The Potts and Ising Models of Statistical Mechanics

Document Sample
The Potts and Ising Models of Statistical Mechanics Powered By Docstoc
					The Potts and Ising Models of
Statistical Mechanics
The Potts Model
The model uses a lattice, a regular,
repeating graph, where each vertex
is assigned a spin. This was
designed to model behavior in
ferromagnets; the spins interact
with their nearest neighbors align in
a low energy state, but entropy
causes misalignment. The edges
represent pairs of particles which
interact.
             The q-state Potts Model
                The spins of the model can be represented as
                 vectors, colors, or angles depending on the
                 application.
                            q=2                     q=3                             q=4

Unit vectors point in the
q orthogonal directions




                                           n  2 n / q, n  0,1, . . . , q  1.

                    q = 2 is the special
                    case Ising Model
            Phase Transitions
               The Potts Model is used to analyze behavior in
                magnets, liquids and gases, neural networks, and
                even social behavior.
               Phase transitions are failures of analycity in the
                infinite volume limit. For ferromagnetics, this is
                the point where a metal gains or loses
                magnetism.

Magnetism
                                        Magnetism
                               N 


            Temperature                               Temperature
     The Hamiltonian
         In mechanics, the Hamiltonian
          corresponds to the energy of a system.
          The lowest energy states exist where all
          spins are in the same direction.
             h( )   J   ( i , j )
                              {i , j }E ( G )



         Where ω is a state of graph G, σi is the spin at vertex i, δ
         is the Kronecker delta function, and J is the interaction
         energy

This measures the number of pairings between vertices
with the same spins, weighted by the interaction energy.
  Ising model Hamiltonian
For the Ising model, the Kronecker delta may be defined
as the dot product of adjacent spins where each spin is
valued as the vector <1,0> or <0,1>.

      1       0
                                                      
                      1
                                 h( )   J                         i  j
 0        1       0                               {i , j}E ( G )
                       0   1
     0        1                   Where J is uniform throughout the graph.

 1        1       0        1
     0        0       1         An edge between two like spins has
                                a value of 1, while an edge between
  1                        1    unlike spins has a value of 0.
          1       1
      0       0       1
                               For this state, h(ω) = -14J
        The Partition Function
   The probability of a particular state:
                              1
                      exp(     h ( ))
        Pr(i  T          kT
                                1
                       exp( h( ))
                          all           kT
                  k=1.38x10-23 joules/Kelvin, the Boltzmann
                 constant
   The denominator is the same for any
    state, and is called the partition function.
                      1                                        J
      Z   exp(       h( ))          Z    exp(              i  j )
         all        kT                     all  {i , j }    kT
          4-cycle example
      Recall the                        1
      partition          exp(
                        all           kT
                                          h( ))
      function:
                                                   H  4J   H  2J   H  2J   H  2J

In this example, the partition
function is:
                                                   H  2J   H  2J   H  2J    H 0
      4J         2J
 2exp( )  12exp( )  2
      kT         kT
                                                    H 0     H  2J   H  2J   H  2J
The probability of an
all blue spin state:
                                 J                 H  2J   H  2J   H  2J   H  4J
                     exp(4         )
                                kT
                   4J             2J
          2 exp(      )  12 exp(    )2
                   kT             kT
Why use another approach?
   The partition function is difficult to
    calculate for graphs of useful size, as the
    number of possible states increases
    rapidly with the number of vertices.
                       |i | is the number of states
        |i | 2   N
                       2 is number of possible spins,
                       and N is the number of vertices

   An effective approach for estimating the
    partition function for large lattices is
    through a transfer matrix. This estimation
    is exact for infinite lattices.
       Ising Model Transfer Matrix
          Define a matrix P such that
                                    1
                   i P j T  exp( J ( i   j ))
                                   kT
                    For like spins:
                
                                         1           0        J
                                 1,0 P    0,1 P    exp( )
                   Unlike spins:        0           1       kT
                                     0           1
                             1,0 P    0,1 P    exp(0)  1
                                     1           0
This is a symmetric 2x2 transfer matrix, describing all
possible combinations of spins between two vertices
                                J        
                           exp( )    1   
                                kT
                                         
                              1
                                        J 
                                   exp( )
                          
                                      kT 
                                          
The exact partition function using a
transfer matrix

 Using the transfer matrix for a
  cycle, the partition function is:
           1

all
   exp( 
          kT
             h( ))    P  P ... P  P
                       
                     all
                           1       2
                                       T
                                               2   3
                                                       T
                                                             N 1   N
                                                                        T
                                                                            N       1
                                                                                     T



 Which can be simplified to become:
  Z    1 P N 1T      As    I the identity matrix
                               n
                                   T
                                           n
       all i

      =Tr (P N )    Since σ1 = σ1 in all possible states

      = i N
        i           Where λ1 and λ2 are the eigenvalues of
      =1N  2 N   the transfer matrix:
                                                    J                                J
                                   1  exp(          ) 1              2  exp(      ) 1
                                                   kT                               kT
Lets check if this works:

   We calculated the partition function
    for a 4-cycle earlier:
                           4J            2J
                     2exp(    )  12exp(    )2
                           kT            kT
 Using the transfer matrix approach,
Z  14  2 4
          J              J
Z  (exp( )  1)  (exp( )  1)
                4               4

         kT             kT
         4J          2J
  =2exp( )  12exp( )  2
         kT          kT
Phase Transitions
   As the partition function is positive, the
    eigenvalue with the largest absolute value must
    also be positive. And, as N  
          Z   i  1
                     N         N

                i
   Failure of analycity occurs where two or more
    eigenvalues have the greatest absolute value.
   This is proven not occur in the one dimensional
    Ising model, where   exp( J )  1 and   exp( J )  1
                           1                   2
                                   kT                 kT

   but does occur in greater dimensions, or in Potts
    Models with more spin possibilities.
Thank You

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:2
posted:6/21/2012
language:
pages:14