Rep:Mod:ha3915cmp

TASK: Show that the lowest possible energy for the Ising model is E\ =\ -DNJ, where D is the number of dimensions and N is the total number of spins. What is the multiplicity of this state? Calculate its entropy.

The total number of edges is equal to 4N/2=2N if there are N sites. The ground state corresponds to all spins being in the same state (all up or all down). The ground state energy is −J for every edge and thus one obtains the total energy to be E\ =\ -DNJ

multiplicity is 2^N

S=Kln2^N

TASK: Imagine that the system is in the lowest energy configuration. To move to a different state, one of the spins must spontaneously change direction ("flip"). What is the change in energy if this happens ()? How much entropy does the system gain by doing so?

-3*(999-1)*J--3*1000*J = 6J

1000!/999!1!

TASK: Calculate the magnetisation of the 1D and 2D lattices in figure 1. What magnetisation would you expect to observe for an Ising lattice with  at absolute zero?

1D = 3-2=1

2D = 13-12 = 1

AT ABS 0 = 1000

TASK: How many configurations are available to a system with 100 spins? To evaluate these expressions, we have to calculate the energy and magnetisation for each of these configurations, then perform the sum. Let's be very, very, generous, and say that we can analyse  configurations per second with our computer. How long will it take to evaluate a single value of ?

2^100

1267650600228229401496.703205376 seconds