Numerically finding Coulomb gap

  1. 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
     
  2. jcsd
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