Statistical study of two-dimensional electron gas within Thomas-Fermi approximation

In this paper we consider the two-dimensional electron gas. To this end we first take into account the electronic coherence length and consider the system as some cells with linear lengths equal to the coherence length. Inside these cells we have an interactive electron gas and outside them we consider a semi-classical dynamics. We obtain the energy functional of the gas inside the cells within the Thomas-Fermi-Dirac approximation and use classical Monte Carlo scheme to investigate the inter-cell dynamics. After each electron injection we let the system to become equilibrated. After these steps we analyze geometrically and algebraically the statistical observables of the system. The most important observation is the decreasing behavior of the energy of the system in terms of temperature in some regions showing the instability of the electron gas in these regions.