Phase Transitions in Equilibrium and Non-Equilibrium Models on Some Topologies

Francisco De Sousa Lima

doi:10.3390/e18030081

Abstract

On some regular and non-regular topologies, we studied the critical properties of models that present up-down symmetry, like the equilibrium Ising model and the nonequilibrium majority vote model. These are investigated on networks, like Apollonian (AN), Barabasi–Albert (BA), small-worlds (SW), Voronoi–Delaunay (VD) and Erdos–Renyi (ER) random graphs. The review here is on phase transitions, critical points, exponents and universality classes that are compared to the results obtained for these models on regular square lattices (SL).

Highlights

Andrade et al [23,24] studied the Ising model (IM) on the undirected Apollonian networks (UAN). They obtained the thermodynamic and magnetic properties, but they found no evidence of a phase transition on UAN for the IM
Unlike the results found by Aleksiejuk et al [13], they showed that the IM on a directed Barabási–Albert networks (DBA) network does not present a phase transition
We presented results for the equilibrium Ising and non-equilibrium majority vote (MV) models on AN, BA and SW networks, ER random graphs and the VD random lattice

Summary

Introduction

Some equilibrium and non-equilibrium models were studied on regularity and non-regularity with a scale-free (SF) and small-worlds (SW) networks [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] to understand the critical properties of these models on some networks. According to the criterion of Grinstein et al [16], non-equilibrium spin systems with two states (±1) on square lattices (SL) may present the same critical exponents of the Ising model (IM) on SL [3]. This criterion was confirmed in some non-equilibrium models [17,18,19,20,21,22] on regular lattices, such as the majority vote (MV) model with states (±1) [17] This presents a continuous phase transition with critical exponents β, γ, ν, similar to those of the IM [3] in agreement with the criterion of Grinstein et al [16]. The DVD random lattices [29] are constructed in the same way as the DAN

Apollonian Networks

ER Random Graphs

BA Networks

SW Networks

VD Random Lattices

Conclusions