# A Hydrodynamic Exercise in Free Probability: Setting up Free Euler Equations

## TL;DR

This paper develops a free probability version of Euler equations for non-commutative vector fields, introducing a cyclic vorticity and conserved quantities, extending classical fluid dynamics concepts to free probability spaces.

## Contribution

It formulates free Euler equations within free probability, introduces cyclic vorticity, and establishes conservation laws in this non-commutative setting.

## Key findings

- Derived free Euler equations for non-commutative vector fields.
- Introduced cyclic vorticity satisfying a vorticity equation.
- Identified conserved quantities in the free probability framework.

## Abstract

For the free probability analogue of Euclidean space endowed with the Gaussian measure we apply the approach of Arnold to derive Euler equations for a Lie algebra of non-commutative vector fields which preserve a certain trace. We extend the equations to vector fields satisfying non-commutative smoothness requirements. We introduce a cyclic vorticity and show that it satisfies a vorticity equation and that it produces a family of conserved quantities.