# Will the equilibrium energy remain same for different temperatures in the 2d-Ising model?

I am doing a numerical simulation of the 2D-Ising model using the Metropolis algorithm. One of the plots that we were asked to make was Energy vs Monte-Carlo Steps for different starting Temperatures. My TA told me that irrespective of the starting temperature we should see the same equilibrium energy after large number of steps.

However, neither am I getting the same equilibrium energies at different temperature nor do I think that they should be the same. Since the Boltzman weight is higher for lower temperatures I should see higher energy values for lower temperatures is what my intuition is telling me.

And that is also the result that I'm getting- Plots. Is my TA wrong or am I missing something here

• The equilibrium state will not depend on the initial condition; it will be given by the Gibbs state at the temperature used in your MC algorithm. This is a general fact about irreducible aperiodic Markov chains with finitely-many states: they always converge to their unique equilibrium distribution, which is entirely determined by the chain's transition probabilities. Apr 6 '18 at 19:03

I think you are confusing the initial temperature with the temperature at which you perform the simulation. I'm not sure what "initial temperature" means in this context, but I'll assume it is the temperature $T_i$ at which a given configuration $i$ was prepared. If that is the case, then what your TA likely meant was that, if you run simulations at temperature $T$, you will get the same (average) energy, regardless of $T_i$ (or, in other words, regardless of the initial configuration).