# Understanding Markov Chains in Metropolis Monte Carlo Ising Simulation

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• LagrangeEuler
In summary, Markov chains are utilized in Monte Carlo Ising simulations by randomly selecting a lattice site and flipping its spin, calculating energy, and moving ahead. The probability distribution of the next lattice site is determined by its displacement from the current one, rather than the previous path taken. The lattice does not retain memory of previously moved spins.
LagrangeEuler
I do not understand where are Markov chains in Monte Carlo Ising simulation. Going randomly from lattice site to lattice site one flip one spin calculates energy and goes ahaid. But where are Markov chain there? Could you please explain me this. Thanks a lot.

The lattice site after a flip has a probability distribution around the previous lattice site.

mathman said:
The lattice site after a flip has a probability distribution around the previous lattice site.
What do you mean by "probability distribution around the previous lattice site"?

The choice of the next lattice site is random. However it is defined in terms of a displacement from the current lattice site.

Thanks. Let me see if I understood you well. So you are now at site ##i## and you fliping spin at that moment. Probablility that after fliping spin will stay in that position depends only of that current position of all spins at the lattice and not of the path that we did before it. Lattice does not remember which spins are moved earlier.

LagrangeEuler said:
Thanks. Let me see if I understood you well. So you are now at site ##i## and you fliping spin at that moment. Probablility that after fliping spin will stay in that position depends only of that current position of all spins at the lattice and not of the path that we did before it. Lattice does not remember which spins are moved earlier.
Correct!

## 1. What is the Metropolis Monte Carlo Ising model?

The Metropolis Monte Carlo Ising model is a mathematical model used to simulate the behavior of a ferromagnetic material at a microscopic level. It is based on the Ising model, which describes the interactions between individual magnetic moments in a material. The Metropolis algorithm is used to simulate the evolution of the system over time, and it is often used to study phase transitions in magnetic materials.

## 2. How does the Metropolis algorithm work in the Ising model?

The Metropolis algorithm is a Monte Carlo method that is used to simulate the behavior of a system over time. In the Ising model, the algorithm randomly selects a spin (magnetic moment) in the material and calculates the energy change that would occur if the spin were to flip. If the energy change is favorable, the spin is flipped. If the energy change is not favorable, the spin may still be flipped with a certain probability. This process is repeated for all spins in the material, resulting in a simulation of the evolution of the system over time.

## 3. What is the significance of the Metropolis Monte Carlo Ising model?

The Metropolis Monte Carlo Ising model is a valuable tool for studying the behavior of ferromagnetic materials, as it allows scientists to simulate and analyze the effects of various parameters on the system. It has been used to study phase transitions, critical phenomena, and other properties of magnetic materials. Additionally, the Metropolis algorithm used in the model has applications in other fields such as statistical mechanics and quantum mechanics.

## 4. What are some limitations of the Metropolis Monte Carlo Ising model?

While the Metropolis Monte Carlo Ising model is a powerful tool, it does have some limitations. One major limitation is that it assumes an infinite lattice, which may not accurately represent real-world materials. Additionally, the model does not take into account the effects of temperature, which can play a significant role in the behavior of magnetic materials. It also does not account for the effects of external magnetic fields.

## 5. How is the Metropolis Monte Carlo Ising model used in research?

The Metropolis Monte Carlo Ising model is used in a variety of research fields, including condensed matter physics, materials science, and statistical mechanics. It is often used to study phase transitions and critical phenomena in magnetic materials, as well as to investigate the behavior of different systems under various conditions. The model has also been used to study other types of materials, such as spin glasses and polymers.

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