[MUSIC] Until now, we've explored non-interacting systems. A magnet is actually a perfect example of an interacting system. Now let's take a magnet, mine is able to of course pull the iron fillings. Now what do you think happens if I heat up the magnet or cool the magnet? Will its magnetization increase or decrease? It turns out that the electrons inside magnetic material is the perfect example of an interacting system. Where the electrons, either like to align with the spin of the neighboring electrons, or anti align. This of course, depends on the properties, of the magnetic material. Now how do we go about describing the magnetization, and its dependence on the temperature? In this module, we will explore this question and develop thermodynamic framework for analyzing this. By the end of this module, you should be able to develop the Ising model and write down the Ising model for an interacting system. Using the Ising Model, develop relations that will quantify the magnetization as a function of temperature. Finally, you should be able to evaluate the fluctuations in the magnetization that emerges near the critical point. So far, our analysis have focused on non-interacting systems. We studied the ideal gas, comprising of indistinguishable, non-interacting bosons. The electron gas, which is indistinguishable, and non-interacting fermions. Crystal fluctuations or phonons, which are distinguishable, but non-interacting quasi-particles. Photons which are non-interacting bosons. Non-interacting systems require the evaluation of a single particle partition function which generally is tractable, either exactly or approximately. However, very few practical problems involve either non-interacting particles or approximately non-interacting particles. Furthermore, a variety of important physical phenomenon are not exhibited in non-interacting systems. For example, phase transitions are a hallmark issue in interacting systems. Our goal here is to discuss the effect of interactions on the thermodynamic behavior of these particles. A very popular model for describing interacting systems is what is known as the Ising model. The Ising model, oddly enough, was not invented by Ising, but by the physicist William Lenz, who gave it as a problem to his graduate student, Ernst Ising. The one dimensional Ising model was solved by Ising himself in his doctoral thesis in 1924. Now, what is the Ising model? It begins by considering a lattice of end sites. The definition of a lattice can include many different geometries. For instance a square lattice, a triangular lattice and so on. And the dimensionality itself can be one dimension, two dimension, and so on. Each site in the lattice contains exactly one spin that is defined by the spin state s i. That is either spin up or spin down. Spin up takes the value of si=+1 and spin down takes the value si=-1. Now the lattice spins are subjected to an external feed h and the spins only interact with their nearest neighbors through a coupling strength j. From this definition the overall system energy can be written as sum of two terms. The first term is a field term which is simply a summation over all the lattice sites. And a second term which is the coupling term is a sum over all nearest neighbor pairs the lattice. This very gentle Ising model can be applied to a variety of problems. For instance magnetic systems, binary alloys, vapor-liquid phase transition, and neuron networks known as the Hadfield Model. The sign of the coupling constant J dictates whether the spins prefer to align with their neighbors, or anti-align with their neighbors. Ferromagnetic state is one where the spins prefer to align with their neighbors, and the suckers went J. Coupling constant is greater than 0. Antiferromagnetic state is one where dispense preferred to anti-align with their neighbors and this occurs when the coupling constant J is less than zero. Let's begin by considering the simplest case of the Ising model, the non-interacting case. How do we impose the constraint of non-interacting? This is done by setting the coupling constant J to be zero. In this case, the partition function for the Ising model where J equals zero is simply given as the single particle partition function raised to the power N. This can be easily evaluated and the partition function goes as the hyperbolic cosine of beta h. A reminder that beta scales inversely with the temperature. A quality of interest, for instance, in a magnet, is the neck magnetization. What is the net magnetization? It's simply a sum over spins in each lattice site. The partition function acts as a generating function, and can be used to evaluate the net magnetization. Now, the net magnetization simply goes as the hyperbolic tangent function. This function says that at large field strengths the net magnetization per unit site goes to plus 1 or minus 1 depending on the sign of the external feed, h. Now what about the average energy? The average energy is simply the magnetization multiplied by the external field h. But what about the Helmholtz free energy? Remember, this is simply given as minus k b t times the logarithm of the partition function. But the average energy and the Helmholtz free energy, we can evaluate the entropy. And the entropy is evaluated as. Let's look at some limiting cases of these quantities. Now what happens when we go to zero temperature? When we go to zero temperature, all the spins line up, and we get the total energy to be simply minus h times N. Since there's only one possible energy state, the entropy tends to zero as temperature tends to zero. Now what about as temperature tends to infinity? Now we go to the case where exactly half the spins are pointing up, and half the spins are pointing down. So the average energy is zero and the entropy is maximal given by NkB log 2. What we learn from this is that the non-interacting Ising model, there is one that has a coupling constant J equals zero. Exhibits a gradual change in the magnetization m with the external feed h. And thus, no phase transition is exhibited in the non-attracting phase. In fact, there is never a phase transition for non-interacting systems.
