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Updated: May 30, 2026

Atomic Layer Deposition of Vanadium Dioxide and a Temperature-dependent Optical Model

Atomic Layer Deposition of Vanadium Dioxide and a Temperature-dependent Optical Model

Published on: May 23, 2018

Vlasov equation for long-range interactions on a lattice.

R Bachelard¹, T Dauxois, G De Ninno

¹University of Nova Gorica, School of Applied Sciences, Ajdovcina, Slovenia. bachelard.romain@gmail.com

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|July 30, 2011

Summary

The Vlasov equation accurately describes Hamiltonian lattice dynamics with long-range interactions. Stability analysis reveals mode-dependent thresholds and growth rates, confirmed by simulations.

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Area of Science:

Statistical mechanics
Computational physics
Nonlinear dynamics

Background:

Hamiltonian systems on lattices with long-range interactions present complex dynamics.
Understanding the stability and behavior of such systems is crucial in various physics fields.

Purpose of the Study:

To demonstrate that the Vlasov equation effectively models the continuum limit of Hamiltonian lattice dynamics.
To analyze the stability of the homogeneous state and derive its dependence on lattice Fourier modes.
To validate theoretical predictions through numerical simulations.

Main Methods:

Linearization of the Vlasov equation around the homogeneous state.
Derivation of a dispersion relation dependent on lattice Fourier modes.
Calculation of stability thresholds and growth rates as functions of mode number.
Explicit analysis of the α-Hamiltonian mean field model (0≤α<1).
Comparison of theoretical results with numerical simulations on finite lattices.

Main Results:

The Vlasov equation provides a valid description for the continuum limit of these systems.
Stability thresholds and growth rates are explicitly dependent on the mode number.
For the α-Hamiltonian mean field model, the mean-field mode dominates exponential growth.
Theoretical predictions align well with numerical simulation outcomes.

Conclusions:

The Vlasov equation is a powerful tool for studying long-range interacting Hamiltonian lattice systems.
Mode analysis is essential for understanding system stability and dynamics.
Numerical simulations confirm the validity of the theoretical framework.