 Landau theory is sort of the general way of looking at the mean field theory approach to second order phase transition. The central idea is that we can think of the help holds free energy as a non-singular function, which therefore we can expand in a power series in both temperature and an order parameter that characterizes whether we're in the broken symmetry low temperature phase or the manifest symmetry high temperature phase. So if I denote that order parameter, which you can think of as being like the magnetization in our spin system example, and we consider the temperature to be close to the critical temperature, so in particular temperature minus critical temperature small compared to one when we divide by the critical temperature. Then the free energy as a function of order parameter and temperature has a constant term independent of the order parameter, which depends on the temperature, but imagine evaluating it at Tc or value close to Tc. Then the important thing is that I'm assuming by the way that there's an order parameter goes to minus order parameter symmetry like in the magnetic model. So there are no odd terms. The next term in the power series expansion in the order parameter is quadratic, and that quadratic term has a coefficient which changes sign at the critical temperature. I'm considering alpha to be positive, so that means that the quadratic term is as a positive coefficient for temperature above critical temperature and negative for T less than Tc. And then there are higher order terms as well of all even power, so the next most important term is side to the fourth. So when the temperature is above the critical temperature, the free energy as a function of order parameter is quadratic and leading order. And right at the critical temperature, this is for tau greater than tau c, right at the critical temperature the free energy as a function of order parameter has no quadratic term, only a quartic one, so it gets very flat. And then as the temperature dips below the critical temperature, the quadratic term has a negative coefficient, it turns down, but then the quartic term wins in the end and it turns over. And so the minimum of the free energy occurs actually at two non-zero values, and so in the low temperature phase the order parameter has a non-zero value. So you can think of that as like the spontaneousization in the magnetic model. So in particular, you can ask how does this value of the order parameter turn on when the temperature is just below the critical temperature, which we can find just by minimizing this function when tau is close to tau c and a little bit less than tau c. And so the minimum occurs when the square of the order parameter is linear in the deviation from the critical temperature. So Landau theory makes the prediction that the order parameter will behave like some constant with either a plus or minus sign, times I'll write minus epsilon, where epsilon is this dimensionless deviation from critical temperature to a power which in general is called beta and according to Landau, beta should be equal to one half. We can also consider the response when we introduce some external field that couples to the order parameter. So consider modifying the Helmholtz free energy by introducing some terms that break the symmetry due to some external field like the magnetic field in the case of our ferromagnetic model. So the leading behavior will then be linear in the order parameter with some coefficient. So you can think of this as being like the applied field in the case of the magnetic model. That breaks the psi goes to minus psi symmetry so that when we're in the broken symmetry phase, when psi zero is nonzero, one or the other of these two minimum will be favored depending on the sign of Landau, whether it's positive or negative. When we fix the order parameter by minimizing this deformed, perturbed Helmholtz free energy, then the external field is derivative of Helmholtz free energy with respect to order parameter with temperature fixed. And then we can consider the susceptibility which tells us in particular we're interested in the case where the temperature is close to the critical temperature and we turn on an external field. How does the system respond? The susceptibility then is the derivative of the order parameter with respect to the applied field with temperature fixed. So if you consider the inverse susceptibility, that's the derivative of lambda with respect to order parameter, the second derivative of Helmholtz free energy with respect to order parameter with temperature fixed. And so if we use our expression for f, now take another derivative, we can write chi inverse, the second derivative as a constant term which goes to zero when tau is tau c. And then the next most important term for chi small is obtained by differentiating twice the quartic term, so that's a quadratic term, plus higher order. When we consider approaching the critical point from the high temperature phase, then in the most probable configuration psi is zero, and so the inverse susceptibility is going to zero at the critical temperature like alpha tau minus tau c. But before in the low temperature phase, we can replace this by psi zero squared, so that's zero for tau greater than tau c, but it's, I wrote it down somewhere here, it's