[Paper Review] Derivation of the Euler equations from many-body quantum mechanics

Bruno Nachtergaele, Horng‐Tzer Yau|ArXiv.org|Oct 19, 2002

Advanced Thermodynamics and Statistical Mechanics11 references3 citations

TL;DR

This paper derives the compressible Euler equations from the exact Heisenberg dynamics of N-body quantum systems with short-range pair interactions, using a quantum entropy method and a novel quantum virial theorem. The key result shows that the macroscopic pressure in the hydrodynamic limit arises directly from quantum statistical mechanics, without semiclassical approximations or intermediate Boltzmann-type descriptions.

ABSTRACT

The Heisenberg dynamics of the energy, momentum, and particle densities for fermions with short-range pair interactions is shown to converge to the compressible Euler equations in the hydrodynamic limit. The pressure function is given by the standard formula from quantum statistical mechanics with the two-body potential under consideration. Our derivation is based on a quantum version of the entropy method and a suitable quantum virial theorem. No intermediate description, such as a Boltzmann equation or semi-classical approximation, is used in our proof. We require some technical conditions on the dynamics, which can be considered as interesting open problems in their own right.

Motivation & Objective

• To rigorously derive the compressible Euler equations from the exact quantum dynamics of fermions with short-range pair interactions.
• To establish that the macroscopic pressure in the hydrodynamic limit is given by the quantum statistical mechanical expression for pressure.
• To avoid intermediate descriptions such as the Boltzmann equation or semiclassical approximations in the derivation.
• To develop a quantum version of the entropy method for non-commuting observables in many-body quantum systems.
• To demonstrate that particle statistics and quantum effects survive in the Euler limit only through the pressure function.

Proposed method

• Use the Heisenberg equation of motion for energy, momentum, and particle density operators in a many-body quantum system.
• Apply a quantum version of the relative entropy method to control the distance between the true quantum state and a local Gibbs state.
• Construct a time-dependent local Gibbs state that matches the solution of the Euler equations via a variational principle.
• Employ a quantum virial theorem to relate microscopic currents to the thermodynamic pressure in quantum statistical mechanics.
• Use the variational inequality for relative entropy to bound expectation values of non-commuting observables without absolute values.
• Introduce cutoff functions and commuting approximations to local conserved quantities to handle non-commutativity in the hydrodynamic limit.

Experimental results

Research questions

• RQ1 Can the compressible Euler equations be derived directly from the exact Heisenberg dynamics of a quantum many-body system without intermediate approximations?
• RQ2 How does the quantum pressure function emerge in the hydrodynamic limit, and what is its relation to quantum statistical mechanics?
• RQ3 To what extent do quantum statistics and non-commutativity affect the macroscopic hydrodynamic behavior?
• RQ4 Can the relative entropy method be extended to non-commuting observables in quantum many-body systems to control convergence?
• RQ5 What role does the quantum virial theorem play in connecting microscopic currents to macroscopic pressure?

Key findings

• The Heisenberg dynamics of energy, momentum, and particle density for N-body fermions with short-range pair interactions converge to the compressible Euler equations in the hydrodynamic limit.
• The pressure function in the Euler equations is given exactly by the standard quantum statistical mechanical formula, without modification.
• Quantum corrections to the pressure survive at the macroscopic scale, and this is the only place where quantum statistics and non-commutativity affect the macroscopic dynamics.
• The derivation does not rely on the Boltzmann equation or semiclassical approximations, providing a direct link from quantum dynamics to hydrodynamics.
• The quantum virial theorem is essential to relate microscopic currents to the thermodynamic pressure in the derivation.
• The relative entropy method is successfully adapted to non-commuting observables by using a variational inequality and careful estimation of error terms.