Vol. 6, No. 3, 2018

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On the effect of phase transition on the manifold dimensionality: application to the Ising model

Elena Lopez, Adrien Scheuer, Emmanuelle Abisset-Chavanne and Francisco Chinesta

Vol. 6 (2018), No. 3, 251–265
Abstract

Fields can be represented in a discrete manner from their values at some locations, the nodes when considering finite element descriptions. Thus, each discrete scalar solution can be considered as a point in N (N being the number of nodes used for approximating the scalar field). Most manifold learning techniques (linear and nonlinear) are based on the fact that those solutions define a slow manifold of dimension n N embedded in the space N. This paper explores such a behavior in systems exhibiting phase transitions in order to analyze the evolution of the local dimensionality n when the system moves from one side of the critical behavior to the other. For that purpose we consider the Ising model.

Keywords
Ising equation, phase transition, manifold learning
Physics and Astronomy Classification Scheme 2010
Primary: 05.10.Ln
Milestones
Received: 6 January 2018
Revised: 20 March 2018
Accepted: 15 May 2018
Published: 26 July 2018

Communicated by Francesco dell'Isola
Authors
Elena Lopez
Institut de Calcul Intensif
École Centrale de Nantes
Nantes
France
Adrien Scheuer
Institut de Calcul Intensif
École Centrale de Nantes
Nantes
France
Institute of Information and Communication Technologies, Electronics and Applied Mathematics
Université catholique de Louvain
Louvain-la-Neuve
Belgium
Emmanuelle Abisset-Chavanne
Institut de Calcul Intensif
École Centrale de Nantes
Nantes
France
Francisco Chinesta
Procédés et Ingénierie en Mécanique et Matériaux
Arts et Métiers ParisTech
Paris
France