Transcript The Model - smcvt.edu

The Potts Model
Laura Beaudin
Saint Michael’s College
The project described was supported by the Vermont Genetics Network
through NIH Grant Number 1 P20 RR16462 from the INBRE program
of the National Center for Research Resources.
Advisors:
Joanna Ellis-Monaghan (Mathematics)
Greta Pangborn (Computer Science)
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Foundations
 The Potts model is used in an
area of mathematical modeling
known as Statistical
Mechanics.
 A Lattice is a Graph with a
Regular Structure
Square lattice
 The Potts model “provides a
framework for relating the
microscopic properties … to
the macroscopic properties
… that can be observed in
everyday life” [4].
Triangular lattice
 The Potts model investigates
phase transitions.
Honeycomb lattice
 Models how nearest neighbor
elements with different spins
interact with one another on a
lattice.
2
The Hamiltonian
 The Hamiltonian
measures the overall
energy of the state of a
system.
The Hamiltonian of a state of a 4X4
lattice with 3 choices of spins (colors)
for each element.
1
H   J   i j
0
{i , j }
 Here J is the energy between
elements,  i is the spin of an
element at position i, and 
assigns a 1 to an edge
between elements with like
spins and a 0 between
elements with different spins.
3
0
0
0
1
0
1
0
1
1
1
1
0
0
1
0
1
0
0
0
0
0
1
H  10 J
The Potts Model Partition
Function
 The Potts model
investigates state on a
lattice.
 The formula for finding the
probability of a particular
state  in the set of all
possible configurations  is
exp(  H ( ))

The Potts model partition function of a
square lattice with two possible spins
on each element.
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
exp(   H ( ))
all states
 
1

where k  1.38 1023 joules/Kelvin and T is the temperature of the system.
kT
 The denominator is the
Potts model partition
function.
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12exp(2 J  )  2exp(4 J  )  2
Approximations

The Potts model partition function is impractical to compute when q  2.
and the lattice is large. Computers can be used to approximate these
calculations.
 Monte-Carlo simulations: creates, combines,
stores random numbers which represent variables
in experiments, to probabilistically generate a
representative random sample of data points
to get a good estimate of the value you are
interested in.
 Metropolis Algorithm: generates probabilities using the following
actions.




Begin with a state. Find the energy of the state.
Make a small change and find the energy of the new state.
If the new energy is lower, the system will move to the new state.
If the new energy is higher, the system will move to the new state with a probability
of
 ( EB  EA ) / T
Where T is the temperature.
pe
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Biological Application
 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
ij

 Energy function is modified by the volume of a cell
and the amount of nutrient.
Results:
Tumors grow exponentially in the
beginning.
The tumor migrated toward the
nutrient. [15]
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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 … the extent to which
inferences can be drawn, from the phenomenon of
collective segregation, about the preferences of individuals,
the strength of those preferences, and the facilities for
exercising them” [13].
Variables:
Preferences of individuals
Size of the neighborhoods
Number of individuals
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Physical Application
 “Foams are of practical
importance in applications as
diverse as brewing, lubrication,
oil recovery, and fire fighting” [9].
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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The Tutte Polynomial
 Encodes structural properties of graphs.
(Bridge)
 Two fundamental graph operations
 Deletion
 Contraction
(Loop)
 Recursive Computation
 T(G; x, y) = T(G-e; x, y) + T(G/e; x, y) if e is not a bridge or a loop.
 T(GH) = T(G)T(H) if G and H share at most one vertex.
 T(G) = xi y j if G has only i bridges and j loops.
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Universality
 The Tutte polynomial is well-defined:
 The polynomial does not depend on the order of deletion and
contraction.
 Universality Theorem:
If f(G) is a function of graphs such that
 f(G) = 1 if G consists of only one vertex and no edges,
 f(G) = af(G-e)+bf(G/e) whenever e is not a loop or a bridge,
 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 f is an evaluation of the Tutte polynomial.
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 x y 
f (G)  a s btT  G; 0 , 0 
 b a
Where s  E(G)  V (G)  k (G), t  V (G)  k (G), x0  f (Bridge), and y0  f (Loop)
Relating the Potts model to
the Tutte Polynomial
 Otherwise the
 Recall that the
Hamiltonian places a
Hamiltonian places a
1 on an edge
0 on an edge
between like spins.
between unlike spins.
Therefore, we can
Therefore, we can
contract these edges.
delete these edges.
Now we can apply the universality theorem to the Potts model partition function.
P(G; q,  )  q k (G )  v 
Where v  exp( J  ) 1.
12
V (G ) k (G )
 qv