alpha over g4 tau c minus tau when tau is less than tau c. So when we multiply by 3g4, this term when tau is less than tau c looks like this term except with the opposite sign and with the coefficient 3. So that land out theory tells us about the inverse susceptibility when we're close to the critical point is that it vanishes like linearly in the deviation from the critical temperature when we're in the high temperature phase, but 2 alpha tau c minus tau when we approach the critical temperature from below. Either way, the susceptibility itself is diverging at the critical point. It's inverse is going to zero. The land out theory says that the susceptibility as we approach the critical temperature goes like epsilon or minus epsilon depending on whether we're coming at it from positive or negative epsilon, so absolute value of epsilon to a negative power called minus gamma and land out says gamma is equal to 1. The inverse susceptibility goes to zero linearly with tau minus tau c. It's still here. I hadn't discovered it until I bumped into it. By the end of the class, knowing my intensity for pacing, I go by it about 20 times. There's also a prediction about the heat capacity, but I guess I'll skip over that. Of course, it also tells us something about what happens when we're on the critical isotherm right at the critical temperature that means that the free energy is constant plus a quart of term according to land out theory. When tau is equal to tau c, the equation which relates the applied field and the order parameter on that critical isotherm is that lambda, derivative of free energy with respect to psi, goes like psi q plus higher order. When we consider the response when the order parameter is small, it's psi cubed, so there's a prediction for the behavior of the equation of state when we're at the critical temperature. The applied field goes like order parameter cubed. In general, the exponent that relates the two is called delta. Land out theory makes a prediction for delta, which is 3. There's a more general statement about the equation of state when tau deviates away from the critical temperature, which is another one of the homework problems. The neat thing about these predictions is that they're universal. They follow for any system as long as our basic assumption is satisfied, which is an extremely mild assumption. The Helmholtz free energy can be viewed as a non-singular function which we can expand in a power series. They apply in particular to both our van der Waals model of the solid liquid gas phase transition and our mean field model of the ferromagnetic phase transition because both have that property that they can be viewed as models in which the Helmholtz free energy is non-singular. It's great to have universal predictions, but there are also wrong predictions. These critical exponents tell us in particular how things behave when the temperature is close to critical temperature. They're universal, but also wrong. The actual situation is that there's a broad class of second order phase transitions which have the same exponents. Exponents really are universal, but data instead of being one-half in practice is something like a third, about 0.32, and gamma is something like 1.3. So universality is a great idea, but there's something wrong with our predictions. But what's nice about the point of view we're currently taking is the predictions followed from such mild assumptions which we've now learned must be wrong. That we can't think of the Helmholtz free energy close to the critical temperature as being a non-singular function. It actually has singular behavior, and in order to make correct predictions we have to understand that singular behavior better. So what is it that goes wrong? Why is Landau theory wrong? Well, it has to do with the fact that our analysis has been aimed at trying to understand the most probable behavior, the value of the order parameter in equilibrium. But what we've ignored are the fluctuations around equilibrium. Usually, as in past discussions, the fluctuations are a small effect. So just looking at average values is fine for getting predictions that agree with experiment. But close to a second-order phase transition, fluctuations are very important. So what we conclude from this failure is that f is singular at critical point, and the qualitative reason why is because fluctuations of order parameter, the order parameter psi, about its mean value, are large for epsilon small. That is, for temperature close to critical temperature. So when we're approaching the critical temperature, like I said, the free energy function gets extremely flat. And you can think of this function as governing fluctuations if in some region the order parameter departs from its mean value. It's not the most probable configuration. So now, usually we don't talk about dynamics when we think about equilibrium properties. But if we do think about dynamics, the free energy functional, if the fluctuation tries to climb up the potential, it's going to make it want to come back down. But it's going to come back down really slowly when the curvature of the free energy as a function of the order parameter is very small. The quadratic term, which is going away at the critical point, is responsible for the restoring force that makes fluctuation relax