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

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, 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

Periodic and Non-periodic Tilings

Archimedean Lattices - Tiling in the Plane

Regular Tilings Triangles Squares Hexagons Random

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

Adjacency Matrix ﾠ A ij = 0 if no link between i and j � � 1 if a link between i and j Euclidean adjacency matrix: � 0 if no link between i and j � A ij = a ij if a link between i and j � a ij = Euclidean distance between i and j

Adjacency Matrices � � 0 1 0 0 0 1 � � 1 0 1 0 0 0 � � 0 1 0 1 � � � � � � 0 1 0 1 0 0 1 0 1 0 � � A H = � � A S = 0 0 1 0 1 0 � � 0 1 0 1 � � � � � � 0 0 0 1 0 1 � 1 0 1 0 � � � � 1 0 0 0 1 0 �

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

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:

Statistical Mechanics of Graphs, cont’d The (Shannon) entropy is : which can be rearranged as: The energy relationships are then:

Statistical Mechanics of Graphs, cont’d The total energy can be written as: and the Helmholtz free energy is:

Hexagonal Arrays 10 Coordinates 1000 Coordinates 11 Links 1446 Links

Statistical Mechanics Results Partition Function and Enthalpy

Results, cont’d Entropy and Free Energy

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

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