# Critical exponents in mean-field classical spin systems

## TL;DR

This paper investigates the critical exponents in mean-field classical spin systems by analyzing their dynamical evolution via the Vlasov equation, revealing significant differences from static analysis and emphasizing the importance of dynamics in understanding phase transitions.

## Contribution

The study demonstrates that critical exponents derived from dynamical phase space evolution differ markedly from static analysis, highlighting the necessity of considering dynamics in mean-field systems.

## Key findings

- Dynamical analysis yields different critical exponents than static methods.
- Static approaches may be insufficient for accurate critical exponent determination.
- Emphasizes the importance of phase space dynamics in phase transition analysis.

## Abstract

For a mean-field classical spin system exhibiting a second-order phase transition in the stationary state, we obtain within the corresponding phase space evolution according to the Vlasov equation the values of the critical exponents describing power-law behavior of response to a small external field. The exponent values so obtained significantly differ from the ones obtained on the basis of an analysis of the static phase-space distribution, with no reference to dynamics. This work serves as an illustration that cautions against relying on a static approach, with no reference to the dynamical evolution, to extract critical exponent values for mean-field systems.