The Geometry of Fluid Membranes: Variational Principles, Symmetries and Conservation Laws
The Geometry of Fluid Membranes: Variational Principles, Symmetries and Conservation Laws
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流体膜的几何形状：变分原理、对称性和守恒定律
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发表时间：
2018
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通讯作者：
Pablo Vázquez
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作者：
J. Guven;Pablo Vázquez
脂质膜在介观尺度上的行为被其几何自由度异常准确地捕捉到。事实上，膜的几何形状常常是膜的物理状态的直接反映。在这一章中，我们将从几个角度考察流体膜的几何学和物理学之间的密切联系。我们首先回顾了用度量和外曲率来描述曲面几何的方法，并根据这两个张量的行为来检查曲面的变形。推导了描述膜平衡的形状方程，描述了溶液的定性行为。接下来，我们来看看能量的欧几里德不变性所隐含的守恒定律，它描述了膜中的应力分布和它的几何形状之间非常简单的关系。这种关系被用来研究膜介导的相互作用。我们展示了如何扩展该几何框架以适应全局和局部约束以及与几何图形耦合的附加材料自由度。应用守恒定律考察了轴对称薄膜对局部外力的响应，并刻画了非平凡的拓扑态。最后，我们来看看对称二维弯曲能的共形不变性，并考察它的一些结果。
The behavior of a lipid membrane on mesoscopic scales is captured unusually accurately by its geometrical degrees of freedom. Indeed, the membrane geometry is, very often, a direct reflection of the physical state of the membrane. In this chapter we will examine the intimate connection between the geometry and the physics of fluid membranes from a number of points of view. We begin with a review of the description of the surface geometry in terms of the metric and the extrinsic curvature, examining surface deformations in terms of the behavior of these two tensors. The shape equation describing membrane equilibrium is derived and the qualitative behavior of solutions described. We next look at the conservation laws implied by the Euclidean invariance of the energy, describing the remarkably simple relationship between the stress distributed in the membrane and its geometry. This relationship is used to examine membrane-mediated interactions. We show how this geometrical framework can be extended to accommodate constraints—both global and local—as well as additional material degrees of freedom coupling to the geometry. The conservation laws are applied to examine the response of an axially symmetric membrane to localized external forces and to characterize topologically nontrivial states. We wrap up by looking at the conformal invariance of the symmetric two-dimensional bending energy, and examine some of its consequences.