[MUSIC] In the two dimensional Ising model, under the mean field approximation we found that for temperatures below the critical temperature, there is a phase transition. But what happens near the critical point? Now we will discuss the physical behavior near the critical point for a magnetic system and a liquid vapor system. Near the critical point, fluctuations become extremely important. The fluctuations of the energy. The density and the magnetization for a system are found using various ensemble manipulations. From the canonical ensemble, we can find the variance of the energy using the following simple equation. With a little bit of algebraic manipulation, we can show that the energy fluctuations are related to the quantity known as specific heat, CV. The standard deviation for the energy fluctuation results in the characteristic magnitude of the energy fluctuations per molecule in terms of the specific heat per molecule. Now, this calculation verifies that the average energy per molecule is a well defined quantity with negligible fluctuations, provided that the heat capacity does not diverge. Similarly, we can show that in the grand canonical ensemble, that is in open system, for a one component system, the variable of the molecule number can be related to the derivative of the specific volume with respect to pressure. Now using the definition of the molar density row, we can rewrite the expression as a variation for density fluctuations. The condition for thermodynamic stability dictates that the derivative of the specific volume with respect to pressure is greater than 0. And in the limit that the volume tends to infinity, the density fluctuations in a stable system are negligible. However, as this derivative tends to 0, this dictates that the density fluctuations in the system diverge. Thus the limit of stability for a one-component system, marked by the derivative pressure with respect to specific volume equals 0 exhibits wild density fluctuations. Now as the system approaches the limit of stability for temperatures less than the critical temperature, density fluctuations become substantial. Leading to the eventual formation of two coexisting phases, spinodal decomposition, as was discussed in phase equilibrium. As the system approaches the critical point, the density fluctuations dominate the physical behavior. Now, this is all very theoretical. Where are these density fluctuations important? Well a phenomenon known as Rayleigh scattering occurs when the length scale of density fluctuations is compatible to the wavelength of light. Where there is a local mismatch in the index of refraction that leads to light scattering. Now why is this important? Where does this play a role? This is an important effect for the age-old question, why is the sky blue? Well, the sky is blue due to Rayleigh scattering where the light is scattered due to local density fluctuations in the fluid. Such density fluctuations occur for a fluid that is near the critical point which is the case for the oxygen and nitrogen gas molecules in the upper atmosphere. The shortest wavelength light, that is the violet end of the spectrum, is scattered the most, leading to an overall perceived color that is light blue. Now, let's revisit the Ising model. In the case of the Ising model, we're interested in the fluctuations in the magnetization. The standard deviation for the magnetization per site can be evaluated as. This suggests that the magnetization is a well-defined quantity in the thermodynamic limit that N tends to infinity. As we approach the critical point, the role of fluctuations dominates the system thermodynamics, as we will proceed to discuss. Analogous to the vapor liquid system, the Ising model exhibits wild fluctuations when the derivative of beta h, with respect to the magnetization M, approaches 0. Let's discuss what happens near the critical point for the Ising model under the mean field approximation. Remember, the equation of state for the mean field approximation to the Ising model, this says that the magnetization per site is given by a self-consistency condition given by, The mean field critical point is given at h, the external field equal 0, and K, which depends on the coupling constant, KC equals 1. Thus we can write the constant K as the critical temperature divided by the actual temperature. Just below t = tc and h = 0, we can expand the magnetization m around m = 0 and we get the following relation. In order to find the fluctuations in the magnetization near the critical temperature, we evaluate the change in magnetization with respect to the field, h bar. For temperatures T greater than the critical temperature, there's only one solution and this is the trivial solution of m equals 0. Therefore, this leads to the following simplification. Now for T less than the critical temperature, we can show using some algebra an analogous expression. Now both of these equations show that the mean field approximation to the Ising model predicts that the fluctuations in the magnetization diverges near the critical temperature. That is, a deviation of the mean field approximation from the exact result derived by [INAUDIBLE]. This lies in the neglect of what is known as correlations near the critical point. And thus the mean field approximation is best used far away from the critical point. Now, for a vapor liquid system governed by a mean field theory, for example, the van der Waals equation of state, the critical behavior is very, very similar. This shows a direct correspondence between the magnetization m to the molar density rho and the external field h to the external pressure p. The critical behavior of the Ising model and of other thermodynamic systems reflects the underlying impact of correlations on the thermodynamic behavior. The mean field approximation does not properly account for these correlations. To summarize, in this module, we learned about a model for treating interacting systems. This is known as the Ising model. We found that there is no phase transition that occurs for the one-dimensional Ising model. And the model exhibits a true phase transition for dimensions D greater than or equal to two. This was argued by physical arguments and through the mean-field approximation. We went on to show that near the critical point, there are wild fluctuations in thermodynamic quantities. Finally, we showed a nearly exact mapping between the conclusions of the Ising model and that of a van der Waals fluid.
