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I'm a computer scientist and new to Ising Model. I've learned that if such a system is left to itself it will converge to its minimum energy state. Here are the questions I have:

  1. As the system is going toward its minimum energy, where does the extra energy go? Does it get transferred to the environment?
  2. If the answer to previous question is "yes", then if we isolate the system from its environment hypothetically so that no energy can escape from the system, would it still evolve to a lower energy state or it would stay at whatever state it is in?
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The total energy of an isolated system can never change. This holds not only for the Ising model, but for all models, and also all real-world systems. This is basically the definition of isolated.

But you have the right intuition: even if some system is in an excited state, and lower-energy states are available, the system must couple to something in order to reach that lower-energy state.

If you can limit the coupling to other parts of the system and to the environment, some high-energy states can be very long lived. See for instance Rydberg atoms.

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