I meet this problem when I read a book about renormalization. There is a classical gas with $N$ particles of mass $m$, which occupies a volume $V$ at temperature $T$. The interaction between the classical particles has the form of \begin{equation} \phi(r_{ij}) = \frac{A}{r_{ij}^{n}} \end{equation} $A$ is a constant, $r_{ij}=|\textbf{q}_i-\textbf{q}_j|$, $n>0$. The aim is to show that the canonical partition function is a homogeneois function, i.e. \begin{equation} Z(\lambda T,\lambda^{-3/n}V)= \lambda^{3N(1/2-1/n)}Z(T,V) \end{equation}

By definition: \begin{equation} Z = \int d\textbf{p}_i\int d\textbf{q}_i\exp(-\beta H), H = \sum_{i=1}^N \frac{\textbf{p}_i^2}{2m}+\sum_{i<j}\phi(r_{ij}) \end{equation}

I can easily evaluate "impulse part of" the integral, let say it is $Z_0$ (partition function of the non-interacting system)

\begin{equation} Z_0 = (2\pi mkT)^{3N/2} \end{equation} The $3$ denotes that the volume is $3D$.

The hard part is the evaluation of the configuration integral, the part which contains the interaction term. I tried to set the parameter $n$ to different values $(n = 1,2)$, and also tried to set the number of particles to some low values, let say $N=2,3,4,\dots$, but eventually I cannot get any usefull result.

  • $\begingroup$ This lecture notes may be helpful: tkm.kit.edu/downloads/TheoryF2012.pdf In particular, the chapter about interacting particles has some details on how to compute this partition function. $\endgroup$ – VictorSeven Jul 21 '17 at 15:12

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