Recall that a Gibbs measure gives a probability distribution on states $x$ of the form
$$ p_X(x) = \frac{1}{Z(\beta)}\exp(-\beta E(x)) $$
As I understand, the function $E$ is interpreted as the energy of the state. I'm wondering if there is a physical interpretation or significance to the characteristic function of $p$:
$$ \phi_X(k) = \mathbb{E}[\exp(ik^tx)] $$
Obviously, if one has $\phi_X$, one can compute moments and other interesting things about the ensemble, but I'm wondering (as a math guy) if $\phi_X$ has a physical significance.
To provide a little more context for my particular problem, I'm really thinking of $x$ as a digital image, and hence $p_X(x)$ is a probability distribution on possible images (see e.g. texture synthesis). In many applications, it's more convenient to derive/work with $\phi_X(k)$ instead of $p_X(x)$, and I'm trying to think of a way to do MCMC using $\phi_X$ instead of $p_X$. A physical intuition for $\phi_X$ might help, if there is one.
