Landau theory: why does a m^3 term implies first order transition phase

In summary, the conversation discusses the addition of m^3 to the Landau free energy and its implications on the observation of a first order transition phase. The concept of m as the magnetic moment and its relation to the order parameter is also mentioned. The conversation concludes with the suggestion to plot a sample free energy and the explanation of a first order transition as a discontinuous jump in the order parameter.
  • #1
IRobot
87
0
Hi,

I am not sure it is the right subcategory to post a question on statistical physics. But anyway, I read a couple of times that adding a m^3 to the Landau free energy implies that we may observe a first order transition phase, but I don't see why. Maybe it does imply some discontinuity in the entropy, latent heat, but I am not seeing that.
 
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  • #2
IRobot said:
Hi,

I am not sure it is the right subcategory to post a question on statistical physics. But anyway, I read a couple of times that adding a m^3 to the Landau free energy implies that we may observe a first order transition phase, but I don't see why. Maybe it does imply some discontinuity in the entropy, latent heat, but I am not seeing that.

What is m? Mass, magnetic moment?
 
  • #3
M is the magnetic moment, the order parameter.
 
  • #4
IRobot said:
Hi,

I am not sure it is the right subcategory to post a question on statistical physics. But anyway, I read a couple of times that adding a m^3 to the Landau free energy implies that we may observe a first order transition phase, but I don't see why. Maybe it does imply some discontinuity in the entropy, latent heat, but I am not seeing that.

Try to make a plot of a sample free energy:
[itex]F=t m^2 + b m^3 + m^4[/itex]
for different values of t and b? In particular, try a fixed t at some finite value and varies b.

When t >> b, the free energy is minimum is at m=0, then as you increase b, at some point, the minimum JUMPS from 0 to some finite value. That, by definition, is a first order transition where the order parameter has a discontinuous jump.
 
  • #5
Thanks, you made it clear ;)
 

FAQ: Landau theory: why does a m^3 term implies first order transition phase

1. What is Landau theory and how does it relate to phase transitions?

Landau theory is a theoretical framework that describes the behavior of a physical system near a phase transition. It is based on the concept of an order parameter, which is a quantity that changes as the system transitions from one phase to another. The m^3 term in the Landau theory represents the third-order coefficient in the expansion of the free energy of the system, and it is used to determine the type of phase transition that will occur.

2. How does the m^3 term imply a first-order phase transition?

The m^3 term in the Landau theory represents the cubic term in the expansion of the free energy. When this term is non-zero, it indicates that the free energy has a local minimum at a non-zero value of the order parameter. This means that there are two possible stable states for the system, one with a non-zero value of the order parameter and one with a zero value. This leads to a first-order phase transition, where the system can switch between these two states at a critical temperature.

3. Can you explain the difference between first-order and second-order phase transitions?

A first-order phase transition, as described by the Landau theory, involves a sudden change in the order parameter at a critical temperature. This is accompanied by a discontinuity in the first derivative of the free energy, resulting in a jump in the system's internal energy. In contrast, a second-order phase transition, also known as a continuous phase transition, involves a gradual change in the order parameter and its derivatives, with no jump in the system's internal energy.

4. How does Landau theory account for other types of phase transitions?

The Landau theory can be extended to include higher-order terms in the expansion of the free energy, which can account for more complex phase transitions such as tricritical or multicritical transitions. By including these additional terms, the theory can describe a wider range of physical systems and their phase transition behaviors.

5. What are some limitations of Landau theory in understanding phase transitions?

While Landau theory provides a useful framework for understanding phase transitions, it has some limitations. For example, it assumes that the order parameter is a smooth function of temperature, which may not always be the case in real systems. It also does not take into account the effects of fluctuations, which can become significant near the critical point. Therefore, Landau theory may not accurately describe all types of phase transitions, and other theories such as renormalization group theory may be needed to fully understand these phenomena.

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