Transcript Slide 1

Patti Bodkin
Saint Michael’s College
Colchester, VT 05439
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Phase Transitions
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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:
 n  2 n / q, n  0,1, . . . , q  1.
q2
q 3
q4
The q  2 case is a special case known as the Ising Model.
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Hamiltonian and
the Kronecker delta Function
Kronecker delta-function is defined as:
0 for a  b
 a ,b  
1 for a  b
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:
H  J   si ,s j
i j 
where si and s j are the states at the i th and j th sites.
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Example:
Consider the following model of a magnet system where each site has two possible
states, positive or negative, an example of the Ising Model
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1
0
0
1
1
0
1
0
1
i j 
0
1
1
1
H ( )  J   si ,s j
1
0
0
1
0
1
1
1
q2
1
1
1
0
1
1
1
1
H ( ) of this system is: 21
0
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Computing the Partition Function…
The probability of a particular system occurring is:
e  H ( )

all states 
e  H ( )
The denominator is the partition function, and is very hard to compute.
In fact, for our model, there are 2 20 possible states to consider for the denominator.
Partition Function of the Potts Model:
P ( q,  ) 

all states 
e  H ( )
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The Dichromatic Polynomial
Recall: The Universality Theorem:
f (G )  a f (G  e)  b f (G / e) whenever e is not a loop or a bridge
and
where GH is either the disjoint union of G and H
f (GH )  f (G ) f ( H )
or where G and H share at most one vertex
then,
f (G)  a|E||V | k (G )b|V | k (G )t (G;
x0 y0
, )
b a
The Dichromatic Polynomial is defined as
Z (G)  Z (G  e)  vZ (G / e)
Clearly Z (G ) satisfies condition 1, with a  1 and
So,
bv.
Z (G) is an evaluation of the Tutte polynomial:
Z G ( q, v )  q
 (q  v)

TG 
, 1 v 
 v

k ( G ) |V | k ( G )
v
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The Potts Partition Function
is an evaluation of the Dichromatic Polynomial
(hence of the Tutte polynomial too!)
PG (q,  )  e  |E| ZG (q, v)
dichrom. poly
Simplified Proof:
Let
P(G )  e  |E| P(G; q,  )
Consider all edges e  ( c, d ) of the system, perform the deletion and contraction steps.
After simplifying, we’ll end up with:
P(G )  P(G  e)  (e   1) P(G / e)
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P(G )  P(G  e)  (e   1) P(G / e)
Recall:
Z (G)  Z (G  e)  vZ (G / e)
Let:
v  (e   1)
P(G)  P(G  e)  (v) P(G / e)
P (G )  Z (G; q, e   1)
Since we defined
P(G )  e  |E| P(G; q,  )
After simplifying we have:
P(G; q,  )  e  |E| Z (G; q, v)
We can now show that the Potts Model is an evaluation of the Tutte Polynomial!!

q

e
1 
P(G; q,  )  e   |E|q k (G ) (e   1)|V | k (G ) T (G; 
,e )
e 1
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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 Networks
There are two ways to develop machines which exhibit “intelligent behavior”;
Artificial Intelligence
Neural Networks
Neural Networks:
Architecture that is based loosely on an animal’s brain.
Learns from a training environment, rather than being preprogrammed.
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 onephase systems and can be easily extended to include drainage.
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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
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