Why does Finite Size Scaling shift the Phase Transition Down?

[georgeD123]

See the title. I'm not sure that this is the right place to post this question, but I'm not sure it fits any better on any of the other boards.

Let's say you have a phase transition. The correlation length will scale as:

ξ = |T_C-T|^ν

This diverges on both sizes of the phase transition. Now, the finite size scaling theory says to replace ξ with L, since that is the largest the correlation length can be in a real system with finite size, and solve for what temperature that occurs at.

You get two temperatures! One above T_C and one below. Obviously, the temperature has to shift down; if it didn't, we would have no phase transitions in bulk materials, since we would be always below the critical temperature. However, this seems like an argument against using finite scaling theory at all, as much as an argument to only consider the lower temperature.

If anybody understands this better than I do, I would love some help understanding!

 

[AndyW]

Hello,

Thank you for your post. I can offer some insight into your question about finite scaling theory and phase transitions.

Firstly, it is important to note that finite scaling theory is a powerful tool for understanding phase transitions in systems with finite size. It allows us to make predictions about the behavior of a system near the critical point, where the correlation length diverges. This is important because in real systems, we cannot have an infinitely large system, so we need to take into account the finite size effects.

Now, to address your concern about the two temperatures obtained from finite scaling theory, one above and one below the critical temperature. This is actually expected and in line with our understanding of phase transitions. The critical temperature, TC, is the temperature at which the correlation length diverges. Above TC, the system is in a disordered phase and below TC, the system is in an ordered phase. However, near the critical point, the correlation length is large and spans both the ordered and disordered regions, leading to two temperatures that satisfy the finite scaling theory equation.

So, it is not an argument against using finite scaling theory, but rather an indication of the complex behavior of systems near the critical point. In fact, this behavior has been observed in many experimental studies of phase transitions.

I hope this helps to clarify your understanding of finite scaling theory and its application in studying phase transitions. If you have any further questions, please don't hesitate to ask. As scientists, we are always happy to help others understand complex concepts.