Convergence of lattice Ising model

In summary, the participants are discussing a program to solve the 2D Ising model of magnetic materials using a system with 10x10 spins at a temperature of 1E-8 K. They are concerned about the time it will take to run the simulation and how to avoid getting stuck in a local equilibrium. One suggestion is to start with a perfectly aligned state and gradually increase the temperature to observe the phase transition. Another option is to start with a random state and heat the system up instead. They also mention the importance of measuring other quantities such as average magnetization and susceptibility in order to observe the phase transition.
  • #1
Leonardo Machado
57
2
Hello everyone.
I'm working on a program to solve 2D Ising model of magnetic materials, using a system with 10x10 spins for simplicity at a temperature of 1E-8 K. I'm using this parameters to get a faster result of m=1 and guarantee it is correct. but...

For now i already pass 300 Monte Carlo's moves and it is in the same position of when it was in 50... is that normal ? I've watch a vídeo on youtube where they took like a 10 million moves to compute a 200x200 system. I'm a bit scared with the time it will cost me.

Thanks.
 
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  • #2
Could I check what your initial state was? If you started with an arbitrary random state and immediately put it at such a low temperature, it is highly likely for the system to become stuck in a local equilibrium. Most of the Monte Carlo moves will be rejected because the extremely low temperature essentially prevents moves that increase the energy of the system from occurring.

Also, it is also worth noting that such Monte Carlo simulations are often measured in terms of lattice sweeps (total number of moves / number of lattice sites) - the idea being that in each update one should allow for every lattice spin to have a chance to flip on average.
 
  • #3
Fightfish said:
Could I check what your initial state was? If you started with an arbitrary random state and immediately put it at such a low temperature, it is highly likely for the system to become stuck in a local equilibrium. Most of the Monte Carlo moves will be rejected because the extremely low temperature essentially prevents moves that increase the energy of the system from occurring.

Also, it is also worth noting that such Monte Carlo simulations are often measured in terms of lattice sweeps (total number of moves / number of lattice sites) - the idea being that in each update one should allow for every lattice spin to have a chance to flip on average.

yes, I've made it totally random and in it first move it was already at such temperature.. So, how should i proceed ? Low the temperature gradually ?
 
  • #4
I usually prefer the alternative - to heat the system up instead. So instead of starting from a random state, I start with the zero temperature limit of perfectly aligned spins. This avoids the problem of getting trapped in local minima.
 
  • #5
Fightfish said:
I usually prefer the alternative - to heat the system up instead. So instead of starting from a random state, I start with the zero temperature limit of perfectly aligned spins. This avoids the problem of getting trapped in local minima.

Not sure if I've got it. So, you start your system at a temperature of approximate 0 ( not exactly zero because of the exponential of delta E / T ), then you heat the system until what temperature ? It cannot pass 2.6..something Kelvin because i want to observe the consequence of phase transition.
 
  • #6
Not sure what it is that you want to achieve but here is a set of pictures of representative states of a Ising system at different temperatures that I've done in the past:
download.png


You can either start with the uniformly aligned state each time or use the final output system state of the lower temperature as the starting state for the next (higher temperature) simulation.

However, it must be noted that these are just representative states of the system. In order to "observe" the phase transition, one has to measure other quantities such as the average magnetization and the susceptibility, which are averaged over the Monte Carlo runs.
 

1. What is the lattice Ising model and how does it relate to convergence?

The lattice Ising model is a mathematical model used to study the behavior of physical systems, specifically those with interacting particles or spins. It consists of a lattice of discrete points, each with an associated spin value. The model is used to investigate the convergence of these spin values to a stable state over time, which can provide insight into the behavior of the system as a whole.

2. How is convergence defined in the lattice Ising model?

In the context of the lattice Ising model, convergence refers to the process of the spin values approaching a stable state over time, where the energy of the system is minimized. This is typically achieved through repeated iterations of updating the spin values based on the interactions with neighboring spins.

3. What factors can affect the convergence of the lattice Ising model?

There are several factors that can impact the convergence of the lattice Ising model, including the size and shape of the lattice, the initial spin configurations, the strength of the interaction between spins, and the temperature of the system. Other factors such as boundary conditions and the type of algorithm used for updating the spin values can also play a role.

4. How is the convergence behavior of the lattice Ising model studied?

The convergence behavior of the lattice Ising model can be analyzed through simulations and computational methods. By tracking the changes in spin values over time, researchers can observe the convergence process and identify patterns or behaviors. The model can also be studied analytically through mathematical techniques such as mean-field theory.

5. What real-world applications does the convergence of the lattice Ising model have?

The lattice Ising model has been applied to various physical systems, including magnetism, phase transitions, and social phenomena. It has also been used in the field of statistical physics to understand the behavior of complex systems and predict their properties. Additionally, the model has been used in computer science and machine learning to develop algorithms for optimization and pattern recognition.

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