In statistical mechanics, a Mean Field Theory defined as solving the reduced representation of physical system, on the other hand Coarse-grained modelling has similar purposes. Are these two approaches the same methodology? What is the convention in the literature?
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The two are closely related, but refer to different aspects of the issue: Coarse-graining is a procedure by which the continuous field is introduced, whereas mean-field theory is description of a system in terms of such a continuous field. In other words, they are in the same relation as the notions of a point particle and Newton laws in Newtonian mechanics.
A good discussion of coarse graining in relation to mean field theory can be found in Goldenfeld's Lectures On Phase Transitions And The Renormalization Group. Note however, that the procedure is by no means unique to strongly correlated systems and phase transitions, although it is the place where it has been studied more in depth. Similar approaches are standard in continuum mechanics and macroscopic electrodynamics:
Macroscopic vs microscopic electric fields
Continuum hypothesis in fluid mechanics
Is there a general math term for the idea behind the WKB and similar methods that assume slowly varying sources?
Update
As @Vokaylop have correctly pointed in the comments, modern use of term "mean field" may occur without any relation (or any direct relation) to coarse-graining. Thus, when using Habbard-Stratonovich transformation or path integrals extremum trajectories we often speak of "averaging out fluctiations" and keeping only the "mean". Indeed, in this context many well-known approximations, such as Hartree-Fock for an isolated site or an atom, appear technically as mean field, even though no averaging over space is implied. One could argue that the term "mean" still traces back to the origins coarse-graining, i.e., envelope approximation or "effective field theory", rather than to taking the mean over fluctuations.
Roger V.
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