New Results for 2D Tilings: Wiener Index & Statistical Mechanics - - PowerPoint PPT Presentation

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New Results for 2D Tilings: Wiener Index & Statistical Mechanics - - PowerPoint PPT Presentation

Dec 20, 2023 304 likes •595 views

New Results for 2D Tilings: Wiener Index & Statistical Mechanics of Graphs ArizMATYC & MAA Southwestern Section Scottsdale, AZ April 9-10, 2010 Forrest H Kaatz Maricopa Community Colleges, AZ University of Advancing Technology, Tempe,

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New Results for 2D Tilings: Wiener Index & Statistical Mechanics of Graphs ArizMATYC & MAA Southwestern Section Scottsdale, AZ April 9-10, 2010

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Forrest H Kaatz Maricopa Community Colleges, AZ University of Advancing Technology, Tempe, AZ Adhemar Bultheel Department of Computer Science K.U.Leuven, Celestijnenlaan 200A, 3001 Heverlee, Belgium Andrej Vodopivec Department of Mathematics IMFM, 1000 Ljubljana, Slovenia Ernesto Estrada Institute of Complexity Science Department of Physics and Department of Mathematics University of Strathclyde, Glasgow G1 1XH, United Kingdom

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Periodic and Non-periodic Tilings

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Archimedean Lattices - Tiling in the Plane

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Regular Tilings

Squares Triangles Hexagons Random

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Methods

Image SXM determines array coordinates Excel macros for distances (adjacency matrix) Maxima used for distance matrices, Wiener Index, and eigenvalues MATLAB used for graphing and statistical mechanics

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Adjacency Matrix

Aij = 0 if no link between i and j 1 if a link between i and j ﾠ

  • Euclidean adjacency matrix:

Aij = 0 if no link between i and j aij if a link between i and j

  • aij = Euclidean distance between i and j
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Adjacency Matrices

AS = 1 1 1 1 1 1 1 1

  • A H =

1 1 1 1 1 1 1 1 1 1 1 1

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Statistical Mechanics of Graphs

The partition function is defined as: where the Hamiltonian H = -A and A is the adjacency matrix, G is a graph, and ß = 1 for an unweighted network

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Statistical Mechanics of Graphs, cont’d

The probability that the system will occupy the jth microstate is given by: where is an eigenvalue of the adjacency matrix We can then define the thermodynamic functions as follows:

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Statistical Mechanics of Graphs, cont’d

The (Shannon) entropy is : which can be rearranged as: The energy relationships are then:

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Statistical Mechanics of Graphs, cont’d

The total energy can be written as: and the Helmholtz free energy is:

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Hexagonal Arrays

10 Coordinates 11 Links 1000 Coordinates 1446 Links

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Statistical Mechanics Results Partition Function and Enthalpy

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Results, cont’d Entropy and Free Energy

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Power Laws

If a relationship has a power fit A log-log plot produces a straight line, slope k

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Summary

Wiener Index used to model real 2D tilings, i.e. porous arrays Porous arrays are increasing (expanding) in size Distributions of normalized link length determined Statistical mechanics functions are fit with power regression Work to be Done Still need a model for the thermodynamic behavior of graphs