Koopman-von Neumann and Weyl-Wigner Phase-Space Formulation of Inviscid Euler Flows

Area of Science:

• Fluid Dynamics
• Mathematical Physics
• Quantum Mechanics

Background:

• Classical fluid dynamics, particularly inviscid Euler flows, presents complex nonlinear behavior.
• Existing models like Liouville (Vlasov) equations have limitations in fully nonlinear regimes.
• The Koopman-von Neumann (KvN) operator offers a linear perspective on nonlinear dynamics.

Purpose of the Study:

• To develop a unified Koopman-von Neumann (KvN) operator and Weyl-Wigner phase-space framework for inviscid ideal (barotropic) Euler flows.
• To reformulate nonlinear fluid dynamics as a linear KvN evolution on an enlarged field phase space.
• To enable the application of quantum mechanics tools to classical fluid systems.

Main Methods:

• Constructing the KvN generator, including the Jacobian term for unitarity.
• Deriving the evolution equation for the Wigner functional.
• Applying Weyl quantization, Moyal ⋆-products, and Wigner functionals to fluid dynamics.
• Analytically solving a one-dimensional Burgers flow using the Wigner solution.

Main Results:

• The framework clarifies the conditions under which the classical Liouville (Vlasov) description is exact (linear/quadratic regimes).
• It identifies when higher-order, quantum-like corrections become significant in fully nonlinear regimes.
• A closed-form Wigner solution for 1D Burgers flow was obtained, reproducing Liouville transport.
• Comparison with kinetic (Vlasov-monokinetic) formulation was performed.

Conclusions:

• The unified KvN and Weyl-Wigner framework provides a novel approach to analyzing fluid dynamics.
• This quantum-inspired method offers insights into nonlinear fluid behavior and corrections.
• The framework is extensible to three-dimensional flows, enhancing its applicability.