Help understanding textbook on Landau Ginzburg theory

In summary, the conversation discusses the Landau Ginzburg theory of phase transitions and the equation b(T) = b'(T-Tc), where b(T) is redefined and the prime symbol indicates a derivative. The context of this equation is within a larger case involving the free energy of a scalar field and the temperature. A question is posed about a coordinate change and the relationship between two functions, f(x) and g(x).
  • #1
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This question is about the Landau Ginzburg of phase transitions which seem to take this classical field theory form.
I don't understand the meaning of the 2nd to last equation
$$b(T) = b'(T -T_c) $$
does that mean b(T) has be redefined in the previous two equations relative to the original definition of ##b(T)##. Does the prime ##\prime## indicate a derivative? Here is the entire case, with ##A## being the free energy of some scalar field m(r), which may be interpreted as magnetization. T is temperature.
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  • #2
It's a coordinate change; both ##b## and ##b'## are functions. Suppose ## f(x) = (x - a)^2 ##. What is ## f(x-a) ## ? Does ##f(x) = f(x-a)## ? If not, can you find ##g(x)## such that ##f(x-a) = g(x) ##?
 

1. What is Landau-Ginzburg theory?

Landau-Ginzburg theory is a mathematical framework used to describe phase transitions in physical systems. It was originally developed by Lev Landau and Evgeny Lifshitz in the 1950s and later extended by V. L. Ginzburg. It is often used in condensed matter physics to study the behavior of materials at low temperatures.

2. What is the main concept behind Landau-Ginzburg theory?

The main concept behind Landau-Ginzburg theory is the idea of an order parameter, which is a measurable quantity that undergoes a sudden change at a phase transition. This theory describes how the order parameter changes as a function of temperature, allowing us to understand the behavior of materials near a phase transition.

3. How is Landau-Ginzburg theory applied in real-world situations?

Landau-Ginzburg theory has been applied in a variety of real-world situations, including superconductors, liquid crystals, and magnetic materials. It has also been used to study the behavior of complex systems such as biological systems and neural networks.

4. What are the mathematical equations used in Landau-Ginzburg theory?

The main mathematical equation used in Landau-Ginzburg theory is the Landau-Ginzburg free energy functional, which describes the energy of a system as a function of the order parameter. This equation can be used to calculate the behavior of the system at different temperatures and to predict phase transitions.

5. Are there any limitations to Landau-Ginzburg theory?

Like any scientific theory, Landau-Ginzburg theory has its limitations. It is most accurate for systems near a second-order phase transition, which is a continuous change in the order parameter. It also assumes that the system is in thermal equilibrium, which may not always be the case in real-world situations.

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