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

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 45; no. 39; pp. 395001 - 10
• Main Authors: Owczarek, A L, Prellberg, T
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 05.10.2012; IOP
• Subjects: Algebra; Catalysis; Catalysts; Dyck paths; Exact sciences and technology; Exact solutions; functional equations; kernel method; Kernels; Mathematical analysis; Mathematical models; Physics; pulling force; q-series; Vesicles; Walls; Deformation; Non linear functional; Functional equations; Mechanical model; Algebraic equation; Two dimensional model; Linear functional; Phase transitions; Kernel method; Exact solution
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/45/39/395001

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.