# Numerically finding Coulomb gap

1. Sep 26, 2013

### Asban

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