Ensemble Averages (flashback).

Classical System:

Phase Space $\{\mathbf{p},\mathbf{q}\}$

Distribution function for the NVT-ensemble:

$$\rho(\{\mathbf{p},\mathbf{q}\})=\frac{1}{Z}\exp(-\beta U(\{\mathbf{p},\mathbf{q}\}))d\{\mathbf{p},\mathbf{q}\}$$

Ensemble Average for the quantity $f(\{\mathbf{p},\mathbf{q}\})$:

 

$$\begin{split} \langle f(\{\mathbf{p},\mathbf{q}\})\rangle &=... ...athbf{q}\} \exp(-\beta U(\{\mathbf{p},\mathbf{q}\}))\end{split}$$ (1)

Unless explicitly stated, below we always assume, that kinetic energy is separable, and therefore, we care about the configurational $\{\mathbf{q}\}$ 3N-dimensional space. The contribution from the momentum part is trivial, and can be taken into account a posteriori.

Brute Force Approach:

We can try to calculate the integrals in (1) directly. It works for simple systems, like the Harmonic oscillator for instance.
$$\begin{multline*} U({\mathbf{p},\mathbf{q}}) =\frac{ kq^2}{2} + \frac{p^2}{2m} \... ...hbf{p},\mathbf{q}}) \rho({\mathbf{p},\mathbf{q}}) \ = \beta^{-1}\end{multline*}$$
Or we can calculate it numerically using e.g. Simpson's rule, if the energy function is more complex.

But if it's a system of the N particles, integrals are over the 6N dimensional space and we would have to evaluate the function at the grid of $\approx 100^{6N}$ points!!!!

Monte Carlo method for calculating integrals:

An integral of a function f(x) on the interval [a,b] can be represented as:

$$\int_a^bf(x)dx = (b-a)\langle f(x)\rangle$$

This is a key to the Monte Carlo method for calculating integrals:

1.

throw n points at random positions x_i within the interval (or region for the multi-dimensional case)

2.

calculate the average

$$\langle f(x)\rangle =\frac{1}{n} \sum_{i=1}^n f(x_i)$$

3.

accuracy of the average is estimated by calculating the deviation:

$$(\delta \langle f(x)\rangle)^2 \approx \frac{1}{n} \sum_{i=1}^n (f(x_i)-\langle f(x)\rangle)^2$$

Fortran code Example 1 :

Here is an example of the Fortran code to calculate integral of the x² from to 1 by Monte Carlo method. Code.