorithm is used to approximately enumerate states in the partition function of this model. The data are used to approximate the free energy in the model, from which estimates of the location of the critical point and crossover exponents are made. The critical point is found to be located atin the square lattice;1.306 ± 0.007, in the cubic lattice.These results are then used to estimate the crossover exponent φ associated with the adsorption transition, giving φ = 0.496 ± 0.009, in two dimensions;0.505 ± 0.006, in three dimensions.In addition, the scaling of these thermodynamic quantities is examined using the numerical data, including the scaling of metric quantities, and the partition and generating functions. In all cases results and numerical values of exponents were obtained which are consistent with estimates in the literature.

Partition and generating function zeros in adsorbing self-avoiding walks

“…The left hand side is positive if ρ > ζ d , and for these values of ρ there is a positive density of zeros in the annulus 1−ρ ≤ |a| ≤ [24]. In the limit d → ∞ a positive fraction of the zeros accumulate on the unit circle in the a-plane.…”

“…. [2] and Monte Carlo simulations in L 3 + give a + c = 1.306(7) [24]. It is known that [18] A + (a) = log µ d , if a ≤ a…”

“…It is thought to have value 1 2 in all dimensions d ≥ 2 and in d = 3 was estimated to be φ = 0.505 (6) [24]. Since there is a bulk entropy contribution of log µ d in L …”

“…In each case the algorithm was used to sample along 10 3 sequences each of length 10 9 iterations (see reference [24] for details), estimating microcanonical data for 0 ≤ v ≤ n where 0 ≤ n ≤ 500 over 10 12 iterations. Since the algorithm estimates approximate values c + .…”

“…The partition function A + n (a) (see equation (7)) was approximated using the approximate enumeration data obtained by the GAS algorithm for adsorbing walks in L 2 + [24]. The locations of zeros of A + n (a) were obtained to high accuracy by using Maple [36].…”