Phase transitions: from the potential to the dynamical structure factor ?

[Gan_HOPE326]

Hi everybody,
I've just started studying phase transitions and symmetry breaking for my PhD and I've met something I'd like to understand better. When we study the local potential vs. the order parameter in a structural phase transition I see there are two limiting cases, the one in which the potential barrier is much greater than thermal fluctuations and the one in which it's much smaller, the first one leading to order-disorder transitions and a dynamical structure factor with 'relaxation' shape, the second one leading to displacive phase transitions and a 'phonon' dynamical structure factor. I can see why this happens (in second case system can oscillate between the two minima thanks to thermal fluctuations, in first case it can't), but how is it derived formally? I want to know because I'd like to make calculations about what happens if, instead, potential energy barrier and thermal fluctuations are of the same order. Thanks for your help!

 

[BigJDubs]

It is difficult to give a formal derivation of this result without having access to the full details of the potential energy landscape. However, the general idea is that in order for a structural phase transition to occur, the thermal fluctuations must be sufficiently large to overcome the potential energy barrier between the two minima of the potential energy surface. If the thermal fluctuations are much smaller than the potential energy barrier, then the system will remain in one of the two minima and no phase transition will occur. On the other hand, if the thermal fluctuations are much larger than the potential energy barrier, then the system can oscillate between the two minima, leading to a displacive phase transition and a 'phonon' dynamical structure factor. In the case when the potential energy barrier and the thermal fluctuations are of the same order, the system can still undergo a phase transition, but the nature of the transition and the corresponding dynamical structure factor will depend on the exact details of the potential energy landscape. In this case, it is necessary to perform numerical simulations or analytic calculations to determine the outcome of the phase transition.