Monte Carlo method in statistical physics

Monte Carlo in statistical physics refers to the application of the Monte Carlo method to problems in statistical physics, or statistical mechanics.

The general motivation to use the Monte Carlo method in statistical physics is to evaluate a multivariable integral. The typical problem begins with a system for which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics. To obtain the mean value of some macroscopic variable, say A, the general approach is to compute, over all the phase space, PS for simplicity, the mean value of A using the Boltzmann distribution:

${\displaystyle \langle A\rangle =\int _{PS}A_{\vec {r}}{\frac {e^{-\beta E_{\vec {r}}}}{Z}}d{\vec {r}}}$.

where ${\displaystyle E({\vec {r}})=E_{\vec {r}}}$ is the energy of the system for a given state defined by ${\displaystyle {\vec {r}}}$ - a vector with all the degrees of freedom (for instance, for a mechanical system, ${\displaystyle {\vec {r}}=\left({\vec {q}},{\vec {p}}\right)}$), ${\displaystyle \beta \equiv 1/k_{b}T}$ and

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