Minimization of free energy in Landau theory

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• physlosopher
In summary, the equilibrium state of a system is determined by minimizing the free energy with respect to its state parameters. This ensures that the system is in its lowest energy state. In the case of zero-field free energy, the equilibrium state is not dependent on the magnetization, but the second partial derivative must be positive for the extremum to be a minimum. Therefore, the minimum of the free energy with respect to the order parameter is sought as an equilibrium condition.
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?

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.

1. What is free energy in Landau theory?

In Landau theory, free energy is a thermodynamic potential that represents the energy of a system at a given temperature and external conditions. It is a function of the order parameter, which describes the state of the system, and is minimized to determine the equilibrium state of the system.

2. Why is minimizing free energy important in Landau theory?

Minimizing free energy is important because it allows us to determine the equilibrium state of a system. In Landau theory, this equilibrium state corresponds to the most stable configuration of the order parameter, and minimizing free energy helps us understand the behavior of the system at different temperatures and external conditions.

3. How is free energy minimized in Landau theory?

In Landau theory, free energy is minimized by finding the values of the order parameter that satisfy the condition that the first derivative of free energy with respect to the order parameter is equal to zero. This results in a set of equations known as the Landau equations, which can be solved to determine the equilibrium state of the system.

4. What is the role of symmetry in minimizing free energy in Landau theory?

Symmetry plays a crucial role in minimizing free energy in Landau theory. The symmetry of the system determines the form of the Landau free energy function, and the symmetry breaking that occurs when the system transitions to a new equilibrium state is reflected in the change of form of the free energy function. Minimizing free energy allows us to understand how this symmetry breaking occurs.

5. How does minimizing free energy in Landau theory relate to phase transitions?

Minimizing free energy is closely related to phase transitions in Landau theory. Phase transitions occur when the equilibrium state of the system changes, and this is reflected in the change of form of the free energy function. By minimizing free energy, we can determine the critical points at which these phase transitions occur and understand the behavior of the system at these points.

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