From Long-Range Order to Complex Networks, an Hamiltonian Dynamics Perspective

Abstract

In this chapter we discuss the influence of a nontrivial network topology on the thermodynamic behavior of an Hamiltonian model defined on it, the XY -rotors model. We first focus on network topology analysis, considering the regular chain and a Small World network, created with the Watt–Strogatz model. We parametrize these topologies via γ, giving the vertex degree k ∝ N ^γ−1 and p, the probability of rewiring. We then define two topological parameters, the average path length ℓ and the clustering coefficient C and we analyze their dependence on γ and p. We conclude this part presenting an algorithm, the tree algorithm, which enhances the calculation speed of ℓ and C. In the second part, we consider the behavior of the XY model on the regular chain and we find two regimes: one for γ < 1. 5, which does not display any long-range order and one for γ > 1. 5 in which a second order phase transition of the magnetization arises. Moreover we observe the existence of a metastable state appearing for γ _c = 1. 5. Finally we illustrate in what conditions we retrieve the phase transition on Small World networks and how its critical energy $\varepsilon _{c}(\gamma,p)$ depends on the topological parameters γ and p.