Normalisation Constant (Ising Spins)

[Aidan1]

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.

Homework Equations

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

The Attempt at a Solution

In my code I have set $P = k\exp{( -\Delta E/T )}$ & 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 $$k\exp{(-\Delta E/T )} + P( \Delta E < 0) + ( 1 - k\exp{(-\Delta E/T )} ) * P(\Delta E > 0) = 1$$ (ie probability of particle flipping spin + probability of not flipping spin = 1 - if $\Delta E < 0$ the probability of flipping is 1) & just got trivial solutions.

Thanks for the help

 

[DrClaude]

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[.