自由刚体的随机耗散欧拉方程
Stochastic Dissipative Euler's equations for a free body
AI总结：
本研究基于热力学一致的随机耗散欧拉方程框架，证明无角动量时自由刚体主轴单位矢量呈各向异性布朗运动，通过MD模拟验证了理论预测，并成功描述了刚体绕不稳定中间轴旋转的非平衡过程。
摘要：
真实固体内部的本征热涨落对自由刚体运动欧拉方程的核心假设——刚体假设提出了挑战。近期，我们以热力学一致的方式提出了欧拉方程的耗散随机版本（发表于《European Journal of Mechanics - A/Solids》第103卷，文章编号105184，2024年）。该框架描述自由刚体的取向和形状演化，纳入了内部热涨落及其伴随的耗散机制。在本研究中，我们证明：在无角动量的情况下，该理论预测刚体的主轴单位矢量会在单位球面上发生各向异性布朗运动，各向异性源于刚体转动惯量的差异。由此得到的主特征向量平衡时间相关函数呈指数衰减。这一理论预测在小尺度刚体的分子动力学（molecular dynamics, MD）模拟中得到了验证。通过对比理论结果与平衡MD模拟结果，我们可以测量取向扩散张量。随后，我们将该信息应用于随机耗散欧拉方程，以描述刚体绕不稳定中间轴旋转的非平衡情形。理论与模拟结果高度吻合，验证了该理论框架的有效性。
英文摘要：
Intrinsic thermal fluctuations within a real solid challenge the rigid body assumption that is central to Euler's equations for the motion of a free body. Recently, we have introduced a dissipative and stochastic version of Euler's equations in a thermodynamically consistent way (European Journal of Mechanics - A/Solids 103, 105184 (2024)). This framework describes the evolution of both orientation and shape of a free body, incorporating internal thermal fluctuations and their concomitant dissipative mechanisms. In the present work, we demonstrate that, in the absence of angular momentum, the theory predicts that principal axis unit vectors of a body undergo an anisotropic Brownian motion on the unit sphere, with the anisotropy arising from the body's varying moments of inertia. The resulting equilibrium time correlation function of the principal eigenvectors decays exponentially. This theoretical prediction is confirmed in molecular dynamics simulations of small bodies. The comparison of theory and equilibrium MD simulations allow us to measure the orientational diffusion tensor. We then use this information in the Stochastic Dissipative Euler's Equations, to describe a non-equilibrium situation of a body spinning around the unstable intermediate axis. The agreement between theory and simulations is excellent, offering a validation of the theoretical framework.
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