# Functional partition function and coarse graining

A possible way to overcome the limitations of mean field theories (i.e. their inability to discuss the local properties of the order parameter of a given system, for example its fluctuations) can be the following: we could regard the profile of the order parameter $\eta ({\vec {r}})$ to be the "degree of freedom" of our system and compute the partition function as a functional integral; in other words from the microscopic configuration ${\mathcal {C}}$ of our system we can obtain $\eta ({\vec {r}})$ with a coarse graining procedure (we will immediately see what we mean by this) and then determine $Z$ as a trace over all the possible configurations of our system, i.e. over all the possible forms of $\eta ({\vec {r}})$ :

$Z=\operatorname {Tr} e^{-\beta {\mathcal {H}}({\mathcal {C}})}=\int {\mathcal {D}}[\eta ({\vec {r}})]\sum '_{\mathcal {C}}e^{-\beta {\mathcal {H}}({\mathcal {C}})}=\int {\mathcal {D}}[\eta ({\vec {r}})]e^{-\beta {\mathcal {H}}_{\text{eff.}}[\eta ({\vec {r}})]}$ where with $\sum '_{\mathcal {C}}$ we mean a sum over all the possible microscopic configurations ${\mathcal {C}}$ compatible with the order parameter profile $\eta ({\vec {r}})$ , and the last step implicitly defines the effective Hamiltonian ${\mathcal {H}}_{\text{eff.}}$ :
$e^{-\beta {\mathcal {H}}_{\text{eff.}}}:=\sum '_{\mathcal {C}}e^{-\beta {\mathcal {H}}({\mathcal {C}})}$ We therefore must understand how to determine $\eta ({\vec {r}})$ ; the idea of coarse graining procedures is the following: for a given microscopic configuration ${\mathcal {C}}$ we average the order parameter $\eta$ over sufficiently wide "blocks", i.e. portions of the system with linear dimension $\ell$ much greater than its microscopic scale, which we call $a$ (in the case of the Ising model, for example, $a$ can be taken as the lattice constant), but still microscopic and in particular much smaller than the correlation length $\xi$ , so that the order parameter is uniform in every block. In other words, coarse graining a system means dividing it into cells of linear dimension $\ell$ , with $\ell$ such that:
$a\ll \ell \ll \xi (T)\leq L$ ($L$ being the linear dimension of our system) and averaging the order parameter $\eta$ in every cell. This way we can obtain an expression for $\eta ({\vec {r}})$ (since $\ell$ is anyway microscopic with respect to the size of the system, so we can regard ${\vec {r}}$ as a continuous variable).