back to order parameter equals 0. So because it's flat at temperature close to critical temperature, fluctuations in xi take a long time to relax. That's called critical slowing down. Usually the system is fluctuating around its most probable configuration, but the fluctuations have a short time scale. So you don't have to wait very long for the fluctuation to relax away. But when you're close to the critical point, it's not true anymore, and a fluctuation can take a long time to decay. In fact, the fluctuations can grow very large in spatial extent as well as temporal extent. Suppose you have a gas, like a water vapor, above the critical temperature, and then you let it cool, and you pass through the critical temperature. What happens? Strange happen while tau passes through tau c? Actually, what happens is it gets all cloudy. It's transparent when the temperature is a little bit above the critical temperature. It's transparent when the temperature is a little bit below. But right near the critical temperature it gets all cloudy. Why does that happen? Well, it happens because there are big fluctuations in density. There are regions that are underdense and overdense, which have a fairly big spatial extent, and the same kind of thing happens in a magnet where you've got big droplets where lots of spins are pointing in the same direction, and other big droplets where they're pointing in the opposite direction. For temperature close to critical temperature, there are large regions, the deviation of the density from its mean value is negative or positive, and those regions get to be hundreds of nanometers in size comparable to the size of the wavelength of visible light, so light gets scattered. So visible light in particular, the kind we see gets scattered, and that's why it gets cloudy. That's called critical opalescence. Opalescence being a fancy word for you can't see through it. So what's happening when the temperature is close to the critical temperature are all these big fluctuations on big scales, and they take a long time to relax. So for temperature comparable to critical temperature, many length scales are important. We have, in other words, to figure out how to calculate the properties of a system in which there's dynamics occurring at all different scales blank, and that makes it hard to compute things. And this was a big problem for decades, but it finally got cracked in the early 70s, in particular due to work by Caddon Offen Wilson, who used an idea which has a scary, fancy name, but it's not really, it's Caddon Off, actually, with a D, has a scary name, but it's not really such a scary concept. We call it the renormalization group. And our idea for tackling a problem which has fluctuations on all these different length scales is to do a kind of coarse-graining of the system. So we have, let's say it's a spin system, like a magnet, like our model of a ferromagnet. So the spins in the spin system interact with their neighbors. So here's the spin system. There are a bunch of spins maybe arranged on a regular lattice. And each one interacts with its neighbors, namely their interactions that make the spins want to line up in the same direction, which is going to give rise to ferromagnetism when the temperature is low enough. And as we start to cool the system down, starting at a high temperature, the spins get more and more correlated. Their tendency to line up with one another is becoming more and more important as the temperature gets lower. So as the temperature approaches the critical temperature, we start to get droplets of spins which all want to line up the same way. That's supposed to be a droplet of spin. And then there are other droplets where many spins are lined up the same way. We get large droplets of correlated spins. So now what do we do? Well, what Kednoff and Wilson say is we should sort of coarse grain the system. We should look at it with a coarser spatial resolution. In other words, we should think of each one of these big droplets, which actually contains many microscopic spins, as like one big spin. And that's essentially what the renormalization group is. It's supposed to replace many spins with sort of an averaged spin. That's what I mean by coarse graining. Now we're talking about going beyond the Landau theory. So we take this droplet in which the spins are lined up in a common direction because the spins have lots of correlations. And we think of that as just one bigger average spin. So now we can describe the system as one in which those big average spins are interacting with one another. So now you can think of each one of the big spins as interacting with its neighbors. So each big spin interacts with neighbors. But now the temperature keeps coming closer and closer to the critical point. And as it does so, the big spins start to get highly correlated with one another. And now there are even bigger spins in which many big spins form a droplet that points kind of in the same direction. And so then we do the whole thing again and replace that by a big, even bigger average spin. And after a while, after we've done this a few times and the temperature's gotten very close to the critical temperature, each time we take another coarse graining step, replacing lots of effective spins by an even bigger average spin, what we