Let $(X,g)$ be a Riemannian manifold on which a group $G$ acts properly by isometries $G\curvearrowright X$. We can define the Poincaré serie for such action $$\mathcal{P}_s(G,X)=\sum_{g\in G^*}e^{-sd(x,g.x)}$$ where the sum runs over the elements other that the identity, and the value is independent on the choice of $x\in X$. When we have the fundamental group of $X$ acting in the obvious way on the unversal cover $\widetilde{X}$, the critical exponent of the serie $$s_0=\inf\{s|\mathcal{P}_s(\Pi(X),\widetilde{X})<\infty\}$$ is called volume entropy of $(X,g)$.
In statistical mechanics the partition function $Z$ of a system is defined in a completely analogous fashion $$Z=\sum_i e^{-\beta E_i}$$ where $i$ runs over all microstates of the systems whose energy is $E_i$. These two concept seem analogous, so here is my Question: Is there a precise way to interpret the Poincaré serie in the statistical mechanical settings? e.g. as the partition function of a system?