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 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 of our system we can obtain with a coarse graining procedure (we will immediately see what we mean by this) and then determine as a trace over all the possible configurations of our system, i.e. over all the possible forms of :

where with we mean a sum over all the possible microscopic configurations compatible with the order parameter profile , and the last step implicitly defines the effective Hamiltonian :
We therefore must understand how to determine ; the idea of coarse graining procedures is the following: for a given microscopic configuration we average the order parameter over sufficiently wide "blocks", i.e. portions of the system with linear dimension much greater than its microscopic scale, which we call (in the case of the Ising model, for example, can be taken as the lattice constant), but still microscopic and in particular much smaller than the correlation length , 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 , with such that:
( being the linear dimension of our system) and averaging the order parameter in every cell. This way we can obtain an expression for (since is anyway microscopic with respect to the size of the system, so we can regard as a continuous variable).

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