In physics, events with significant differences in size (e.g., waves in the ocean and the behavior of their individual water molecules) have little effect on each other. Thus, we can independently study the corresponding physical properties of each order of magnitude. This scale independence is precisely why we use fluid mechanics to model ocean waves, ignoring the behavior of individual water molecules. In other words, the theory succeeds because physics at different scales can be modeled with different theoretical frameworks.
Figure 1: The behavior of waves and water molecules can be explained by different physical models. This scale independence gives physical theory explanatory power.
However, there is a phenomenon called a critical phenomenon, in which events of different sizes have the same importance. The example Wilson gives is the liquid-gas tipping point.
Figure 2: Pressure-temperature phase diagram of a pure substance (Wikipedia). The critical points in water are 647.096 K and 217.75 atm.
As we approach the gas-liquid critical point, the physical properties of the two phases become more and more similar. At the critical point, they become a single, undifferentiated liquid phase. Liquids exhibit density fluctuations "at all possible scales." To quote Wilson:
These fluctuations occur in the form of droplets, droplets and bubbles that are completely dispersed together, from individual molecules to specimen volumes, droplets and bubbles of various sizes. It is at the tipping point that the scale of the largest fluctuation becomes infinite, but the smaller fluctuation does not weaken in the slightest. Any theory describing water close to a critical point must consider the entire spectrum. —Kenneth Wilson (1979)
<h1 class="pgc-h-arrow-right" data-track="7" > another example</h1>
Ferromagnet is a magnetic material that produces magnetic domains. In these magnetic domains, the magnetic fields of individual atoms are aligned. However, the magnetic field direction for each magnetic domain is random. Therefore, the net magnetic field is zero. However, when there is an applied magnetic field, the magnetic field of all magnetic domains is consistent with the direction of the applied magnetic field, resulting in an increase in the applied magnetic field.
The so-called magnetic domain refers to the small magnetized regions with different directions of differentiation in the process of spontaneous magnetization of ferromagnetic materials to reduce static magnetic energy, and each region contains a large number of atoms, and the magnetic moments of these atoms are neatly arranged like small magnets, but the direction of atomic magnetic moments is different between adjacent different regions.
Figure 3: No outfield (left) with outfield B (center) (Wikipedia).
Another way to make a ferromagnet exhibit an external macroscopic magnetic field is to reduce its temperature. Below a certain critical temperature, rotational invariance is spontaneously broken, and even in the absence of an applied magnetic field, a macroscopic magnetic field appears.
Here, the magnetization intensity M is equal to the average magnetic moment of all atoms in a magnetic domain that is much larger than the typical scale at which the relevant microphysical processes occur:
Equation 1: The average magnetic moment in an area is much larger than the typical scale of the corresponding microphysics.
When the temperature rises above this critical temperature, macromagnetization disappears. This shift is actually extremely drastic. When | When the M| is close to 0, the function | The slope of the M(T) | is infinite.
Figure 4
Ferromagnetism is the result of the interaction of electron spins with the coulomb repulsion exchange.
Let me clarify two concepts first. First of all, what does spin mean? Broadly speaking, spin is the intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. Although the definition of spin is an object of quantum mechanics (there is no concept of spin in classical physics), people often describe a spin particle as a small gyroscope that rotates around its own axis.
Figure 5: Rotating the electron error but a description of the image
Second, what is an exchange interaction? They are quantum mechanical effects that occur between the same particles, such as electrons. When two particles are exchanged, the wave function of the same particle either remains unchanged (symmetrical) or alters the sign (antisonym), and these effects are the result of such a fact.
Figure 6: Symmetric and antisymmetric wave functions
As in the water vapor critical point, several length scales are involved at the critical point. As shown in the figure below, it describes some hypothetical solids. Each square corresponds to the direction of rotation of the individual atoms in the solid (more specifically, to the magnetic moment).
We choose:
The black square represents the "up" rotation
The white square represents the "down" rotation
The figure above shows solids above critical temperatures. There, the system is disordered. In the middle figure, when the temperature decreases, a wider range of plaques begin to appear. The third figure shows the system at a critical point (called the Curie temperature). We see plaques "expanding to infinite" scales, but fluctuations on smaller scales continue to exist. Therefore, in this case, it is necessary to include all length scales to build a theoretical model of the ferromagnet.
Figure 7: Patterns of magnetic momentum in solids at three different temperatures
<h1 class="pgc-h-arrow-right" data-track="27" > why study critical phenomena? </h1>
There are three main reasons why key phenomena are particularly appealing:
Physicists have not yet fully understood the underlying microscopic phenomena
Different physical systems exhibit very similar behaviors as they approach a tipping point. A well-known example is the similarity between ferromagnets and simple fluids as they approach a critical point. In fact, for several seemingly different sets of systems, the values of the tipping point exponents are equal.
According to Stanley, the third reason is reverence. "We wanted to know how the spin 'knows' to be suddenly aligned when we approached the critical temperature," he said. How did spins spread their relevance so widely throughout the system?
< h1 class="pgc-h-arrow-right" data-track="32" > distribution function</h1>
For example, in order to study the thermal equilibrium properties of a ferromagnet at a certain temperature T, in principle we should write its distribution function. The system's distribution function Z describes the statistical properties of the system when it is in (thermodynamic) equilibrium, which is expressed in terms of the system's Hamiltonian quantity H. Most of the thermodynamic variables of a system, including total and free energy, entropy, pressure, magnetization, etc., can be written as a coordination function (or its derivative).
