1. The Potts Model Laura Beaudin Saint Michael’s College The project described was supported by the Vermont Genetics Network through NIH Grant Number 1 P20 RR16462 from the INBRE program of the National Center for Research Resources. Advisors: Joanna Ellis-Monaghan (Mathematics) Greta Pangborn (Computer Science) 1

2. Foundations  The Potts model is used in an area of mathematical modeling known as Statistical Mechanics.  A Lattice is a Graph with a Regular Structure  Models how nearest neighbor elements with different spins interact with one another on a lattice.  The Potts model “provides a framework for relating the microscopic properties … to the macroscopic properties … that can be observed in everyday life” [4].  The Potts model investigates phase transitions. Square lattice Triangular lattice Honeycomb lattice 2

3. The Hamiltonian  The Hamiltonian measures the overall energy of the state of a system.  Here J is the energy between elements, is the spin of an element at position i, and assigns a 1 to an edge between elements with like spins and a 0 between elements with different spins. 1 1 1 1 1 11 1 1 100 0 0 0 0 0 0 0 0 0 00 0 The Hamiltonian of a state of a 4X4 lattice with 3 choices of spins (colors) for each element. 3

4. The Potts Model Partition Function  The Potts model investigates state on a lattice.  The formula for finding the probability of a particular state in the set of all possible configurations is  The denominator is the Potts model partition function. The Potts model partition function of a square lattice with two possible spins on each element. 4

5. Approximations  The Potts model partition function is impractical to compute when and the lattice is large. Computers can be used to approximate these calculations.  Monte-Carlo simulations: creates, combines, stores random numbers which represent variables in experiments, to probabilistically generate a representative random sample of data points to get a good estimate of the value you are interested in.  Metropolis Algorithm: generates probabilities using the following actions.  Begin with a state. Find the energy of the state.  Make a small change and find the energy of the new state.  If the new energy is lower, the system will move to the new state.  If the new energy is higher, the system will move to the new state with a probability of 6 Where is the temperature.

6. Biological Application  This model was developed to see if tumor growth is influenced by the amount and location of a nutrient.  Energy function is modified by the volume of a cell and the amount of nutrient. Results: Tumors grow exponentially in the beginning. The tumor migrated toward the nutrient. [15] 7

7. Sociological Application  The Potts model may be used to “examine some of the individual incentives, and perceptions of difference, that can lead collectively to segregation … the extent to which inferences can be drawn, from the phenomenon of collective segregation, about the preferences of individuals, the strength of those preferences, and the facilities for exercising them” [13]. Variables: Preferences of individuals Size of the neighborhoods Number of individuals 8

8. Physical Application  “Foams are of practical importance in applications as diverse as brewing, lubrication, oil recovery, and fire fighting” [9].  The energy function is modified by the area of a bubble. Results: Larger bubbles flow faster. There is a critical velocity at which the foam starts to flow uncontrollably. 9

9. The Tutte Polynomial  Encodes structural properties of graphs.  Two fundamental graph operations  Deletion  Contraction  Recursive Computation  T(G; x, y) = T(G-e; x, y) + T(G/e; x, y) if e is not a bridge or a loop.  T(GH) = T(G)T(H) if G and H share at most one vertex.  T(G) = if G has only i bridges and j loops. e Delete e Contract e G G-e G/e 10 (Bridge) (Loop)

10. Universality  The Tutte polynomial is well-defined:  The polynomial does not depend on the order of deletion and contraction.  Universality Theorem: If f(G) is a function of graphs such that  f(G) = 1 if G consists of only one vertex and no edges,  f(G) = af(G-e)+bf(G/e) whenever e is not a loop or a bridge,  f(GH) = f(G)f(H) where GH is either the disjoint union of G and H or where G and H share at most one vertex, then f is an evaluation of the Tutte polynomial. 11

11. Relating the Potts model to the Tutte Polynomial  Recall that the Hamiltonian places a 0 on an edge between unlike spins. Therefore, we can delete these edges.  Otherwise the Hamiltonian places a 1 on an edge between like spins. Therefore, we can contract these edges. Now we can apply the universality theorem to the Potts model partition function. 12

12. Example =+=++= +++= Solution using the Universality Theorem Same as slide 4! The square lattice of slide 4. 13

13. [1] Albert, M. S. Kiskowski, M. A. Glazier, J. A. and Jiang, Y. http://math.lanl.gov/~yi/Papers/CArev.pdf. On Cellular Automaton Approaches to Modeling Biological Cellshttp://math.lanl.gov/~yi/Papers/CArev.pdf [2] Ashkin, Julius; Teller, Edward. Statistics of Two Dimensional Lattices with Four Components, Physics Review, 64, pp. 178- 184. (1943) [3] Bodkin, Patricia; Cox, Mary; Ellis-Monaghan, Jo; Sherman, Whitney. Contraction-Deletion Invariants: The Tutte Polynomial in Engineering, Biology, and Physics. Manuscript. June 2006. [4] Chandler, David. Introduction to Modern Statistical Mechanics. Oxford University Press. [5] Chang, Shu-Chiuan; Shrock, Robert. Exact Partition Function for the Potts Model with Next-Nearest Neighbor Coupling on Arbitrary-Length Ladders. International Journal of Modern Physics B, Vol. 15, No. 5. 2001 443-478. World Scientific Publishing Company. [6] Ellis-Monaghan, Jo. From Potts to Tutte and back again … A graph theoretical view of statistical mechanics. PowerPoint Presentation at Middlebury College. October 2005. [7] http://saeta.physics.hmc.edu/courses/p170/Metropolis.pdf#search='the%20metropolis%20algorithm%2C %20%20when%20studying%20systems%20with%20a%20great%20many%20particles%2C%20it'. The Metropolis Algorithm; Statistical Systems and Simulated Annealing.http://saeta.physics.hmc.edu/courses/p170/Metropolis.pdf#search='the%20metropolis%20algorithm%2C %20%20when%20studying%20systems%20with%20a%20great%20many%20particles%2C%20it [8] Huang, Kerson. Statistical Mechanics. Wiley John & Sons, Inc. 1990 [9] Jiang, Yi; Glazier, James A. Extended large-Q model simulation of foam drainage. Department of Physics, University of Notre Dame, Notre Dame, Indianna. 1996. [10] Kroemer, Herbert; Kittel, Charles. Thermal Physics (2 nd ed.) W.H. Freeman Company. 1980. [11] Meyer, Peter. Coputational Studies of Pure and Dilute Spin Models. 2000 [12] Sanyal, Soma; Glazier, James A. Viscous instabilities in flowing foams: A Cellular Potts Model approach. [13] Schelling, Thomas C. Dynamic Models of Segregations. Havard University. Journal of Mathematical Sociology. [14] Schulze, Christian. Potts-Like Model for Ghetto Formation in Multi-Cultural Societies. Internatoinal Journal of Modern Physics C. Vol. 16, No. 3 (2005) 351-355 [15] Sun, L. Chang, Y. F. Cai, X. A Discrete Simulation of Tumor Growth Concerning Nutrient Influence. The Institute of Particle Physics, Central China Normal University. Wuhan, Hubei. August 2003. [16] Winston, Wayne L. Operations Research: Applications and Algorithms. Second Edition. Duxbury Press. Belmont California. 1991. [17] Welsh, D. J. A.; Merino, C. “The Potts model and the Tutte polynomial.” American Institute of Physics. 2000. References 5