Graph Theory and Complex Systems
in Statistical Mechanics
Jo Ellis-Monaghan
e-mail: jellis-monaghan@smcvt.edu
website: http://academics.smcvt.edu/jellis-monaghan
3/16/2016
1
Getting by with a little (a lot of!) help
from my friends….
•
•
•
•
•
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
2
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.
3
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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4
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
9
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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3
H  10J
10
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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11
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  4J
H  2J
H  2J
H  2J
H  2J
H  2J
H  2J
H 0
H 0
H  2J
H  2J
H  2J
H  2J
H  2J
H  2J
H  4J
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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12
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
13
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
P  A
 H  B   H  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.
21
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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8
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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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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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13
+
=
+
+
=
+
=
x3  x 2  x  y
25
Does the order of contraction/deletion
matter?
vs.
The Dichromatic Polynomial:
Z  G; u , v  

u
k  A
v
A
A E  G 
Can show (by induction on the number of edges) that
u
 uv

T  G;
, v  1   Z  G; u , v 
v


k G  V k G 
v
k(A) = 2
|A| = 2
TheTutte polynomial is independent of the order in
which the edges are deleted and contracted.
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Universality
THEOREM: (various forms—Brylawsky, Welsh, Oxley, etc.)
If f is a function of graphs such that
a) f(G) = a f(G-e) + b f(G/e) whenever e is not a loop or bridge, and
b) f(GH) = f(G)f(H) where GH is either the disjoint union of G and H
or where G and H share at most one vertex.
Then,
 x y 
E  V  k  G  V k  G 
f (G )  a
b
T  G; 0 , 0 , where E , V , and k  G 
b a 

are the number of edges, vertices, and components of G, respectively,
and where
f(
) = x0 , and f (
) = y0.
Thus any graph invariant that reduces with a) and b) is an
evaluation of the Tutte polynomial.
Proof: By induction on the number of edges.
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The q-state Potts Model Partition Function
is an evaluation of the Tutte Polynomial
For a loop, note that there are q states, and since both endpoints of a loop necessarily have the
same value, the Hamiltonian of each state is 1. Thus,
P  L; q,   

exp      J 1   q exp   J 
q states
For a single bridge, there are q states where the spins on the endpoints are equal, giving a Hamiltonian
of 1. Then there are q(q-1) states where the spins on the endpoints are different, giving a Hamiltonian
of 0. Thus,
P  B; q,     q  q  1 exp   K  0   q exp   J 1   q  exp   J   q  1
If we let v  e J  1, then:
 qv

P(G; q, v)  q  v 
T  G;
,1  v 
v


The Potts Model Partition Function is a polynomial in q!!!
k (G )
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V ( G ) k ( G )
(Fortuin and Kasteleyn, 1972)
28
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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A different point of view
Another reason to believe that P(G; q, v)  Z  G; q, v  
If v  1  e  J  , then  e
states
  H  w


1
What might we be counting here?
Inspiration: A is the set of edges labeled with
1, so these are exactly the number of edges
contributing to the Hamiltonian, so give the
exponent of v.
0
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v
A
0
0
1
1
1
0
0
0
0
1
0
0
A
A E
0
1
q
1
0
v
k ( A)
0
1
However, this then means that all the vertices
in a component of A must have the same spin,
so q# components ways to assign the spins (sort
of…).

0
1
k  A
A E  G 
  v  a, b   1 
states edges

q
0
1
30
Thermodynamic Functions
Important thermodynamic functions such as internal energy,
specific heat, entropy, and free energy may all be derived from the
Potts Model partition function, Z  Z  G; q, v 
U=Internal Energy (sum of the potential and kinetic energy):
1
U   h   exp    h      ln  Z 
Z

C=Specific heat (energy to raise a unit amount of material one degree):
U
C
T
S=Entropy (a measure of the randomness and disorder in a system):
 ln  Z 
S  
  ln  Z 