The allocation function is:
Equation 2: The partition function has temperature T and microscopic Hamiltonian amount H.
H is the microscopic Hamiltonian volume. We can also write Z as:
Equation 3: A distribution function expressed in free energy G.
G is Gibbs Free Energy. The latter is important because it helps to identify the equilibrium state of the system. This is because if we keep the system at constant temperature and pressure, gibbs is minimal when the system is in equilibrium.
For non-zero temperatures, Z appears to be a smoothing function of T, except for non-analytical behavior at critical temperatures.
<h1 class="pgc-h-arrow-right" data-track="40" > Ginzburg–Landau theory</h1>
However, in the case of most complex systems, Z cannot be computed and therefore cannot be analyzed using microscopic Hamiltonian quantities.
Two prominent Soviet physicists, Lev Landau and Vitaly Ginzburg, argue that another way to represent free energy G in terms of magnetization intensity is to consider the symmetry of G to M. The magnetization intensity is often referred to as the sequence parameter.
Figure 8: Lev Landau and Vitaly Ginzburg
The mathematical form of M disappearing below critical temperature is experimentally known as:
Equation 4: The mathematical form of M disappearing.
Figure 9: Spontaneous magnetization vs. size. The curves are iron (x), nickel (o), cobalt (A) and magnetite (+)
For example, if M is constant in x, rotational invariance will limit the free energy G to:
Equation 5: Gibbs free energy G of the rotational invariant system of volume V, expressed in magnetization intensity M.
The anterior factors a, b are unknown, but we assume that they are smooth, well-behaved functions of temperature T (no singularities or discontinuities). Suppose, according to landau and Günzburg's theory, a disappears at a critical temperature, and naturally, when approaching this temperature,
Equation 6: Temperature dependence of the front factor a.
<h1 class="pgc-h-arrow-right" data-track="51" > breaks the continuous symmetry</h1>
How do I calculate the minimum value of a function, such as G(M)? Considering Tue's case, M has two dimensions:
Equation 7: Gibbs free energy G for two-dimensional systems.
The minimum value occurs, for example:
Equation 8: One of the infinitesimals of the Gibbs free energy G of the two-dimensional system.
where the second component can be equal to any value on the circular cardinality of the potential energy shown below (M_2=0 is just a convenient choice).
Figure 10: The famous bottle bottom potential energy
For temperatures above the critical temperature, the minimum value of G appears at M=0, but for temperatures below the critical temperature, there is a new minimum (derived from above):
Equation 9: The minimum value below the critical temperature.
We see rotational symmetry spontaneously breaking down, with non-analytics emerging. This is an example of a second-order phase transition, which is a critical phenomenon.
This G is too simple and we must consider M. Landau and Ginzburg propose the following generalizations:
Equation 10: Gibbs free-energy G of spatial variation magnetization.
We can rescale M so that the coefficient of the first term is equal to 1.
When the external magnetic field is present and above the critical temperature, G becomes:
Equation 11: Gibbs free energy G with varying spatial magnetization intensity in the presence of an external magnetic field H.
When H ≠ 0, non-resolution disappears and | M| become a regular function of temperature.
For small M, minimize G to get the following differential equation:
Equation 12: Differential equation obtained by minimizing G for small M.
There are the following solutions:
Equation 13: The solution of Equation 12.
To get k points:
After integrating k, equation 13 is obtained.
Now consider that the magnetic field H is at x = 0, which generates magnetization M(0) at the origin. When x≠0, what is the magnetization intensity M(x)?
Here, the concept of related functions is important. Correlation functions measure the order in the system, how microscopic variables at different locations relate to each other, and how they change from one another on average (across space and time). In our case:
Equation 15: Correlation function.
What is the behavior of C(x) or M(x) for large | x|? In other words, how do these quantities decay?
The attenuation formula is as follows:
Equation 16: Attenuation of C(x) when T approaches critical temperature:
Figure 11: Experiments in which the correlated lengths diverge with temperature (critical temperature is set to 1)
This suggests that when T approaches critical temperature, the value of ξ decayed by the correlation function has a correlation length, and that this length diverges (tends to infinity).
Using the Landau-Ginzburg theory for calculations, we found:
Equation 17: The correlation length and exponent ν.
Using Equation 14, we find that the value of the exponent ν is 1/2.
<h1 class="pgc-h-arrow-right" data-track="84" > critical exponent, scale law, and universality</h1>
Critical indices such as ν and β define the properties of many physical quantities (including heat capacity, susceptibility, etc.) at critical points.
But why are key indices so important? The combination of these indices gives the law of scale, which is a kind of universality. Experiments have found that some systems with completely different critical temperatures have the same scale index, which is a combination of critical indices.
For example, using the critical index we found above, we get the so-called Fisher scale:
Equation 18: Fisher scale, one of the general indices.
Figure 12 shows another example of a gas-liquid coexistence region.
Figure 12: Ratio of temperature T and critical temperature to reduced density when different substances coexist with gas and liquid
The relationship between the indices is one of the two manifestations of the so-called scale hypothesis. The second is what Stanley calls a "data crash." Following Stanley's method, consider a uniaxial ferromagnet. The magnetization intensity M depends on H and the reduction temperature ε:
Equation 19: M's dependence on H and reduction temperature.
For example, the relationship between these two quantities of five different substances is shown in the following figure:
Another interesting property found in critical states is universality. Figure 13 above is an example: since the five materials have the same exponential and scaling functions, they belong to the same universal class.
Using the concept of a reorganized group, a more complete theory of critical phenomena can be obtained.
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