Thomas-Fermi's Equation at Any Temperatures

Abstract

Combining statistical-mechanical relations with the theorem from thermodynamics that the total free energy of a continuous medium at a given temperature should take a minimum value if there is no external work done on the system, Thomas-Fermi's equation at any temperatures is derived. The main space of the paper is devoted to a problem to find a quantity corresponding to the thermodynamic potential per electron and to show that it is constant everywhere in the medium when there is an electric field produced by the charged particles themselves