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Q

  • How to evaluate a multivariate integral with a Gaussian weight function?

$$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm f}\left(x_{1},x_{2},\ldots,x_{n}\right)\, {\rm d}x_{1}\,{\rm d}x_{2}\ldots{\rm d}x_{n} $$

Where $\displaystyle{% {\rm f}(x_{1},x_{2},\ldots,x_{n}) =\prod_{j} {1 \over \sqrt{1 + {\rm i}\,b\left(x_{j}^2-x_{j+1}^2\right)^2}}}$.

I need a hint to solve this integral and this is how I proceeded:

$$ \mathcal{Z_{n}}= \int_{-\infty}^{\infty}\prod_{j=1}^{n}{\rm d}x_{j}\, \exp\left(% -\,{a \over 2}\sum_{j = 1}^{n}x_{j}^{2} - {1 \over 2} \log\left(1 + {\rm i}\,b\left[x_{j}^{2} - x_{j + 1}^{2}\right]^{2}\right)\right) $$

Now the integral is of the form of the canonical partition function integrated over the configuration space. Hence the integral can be identified as an $n-$particle partition of the canonical ensemble, which is given by $$ \mathcal{Z}_{n}= \int_{-\infty}^{\infty}\prod_{j = 1}^{n}{\rm d}x_{j}\, {\rm e}^{-\beta{\cal H}}, $$

Where $$\mathcal{H}=\Bigg(-\frac{a}{2}\sum_{j=1}^{n}x_{j}^2-\frac{1}{2}\log{\Big(1+i b(x_{j}^2-x_{j+1}^2)^{2}\Big)\Bigg)}.$$

Then I got stuck. How to proceed? Thanks.

Sijo Joseph
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  • Cross-posted to http://physics.stackexchange.com/q/102296/2451 – Qmechanic Mar 06 '14 at 14:46
  • A usual approximation is the "static one" where we look for stationary points of ${\cal H}$. However, it will be fine for some range of the $a$ and $b$ parameters. Also, Feynman Statistical Mechanics book discusses some sort of technique to express ${\cal Z}_{n + 1}$ in terms of ${\cal Z}_{n}$. Good luck !!!. – Felix Marin Mar 14 '14 at 03:11

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