T  G;
,1  v 
v


Example
The square lattice of slide 4.
=
+
+
=
+
+
+
=
+
=
x3  x 2  x  y
Solution using the Universality Theorem
P(G; q,  )  q k (G )  v 
P(G; 2,  )  21  e J 
Same as slide 4!
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V (G ) k (G )
qv


T  G;
,1  v 
v


3
2
J
J

3 e
 1   e  1   eJ   1  J  
 1  J 
   J
   J
e 
 e  1   e  1   e  1 

12exp(2 J  )  2exp(4 J  )  2
References
[1] Albert, M. S. Kiskowski, M. A. Glazier, J. A. and Jiang, Y. http://math.lanl.gov/~yi/Papers/CArev.pdf. On Cellular
Automaton Approaches to Modeling Biological Cells
[2] Ashkin, Julius; Teller, Edward. Statistics of Two Dimensional Lattices with Four Components, Physics Review, 64, pp. 178184. (1943)
[3] Bodkin, Patricia; Cox, Mary; Ellis-Monaghan, Jo; Sherman, Whitney. Contraction-Deletion Invariants: The Tutte Polynomial
in Engineering, Biology, and Physics. Manuscript. June 2006.
[4] Chandler, David. Introduction to Modern Statistical Mechanics. Oxford University Press.
[5] Chang, Shu-Chiuan; Shrock, Robert. Exact Partition Function for the Potts Model with Next-Nearest Neighbor Coupling on
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Publishing Company.
[6] Ellis-Monaghan, Jo. From Potts to Tutte and back again … A graph theoretical view of statistical mechanics. PowerPoint
Presentation at Middlebury College. October 2005.
[7] http://saeta.physics.hmc.edu/courses/p170/Metropolis.pdf#search='the%20metropolis%20algorithm%2C
%20%20when%20studying%20systems%20with%20a%20great%20many%20particles%2C%20it'. The Metropolis
Algorithm; Statistical Systems and Simulated Annealing.
[8] Huang, Kerson. Statistical Mechanics. Wiley John & Sons, Inc. 1990
[9] Jiang, Yi; Glazier, James A. Extended large-Q model simulation of foam drainage. Department of Physics, University of Notre
Dame, Notre Dame, Indianna. 1996.
[10] Kroemer, Herbert; Kittel, Charles. Thermal Physics (2nd ed.) W.H. Freeman Company. 1980.
[11] Meyer, Peter. Coputational Studies of Pure and Dilute Spin Models. 2000
[12] Sanyal, Soma; Glazier, James A. Viscous instabilities in flowing foams: A Cellular Potts Model approach.
[13] Schelling, Thomas C. Dynamic Models of Segregations. Havard University. Journal of Mathematical
Sociology.
[14] Schulze, Christian. Potts-Like Model for Ghetto Formation in Multi-Cultural Societies. Internatoinal Journal of Modern
Physics C. Vol. 16, No. 3 (2005) 351-355
[15] Sun, L. Chang, Y. F. Cai, X. A Discrete Simulation of Tumor Growth Concerning Nutrient Influence.
The Institute of Particle Physics, Central China Normal University. Wuhan, Hubei. August 2003.
[16] Winston, Wayne L. Operations Research: Applications and Algorithms. Second Edition. Duxbury Press.
Belmont California. 1991.
[17] Welsh, D. J. A.; Merino, C. “The Potts model and the Tutte polynomial.” American Institute of Physics. 2000.
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