I am solving a physics problem and ended up with the following integral:
$\int_{-\infty}^{\infty} exp(-\beta\sum_{i=0}^{N}|\vec{r}_{i+1}-\vec{r}_{i}|^2)dr_{0}dr_{1}...dr_{N+1}$ where $\beta\,$ is just a constant, and I am having some difficulty with evaluating the integral, I tried to do the following substitution $u = \vec{r}_{i+1}-\vec{r}_{i}$ but was not very helpful. any help would be appreciated.
Edit: The physics problem I am solving is:
The Hamiltonian of (+2) interacting classical particles, that are enclosed in a cube of volume at temperature , is given by:
$H = \sum_{i=0}^{N+1} \frac{|\vec{P_{i}}|^2}{2m} +\frac{1}{2}mw^2 \sum_{i=0}^{N}|\vec{r}_{i+1}-\vec{r}_{i}|^2$
Assuming that $\lt{|\vec{r}_{i+1}-\vec{r}_{i}|^2\gt} \,\, \lt\lt \sqrt[3]{V}$ for $0\le i\le N+1,$ Compute the following:
a) The partition function of the sysytem
b) $<|⃗_{+1}−⃗_{0}|^2>$
My Approach:
$Z = \sum exp(-\beta H), where \,\beta = K_{B}T\,$ but in this case since $\lt{|\vec{r}_{i+1}-\vec{r}_{i}|^2\gt} \lt\lt \sqrt[3]{V}$ we can approximate the sum as an integral, So we will have: $\int exp(-\beta\sum_{i=0}^{N+1} \frac{|\vec{P_{i}}|^2}{2m}dP_{i})\int exp(-\beta\sum_{i=0}^{N}|\vec{r}_{i+1}-\vec{r}_{i}|^2 dr_{i})$