A derivation of the isothermal quantum hydrodynamic equations using entropy minimization

Isothermal quantum hydrodynamic equations of order O(ℏ²) using the quantum entropy minimization method recently developed by Degond and Ringhofer are derived. The equations have the form of the usual quantum hydrodynamic model including a correction term of order O(ℏ²) which involves the vorticity. If the initial vorticity is of order O(ℏ), the standard model is obtained up to order O(ℏ⁴). The derivation is based on a careful expansion of the quantum equilibrium obtained from the entropy minimization in powers of ℏ².