Landau theory

Landau theory plays a key role in our understanding of phase transitions in statistical mechanics and thermodynamics. Developed by Lev Landau, this theoretical framework provides a coherent and unified description of the behavior of substances undergoing phase changes. From a liquid becoming a gas or a ferromagnet losing its magnetism, Landau theory provides key insights into these phenomena using a mix of mathematics and physics.

Basic concepts of phase transition

Before delving into the intricacies of Landau theory, it is necessary to understand the basic ideas behind phase transitions. A phase transition is a process in which a substance changes its phase, such as from solid to liquid or liquid to gas. These transitions usually involve an abrupt change in some property of the substance, such as density, magnetism or structure.

There are different types of phase transitions, including first-order and second-order transitions.

• First-order transitions: These involve latent heat, meaning that energy is absorbed or released when the transition occurs. Examples include melting ice and boiling water.
• Second-order transitions: Also known as continuous transitions, these do not involve latent heat. Instead, they are marked by a continuous change in the order parameter, a concept we will look at in more detail in the context of Landau theory.

Introduction to Landau theory

Landau theory provides a framework for describing continuous (second-order) phase transitions. It does so by focusing on the order parameter, a measure that characterizes the degree of order in a system. For example, in a ferromagnet, the order parameter could be the magnetization.

A key assumption in Landau theory is that close to the critical point of a phase transition, the free energy of a system can be expanded as a power series in the order parameter. This expansion helps to understand how the free energy changes when a system undergoes a phase transition.

Mathematical framework of Landau theory

Let us explore the mathematical formulation of Landau theory. We begin by defining the free energy, F, as a function of the order parameter, (phi) For simplicity, assume that there is no external field acting on the system. The Landau free energy expansion takes the form:

F(phi) = a_0 + a_2 phi^2 + a_4 phi^4 + a_6 phi^6 + ldots

Here, a_0, a_2, a_4, and a_6 are coefficients, and their values depend on the temperature. The important part of this expansion is that it only involves even powers of the order parameter. This is because the free energy under the transformation (phi to -phi) must be symmetric.

Key predictions of Landau theory

An important aspect of Landau theory is its ability to predict important phenomena that occur near the critical point of a continuous phase transition. By analyzing the coefficients in the free energy expansion, we can conclude whether or not a phase transition occurs and what its nature is.

In many systems, the coefficient a_2 changes sign at the critical temperature T_c. This sign change is indicative of a phase transition. Below the critical temperature, a_2 is negative, leading to a non-zero order parameter, indicating an ordered phase. Above the critical temperature, a_2 is positive, and the order parameter is zero, corresponding to a disordered phase.

Example: ferromagnetism

Let's consider a classic example: ferromagnetism. In a ferromagnetic material, the order parameter is the magnetization, M Above the Curie temperature T_c, the material is paramagnetic, and the magnetization is zero. Below T_c, the material becomes ferromagnetic, and a spontaneous magnetization appears.

The Landau free energy for a ferromagnet can be written as:

F(M) = a_0 + a_2(T) M^2 + a_4 M^4

At the critical temperature, a_2(T_c) = 0 For temperatures below T_c, the coefficient a_2(T) is negative. The equilibrium state of the system corresponds to minimizing free energy. To find the equilibrium value of the magnetization, we take the derivative of F with respect to M and set it to zero:

(frac{partial F}{partial M} = 2a_2(T)M + 4a_4M^3 = 0)

Solving this equation gives the behaviour of the magnetization across the transition.

Landau theory and critical exponents

One of the remarkable achievements of Landau theory is its ability to predict critical exponents, which describe how physical quantities behave near the critical point. Critical exponents are essential for understanding the nature of phase transitions.

In the context of classical Landau theory, these critical exponents can be obtained from the free energy expansion. For example, the sensitivity (chi) of a system, which measures how the order parameter responds to an external field, is related to the critical exponent (gamma) as follows:

(chi sim |T - T_c|^{-gamma})

Similarly, the order parameter itself follows a power law with respect to temperature, denoted by the critical exponent (beta) :

(phi sim (T_c - T)^{beta})

Limitations of Landau theory

While Landau theory provides many valuable insights, it has its limitations. One important limitation is that it is a mean-field theory, which means that it does not take into account fluctuations at the microscopic level. As a result, near the critical point, where fluctuations become essential, Landau theory's predictions for the critical exponents can deviate from the observed values.

Additionally, Landau theory is most effective for systems with symmetry-breaking phase transitions. For transitions with more complex order parameters or those affected by disorder and randomness, other theoretical approaches such as renormalization group theory may be more appropriate.

Conclusion

Landau theory remains an elegant and powerful tool for understanding continuous phase transitions. By expanding the free energy as a power series in the order parameter and analyzing the resulting coefficients, Landau theory sheds light on the behavior of systems undergoing phase changes. Although it has its limitations, the insights it provides remain foundational in the study of phase transitions, helping physicists to explore and untangle the complexities of the physical world.

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