Phase transitions from ising model

[coolnessitself]

I'm having trouble understanding how changing parameters in the ising model leads to first order phase transitions. I understand how the intersection of
$\frac{kT}{2nJS} (\eta - \frac{g \mu_o H_o}{kT} ) = B_s(\eta)$
where $B_s(\eta)$ is the Brouillion func leads (in the absence of a field H) to one stable state for high T and low T, and how if T is in between this there are three solutions, leading to symmetry breaking. What I don't get is what happens with a magnetic field. Say T is chosen such that we've got three solutions still. The linear component will shift, so one solution will be more stable than the other, but how can you tell which one?

 

[Evalesco]

It sounds like you have a good understanding of the Ising model and how it can lead to first order phase transitions. With the introduction of a magnetic field, the linear component will shift and one solution will be more stable than the other. The way to tell which one is more stable is to look at the relative strength of the magnetic field compared to the temperature. If the magnetic field is stronger than the temperature, then the solution that is closer to the linear component will be more stable. If the temperature is stronger than the magnetic field, then the solution furthest from the linear component will be more stable.

 

[charng]

First, let's review the basics of the Ising model. This model is used to study the behavior of interacting spins in a lattice, where each spin can have only two possible states (up or down). The energy of the system is determined by the interactions between neighboring spins and an external magnetic field.

Now, when we talk about phase transitions in the Ising model, we are referring to a sudden change in the behavior of the system as a function of a particular parameter, such as temperature or magnetic field. In the absence of a magnetic field, the phase transition occurs at a critical temperature, where the system switches from a disordered state (high temperature) to an ordered state (low temperature).

However, when a magnetic field is present, the phase transition becomes more complex. As you mentioned, at a particular temperature, there are three possible solutions for the energy of the system. The linear component in the equation you provided represents the energy of the system without the magnetic field, while the second term accounts for the energy due to the magnetic field.

So, when the magnetic field is present, the three solutions correspond to three different energy levels of the system. The one with the lowest energy will be the most stable state. This means that as you increase the magnetic field, the system will transition from a disordered state to one of the two ordered states, depending on which one has the lowest energy.

In summary, changing parameters in the Ising model, such as temperature and magnetic field, can lead to phase transitions because they affect the energy levels of the system and can cause the system to switch from one stable state to another. The specific behavior of the system will depend on the values of these parameters and the interactions between the spins.