On the failure of mean-field theories near a critical point

TL;DR

This paper challenges traditional views on mean-field theories' failure at critical points, proposing that divergence of fluctuations is due to loss of statistical independence and that mean-field theories miss long-range correlations, explaining their limitations.

Contribution

It offers a new perspective linking divergence of fluctuations to breakdown of statistical independence and explains mean-field failures via Wilsonian renormalization group insights.

Findings

Diverging fluctuations stem from loss of statistical independence.

Mean-field theories reproduce fluctuations but not critical exponents.

Failure to capture long-range correlations explains critical index discrepancies.

Abstract

It is well known that mean-field theories fail to reproduce the experimentally known critical exponents. The traditional argument which explain this failure of mean-field theories near a critical point is the Ginsburg criterion in which diverging fluctuations of the order parameter is the root cause. We argue, contrary to the above mentioned traditional view, that diverging fluctuations in real physical systems near a critical point are genuine consequence of the breakdown of the property of statistical independence, and are faithfully reproduced by the mean-field theory. By looking at the problem from the point of view of "statistical independence" the divergence of fluctuations in real physical systems near criticality becomes immediately apparent as a connection can be established between diverging correlation length and diverging fluctuations. To address the question of why mean-field theories, much successful qualitatively, fail to reproduce the known values of critical indices we argue, using the essential ideas of the Wilsonian renormalization group, that mean-field theories fail to capture the long length scale averages of an order parameter near a critical point.