Stochastic Dissipative Euler's equations for a free body

TL;DR

This paper extends Euler's equations to include stochastic and dissipative effects due to thermal fluctuations, predicting anisotropic Brownian motion of a free body's principal axes, validated by molecular dynamics simulations.

Contribution

It introduces a thermodynamically consistent stochastic dissipative Euler's equations that describe both orientation and shape evolution of a free body under thermal fluctuations.

Findings

Principal axes undergo anisotropic Brownian motion on the unit sphere.

Equilibrium correlation functions decay exponentially.

Theory accurately predicts non-equilibrium spinning behavior.

Abstract

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.