Using metropolis algorithm for 2D ising model

[vitaniarain]

Hi I'm looking for some help with trying to understand how to use the metropolis algorithm for the 2D ising model. In the problem i am trying to solve, the Hamiltonian is simply H= - sum(Si).

I am given a probability function= exp[-H/T] / Z(T) where Z is the partition function which i found analytically and I am asked to use the metropolis algorithm in order to produce N samples representative of the probability function given.

I know more or less how this is supposed to work but I am finding it very hard applying it. Could someone give me a step by step explanation of how to do this? thanks!

 

[mmwave]

Hello, thank you for reaching out for help with understanding the metropolis algorithm for the 2D ising model. The metropolis algorithm is a Monte Carlo simulation method used to sample from a probability distribution. In your case, you are trying to sample from the probability function given by exp[-H/T] / Z(T), where H is the Hamiltonian and T is the temperature.

The first step in using the metropolis algorithm is to initialize a system with a set of spin values, which in this case would be the values of Si. These spin values can be randomly assigned or chosen based on a specific configuration.

Next, you will need to calculate the energy of the system using the given Hamiltonian. In this case, the energy would be the sum of all the spin values.

Once the initial energy is calculated, the metropolis algorithm involves randomly flipping one of the spin values and recalculating the energy of the system. If the new energy is lower than the previous energy, the new spin configuration is accepted. However, if the new energy is higher, it is accepted with a probability of exp[-ΔE/T], where ΔE is the difference between the new and previous energies. This step is important as it allows for the exploration of higher energy states, which can lead to a more accurate representation of the probability distribution.

This process is repeated for a large number of iterations, with each iteration resulting in a new spin configuration. These configurations are then used to calculate the average energy and other thermodynamic properties of the system.

To summarize, the steps for using the metropolis algorithm for the 2D ising model are as follows:

1. Initialize a system with a set of spin values.
2. Calculate the energy of the system using the given Hamiltonian.
3. Randomly flip one of the spin values and recalculate the energy.
4. Accept the new spin configuration with a probability of exp[-ΔE/T].
5. Repeat steps 3 and 4 for a large number of iterations.
6. Use the resulting spin configurations to calculate the average energy and other thermodynamic properties.

I hope this explanation helps in understanding the metropolis algorithm for the 2D ising model. If you have any further questions, please do not hesitate to ask. Good luck with your problem!