Detecting phase transitions in collective behavior using manifold's curvature

Kelum Gajamannage; Erik M. Bollt; Kelum Gajamannage; Erik M. Bollt

doi:10.3934/mbe.2017027

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 2: 437-453. doi: 10.3934/mbe.2017027

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Detecting phase transitions in collective behavior using manifold's curvature

Kelum Gajamannage ^,,
Erik M. Bollt

Department of Mathematics, Clarkson University, Potsdam, NY-13699, USA

Received: 23 September 2015 Revised: 19 July 2016 Published: 01 April 2017
MSC : Primary: 53C15, 53C21; Secondary: 58D15

If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds. Therein, we firstly utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Secondly, we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the Phase Transition Detection (PTD) method using one particle simulation and three real world examples.
- Phase transition,
- manifold,
- collective behavior,
- dimensionality reduction,
- curvature
Citation: Kelum Gajamannage, Erik M. Bollt. Detecting phase transitions in collective behavior using manifold's curvature[J]. Mathematical Biosciences and Engineering, 2017, 14(2): 437-453. doi: 10.3934/mbe.2017027

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Abstract

If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds. Therein, we firstly utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Secondly, we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the Phase Transition Detection (PTD) method using one particle simulation and three real world examples.

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