Order parameter is used to describe second order phase transition. It seems that in some papers it is used in the first order phase transitions. Can first order phase transition have an order parameter? If so, how can we define the order parameter in liquid-gas transition (first order)?
[Physics] Can a first order phase transition have an order parameter
condensed-matterphase-transitionstatistical mechanics
Related Solutions
In the case of the second order transition it can be described by any potential of your choice. The choice then depends upon the thermodynamic variables you need, rather than upon the transition order. However, there is the so-called, theorem about small perturbations. You may find it in Landau and Lifshitz, Theoretical Physics, V. 5 Statistical Mechanics, §15 "The free energy and thermodynamic potential". This theorem states that a small perturbation of a system makes the equal contribution to the any one of the thermodynamic potentials, independent upon this potential. In other words all the peerturbations are equal to one another. The phase transition free energy falls under the condition of this theorem: it is small with respect to the basis free energy. Therefore, it is all the same, what potential to use.
In the case of the first order transition the situation is basically the same, as soon as we discuss the transition into a new phase of a whole body. So you choose the potential whose variables are more suitable for you.
The important difference only arises in the case when a limited portion of the medium transforms into a new phase, while the rest of the body stays in the old one. It is different, since you have simulutaneously the jump of the solid volume and of the number of particles under the first order transition. So you cannot fix the volume and the number of particles simultaneously. In this case it is illegal to use, say, the free energy, whose variables are temperature, volume and the number of particles. You need instead to use the so-called, omega-potential with the variables temperature, volume and the chemical potential. You may find a discussion of this point in the same Landau and Lifshitz book, in §147 "Effective Hamiltonian". arguments become important in the fluctuation theory of phase transitions of even the second order, since a fluctuation is like a local phase transition with the spatial size equal to the correlation radius.
Here it is important to know, however, that in literature people often do not pay attention to the choice of the potential and unrigorously call any such a potential "the free energy". It is no problem unless one starts to calculate the thermodynamc parameters as the derivatives of such a free energy, where it matters which potential to take.
The key is: Landau theory doesn't assume the order parameter is small. All it assumes is that the free energy is analytic in the order parameter. One then usually expands this free energy up to some order (which is possibly by definition of 'analytic'). It is key to realize that expanding a function in a variable to some order does not mean this variable has to be small! It just means that terms we throw away have to be small, which is a different thing.
Let's take an example. Suppose we have this somewhat unusual looking free energy handed to us, which is indeed analytic: $$\boxed{F(\phi,T) = \phi^2 - 2 + e^{-\frac{\phi^4}{T}} + \cosh(\phi^3)}$$ For high temperatures, the minimum of the free energy selects $\phi = 0$. Around $T \approx .3$, there is a first order transition to $\phi \neq 0$. The following two graphs give the intuitive picture (the x-axis is the order parameter, the y-axis the free energy):
In Landau theory one usually expands these free energies. For example if we expand it to 8th order, we get $$F(\phi,T) = \phi^2 - \frac{\phi^4}{T} + \frac{\phi^6}{2} + \frac{\phi^8}{2T^2}$$ To this order, the graph for $T = .25$ looks as follows:
So we see that this already gives a good representation of our free energy in the region $-1 \leq \phi \leq 1$. This is because despite $\phi$ not being small, the terms we have thrown away are.
Note that if one is not interested in quantitative details but rather just wants the intuitive picture, then one can note that $F(\phi,T) = \phi^2 - \frac{\phi^4}{T} + \frac{\phi^6}{2}$ already displays the same qualitative behaviour. Moreover to this order it is easy to solve exactly and one obtains $T_c = \frac{1}{\sqrt{2}} \approx .7$ which is not a great quantitative match to the more exact $T_c = .3$, but the same physics is at play.
Best Answer
Yes, there may still be some order parameters in the presence of first-order transitions. But much like free energy, the order parameter is discontinuous at the transition point. For the liquid/gas phase transitions, the relevant order parameter is the difference between the densities.