Analytical results on the magnetization of the Hamiltonian Mean Field model

TL;DR

This paper provides rigorous analytical results on the magnetization dynamics of the Hamiltonian Mean-Field model, exploring out-of-equilibrium phase transitions and validating findings with simulations.

Contribution

It introduces a Hamiltonian formalism to derive algebraic results for the system's macroscopic observables, offering new insights into long-range interaction dynamics.

Findings

Analytical expressions for magnetization evolution

Validation of theory with N-body simulations

Reinterpretation of phase transition mechanisms

Abstract

The violent relaxation and the metastable states of the Hamiltonian Mean-Field model, a paradigmatic system of long-range interactions, is studied using a Hamiltonian formalism. Rigorous results are derived algebraically for the time evolution of selected macroscopic observables, e.g., the global magnetization. The high and low energy limits are investigated and the analytical predictions are compared with direct $N$-body simulations. The method we use enables us to re-interpret the out-of-equilibrium phase transition separating magnetized and (almost) unmagnetized regimes.