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Normalisation Constant (Ising Spins)

  1. Dec 1, 2015 #1
    1. The problem:

    Building a (probably very simple) computational model for Ising Spins - particles on a lattice with spin up and spin down, only nearest neighbour interactions. I can't for the life of me figure out the renormalisation constant for the probability of a given particle flipping it's spin.

    2. Relevant equations

    We are given that the probability of a given particle flipping its spin goes as [tex] P \propto \exp{( -\Delta E/T )} [/tex] for [itex] \Delta E > 0 [/itex] where [itex] \Delta E [/itex] is the change in energy if the particle flips its spin and T is the temperature of the system.

    3. The attempt at a solution

    In my code I have set [itex] P = k\exp{( -\Delta E/T )} [/itex] & set k = 1 for now while I check other aspects of the code run okay. I can't figure out an expression for k though - I am aware something's going to need to add up to 1 - tried saying [tex] k\exp{(-\Delta E/T )} + P( \Delta E < 0) + ( 1 - k\exp{(-\Delta E/T )} ) * P(\Delta E > 0) = 1 [/tex] (ie probability of particle flipping spin + probability of not flipping spin = 1 - if [itex] \Delta E < 0 [/itex] the probability of flipping is 1) & just got trivial solutions.

    Thanks for the help
     
    Last edited: Dec 1, 2015
  2. jcsd
  3. Dec 2, 2015 #2

    DrClaude

    User Avatar

    Staff: Mentor

    The standard approach is to take ##k=1##, i.e., unconditionally flip if ##\Delta E <0##, and flip with probability ##\exp(-\Delta E/ k_\mathrm{B} T)## for ##\Delta E >0## by comparing the value of that exponential to a uniform random number in the range [0,1[.
     
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