Shear viscosity: velocity gradient as a constraint on wave function

TL;DR

This paper introduces a quantum-mechanical approach to shear viscosity by constraining the wave function with a velocity gradient, deriving a microscopic response that aligns with macroscopic observations without temporal coarse-graining.

Contribution

It develops a novel method to define microscopic response to velocity gradients via constrained wave functions, linking quantum dynamics to macroscopic shear viscosity.

Findings

Microscopic response defined through constrained wave functions

Macroscopic shear viscosity derived without temporal coarse-graining

Dissipation depends on initial occupation probabilities squared

Abstract

By viewing a velocity gradient in a fluid as an internal disturbance and treating it as a constraint on the wave function of a system, a linear evolution equation for the wave function is obtained from the Lagrange multiplier method. It allows us to define the microscopic response to a velocity gradient in a pure state. Taking a spatial coarse-graining average over this microscopic response and averaging it over admissible initial states, we achieve the observed macroscopic response and transport coefficient. In this scheme, temporal coarse-graining is not needed. The dissipation caused by a velocity gradient depends on the square of initial occupation probability, whereas the dissipation caused by a mechanical perturbation depends on the initial occupation probability itself. We apply the method of variation of constants to solve the time-dependent Schrodinger equation with constraints. The various time scales appearing in the momentum transport are estimated. The relation\ between the present work and previous theories is discussed.