A derivation of the isothermal quantum hydrodynamic equations using entropy minimization

Dedicated to Prof. Herbert Gajewski in occasion of his 65th birthday Isothermal quantum hydrodynamic equations of order ( $\hbar$ 2 ) 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 ( $\hbar$ 2 which involves the vorticity. If the initial vorticity is of order ( $\hbar$ 4 , the standard model is obtained up to order . The derivation is based on a careful expansion of the quantum equilibrium obtained from the entropy minimization in powers of $\hbar$ 2 .