Calculating Properties by Integration

Having established that we can indeed explore configurational phase space and derive useful thermodynamic properties, let us consider how we might achieve this in practice. For example, the average potential energy can, in principle at least, be determined by evaluating the integral  

This is a multidimensional integral over the 3N degrees of freedom of the N particles in the system. p(r ) is the probability of obtaining the configuration r and is given by 

Fig 8 1 Evaluation of a one-dimensional integral using the trapezium rule The area under the curve is approximated as the sum of the areas of the trapeziums 

To calculate the partition function for a system of N atoms using this simple Monte Carlo integration method would involve the following steps  

Obtain a configuration of the system by randomly generating 3N Cartesian coordinates, which are assigned to the particles. 

---