The Increasingly Popular Potts Model
or
A Graph Theorist Does Physics (!)
Jo Ellis-Monaghan
e-mail: jellis-monaghan@smcvt.edu
website: http://academics.smcvt.edu/jellis-monaghan
3/16/2016
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Getting by with a little (a lot of!) help
from my friends….
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Patrick Redmond (SMC 2010)
Eva Ellis-Monaghan (Villanova 2010)
Laura Beaudin (SMC 2006)
Patti Bodkin (SMC 2004)
Whitney Sherman (SMC 2004)
This work is supported by the
Vermont Genetics Network
through NIH Grant Number 1
P20 RR16462 from the INBRE
program of the National Center
for Research Resources.
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Mary Cox (UVM grad)
Robert Schrock (SUNY Stonybrook)
Greta Pangborn (SMC)
Alan Sokal (NYU)
Isaac Newton Institute for
Mathematical Sciences
Cambridge University, UK
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Applications of the Potts Model
● Liquid-gas transitions
● Foam behaviors
● Magnetism
● Biological Membranes
● Social Behavior
● Separation in binary alloys
● Spin glasses
● Neural Networks
● Flocking birds
● Beating heart cells
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These are all complex
systems with nearest
neighbor interactions.
These microscale
interactions determine the
macroscale behaviors of the
system, in particular phase
transitions.
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Ernst Ising 1900-1998
Ising, E. Beitrag zur Theorie des Ferromagnetismus. Zeitschrift fr Physik 31 (1925), 253-258.
72,500
Articles on
‘Potts
Model’
found by
Google
Scholar
http://www.physik.tu-dresden.de/itp/members/kobe/isingconf.html
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The Ising Model
Consider a sheet of metal:
It has the property that at low temperatures it is magnetized,
but as the temperature increases, the magnetism “melts away”*.
We would like to model this behavior. We make some
simplifying assumptions to do so.
– The individual atoms have a “spin”, i.e., they act like little bar magnets,
and can either point up (a spin of +1), or down (a spin of –1).
– Neighboring atoms with the same spins have an interaction energy,
which we will assume is constant.
– The atoms are arranged in a regular lattice.
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*Mathematicians should NOT attempt this at home… 5
One possible state of the lattice
A choice of ‘spin’ at each lattice point.
q2
Ising Model has a
choice of two possible
spins at each point
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The Kronecker delta function and the
Hamiltonian of a state
Kronecker delta-function is defined as:
0 for a  b
 a ,b  
1 for a  b
The Hamiltonian of a system is the sum of the energies on edges with
endpoints having the same spins.
H    J   a, b 
edges
where a and b are the endpoints of the edge, and J is the energy of the edge.
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The energy (Hamiltonian) of the state
Endpoints have the same spins, so δ is 1.
1
0
0
1
1
0
1
0
edges
1
0
0
0
0
H ( w)    J  si , s j
0
1
1
1
0
1
0
0
0
0
Endpoints have different spins, so δ is 0.
0
0
0
0
0
0
0
H ( w) of this system is -10J
1
A state w with the value of δ marked on each edge.
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The Potts Model
Now let there be q possible states….
Orthogonal vectors,
with δ replaced by dot
product
q2
q3
q4
Colorings of the points
with q colors
Healthy
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Sick
Necrotic
States pertinent to the
application
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More states--Same Hamiltonian
 The Hamiltonian still
measures the overall energy
of the a state of a system.
The Hamiltonian of a state of a 4X4
lattice with 3 choices of spins (colors)
for each element.
1
0
H ( w)    J  si , s j
0
0
0
0
0
1
1
1
edges
1
0
1
1
0
0
0
1
0
0
1
0
0
1
(note—qn possible states)
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H  10J
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Probability of a state
The probability of a particular state S occurring depends on the
temperature, T
(or other measure of activity level in the application)
--Boltzmann probability distribution--
PS  
exp(  H ( S ))

exp(   H (S))
all states S
1

where k  1.38 1023 joules/Kelvin and T is the temperature of the system.
kT
The numerator is easy. The denominator, called the
Potts Model Partition Function,
is the interesting (hard) piece.
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Example
Minimum Energy States
PS  
The Potts model partition function of a
square lattice with two possible spins
exp(  H ( S ))

exp(   H (S))
H  4 J
H  2 J
H  2 J
H  2 J
H  2 J
H  2 J
H  2 J
H 0
H 0
H  2 J
H  2 J
H  2 J
H  2 J
H  2 J
H  2 J
H  4 J
all states S
P  all red  
exp(4 J )
12exp(2 J )  2exp(4 J )  2
12exp(2 J  )  2exp(4 J  )  2
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Probability of a state occurring
depends on the temperature
P(all red, T=0.01) = .50 or 50%
P(all red, T=2.29) = 0.19 or 19%
P(all red, T = 100, 000) = 0.0625 = 1/16
Setting J = k for convenience, so
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P  all red  
exp(4 / T )
12exp(2 / T )  2exp(4 / T )  2
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Effect of Temperature
• Consider two different states A and B, with H(A) < H(B). The relative
probability of the two states is:
P  A
P B
e   H ( A)


