# Free Energy calculation in Monte Carlo simulations

1. Apr 11, 2012

### kaesekuchen86

Hello everyone,

I am currently working on a classical MC simulation of helimagnets and I try to calculate the experimentally measured phase diagram.

Up to now I am able to reproduce a large piece of the phase diagram by using a single site Metropolis update and simulated annealing. Unfortunately the simulated annealing suffers a hysteresis effect: For low temperatures, the system should undergo a first order transition from one ordered state A to another ordered state B but it does not...
These two ordered states are largely separated in phase space in the sense that "flipping one spin here and one spin there" is not enough to drive the transition...
I also tried to reach the point of interest in the phase diagram by taking a different way (lowering magnetic field at constant temperature) which succeeded in the sense that I found the desired state (that is measured in experiments), hence I assume I have indeed a hysteresis problem.

I compared the energy of the two states and found that the energy of the state which should not be realized at the point of interest is slightly (about <1%) smaller than the energy of the state which should be realized at this point. But since energy is not all, I would like to calculate the free energies F = E - TS of the two states and compare them.

Does anyone know whether this is possible and how it can be done?

I am very sure that my Code is correct (despite the problem mentioned above) since i did a lot of testing.

Here is s sketch of the phase diagrams:

It should look like:
Code (Text):

H
^
|
|-----------------------
|  phase B | phaseA |
|              |           |
---------------------------->T

and in my calculation it looks like:
Code (Text):

H
^
|
|-----------------------
|        phaseA        |
|                         |
---------------------------->T

2. Apr 12, 2012

### Timo

Calculating (relative) free energies is a fairly standard task in the simulation of phase transitions. I do not have the time to really think about your concrete problem, but maybe the following paper is a helpful starting point for you: Simulation techniques for calculating free energies, M. Müller and J.J. dePablo (http://www.springerlink.com/content/v43112678811n031/ ) - but note the techniques discussed therein may be more advanced than what you are looking for.

3. Apr 16, 2012

### kaesekuchen86

thank you for the hint. Unfortunately the paper does not mention my problem:

I have two states and want to compare their free free energies at a given temperature.

Does anyone know a (MC) method for that?