Introduction to Mathematical Physics/Statistical physics/Some numerical computation in statistical physics

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In statistical physics, mean quantities evaluation can be done using by Monte--Carlo methods. in this section, a simple example is presented.


Let us consider a Ising model. In this spin system, energy can be written:

E=-J\sum_i\sum_k S_iS_k-B\sum S_k

The following Metropolis algorithm [ma:compu:Stauffer93], [ma:compu:Koonin90] is used \index{Metropolis} to simulate probabilities exp(-E/k_BT):

  1. select spin S_k to consider.
  2. evaluate variation of energy \Delta E=E_{new}-E_{old} associated to a possible split of spin S_k.
  3. compare a random number z between zero and one with probability p=exp(-\Delta E/k_BT).
  4. split spin number k (that is do S_k=-S_k) i=f and only if z<p.
  5. use the obtained configuration to compute mean quantities.