Numerically finding Coulomb gap

[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

 

[eljose79]

Hello Ofek,

Welcome to the world of numerical analysis! The method used to find the local probability density of energy P(E) in this scenario is called the self-consistent field method. This method is commonly used in physics and chemistry to solve problems involving many interacting particles, such as electrons in a material.

In this case, the mean-field equation of energy is used to calculate the average energy of each site, taking into account the interactions between neighboring sites through the Coulomb potential energy. The initial energies at each site are random and the equation is iteratively solved until it reaches a stable solution. This means that the energies at each site no longer change significantly with each iteration.

To find the local probability density of energy P(E), the equations are solved for many instances, each with different initial random energies. The results are then averaged, giving a more accurate representation of the probability density. This method is often used in physics and chemistry because it allows for a more realistic and accurate description of the system.

As for references, there are many textbooks and online resources that explain the self-consistent field method in detail. Some good starting points include "Introduction to Computational Chemistry" by Frank Jensen and "Numerical Recipes: The Art of Scientific Computing" by William H. Press et al. Additionally, there are many research papers and articles available online that discuss the method in various contexts.

I hope this helps to clarify the method used to find the local probability density of energy in your scenario. If you have any further questions, please don't hesitate to ask.