GEC > Code > Deterministic Ising Dynamics

Statistical Physics

Deterministic Ising Dynamics

Gavin E. Crooks



Options

Glauber Dynamics
This is the standard stochastic Monte Carlo Ising dynamics. The sliders at the bottom control the temperature and external field.
CA Dynamics
A new, energy conserving, deterministic, reversible, Cellular Automata Ising dynamics. The sliders are irrelevant.
Periodic Boundaries
Standard toroidal boundary condition. The left edge is joined to the right edge, and the top edge to the bottom.
Fixed Boundaries
All edge spins are always kept in the down (black) state. Try warming the system up using the Glauber dynamics, and then watch the evaporative cooling of the spin conserving CA dynamics with fixed boundary conditions.

Deterministic Ising Dynamics

This Java applet demonstrates a deterministic, reversible, Cellular Automata Ising dynamics that conserves both the total energy and the total number of up versus down spins. This is in contrast to the Creutz Cellular Automaton, which does not conserve spin.

At each timestep the lattice is partitioned into a checkerboard of 2x2 blocks. This partitioning differs at each time step, and has a periodicity of 4. 

         T=0    1     2     3
         ##..  #..#  ..##  .##.
         ##..  .##.  ..##  #..#
         ..##  .##.  ##..  #..#
         ..##  #..#  ##..  .##.
At each time step only those spins within black ("#") 2x2 block are allowed to change. All neighboring spins are labelled white, and are fixed for this time step.

The following table shows the single time step dynamics within each block. Up spins are labeled '1' and down spins '0', and other possible moves are given by quarter turn rotations, and inversion of '1' and '0'. 

          00 <-> 00    01 <-> 00    11 <-> 00    10 <-> 01   
          00     00    00     10    00     11    01     10
This is basically the Margolus-Toffoli billiard ball dynamics. However, this dynamics is only applied if the move would conserve the Ising energy. (i.e., the number of up spins next to down spins remains the same.) Therefore, both the total energy and the total spin is conserved.

Disclaimer

But does it actually work? Does this dynamics reproduce the correct equilibrium statistic for the Ising model? I don't know for sure. One of these days I might actually get around to checking.

Beware of bugs in the above code; I have only proved it correct, not tried it.
Donald E. Knuth

References

Creutz, Michael, "Deterministic Ising Dynamics," Annals of Physics 167 (1986), 62--76

©2002 Gavin E. Crooks · gec@threeplusone.com