# Time reversal symmetry breaking in two-dimensional non-equilibrium viscous fluids

## TL;DR

This paper investigates how time reversal symmetry breaking affects transport properties like odd viscosity in two-dimensional non-equilibrium viscous fluids, considering internal spin effects and deriving related Green-Kubo relations.

## Contribution

It provides the most general linear constitutive relations for stress and couple stress in isotropic 2D fluids with spin, and derives Green-Kubo formulas linking transport coefficients to fluctuation correlations.

## Key findings

- Odd viscosity requires time reversal symmetry breaking without internal spin.
- Non-zero odd viscosity can occur with preserved time reversal symmetry if internal spin is present.
- Breakdown of equipartition leads to decoupling of rotational viscosities.

## Abstract

We study the rheological signatures of departure from equilibrium in two-dimensional viscous fluids with and without internal spin. Under the assumption of isotropy, we provide the most general linear constitutive relations for stress and couple stress in terms of the velocity and spin fields. Invoking Onsager's regression hypothesis for fluctuations about steady states, we derive the Green-Kubo formulae relating the transport coefficients to time correlation functions of the fluctuating stress. In doing so, we verify the claim that one of the non-equilibrium transport coefficients, the odd-viscosity, requires time reversal symmetry breaking in the case of systems without internal spin. However, the Green-Kubo relations for systems with internal spin also show that there is a possibility for non-vanishing odd viscosity even when time reversal symmetry is preserved. Furthermore, we find that breakdown of equipartition in non-equilibrium steady states results in the decoupling of the two rotational viscosities relating the vorticity and the internal spin.