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Why $$ -\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g} $$type PDE is called 'mean-field equation'? It's closely related to moser-trudinger inequality, there are many classical papers on it google scholar.

What I want to ask is that why this equation is called 'mean-field equation'? Its backgrounds are almost all about conformal geometry (prescribed curvature problem) and physics, but it seems that these backgrounds have nothing to do with the description of 'mean-field theory' I searched on google mean-field theory.

If you know about its history, I will be very glad.

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In my understanding, these equations (like the prescribed Gauss curvature differential equation, etc) also arise from taking an infinite number of stochastic random variables, and then averaging over their behavior, which is exactly what we do when we study mean field theories.

An example of this is this paper by Michael Kiessling.

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  • $\begingroup$ Thanks for your paper! This is the first time I learned about this. $\endgroup$
    – Elio Li
    Commented Dec 13, 2023 at 5:25

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