Exact solution of a model of a vesicle attached to a wall subject to mechanical deformation
arXiv:1111.3323 · doi:10.1088/1751-8113/45/39/395001
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.
extended revision, 12 pages, 6 figures
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