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Patti BodkinSaint Michael’s CollegeColchester, VT 05439

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Phase Transitions

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Different States of a Model

2 / , 0,1, . . . , 1.n n q n q

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. 2q

2q 3q 4q

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Hamiltonian and the Kronecker delta Function

,

0 for

1 for a b

a b

a b

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: ,i js s

i jH J

where and are the states at the and sites.th thi js s i j

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01

00

000 0

00

0

1 1111

1 1 1 1

111

1 1

111

1 1 1

Example:

of this system is:( )H 21

Consider the following model of a magnet system where each site has two possible states, positive or negative, an example of the Ising Model

2q

,( )i js s

i jH J

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Computing the Partition Function…

The probability of a particular system occurring is:

( )

( )

all states

H

He

e

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. 202

( )

all states ( , ) HP q e

Partition Function of the Potts Model:

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Recall: The Universality Theorem:

( ) ( ) ( / )

and

( ) ( ) ( )

f G a f G e b f G e

f GH f G f H

The Dichromatic Polynomial

where is either the disjoint union of and or where and share at most one vertex

GH G HHG

e whenever is not a loop or a bridge

| | | | ( ) | | ( ) 0 0( ) ( ; , )E V k G V k G x yf G a b t G

b a then,

( ) ( ) ( / )Z G Z G e vZ G e

The Dichromatic Polynomial is defined as

Clearly satisfies condition 1, with and .( )Z G 1a b v

So, is an evaluation of the Tutte polynomial:( )Z G

( ) | | ( ) ( )( , ) , 1k G V k G

G Gq v

Z q v q v T vv

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The Potts Partition Function is an evaluation of the Dichromatic

Polynomial(hence of the Tutte polynomial too!)| |

.

( , ) ( , )EG G

dichrom poly

P q e Z q v

Simplified Proof:

Let| |( ) ( ; , )EP G e P G q

Consider all edges of the system, perform the deletion and contraction steps. After simplifying, we’ll end up with:

( ) ( ) ( 1) ( / )P G P G e e P G e

( , )e c d

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( ) ( ) ( ) ( / )P G P G e v P G e

( ) ( ; , 1)P G Z G q e

We can now show that the Potts Model is an evaluation of the Tutte Polynomial!!

| | ( ) | | ( ) 1( ; , ) ( 1) ( ; , )

1E k G V k G q e

P G q e q e T G ee

( ) ( ) ( / )Z G Z G e vZ G e Recall:

( 1)v e Let:

| |( ; , ) ( ; , )EP G q e Z G q v

Since we defined| |( ) ( ; , )EP G e P G q

After simplifying we have:

( ) ( ) ( 1) ( / )P G P G e e P G e

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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

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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 )

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Magnet Model Phase Transition

Cold Temperature

Hot Temperature

Images taken from applet on:http://bartok.ucsc.edu/peter/java/ising/keep/ising.html

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Neural NetworksThere 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.

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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.

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ResourcesModern Graph Theory, Béla Bollobás

“The Potts Model”, F. Y. Wu

“Chromatic Polynomial, Potts Model and All That”, Alan D. Sokal

Jo Ellis-Monaghan