Effect of sample size when using periodic boundary conditions in 2D Ising model

  1. Hi,

    I'm currently using the Monte Carlo Metropolis algorithm to investigate the 2D Ising model.

    I have an NxN lattice of points with periodic boundary conditions imposed. I was wondering if anyone could explain why the sharpness of the phase transition is affected by the size of N?
    I.e. if N is small I get a slow transition and as N is increased, the transition approaches a step function.

    I don't understand why this is as I am only considering nearest neighbour interactions and by using periodic boundary conditions surely I am effectively modelling an infinite lattice? So why does the size of the unit cell affect my results?

    Thanks!
     
  2. jcsd
  3. You are experiencing critical opalescence. As you approach the phase transition, the correlation length increases exponentially. When this domain size reaches the size of your simulation, then the simulation breaks down, i.e. does not describe the physics correctly anymore.

    http://en.wikipedia.org/wiki/Ising_critical_exponents

    Try and find a definition of the correlation length and calculated that on your grid. Then compare the temperature dependence for different grid sizes to what one would expect for an infinite lattice.
     
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