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

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

The following Metropolis algorithm Template:IMP/cite, Template:IMP/cite is used \index{Metropolis} to simulate probabilities $exp(-E/k_{B}T)$:

1. select spin $S_k$ to consider.
2. evaluate variation of energy $\mathrm{\Delta }E=E_{new}-E_{old}$ associated to a possible split of spin $S_k$.
3. compare a random number $z$ between zero and one with probability $p=exp(-\mathrm{\Delta }E/k_{B}T)$.
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.

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