Minimization of free energy in Landau theory

[physlosopher]

TL;DR Summary

Why do we look for equilibrium states by minimizing free energy with respect to the order parameter?

Thanks in advance for any insight!

Following Pathria's discussion of phase transitions, I'm getting tripped up on the discussion of Landau's theory. Pathria begins with a zero-field free energy $\psi = A/NkT$ where $A$ is the Helmholtz free energy.

He proceeds to characterize the equilibrium state by minimizing this free energy (near the critical temperature) with respect to an order parameter $m$. I'm thinking in terms of an Ising magnet, for example, with the order parameter being the average magnetization per spin. In this case $dA = -SdT + HdM$ where $M$ is the net magnetization.

It's not obvious to me a priori that minimizing a free energy with respect to a state variable should be the equilibrium condition - my understanding is that equilibrium states are those that, at particular values of the state variables (here T and M) minimize the free energy (here Helmholtz) with respect to something like an internal constraint on the system, for example in the case of microstates having many different possible statistical distributions that all share the same state variables T and M. The minimization is then with respect to that distribution - it is the assignment of probabilities to the individual microstates that minimizes the free energy. Am I thinking about this correctly?

With this in mind, why are we characterizing equilibrium by minimizing this particular free energy with respect to the order parameter? Are we thinking of it as some kind of internal constraint rather than as a state variable? One thing that immediately jumps out at me is that the partial $(\frac {\partial A}{\partial M})_{T} = H$. So in the case of $H = 0$, which is the one being discussed, it does happen to be that $A$ is extremized with respect to the magnetization. Then the second partial would relate to the susceptibility, which should be positive, making the extremum a minimum. Is that all there is to it, or was there some a priori reason to look for a minimum of the free energy?

 

[gtoguy97]

The answer to your question is yes, you are thinking about this correctly. The extremization of the free energy with respect to the order parameter is indeed an equilibrium condition. This is because the free energy is a thermodynamic potential that measures the energy of a system as a function of its state parameters (e.g. temperature and magnetization). The minimum of this function is the equilibrium state of the system, as it corresponds to the configuration with the lowest energy. In other words, by minimizing the free energy with respect to the order parameter, we are effectively seeking the lowest energy configuration of the system. In the case of the zero-field free energy discussed by Pathria, the partial derivative with respect to the magnetization is zero, which implies that the equilibrium state is not dependent on the magnetization. However, the second partial derivative relates to the susceptibility, which must be positive in order for the extremum to be a minimum. This is why we seek the minimum of the free energy with respect to the order parameter – it ensures that the system is in its lowest energy state.