Orientation Asymmetric Surface Model for Membranes: Finsler Geometry Modeling

Abstract

We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping $\mathbf{r}$ from a two dimensional parameter space $M$ to the three dimensional Euclidean space ${\mathbf{R}}^3$. The metric variable $g_{ab}$, which is always fixed to the Euclidean metric ${\delta }_{ab}$, can be extended to a more general non-Euclidean metric on $M$ in the continuous model. The problem we focus on in this paper is whether such an extension is well-defined or not in the discrete model. We find that a discrete surface model with nontrivial metric becomes well-defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in $M$ depends on the direction. It is also shown that the discrete FG model is orientation assymetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some assymetries for orientation changing transformations.

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References in corpus (5)

• Crumpling transition and flat phase of polymerized phantom membranes
• First order phase transitions in polymerized phantom membranes
• Self-contact and instabilities in the anisotropic growth of elastic membranes
• Universal behavior of crystalline membranes: crumpling transition and Poisson ratio of the flat phase
• Finsler geometry modeling of phase separation in multi-component membranes