Generalized hydrodynamics revisited

Abstract

During the past decade a number of attempts to formulate a continuum description of complex states of matter have been proposed to circumvent more cumbersome many-body and simulation methods. Typically these have been quantum systems (e.g., electrons) and the resulting phenomenologies collectively often called "quantum hydrodynamics". However, there is extensive work from the past based in non-equilibrium statistical mechanics on the microscopic origins of macroscopic continuum dynamics that has not been exploited in this context. Although formally exact, its original target was the derivation of Navier-Stokes hydrodynamics for slowly varying states in space and time. The objective here is to revisit that work for the present interest in complex quantum states -possible strong degeneracy, strong coupling, and all space-time scales. The result is an exact representation of generalized hydrodynamics suitable for introducing controlled approximations for diverse specific cases, and for critiquing existing work. * ij (r, t ′ |y) ∂ i u j + Ψ 0j (r|y(t))t * ij (r, t|y) (S126)

Retracted: Local pressure for inhomogeneous fluids

“…The volume derivative can be calculated directly (e.g., using length scaling 10,21 ) to get where K is the kinetic energy operator and is the virial operator (for the internal forces) …”

“…Consider now a general nonequilibrium state. The macroscopic hydrodynamic equations have their origins in averages of the underlying microscopic conservation laws for number density, energy density, and momentum density, {〈 n ( r , t )〉, 〈 e ( r , t )〉, 〈 p ( r , t )〉} 10,11,22 . In particular the hydrodynamic equation resulting from the conservation law for the momentum density follows from the nonequilibrium average of Equation ) where the brackets now denote a nonequilibrium average and ρ N ( t ) is a solution to the Liouville–von Neumann equation.…”

“…It is shown elsewhere 10,11 that the second term of Equation ) describes the dissipative processes of the system while the first term characterizes the “perfect fluid” (e.g., Euler) dynamics. The latter is entirely determined by its functional dependence on the conjugate fields and reduces to the equilibrium pressure tensor of the last section in the case of uniform β ( r , t ).…”

“…for this state can be defined from the associated grand potential and an associated local pressure identified. 10,11 However, in this case, for spatially varying β(r), there is no longer the flexibility to modify the thermodynamic local pressure to be equal to that from the local equilibrium average of the stress tensor. Consequently, it would seem that the equation of state for hydrodynamics is not the same as that for local equilibrium thermodynamics.…”

“…The associated ensemble is similar to the equilibrium ensemble. A “thermodynamics” for this state can be defined from the associated grand potential and an associated local pressure identified 10,11 . However, in this case, for spatially varying β ( r ), there is no longer the flexibility to modify the thermodynamic local pressure to be equal to that from the local equilibrium average of the stress tensor.…”