# Numerically finding Coulomb gap

by Asban
Tags: coulomb, numerically
 P: 7 Hello, I'm new in the realm of numerical analysis. I need to find the local probability density of energy P(E) of a single electron on site i from a self-consistent equation for the energy (mean field equation of energy). E_i=\epsilon_i+\sum_j(\frac{1}{1+e^{E_j/T}}-\frac{1}{2})\frac{e^2}{r_{ij}} \epsilon_i - energy at site i with random values between -1/2 to 1/2 E_j - average energy of site j E_i - average energy of site j r_{nn} - average distance of nearest neighbors \frac{e^2}{r_{nn}} - coulomb potential energy between nearest neighbors There are 10,000 sites The sites were uniformly distributes on a 2D sample The values of temperature and r_{nn} are \frac{e^2}{r_{nn}T}=20 Now the only explanation of how probability density ( P(E) ) was found is: 1."...starting with a random set of energies and evolving them iteratively within the mean-field model" 2."solving the equations for many instances and averaging over them" I understand the physics of this equation and how they got to it and I'm searching for a good refrence that explains this general numerical method in details or someone that can explain that method. Most regards Ofek

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