We prove existence and uniqueness of Boltzmann distributed quantum steady states of an electron ensemble which moves under the actions of the self-consistent Coulomb potential and of an external potential. The case of the particles confined to a three-dimensional bounded domain and the whole space case in R3 is analyzed. Also, we prove the existence of the classical limit. As the Planck constant tends to zero we obtain a Maxwellian eqilibrium solution of the Vlasov-Poisson problem.
ASJC Scopus subject areas
- Applied Mathematics