Is the mean-field self-consistent-equation approach used to study, e.g., the magnetization of an Ising model able to take into account finite-size effects, or is it written, so to say, directly in the thermodynamic limit? What is the interplay of this approach with finite-size effects? Thank you very much!
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1$\begingroup$ The mean field approximation forgets completely about the geometry of the original system (apart from keeping, in a rather superficial way, the coordination number). From this point of view, there is no way to extract finite-size effect (it cannot even distinguish between two graphs with the same coordination number). Moreover, it is easy to prove that spontaneous magnetization is impossible in finite systems. Since mean-field theory claims the existence of an ordered phase, this indicates that it can only provide relevant information about infinite systems. $\endgroup$– Yvan VelenikCommented May 17 at 15:18
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1$\begingroup$ Of course, since there exist rigorous bounds between mean-field quantities and the corresponding quantities for the model on $\mathbb{Z}^d$ (in the thermodynamic limit), say, and that it is possible to obtain information about finite-size effects in the latter systems, one might deduce some bounds relevant to finite systems. $\endgroup$– Yvan VelenikCommented May 17 at 15:21
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1$\begingroup$ For more about the bounds I mention above, you can have a look at the references in Section 2.5.4 in this book. $\endgroup$– Yvan VelenikCommented May 17 at 15:24
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1$\begingroup$ I would say (in the same spirit as Yvan): mean field system are often coming from a saddle point approximation which implies (under any reasonable physics) a thermodynamic limit. A nice consequence (again already said by Yvan) is that with mean field you can obtain discontinuity in the free energy, something that you cannot have in finite size system $\endgroup$– SyroccoCommented May 17 at 15:38
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