Vlasov equation for long-range interactions on a lattice

TL;DR

This paper demonstrates that the Vlasov equation effectively describes the continuum limit dynamics of long-range interacting Hamiltonian systems on a lattice, enabling stability analysis and comparison with numerical simulations.

Contribution

It introduces a Vlasov-based framework for lattice systems with long-range interactions, deriving mode-dependent stability thresholds and growth rates, and validates predictions with simulations.

Findings

Vlasov equation accurately models lattice dynamics in the continuum limit.

Homogeneous state stability depends on Fourier mode number.

Zero mean-field mode dominates exponential growth in the α-HMF model.

Abstract

We show that, in the continuum limit, the dynamics of Hamiltonian systems defined on a lattice with long-range couplings is well described by the Vlasov equation. This equation can be linearized around the homogeneous state and a dispersion relation, that depends explicitly on the Fourier modes of the lattice, can be derived. This allows one to compute the stability thresholds of the homogeneous state, which turn out to depend on the mode number. When this state is unstable, the growth rates are also function of the mode number. Explicit calculations are performed for the $\alpha$-HMF model with $0 \leq \alpha <1$, for which the zero mean-field mode is always found to dominate the exponential growth. The theoretical predictions are successfully compared with numerical simulations performed on a finite lattice.