# 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 ${\displaystyle \eta ({\vec {r}})}$ 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 ${\displaystyle {\mathcal {C}}}$ of our system we can obtain ${\displaystyle \eta ({\vec {r}})}$ with a coarse graining procedure (we will immediately see what we mean by this) and then determine ${\displaystyle Z}$ as a trace over all the possible configurations of our system, i.e. over all the possible forms of ${\displaystyle \eta ({\vec {r}})}$:

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