Mean field theory is a tractable framework for analyzing parameters of a continuum or an infinite number of identical micro-particles or agents. The former has been treated extensively in statistical mechanics and the latter is a topic of recent attention in the area of mean field games. In all the technical literature, why is it a valid assumption to neglect the collisions among the micro-subsystems, especially when the region of motion is one dimensional?
$\begingroup$
$\endgroup$
5
-
$\begingroup$ I don't know about games, but in statistical physics mean field theory typically fails in one dimension. For example, it predicts a phase transition for the Ising model when there's none $\endgroup$– John DonneCommented Mar 10, 2020 at 8:47
-
$\begingroup$ In a lot of cases it is not a valid assumption to neglect fluctuations, especially in low dimension. $\endgroup$– dan-rosCommented Mar 10, 2020 at 17:50
-
$\begingroup$ Are the fluctuations you mention related to the issue of collisions raised in the question? $\endgroup$– kbakshi314Commented Mar 10, 2020 at 17:55
-
$\begingroup$ Yes. Well, mean-field theory includes some of the collisions/interactions already. But not all. Without including any interactions you wouldn't see a phase transition. Including higher orders of the interaction corresponds to including fluctuations in mean-field theory. For that you would need to look at the Renormalization Group. $\endgroup$– dan-rosCommented Mar 10, 2020 at 18:10
-
$\begingroup$ @tonydo, I am not sure why you say that one cannot observe phase-transitions in the absence of interactions in the theoretic model. Unless I am mistaken, I think bifurcations and phase-transitions are synonyms which are used in the context of systems devoid of interactions, such as, for instance, single particle 1D dynamics. $\endgroup$– kbakshi314Commented Mar 14, 2020 at 22:21
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
For the statistical physics case, mean-field theory gives you a sort of self-consistent scheme to determine for example the order parameter at a given temperature, but it neglects fluctuations, which can sort of destabilize phase transitions.
In low dimensions these fluctuations become very important. In the 1D Ising case, they completely destabilize the mean field predicted phase transition.
In a little higher dimensions, mean-field theory works ok, but gets, for example, the critical exponents wrong, due to the diverging correlation length of the fluctuations around the critical points.
Above the so-called upper critical dimension of a theory, fluctuations generally become less important and mean-field theory becomes exact, even at critical points.
A great reference for this is Goldenfeld's 'Lectures on phase transitions'.
