Lattice of 1D anharmonic oscillators (Cannonical Ensemble)

1. Oct 19, 2013

dipole

1. The problem statement, all variables and given/known data

I have a system of N non-interacting anharmonic oscillators whose potential energy is given by,

$$V(q) = cq^2 -gq^3 -fq^4$$

where $c,f,g > 0$ and $f,g$ are small.

2. Relevant equations

The Hamiltonian is given by,

$$H = \sum_{i=1}^N \big ( \frac{p^2_i}{2m} + V(q_i) \big )$$

And the corresponding partition function is,

$$Z = \int \prod_{i=1}^N\frac{dq_idp_i}{h}e^{-\beta H}$$

3. The attempt at a solution
I'm trying to calculate the partition function, from which everything else will essentially follow. Substituting H into my integral, and re-arranging things a bit, I find,

$$Z = \frac{1}{h^N}\prod_{i=1}^N \int dq_ie^{-\beta V(q_i)} \int dp_i e^{-\beta\frac{p^2_i}{2m}}$$

The integral over the $p_i$'s is easy, it's the one over the $q_i$ which has me stuck, because it seems like it has to diverge... written out in full,

$$\int dq_ie^{-\beta V(q_i)} = \int dq_ie^{-\beta(cq^2 -gq^3-fq^4)}$$

The problem wants things in leading orders of $f,g$, but I still don't see how the integral is not going to explode since I'm integrating over all possible phase-space... :(

Can someone show me what I'm missing here?