The Ginzburg-Pitaevskii equation and microscopic quantum hydrodynamics

Biswas, A. C. ; Shenoy, S. R. (1977) The Ginzburg-Pitaevskii equation and microscopic quantum hydrodynamics Physica B+C, 90 (2). pp. 265-268. ISSN 0378-4363

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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)}νs(r)}| ≪ 1, where νs(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.

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