Hamiltonian structure of 2D fluid dynamics with broken parity

TL;DR

This paper explores the conditions under which two-dimensional isotropic fluids with broken parity symmetry can be described by Hamiltonian dynamics, focusing on transport coefficients like odd viscosity and their relation to energy conservation.

Contribution

It identifies specific conditions on transport coefficients that enable dissipationless, Hamiltonian fluid dynamics in parity-breaking 2D fluids, clarifying the link between microscopic Hamiltonian systems and macroscopic hydrodynamics.

Findings

Certain transport coefficients correspond to dissipationless, Hamiltonian dynamics.

Not all parity-breaking coefficients lead to energy conservation.

Energy conserving but non-Hamiltonian dynamics can be realized with nonholonomic constraints.

Abstract

Isotropic fluids in two spatial dimensions can break parity symmetry and sustain transverse stresses which do not lead to dissipation. Corresponding transport coefficients include odd viscosity, odd torque, and odd pressure. We consider an isotropic Galilean invariant fluid dynamics in the adiabatic regime with momentum and particle density conservation. We find conditions on transport coefficients that correspond to dissipationless and separately to Hamiltonian fluid dynamics. The restriction on the transport coefficients will help identify what kind of hydrodynamics can be obtained by coarse-graining a microscopic Hamiltonian system. Interestingly, not all parity-breaking transport coefficients lead to energy conservation and, generally, the fluid dynamics is energy conserving but not Hamiltonian. We show how this dynamics can be realized by imposing a nonholonomic constraint on the Hamiltonian system.