F=Total free energy:
F  U  TS  T ln  Z 
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Phase Transitions
For convenience, work with the dimensionless, reduced free energy:
f   F
Reduced free energy per unit volume (or in this context, per vertex):
f  G; q ,   
1
ln  P1  G; q,   
V G 
For a fixed graph G, this is clearly an analytic function in both q and T. Any
failures of analyticity can only occur in the infinite volume limit.
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The infinite volume limit
Let {G} be an increasing sequence of finite graphs (e.g. lattices).
The (limiting) free energy per unit volume is:
f G  q, v   lim Vn
n 
1
log P  Gn ; q, v 
Phase transitions (failure of analyticity) arise in the infinite
volume limit.
Energy
n 
Energy
Temp
Temp
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The antiferromagnetic model at
zero temperature
J is negative in the antiferromagnetic model, so minimal energy states have a maximum
number of zeros in the summation, i.e. every edge has endpoints with different spins. If we
think of the spins as colors, a minimum energy state is then just a proper coloring of the graph.
Now low energy  lots of edges with
different spins on endpoints.
At zero temperature, low energy states
prevail, i.e. we really need to consider
states where the endpoints on every
edge are different.
Such a state corresponds to a proper
coloring of a graph.
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A
E
B
D
C
34
Chromatic polynomial
The Chromatic Polynomial counts the ways to vertex color a
graph: C(G, n ) = # proper vertex colorings of G in n colors.
G
G-e
G/e
+
=
Recursively: Let e be an edge of G . Then,
C  G; n   C (G  e; n)  C  G / e; n 
C; n  n
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Example
=
-
=
n(n-1)2 +
+
= n(n-1)2 +n(n-1) + 0 = n2 (n-1)
Since a contraction-deletion invariant, the chromatic polynomial
is an evaluation of the Tutte polynomial:
C  G; x  
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 1 V k  G  x k  G T  G;1  x,0 
36
The connection
Note:
C  G; x    1
V G  k G 
x
k G 
Recall:
P1  G; q,    q k G 
with
V G  k G 
t  G;1  x,0 
 q 

t  G;
, v  1



v  exp  K  1
These are the same function precisely when v  1
that is, when   
which is exactly the zero-temperature model.
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Another connection
Consider the summands of

P1  G; q,     exp  J    i ,  j 
T  ,0and hence
as

 
A summand is 0 except precisely when
   ,    0
i
j
in which case it is 1. Thus
P1  G  simply counts the number of proper colorings of G
with q colors.
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Zeros of the chromatic polynomial
In the infinite volume limit, the ground state entropy per
vertex of the Potts antiferromagnetic model becomes:
1
S   lim
ln  C  Gn ; q  
n V G
 n
Thus, phase transitions correspond to the
accumulation points of roots of the chromatic
polynomial in the infinite volume limit.
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Locations of Zeros
Mathematicians originally focused on the real zeros of
the chromatic polynomial (the quest for a proof of the
4-color theorem…)
Physicists have changed the focus to the locations of
complex zeros, because these can approach the real
axis in the infinite limit.
Now an emphasis on ‘clearing’ areas of the complex
plane of zeros.
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40
Some zeros
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(Robert Shrock)
41
What happens if….
• Interaction energy depends on the edge?
• Depends on whether the edge is contracted or
deleted?
• Depends on whether the edge is a bridge or a loop?
• What if no longer multiplicative? (!)
• There is an external field?
Fortuin and Kasteleyn, Traldi, Zaslavsky, Bollobas and
Riordan, Sokal, E-M and Traldi, E-M and
Zaslavsky….
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Life gets interesting…
• No longer necessarily get a well-defined
function.
There are necessary and sufficient conditions on the
relations among the edge-weights to guarantee this.
These conditions essentially live on three small
graphs:
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43
Can lose mulitiplicativity….
• Essential characteristics encoded by
contraction/deletion
• Still need relations to assure well-defined
• BUT…
Retain universality properties and that a
function is determined by action on ‘smallest’
objects (all discrete matroids not just a single
bridge/loop).
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44
Thank you for your attention!
• Questions?
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45