Monte Carlo method in 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

${\displaystyle Z=\int _{PS}e^{-\beta E_{\vec {r}}}d{\vec {r}}}$

is the partition function.

One possible approach to solve this multivariable integral is to exactly enumerate all possible configurations of the system, and calculate averages at will. This is done in exactly solvable systems, and in simulations of simple systems with few particles. In realistic systems, on the other hand, an exact enumeration can be difficult or impossible to implement.

For those systems, the Monte Carlo integration (and not to be confused with Monte Carlo method, which is used to simulate molecular chains) is generally employed. The main motivation for its use is the fact that, with the Monte Carlo integration, the error goes as ${\displaystyle 1/{\sqrt {N}}}$, independently of the dimension of the integral. Another important concept related to the Monte Carlo integration is the importance sampling, a technique that improves the computational time of the simulation.

In the following sections, the general implementation of the Monte Carlo integration for solving this kind of problems is discussed.

An estimation, under Monte Carlo integration, of an integral defined as

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

is

${\displaystyle \langle A\rangle \simeq {\frac {1}{N}}\sum _{i=1}^{N}A_{{\vec {r}}_{i}}e^{-\beta E_{{\vec {r}}_{i}}}/Z}$

where ${\displaystyle {\vec {r}}_{i}}$ are uniformly obtained from all the phase space (PS) and N is the number of sampling points (or function evaluations).

From all the phase space, some zones of it are generally more important to the mean of the variable ${\displaystyle A}$ than others. In particular, those that have the value of ${\displaystyle e^{-\beta E_{{\vec {r}}_{i}}}}$ sufficiently high when compared to the rest of the energy spectra are the most relevant for the integral. Using this fact, the natural question to ask is: is it possible to choose, with more frequency, the states that are known to be more relevant to the integral? The answer is yes, using the importance sampling technique.

Lets assume ${\displaystyle p({\vec {r}})}$ is a distribution that chooses the states that are known to be more relevant to the integral.

The mean value of ${\displaystyle A}$ can be rewritten as

${\displaystyle \langle A\rangle =\int _{PS}p^{-1}({\vec {r}}){\frac {A_{\vec {r}}}{p^{-1}({\vec {r}})}}e^{-\beta E_{\vec {r}}}/Zd{\vec {r}}=\int _{PS}p^{-1}({\vec {r}})A_{\vec {r}}^{*}e^{-\beta E_{\vec {r}}}/Zd{\vec {r}}}$,

where ${\displaystyle A_{\vec {r}}^{*}}$ are the sampled values taking into account the importance probability ${\displaystyle p({\vec {r}})}$. This integral can be estimated by

${\displaystyle \langle A\rangle \simeq {\frac {1}{N}}\sum _{i=1}^{N}p^{-1}({\vec {r}}_{i})A_{{\vec {r}}_{i}}^{*}e^{-\beta E_{{\vec {r}}_{i}}}/Z}$

where ${\displaystyle {\vec {r}}_{i}}$ are now randomly generated using the ${\displaystyle p({\vec {r}})}$ distribution. Since most of the times it is not easy to find a way of generating states with a given distribution, the Metropolis algorithm must be used.

On this section, the implementation will focus on the Ising model. Lets consider a two-dimensional spin network, with L spins (lattice sites) on each side. There are naturally ${\displaystyle N=L^{2}}$ spins, and so, the phase space is discrete and is characterized by N spins, ${\displaystyle {\vec {r}}=(\sigma _{1},\sigma _{2},...,\sigma _{N})}$ where ${\displaystyle \sigma _{i}\in \{-1,1\}}$ is the spin of each lattice site. The system's energy is given by ${\displaystyle E({\vec {r}})=\sum _{i=1}^{N}\sum _{j\in viz_{i}}(1-J_{ij}\sigma _{i}\sigma _{j})}$, where ${\displaystyle viz_{i}}$ are the set of first neighborhood spins of i and J is the interaction matrix (for a ferromagnetic ising model, J is the identity matrix). The problem is stated.

On this example, the objective is to obtain ${\displaystyle \langle M\rangle }$ and ${\displaystyle \langle M^{2}\rangle }$ (for instance, to obtain the magnetic susceptibility of the system) since it is straightforward to generalize to other observables. According to the definition, ${\displaystyle M({\vec {r}})=\sum _{i=1}^{N}\sigma _{i}}$.

The method thus neglects dynamics, which can be a major drawback, or a great advantage. Indeed, the method can only be applied to static quantities, but the freedom to choose moves makes the method very flexible. An additional advantage is that some systems, such as the Ising model, lack a dynamical description and are only defined by an energy prescription; for these the Monte Carlo approach is the only one feasible.

The great success of this method in statistical mechanics has led to various generalizations such as the method of simulated annealing for optimization, in which a fictitious temperature is introduced and then gradually lowered.

• Monte Carlo integration
• Metropolis algorithm
• Importance sampling
• Quantum Monte Carlo
• Monte Carlo molecular modeling

• Allen, M.P. & Tildesley, D.J. (1987). Computer Simulation of Liquids. Oxford University Press. ISBN 0-19-855645-4.
• Frenkel, D. & Smit, B. (2001). Understanding Molecular Simulation. Academic Press. ISBN 0-12-267351-4.
• Binder, K. & Heermann, D.W. (2002). Monte Carlo Simulation in Statistical Physics. An Introduction (4th ed.). Springer. ISBN 3-540-43221-3.
• Spanier, Jerome; Gelbard, Ely M. (2008). "Importance Sampling". Monte Carlo Principles and Neutron Transport Problems. Dover. pp. 110–124. ISBN 978-0-486-46293-6.