Exact solution of a model of a vesicle attached to a wall subject to mechanical deformation

TL;DR

This paper provides an exact analytical solution for a vesicle model attached to a wall under mechanical deformation, revealing phase transition behavior influenced by area weighting and force application.

Contribution

It introduces a novel q-deformed algebraic and functional equation approach to model vesicle deformation, extending previous Dyck-path models.

Findings

Identifies a non-trivial phase transition under applied force

Shows phase transition vanishes with non-unity area weight

Provides exact solutions using two different mathematical approaches

Abstract

Area-weighted Dyck-paths are a two-dimensional model for vesicles attached to a wall. We model the mechanical response of a vesicle to a pulling force by extending this model. We obtain an exact solution using two different approaches, leading to a q-deformation of an algebraic functional equation, and a q-deformation of a linear functional equation with a catalytic variable, respectively. While the non-deformed linear functional equation is solved by substitution of special values of the catalytic variable (the so-called "kernel method"), the q-deformed case is solved by iterative substitution of the catalytic variable. Our model shows a non-trivial phase transition when a pulling force is applied. As soon as the area is weighted with non-unity weight, this transition vanishes.