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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$\begingroup$ Coarse graining is reducing the number of variable of a system by first changing the representation of a problem and then doing sole kind of approximation, usually the approximation corresponds to assuming that the fast degrees of freedom act as a random noise (going from $N=10^{23}$ molecules to stochastic navier stokes for example). Mean field is an approximation from which you start with a complex coarse grained model and you (most of the time) ignore some kind of correlations in the fluctuations. This sometime corresponds to just putting any kind of noise to 0 for example $\endgroup$– SyroccoCommented Dec 9 at 8:18
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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.
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1$\begingroup$ I disagree with the usage of mean field theory. The term "mean field" is often applied to any approximation that replaces some operator/fluctuating variable with its averaged value in order to gain a self-consistency condition. You can do so directly with a lattice model without any coarse-grained continuous field, and people still call that a mean field whatever. To me, the defining quality appears to be only mean, but not field. $\endgroup$– VokaylopCommented Dec 9 at 12:46
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$\begingroup$ @Vokaylop good point - we may talk about "mean" not over space, but in the sense of the averaging out the fluctuations, as in Habbard-Stratonovich... (which is a generalized version of a quasiclassical trajectory... which can be veiwed as coarse-graining of space - but this is a rather stretched argument.) However, in your example of lattice sites the field will be smoothly varying from one site to another - it is coarse-graining/envelope approximation, even if swept under the carpet. $\endgroup$– Roger V.Commented Dec 9 at 13:04
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$\begingroup$ @Vokaylop I expanded the answer to take your comment into account. $\endgroup$– Roger V.Commented Dec 9 at 13:17
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$\begingroup$ I brought up lattice model because the mean value is usually also defined only in lattice sites, and not a continuous field by definition. And it is also not necessarily slow-varying. Staggered magnetization of an AFM has the shortest possible length scale. $\endgroup$– VokaylopCommented Dec 9 at 14:49
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$\begingroup$ But, to be honest, I rather dislike the prevailing usage. Or indeed this whole "condensed matter lattice model being called QFT" thing. People claim that they are doing field theory but I see no bloody field at all... $\endgroup$– VokaylopCommented Dec 9 at 14:52