Statistical Models on Spherical Geometries

TL;DR

This paper investigates critical phenomena in hyper-spherical geometries using a random walk model, revealing phase transition behaviors and polymer adsorption characteristics near boundaries.

Contribution

It introduces a novel analysis of statistical systems on hyper-spheres, including phase diagrams and adsorption transitions, expanding understanding of geometry-dependent critical behavior.

Findings

Identified a phase diagram with second and first order transitions separated by a tricritical point.

Demonstrated exponential decay of adsorbed monomers near the critical adsorption energy.

Analyzed the properties of a random walk on hyper-spheres in the context of statistical physics.

Abstract

We use a one-dimensional random walk on $D$-dimensional hyper-spheres to determine the critical behavior of statistical systems in hyper-spherical geometries. First, we demonstrate the properties of such a walk by studying the phase diagram of a percolation problem. We find a line of second and first order phase transitions separated by a tricritical point. Then, we analyze the adsorption-desorption transition for a polymer growing near the attractive boundary of a cylindrical cell membrane. We find that the fraction of adsorbed monomers on the boundary vanishes exponentially when the adsorption energy decreases towards its critical value.