Collisionless kinetic theory of rolling molecules

TL;DR

This paper develops a collisionless kinetic theory for molecules with nonholonomic rolling constraints, revealing fundamental differences from traditional statistical physics and providing a Hamiltonian-based framework.

Contribution

It introduces a novel kinetic theory for nonholonomic rolling molecules, addressing challenges in applying standard statistical methods to such systems.

Findings

Invariant measure does not hold for nonholonomic systems.

A Hamiltonian variational approach yields a consistent kinetic theory.

Exact solutions and a cold fluid closure are derived.

Abstract

We derive a collisionless kinetic theory for an ensemble of molecules undergoing nonholonomic rolling dynamics. We demonstrate that the existence of nonholonomic constraints leads to problems in generalizing the standard methods of statistical physics. In particular, we show that even though the energy of the system is conserved, and the system is closed in the thermodynamic sense, some fundamental features of statistical physics such as invariant measure do not hold for such nonholonomic systems. Nevertheless, we are able to construct a consistent kinetic theory using Hamilton's variational principle in Lagrangian variables, by regarding the kinetic solution as being concentrated on the constraint distribution. A cold fluid closure for the kinetic system is also presented, along with a particular class of exact solutions of the kinetic equations.