Approximation of Thermal Equilibrium for Quantum Gases with Discontinuous Potentials and Application to Semiconductor Devices

We derive an approximate solution valid to all orders of h to the Bloch equation for quantum mechanical thermal equilibrium distribution functions via asymptotic analysis for high temperatures and small external potentials. This approximation can be used as initial data for transient solutions of the quantum Liouville equation, to derive quantum mechanical correction terms to the classical hydrodynamic model, or to construct an effective partition function in statistical mechanics. The validity of the asymptotic solution is investigated analytically and numerically and compared with Wigner's O(h²) solution. Since the asymptotic analysis results in replacing second derivatives of the potential in the correction to the stress tensor in the original O(h²) quantum hydrodynamic model by second derivatives of a smoothed potential, this approach represents a definite improvement for the technologically important case of piecewise continuous potentials in quantum semiconductor devices. quantum gases, quantum hydrodynamics, nonlinear PDEs, conservation laws, semiconductor device simulation.