The Ginzburg-Pitaevskii equation and microscopic quantum hydrodynamics

Abstract

We show that for local equilibrium the microscopically derived phase and continuity equations, neglecting certain terms, yield an equation similar to the Ginzburg-Pitaevskii equation at absolute zero. The complex wave function is ψ = √p(r) × exp{i( < Φ(itr) > + μt/ℏ} where μ is the uniform chemical potential in strict equilibrium. A length scale ξ(0) = ℏ/√nρ₀V₀ appears naturally, where V₀ is the spatially integrated potential. With experimental values for the uniform total number density ρ₀ and a hard sphere radius of 1 Å, ξ(0)~1.35 Å, comparable to the phenomenological GP length ~4 Å. Deviations of macroscopic quantities from their equilibrium values must be small, e.g.|ρ(r)-ρ|/ρ₀ ≪ 1 and we must have 2mξ²(0)/hρ₀)| Δ · {j(r)-ρ(r)}ν_s(r)}| ≪ 1, where ν_s(r) = (ℏ/m Δ < Φ(r) >. In the above, j(r), ρ(r) and < φ(r) > are the total number current and number density, and the mean phase, respectively, in local equilibrium.