get looks like what we had in the previous step. We say that eventually, after we coarse-grained a few times, after many coarse-graining, the coarse-grained system looks just like the system before coarse-graining. So in that case, we say the system scales more very close to the critical temperature. The system has kind of a scale invariance or self-similarity. If you look at it in a more and more coarse-grained way, it still looks the same as it did originally. That's because it's got these fluctuations on all different scales. And when we average out the fluctuations on a small scale, we still have the fluctuations on a bigger scale. So it looks like the same as it did before. That's what I mean by scaling. But it looks the same only after some redefinitions of the parameters. So the new system, after you do the coarse-graining, looks like the one before you did the coarse-graining, but you have to tweak the temperature a little bit and tweak the order parameter a little bit so it'll look just the same. That's because the big average spin is the correlations between the spin on the droplets on bigger scales aren't quite as strong as the correlations on smaller scales. Or if I think about the response to an external field, I get a stronger response to the spins in little cells than I do from the average spin in a big cell because some of those spins are pointing down even though most of them are pointing up, some of them are pointing down. So the average behavior of the big spin in response to an external field isn't quite as strong as for the little spin. So when we do this coarse-graining, we describe average behavior in a big cell, these boxes I'm calling cells, just a box that contains a lot of spins. And it has a volume for the big cell which is rescaled by some factor omega compared to the volume of a little cell. The big cell is bigger than the little cell. And the system after a coarse-grained looks like the original one except I have to change the strength of the external field. Like I did last time, it's a little bit simpler to discuss this in terms of Gibbs free energy instead of Helmholtz free energy. And I wrote down this scaling hypothesis last time which looks like this. And as the external field, epsilon is the dimensionless deviation from the critical temperature. So omega is like a volume factor. Since the Gibbs free energy is extensive, the Gibbs free energy in a big cell, which is omega times bigger, goes like omega times the Gibbs free energy in a small cell. So you can think of this as the Gibbs free energy in a small cell per unit volume times the total volume. And then there's a factor of omega because we change the volume of the cell. And then what this says is if we want the physics to look exactly the same, we have to make a little adjustment by some factor depending on the ratio, the volume of the big cell to the little cell to the external field and to the deviation from the critical temperature. So we have to adjust the external field for the reason I said that the average spin in the big cell is a little bit softer than the average spin in the little cell because not all the cell spins are pointing in the same direction. And we rescale the temperature because the big cells have weaker correlations than the little cells because they're bigger and therefore in effect it looks like we're further away from the critical temperature after we do the rescaling. And that's where this type of scaling hypothesis comes from, question. Closer to the temperature it gets more line maybe. Yeah, so the idea is that as you're approaching the critical temperature there's some characteristic scale over which things are strongly correlated. And it gets longer and longer as you approach the critical temperature. Exactly, you're moving farther away. Sorry? Okay, so the correlation length is getting bigger and bigger when you get close to the critical temperature. But now suppose you do this coarse graining then in terms of the new variables now you're talking about the correlations between the cells and the correlations between the cells have a shorter length than the original correlation length. In other words, in terms of the number of cells you have to travel before the correlations decay that's smaller for the big cells than for the little cells. You compensate for that by saying, well, I just moved a little further away from the critical point where the correlations aren't quite as strong. And that's why you rescale the temperature. Okay. So anyway, it's a long story how you can actually calculate the p's and q's from some more sophisticated description of the fluctuations near the critical point than Landau theory which we won't go into but as you've seen in the homework or we'll see by 5 p.m. Just from this scaling hypothesis you can learn a lot since you can predict the behavior of various critical exponents in terms of the p and the q involving that describe how the Gibbs free energy scales and because you can extract more predictions than the number of parameters p and q there are some relations among those quantities that you can extract that's what you do in the homework. So the big lesson is that Landau theory doesn't work because of the large fluctuations and that gives rise to the singular behavior and to understand the singular behavior