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Graph Theory and Complex Systemsin Statistical Mechanics

e-mail: jellis-monaghan@smcvt.edu

website: http://academics.smcvt.edu/jellis-monaghan

Jo Ellis-Monaghan

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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.

• Mary Cox (UVM grad)

• Robert Schrock (SUNY Stonybrook)

• Greta Pangborn (SMC)

• Alan Sokal (NYU)

Isaac Newton Institute for Mathematical SciencesCambridge 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

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

http://www.physik.tu-dresden.de/itp/members/kobe/isingconf.html

72,500 Articles on ‘Potts Model’ found by Google Scholar

Ising, E. Beitrag zur Theorie des Ferromagnetismus. Zeitschrift fr Physik 31 (1925), 253-258.

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The Ising ModelConsider 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.*Mathematicians should NOT attempt this at home…

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One possible state of the latticeA choice of ‘spin’ at each lattice point.

2q

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

,

0 for

1 for a b

a b

a b

Kronecker delta-function is defined as:

The Hamiltonian of a system is the sum of the energies on edges with

endpoints having the same spins.

,edges

H J a b

where and are the endpoints of the edge, and is the energy of the edge.a b J

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The energy (Hamiltonian) of the state

of this system is ( )H w -10J

,( )i js s

edgesH w J

A state w with the value of δ marked on each edge.

0

10

0

000 0

00

0

1

1

1 111

0

11

1

0

0

00

0 0

0 0 0

0Endpoints have different spins, so δ is 0.

Endpoints have the same spins, so δ is 1.

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The Potts ModelNow let there be q possible states….

2q 3q 4q

Orthogonal vectors, with δ replaced by dot product

Colorings of the points with q colors

States pertinent to the application

Healthy Sick Necrotic

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More states--Same Hamiltonian The Hamiltonian still

measures the overall energy of the a state of a system.

10H J

1

1

1

1

1

1 1

1

1

1 0 0

0

0

0

0

0

0

0

0

0

00

0

The Hamiltonian of a state of a 4X4 lattice with 3 choices of spins (colors) for each element.

3

,( )i js s

edgesH w J

(note—qn possible states)

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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--

Probability of a state

all states

exp( ( ))

exp( ( ))

H SP S

H

S

S

4

231 where 1.38 10 joules/Kelvin and is the temperature of the system.k T

kT

The numerator is easy. The denominator, called the Potts Model Partition Function, is the interesting (hard) piece.

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Example

all states

exp( ( ))

exp( ( ))

H SP S

H

S

S

The Potts model partition function of a square lattice with two possible spins

12exp(2 ) 2exp(4 ) 2J J

4

4H J 2H J 2H J 2H J

2H J 2H J 2H J

2H J 2H J 2H J

2H J 2H J 2H J 4H J

0H

0H

exp(4 )

12exp(2

all r

) 2exp(4 ) 2

edP

J

J J

Minimum Energy States

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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 exp(4 / )

12exp(2 /all red

) 2exp(4 / ) 2

T

PT T

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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:

• 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.

( ) ( )

all states all states

H A H B

H H

P A e e

P Be e

S S

S S

( )

( ), where 0.

H A D

kTH B

D

kTee

e D H A H Be

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Ising Model at different temperatures

Cold Temperature Hot Temperature

Images from http://bartok.ucsc.edu/peter/java/ising/keep/ising.htmlhttp://spot.colorado.edu/~beale/PottsModel/MDFrameApplet.html

Critical Temperature

Here H is

and energy is

i js s

# of squares

H

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Monte Carlo Simulations

?

http://www.pha.jhu.edu/~javalab/potts/potts.html

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Monte Carlo Simulations

B (old)

A (change to new)

B (stay old)

Generate a random number r between 0 and 1.

P Ar

P B

P Ar

P B

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Capture effect of temperature

High Temp Low Temp

H(B) < H(A)

B is a lower energy state than A

exp(‘-’/kT) ~1 exp(‘-’/kT) ~ 0

> r, so change states. < r, so stay in low energy state.

H(B) > H(A)

B is a higher energy state than A

exp(‘+’/kT) ~1 exp(‘+’/kT) ~1

> r, so change states. > r, so change to lower energy state.

Given r between 0 and 1, and that , with B the current state and A the one we are considering changing to, we have:

exp

P A H B H A

P B kT

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Foams “Foams are of practical

importance in applications as

diverse as brewing, lubrication,

oil recovery, and fire fighting”.

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

2

{ , }

(1 ) ( )i j n n

i j n

H J a A

9

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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://mathdl.maa.org/mathDL?pa=mathNews&sa=view&newsId=392

http://www.lactamme.polytechnique.fr/Mosaic/images/ISIN.41.16.D/display.html

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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.

Energy function is modified by the volume of a cell and the amount of nutrients.

