Theoretical dependency of moments on parameter of a Boltzmann distribution

Assume $$X\sim \frac{e^{-\beta Nf(x)}}{Z_{\beta}}$$ where $$f(x) = -hx -x^2 + \frac{1}{\beta N}(1-x)\ln(1-x) + (1+x)\ln(1+x)$$ and $$Z_{\beta}$$ is the appropriate normalization factor. The support of $$X$$ is the lattice $$\Gamma_N = \{-1,-1+2/N,\ldots,1-2/N,1\}$$ and $$N$$ is some large positive integer and $$\beta>1$$.

I would like to know if there's any way to characterize the first three moments of $$X$$, that is

$$\mathbb{E}[X], \mathbb{E}[X^2],\mathbb{E}[X^3]$$

as functions of $$\beta$$ and whatever else comes out, but in particular I'm interested in the overall behaviour when $$\beta$$ grows large.

If you are familiar with statistical physics this is the Curie Weiss model (or a mean field Ising model).

Since this model constitutes an exponential family, the moments can be derived from the moment generating function of the sufficient statistic: $$\Psi(u)=\mathbb E_\beta[e^{u f(X)}]=\int e^{u f(x)-\beta N f(x)-\log Z_\beta} \text{d} x=e^{\log Z_{\beta-u/N} -\log Z_\beta}$$ Then $$\Bbb E_\beta[f(X)^k] = \frac{\text{d}^k}{\text{d}u^k}\Psi(u) =e^{-\log Z_\beta}\frac{\text{d}^k}{\text{d}u^k} e^{\log Z_{\beta-u/N}}$$ Obviously, this does not answer the question, but I do not believe there is a generic formula for the moments of $$X$$ itself since this depends on the choice of the representation of $$X$$.
• I'm afraid so. I also wonder what these moment genereting formula are useful for, in the end they will just be a way to write out the definition of the moment $\sum_x g(x) \frac{e^{-\beta f(x)}}{Z_{\beta}}$ where do they ever come in handy? – Three Diag Apr 23 at 14:56