Shear viscosity: velocity gradient as a constraint on wave function

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.