Abstract
We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping r from a two-dimensional parameter space M to the three-dimensional Euclidean space R 3 . The metric variable g a b , which is always fixed to the Euclidean metric δ a b , 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 a 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 asymmetric 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 asymmetries for orientation-changing transformations.
Highlights
Biological membranes including artificial ones, such as giant vesicles, are understood as two-dimensional surfaces [1]
The well-known surface model for membranes is statistical mechanically defined by using a mapping r from a two-dimensional parameter space M to R3 [2]
The discrete model is defined on dynamically-triangulated surfaces in R3, and the model is aimed at describing properties of fluid membranes, such as lipid bilayers
Summary
Biological membranes including artificial ones, such as giant vesicles, are understood as two-dimensional surfaces [1]. In [13], the anisotropic morphologies of membranes are studied, and the notion of the multi-component is essential for scalar functions, which are used to define the metric function on the triangles [14] It is still unclear whether the non-Euclidean metric can be assumed or not for discrete models. The dynamical triangulation for the DT models is simulated by the bond-flip technique as one of the Monte Carlo processes on triangulated lattices [22,23,24], while the FC surface models are defined on triangulated lattices without the bond flips According to this classification, the discrete models in this paper belong to the DT surface models and correspond to fluid membranes, because the dynamical triangulation is assumed in the partition function, which will be defined, just like in the model of [13].
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