Finite size effects at first order phase transitions

[_sandro_]

Hi.

I am trying to understand some features related to the order of a phase transition. It is known that there are finite size effects in a finite system. The finite size scaling theory provides relations between some quantities with the length of the system L.

At second order phase transitions, finite size effects are due to the fact that the correlation length ε is about the same size of the system L in the transition point T_c of the control parameter as, for example, the temperature in a Ising-like system. However, it is not clear for me the relation between the correlation length and the system size at first order phase transitions.

I have read that at first order phase transitions there are no critical exponents because the correlation length never increase up to the size of the system. In contrast, I read today that there are some kind of weakly first order phase transitions in finite systems where critical exponents can be defined and metastable states are seen with power law distributed domains.

To sum up, if there is a weakly first order phase transition, can one use the cumulant method to find the transition point T_c?

Thanks.

 

[AndyW]

Greetings,

I can provide some insights into your questions about the order of phase transitions and the relationship between the correlation length and system size.

Firstly, let's define what we mean by a phase transition. A phase transition is a physical phenomenon where a system undergoes a sudden change in its properties, such as its density, magnetization, or electrical conductivity. This change is usually triggered by a small variation in a control parameter, such as temperature or pressure.

Now, let's focus on second order phase transitions. These transitions are characterized by a continuous change in the system's properties without any sudden jump. At the transition point, there is a critical value of the control parameter where the correlation length, which is a measure of how strongly correlated the particles in the system are, diverges. This means that the correlation length becomes infinite at the transition point, leading to the appearance of critical exponents. These exponents describe the behavior of the system near the transition point and can be used to classify different types of second order phase transitions.

On the other hand, first order phase transitions are characterized by a discontinuous change in the system's properties. This means that the correlation length does not diverge at the transition point, and there are no critical exponents. However, as you mentioned, there can be weakly first order phase transitions in finite systems, where the correlation length is not infinite but can still be comparable to the system size. In these cases, critical exponents can be defined, but they may differ from those in a true second order phase transition.

Regarding the use of the cumulant method to find the transition point T_c, it is not a straightforward approach for first order phase transitions. This method relies on the behavior of the fourth-order cumulant, which is expected to diverge at the transition point for second order phase transitions. However, for first order phase transitions, the fourth-order cumulant may not show a significant change at the transition point, making it difficult to determine T_c accurately.

In conclusion, the order of a phase transition is determined by the behavior of the correlation length and the presence of critical exponents. While second order phase transitions are well-defined, first order phase transitions can vary depending on the system size and can exhibit weakly first order behavior. The cumulant method may not be suitable for determining the transition point in such cases. I hope this helps clarify some of your questions. Keep exploring and learning about phase transitions, and good