流体可变形表面形态动力学中的规范自由度与客观率：Jaumann率与物质导数
Gauge Freedom and Objective Rates in the Morphodynamics of Fluid Deformable Surfaces: the Jaumann Rate vs. the Material Derivative
AI总结：
本文澄清了流体可变形表面形态动力学中混淆的Eulerian与Lagrangian视角及三种规范自由度，证明客观率仅限物质导数与Jaumann率，并分别阐述了其适用场景与应用示例。
摘要：
流体可变形表面的形态动力学描述与一系列生物和软物质现象相关，涵盖被动或主动、有序或拓扑的材料。然而，正确流体力学方程的严谨几何表述一直不够清晰，其中客观率被证明是一个核心且存在争议的问题。我们认为，这是由于在描述流体可变形表面时混淆了几个必须加以澄清的重要概念。这些概念包括流体运动的 Eulerian 和 Lagrangian 视角，以及三种不同类型的规范自由度：环境空间中的规范自由度；表面参数化中的规范自由度；以及表面上标架场选择中的规范自由度。我们澄清了这些概念，并证明了流体可变形表面中的客观率是时间导数，它们在第一种规范自由度下保持不变，并且保持环境度量的结构。后一条件将可能无限多的客观率缩减为仅有的两种：物质导数和 Jaumann 率。物质导数在 Galilean 群下保持不变，因此适用于速度，其率反映了动量守恒。Jaumann 导数在所有时间依赖的等距变换下保持不变，因此适用于局部序参量或对称性破缺变量，例如 nematic $Q$-tensor。我们提供了在两种不同标架场下物质率和 Jaumann 率的示例，这些标架场与当前可变形表面流体力学的应用密切相关。
英文摘要：
Morphodynamic descriptions of fluid deformable surfaces are relevant for a range of biological and soft matter phenomena, spanning materials that can be passive or active, as well as ordered or topological. However, a principled, geometric formulation of the correct hydrodynamic equations has remained opaque, with objective rates proving a central, contentious issue. We argue that this is due to a conflation of several important notions that must be disambiguated when describing fluid deformable surfaces. These are the Eulerian and Lagrangian perspectives on fluid motion, and three different types of gauge freedom: in the ambient space; in the parameterisation of the surface, and; in the choice of frame field on the surface. We clarify these ideas, and show that objective rates in fluid deformable surfaces are time derivatives that are invariant under the first of these gauge freedoms, and which also preserve the structure of the ambient metric. The latter condition reduces a potentially infinite number of possible objective rates to only two: the material derivative and the Jaumann rate. The material derivative is invariant under the Galilean group, and therefore applies to velocities, whose rate captures the conservation of momentum. The Jaumann derivative is invariant under all time-dependent isometries, and therefore applies to local order parameters, or symmetry-broken variables, such as the nematic $Q$-tensor. We provide examples of material and Jaumann rates in two different frame fields that are pertinent to the current applications of the fluid mechanics of deformable surfaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
