Non-mean-field Critical Exponent in a Mean-field Model : Dynamics versus Statistical Mechanics

TL;DR

This paper reveals that in mean-field models, the susceptibility's critical exponent differs from classical predictions due to Casimir invariants, with numerical simulations confirming the theoretical findings.

Contribution

It demonstrates that Casimir invariants cause a deviation from classical mean-field critical exponents in the Hamiltonian mean-field model.

Findings

Susceptibility exponent is half of the magnetization exponent in quasistationary states.

Numerical simulations confirm the theoretical prediction.

Casimir invariants trap the system, affecting critical behavior.

Abstract

The mean-field theory tells that the classical critical exponent of susceptibility is the twice of that of magnetization. However, the linear response theory based on the Vlasov equation, which is naturally introduced by the mean-field nature, makes the former exponent half of the latter for families of quasistationary states having second order phase transitions in the Hamiltonian mean-field model and its variances. We clarify that this strange exponent is due to existence of Casimir invariants which trap the system in a quasistationary state for a time scale diverging with the system size. The theoretical prediction is numerically confirmed by $N$-body simulations for the equilibrium states and a family of quasistationary states.