1. 1 Patti Bodkin Saint Michael’s College Colchester, VT 05439

2. 2 Phase Transitions

3. 3 Different States of a Model The Potts Model is often referred to as the q-state Potts Model where the spins in the system can have the value of one of the q equally spaced angles: The case is a special case known as the Ising Model.

4. 4 Hamiltonian and the Kronecker delta Function Kronecker delta-function is defined as: for two “nearest neighbor” sites, a and b. The Hamiltonian of a system is the sum of the changes in states of all of the sites. It is defined as:

5. 5 Example: of this system is: Consider the following model of a magnet system where each site has two possible states, positive or negative, an example of the Ising Model

6. 6 Computing the Partition Function… The probability of a particular system occurring is: The denominator is the partition function, and is very hard to compute. In fact, for our model, there are possible states to consider for the denominator. Partition Function of the Potts Model:

7. 7 Recall: The Universality Theorem: The Dichromatic Polynomial where is either the disjoint union of and or where and share at most one vertex whenever is not a loop or a bridge then, The Dichromatic Polynomial is defined as Clearly satisfies condition 1, with and. So, is an evaluation of the Tutte polynomial:

8. 8 The Potts Partition Function is an evaluation of the Dichromatic Polynomial (hence of the Tutte polynomial too!) Simplified Proof: Let Consider all edges of the system, perform the deletion and contraction steps. After simplifying, we’ll end up with:

9. 9 We can now show that the Potts Model is an evaluation of the Tutte Polynomial!! Recall: Let: Since we defined After simplifying we have:

10. 10 Where would you find the Potts Model being used today? ● Atoms ● Animals ● Protein Folds ● Biological Membranes ● Social Behavior ● Phase separation in binary alloys ● Spin glasses ● Neural Networks ● Flocking birds ● Beating heart cells

11. 11 The Ising Model and Magnets At a low temperature, a sheet of metal is magnetized At high temperatures, the metal becomes less magnetized. The magnetism of a sheet of metal as it goes through temperature phase transitions can be modeled with the Ising model (Potts Model with q =2 )

12. 12 Magnet Model Phase Transition Cold Temperature Hot Temperature Images taken from applet on: http://bartok.ucsc.edu/peter/java/ising/keep/ising.html

13. 13 Neural Networks There are two ways to develop machines which exhibit “intelligent behavior”; Artificial Intelligence Neural Networks Architecture that is based loosely on an animal’s brain. Learns from a training environment, rather than being preprogrammed. Neural Networks: John Hopfield showed that a highly interconnected network of threshold logic units could be arranged by considering the network to be a physical dynamic system possessing an “energy.” “Associative Recall” is where a net is started in some initial random state and goes on to some stable final state. The process of Associate Recall parallel the action of the system falling into a state of minimal energy. The mathematics of these systems is very similar to the Ising Model of magnetic phenomena in materials.

14. 14 Applications of large Q-Potts Model The extended large Q-Potts Model “captures effectively the global features of tissue rearrangement experiments including cell sorting and tissue engulfment. The large Q-Potts Model “simulates the coarsening of foams especially in one- phase systems and can be easily extended to include drainage.

15. 15 Resources Modern Graph Theory, Béla Bollobás “The Potts Model”, F. Y. Wu “Chromatic Polynomial, Potts Model and All That”, Alan D. Sokal Jo Ellis-Monaghan