The pressure exerted by adsorbing directed lattice paths and staircase polygons

TL;DR

This paper analyzes the entropy-induced pressure exerted by adsorbing directed lattice paths and staircase polygons on a wall, deriving asymptotic expressions for different phases and path models.

Contribution

It provides the first asymptotic formulas for the pressure exerted by adsorbing directed paths and staircase polygons in the first quadrant.

Findings

Pressure vanishes in the desorbed phase for long paths.

Non-zero pressure exists in the adsorbed phase.

Asymptotic approximations are derived for finite-length paths and polygons.

Abstract

A directed path in the vicinity of a hard wall exerts pressure on the wall because of loss of entropy. The pressure at a particular point may be estimated by estimating the loss of entropy if the point is excluded from the path. In this paper we determine asymptotic expressions for the pressure on the X-axis in models of adsorbing directed paths in the first quadrant. Our models show that the pressure vanishes in the limit of long paths in the desorbed phase, but there is a non-zero pressure in the adsorbed phase. We determine asymptotic approximations of the pressure for finite length Dyck paths and directed paths, as well as for a model of adsorbing staircase polygons with both ends grafted to the X-axis.