e
all states S
  H ( A)
e
  H ( B)  e
e

D
kT
 H S 
e  H ( B )

e
 H S 
all states S
D
kT
 e , where D  H  A   H  B   0.
• At high temperatures (i.e., for kT much larger than the energy difference
|D|), the system becomes equally likely to be in either of the states A or B that is, randomness and entropy "win". On the other hand, if the energy
difference is much larger than kT (very likely at low temperatures), the
system is far more likely to be in the lower energy state.
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Ising Model at different temperatures
Cold Temperature
Hot Temperature
Here H is
s s
i j
and energy is
H
# of squares
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Critical Temperature
Images from http://bartok.ucsc.edu/peter/java/ising/keep/ising.html
http://spot.colorado.edu/~beale/PottsModel/MDFrameApplet.html
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Monte Carlo Simulations
?
http://www.pha.jhu.edu/~javalab/potts/potts.html
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Monte Carlo Simulations
Generate a random number r between 0 and 1.
P  A
P B
r
B (stay old)
P  A
B (old)
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P B
r
A (change to new)
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Capture effect of temperature
 H  B   H  A 
P  A
 exp 
 , with B
Given r between 0 and 1, and that P  B 
kT


the current state and A the one we are considering changing to, we have:
High Temp
H(B) < H(A)
exp(‘-’/kT) ~1
B is a lower energy state
> r, so change states.
than A
H(B) > H(A)
B is a higher energy
state than A
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Low Temp
exp(‘-’/kT) ~ 0
< r, so stay in low
energy state.
exp(‘+’/kT) ~1
exp(‘+’/kT) ~1
> r, so change states.
> r, so change to lower
energy state.
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Foams
 “Foams are of practical
importance in applications as
diverse as brewing, lubrication,
oil recovery, and fire fighting”.
H   J (1   i j )    (an  An ) 2
{i , j }
n
 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
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A personal favorite
Y. Jiang, J. Glazier, Foam
Drainage: Extended Large-Q Potts
Model Simulation
We study foam drainage using the
large-Q Potts model... profiles of
draining beer foams, whipped
cream, and egg white ...
Olympic Foam:
http://www.lactamme.polytechnique.fr/Mosaic/images/ISIN.41.16.D/display.html
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http://mathdl.maa.org/mathDL?pa=mathNe
ws&sa=view&newsId=392
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Life Sciences Applications
 This model was developed to see if tumor growth is
influenced by the amount and location of a nutrient.


H   J ( ij ) ( i ' j ' ) 1   ij i ' j '    (   VT ) 2  Kp(i, j )
ij
i j 

 Energy function is modified by the volume of a cell and
the amount of nutrients.
Results:
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Tumors grow exponentially in the
beginning.
The tumor migrated toward the
nutrient.
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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 …”.
 (T. C. Schelling won the 2005 Nobel prize in economics for this
work)
Variables:
Preferences of individuals
Size of the neighborhoods
Number of individuals
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What’s a nice graph theorist doing with
all this physics?
• If two vertices have different spins, they don’t interact, so
there might as well not be an edge between them (so delete it).
• If two adjacent vertices have the same spin, they interact with
their neighbors in exactly the same way, so they might as well
be the same vertex (so contract the edge)*.
Delete e
e
G
*with
G-e
Contract e
a weight for the interaction energy
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G/e
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Bridges and Loops
bridges
Not a bridge
A bridge is an edge
whose deletion separates
the graph
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A loop
A loop is an edge with
both ends incident to the
same vertex
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Tutte Polynomial
(The most famous of all graph polynomials)
Let e be an edge of G that is neither a bridge nor a loop. Then,
T  G; x, y   T (G  e; x, y )  T  G / e; x, y 
And if G consists of i bridges and j loops, then
T  G; x, y   xi y j
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Example
The Tutte polynomial of a cycle on 4 vertices…
=
+
+
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+
=
+
+
=
+
=
x3  x 2  x  y
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The q-state Potts Model Partition Function
is an evaluation of the Tutte Polynomial!
If we let v  e J  1, and have q states, then:

all states S
exp(   H (S))  q k ( G )  v 
V ( G ) k ( G )
 qv

T  G;
,1  v 
v


The Potts Model Partition Function is a polynomial in q!!!
Fortuin and Kasteleyn, 1972
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Example
The Tutte polynomial of a 4-cycle:
x3  x 2  x  y
Compute Potts model partition function from the Universality Theorem result:
P(G; q, v)  q
Let q = 2 and v  e
21  e J 
J
 qv

T  G;
,1  v 
v


k (G ) V (G ) k (G )
v
1
 e J   1 3  e J   1  2  e J   1 

J
 1  J 
   J
   J
e 
 e  1   e  1   e  1 

3
12 exp(2 J  )  2 exp(4 J  )  2
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Thank you for attending!
• Questions?
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