Microcanonical Simulations of Adsorbing Self-Avoiding Walks

TL;DR

This paper uses a numerical GAS algorithm to study the adsorption transition of self-avoiding walks in two and three dimensions, estimating critical points, crossover exponents, and scaling behaviors.

Contribution

It introduces a microcanonical simulation approach with the GAS algorithm to estimate critical parameters and scaling exponents for adsorbing self-avoiding walks.

Findings

Critical point in square lattice: 1.779 ± 0.003

Critical point in cubic lattice: 1.306 ± 0.007

Crossover exponent φ ≈ 0.5 in both dimensions

Abstract

Linear polymers adsorbing on a wall can be modelled by half-space self-avoiding 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 at \begin{equation} a_c^+ = \cases{ 1.779 \pm 0.003, & \hbox{in the square lattice}; \\ 1.306 \pm 0.007, & \hbox{in the cubic lattice}. } \end{equation} These results are then used to estimate the crossover exponent $\phi$ associated with the adsorption transition, giving \begin{equation} \phi = \cases{ 0.496 \pm 0.009, & \hbox{in two dimensions}; \\ 0.505 \pm 0.006, & \hbox{in three dimensions}. } \end{equation} 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.