1. 1 The Potts model Mike Sinclair

2. 2 Overview Potts model –Extension of Ising model –Uses interacting spins on a lattice –N-dimensional (spin states > 2) –Used for studying phase changes or transition states in chemistry and physics; used for studying cell aggregation and limb morphogenesis in biology

3. 3 Set up of the problem Each spin state at intersection of lattice Random probability of initial position Time-step Expectation: like spin states “aggregate” together

4. 4 More complex example This is a q = 5 spin state After several cycles, some aggregation has taken place Expectation: like spin states “aggregate” together

5. 5 Real world problem This is an example of using the Potts model A chemical pre- pattern is formulated in the simulation A random cellular system is established and allowed to run over n time-steps

6. 6 Real world problem Energy configurations yield limb morphogenesis The aggregation follows the chemical pre-pattern Equilibrium is established and model stops

7. 7 Comments Initial conditions have random positions of spin states Pre-patterns formulate energy interactions of each particle Equilibrium reached after n time-steps and aggregation is achieved

8. 8 Conclusions The Potts model is used extensively in biology, chemistry, and physics. It is an extension of the Ising model using more than 2 spin states. It can be visualized by a modeling large numbers of particles. Used for studying cell aggregation and limb morphogenesis.