Results: Tumors grow exponentially in the

beginning.

The tumor migrated toward the

nutrient.

' ' ' '

2( ) ( ) 1 ( ) ( , )

ij i j ij i j Tij i j

H J V Kp i j

7

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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 individualsSize of the neighborhoodsNumber of individuals

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)*.

*with a weight for the interaction energy

e

Delete e

Contract eG

G-e

G/e

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Tutte Polynomial(The most famous of all graph polynomials)

; , ( ; , ) / ; ,T G x y T G e x y T G e x y

; , i jT G x y x y

Let e be an edge of G that is neither a bridge nor a loop. Then,

And if G consists of i bridges and j loops, then

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Example

3 2x x x y

= + = + + =

+ + + =

The Tutte polynomial of a cycle on 4 vertices…

13

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Does the order of contraction/deletion matter?

The Dichromatic Polynomial:

Can show (by induction on the number of edges) that

TheTutte polynomial is independent of the order in which the edges are deleted and contracted.

; , k A A

A E G

Z G u v u v

; , 1 ; ,k G V k G u vu v T G v Z G u v

v

vs.

k(A) = 2

|A| = 2

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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.

Universality

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

b a

, , and E V k G

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,

, where

are the number of edges, vertices, and components of G, respectively, and where

f ( ) = x0 , and f ( ) = y0.

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The q-state Potts Model Partition Function is an evaluation of the Tutte Polynomial

If we let , then:

( ) ( )( )( ; , ) ; ,1V G k Gk G q v

P G q v q v T G vv

(Fortuin and Kasteleyn, 1972)

1Jv e

The Potts Model Partition Function is a polynomial in q!!!

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,

states

; , exp 1 expq

P L q J q J 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,

; , 1 exp 0 exp 1 exp 1P B q q q K q J q J q

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Example

3 2x x x y

( ) ( )( )( ; , ) ; ,1V G k Gk G q vP G q v q v T G v

v

3 2

31 1 1 12 1

1 1 1

J J JJ J

J J J

e e ee e

e e e

Compute Potts model partition function from the Universality Theorem result:

12exp(2 ) 2exp(4 ) 2J J

The Tutte polynomial of a 4-cycle:

Let q = 2 and 1Jv e

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A different point of view

states states

If 1 , then , 1H wJ

edges

v e e v a b ( ) Ak A

A E

q v

1

1

1

1

1

1 1

1

1

1 0 0

0

0

0

0

0

0

0

0

0

00

0

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.

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…).

( ; , ) ; , k A A

A E G

P G q v Z G q v q v

Another reason to believe that

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Important thermodynamic functions such as internal energy, specific heat, entropy, and free energy may all be derived from the Potts Model partition function,

U=Internal Energy (sum of the potential and kinetic energy):

C=Specific heat (energy to raise a unit amount of material one degree):

S=Entropy (a measure of the randomness and disorder in a system):

F=Total free energy:

Thermodynamic Functions

; ,Z Z G q v

1expU h h

Z ln Z

UC

T

lnln

ZS Z

lnF U TS T Z

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Phase Transitions

For convenience, work with the dimensionless, reduced free energy:

Reduced free energy per unit volume (or in this context, per vertex):

f F

1

1; , ln ; ,f G q P 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:

Phase transitions (failure of analyticity) arise in the infinite volume limit.

1, lim log ; ,G n n

nf q v V P G q v

TempTemp

EnergyEnergyn

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The antiferromagnetic model at zero temperature

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.

A

E

D C

B

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.

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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 ;

Chromatic polynomial

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Example

-

= n(n-1)2 +n(n-1) + 0 = n2 (n-1)

n(n-1)2 + += =

;C G x 1 ;1 ,0V k G k Gx T G x

Since a contraction-deletion invariant, the chromatic polynomial

is an evaluation of the Tutte polynomial:

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The connectionNote:

; 1 ;1 ,0V G k G k GC G x x t G x

Recall:

1 ; , ; , 1V G k Gk G q

P G q q t G v

with

exp 1v K

These are the same function precisely when 1v

that is, when

which is exactly the zero-temperature model.

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Another connectionConsider the summands of

1 ; , exp ,i jP G q J as 0T , and hence

A summand is 0 except precisely when

, 0i j in which case it is 1. Thus

1P G simply counts the number of proper colorings of G with q colors.

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Zeros of the chromatic polynomial

1lim ln ;nn

n

S C G qV G

Thus, phase transitions correspond to the accumulation points of roots of the chromatic

polynomial in the infinite volume limit.

In the infinite volume limit, the ground state entropy per vertex of the Potts antiferromagnetic model becomes:

04/18/23 40

Locations of ZerosMathematicians 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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Some zeros

(Robert Shrock)

04/18/23 42

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:

04/18/23 44

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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Thank you for your attention!

• Questions?