The Cluster expansion approximation

The Cluster expansion approximation

The cluster expansion is an expansion that can be used to calculate the partition function for a system of non-interacting particles. Suppose that we have a system of $N$ particles and that the interaction potential between particles $i$ and $j$ is given by $v_{ij}$. The partition function of this system is then: $$ Z = \frac{1}{N!} \left( \frac{2\pi m}{\beta h^2} \right)^\frac{3N}{2} \int \int \dots \int \textrm{d}x_1 \textrm{d}x_2 \dots \textrm{d}x_{3N} \prod_{i=2}^N \prod_{j=1}^{i-1} 1 + f_{ij} $$ where $f_{ij} = e^{-\beta v_{ij}}-1$. We arrive at the cluster expansion by recognising that the integral over the spatial coordinates above can be rewritten as the following sum of integrals: $$ \int \int \dots \int \textrm{d}x_1 \textrm{d}x_2 \dots \textrm{d}x_{3N} \prod_{i=2}^N \prod_{j=1}^{i-1} 1 + f_{ij} = \int \int \dots \int \textrm{d}x_1 \textrm{d}x_2 \dots \textrm{d}x_{3N} + \int \int \dots \int \textrm{d}x_1 \textrm{d}x_2 \dots \textrm{d}x_{3N} \sum_{i=2}^N \sum_{j=1}^{i-1} f_{ij} + \dots $$ If we take the first two terms in this expansion only and thus assume that the particles are weakly interacting it is possible to show that when we take suitable derivatives of the partition function that emerges by solving the integral we get the van der Waals equation of state: $$ \left[ P + \frac{B}{2} \left(\frac{N}{V}\right)^2 \right] \left( V - \frac{AN}{2} \right) = Nk_B T $$

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