which is encapsulated by the scaling hypothesis you have to do a more difficult calculation that you won't learn how to do until you take more advanced courses. Mean field theory is or Landau theory since Landau theory is just kind of a formal way of looking at mean field theory should work if the fluctuations are not so big if they're not so important. The idea of mean field theory if you think about our ferromagnetic model for example is that each spin is responding to the average behavior of all the other spins or to the average of all the other spins but that's not really what happens because of the fluctuations. Different spins are responding in different ways because their neighbors are fluctuating in different ways but averaging over all the other spins is a more and more reasonable thing to do if you consider higher dimensional systems or for higher dimension or higher spatial dimension d Landau should work better the idea is just that each spin has more neighbors and the one dimensional system it's only got two and the two dimensional system on a square lattice it has four on a cubic lattice it has six and so on so each one of those neighbors is equally important in determining what a particular spin wants to do if it has more neighbors then it's a more sensible approximation to say that it's responding to the average of many other spins which is kind of interesting it's true that if you consider the dimension going to infinity then Landau theory works and then there are lots and lots of neighbors but you don't actually have to go to dimension equals infinity there's some finite dimension which is called the upper critical dimension and when you're above that dimension Landau theory prediction is worth the exponents predicted by Landau theory Bayt equals one-half, Gamma equals one so on are actually correct so Landau predictions beta, Gamma, etc work for spatial dimension greater than or equal to some so-called upper critical dimension and that turns out to be four so Landau was extremely unlucky that the world is three-dimensional yes sorry? nothing sometimes an extra dimension can appear, for example in the following way it's a very good question the answer isn't quite nothing but for example you could have a spin which has many components so that it's the spin itself is like a little vector pointing in d-dimensional space and I shouldn't call it d, an n-dimensional space where n is very large and so that sort of behaves like an extra spatial dimension but the reality is that for most purposes we live in three dimensions and this is bad news for Landau because dimensions don't work if we had lived in four or more dimensions he would be even more famous he's pretty famous as it is how do we know? well, it's a mathematical statement so we can it's a theorem okay? theorem it's not a statement about comparing with experiment is this the first theorem? I mean what do you guys think I'm doing here every day? I'm deriving things this is what's different about this one it's a theorem I'm just telling you about and not proving because it's interesting okay now there's also a converse statement which is when the dimension gets lower then Landau theory is worse or in other words the fluctuations are bigger and have more important effects and so it turns out that if you go to a low enough dimension the fluctuations are so big that you can't have an ordered phase at all so for example if you consider a magnetic model so the converse is fluctuations have become more important for lower dimension in fact so important when the dimension is low enough that spontaneous magnetization or in other words order parameter not equal to zero in the most probable configuration cannot occur for d less than or equal to some d lower called the lower critical dimension and d lower well actually here it depends on the symmetries of the system so if you consider a uniaxial magnet where the spin either points up or down or an order parameter Xi which either takes a positive Xi zero or negative Xi zero value in the low temperature phase the lower critical dimension is one we're talking about an axial magnet but turns out to be two if we have a rotationally invariant magnet so if you have a one dimensional magnet a line of spins so I'm saying that in that case there's actually no long range order except at exactly zero temperature cannot occur I should say for any temperature greater than zero because no matter how small the temperature is there are always fluctuations which are big enough to wipe out long range order a one dimensional uniaxial magnet is disordered not spontaneously magnetized for any temperature greater than zero the idea is this you got a bunch of spins in a line that's what I mean by a one dimensional magnet and they interact with one another the neighbors want to line up the same way so at very low temperature you might have droplets of spins which all line up the same way so they're happy but then at some point there's a misalignment and then another droplet occurs where there are many spins turned the other way there's a droplet of up spins and so on so I'll call that a droplet of spins that all point the same way what's the energy cost of a droplet in which here the spins in the middle are pointing in the opposite direction of the spins on the left and the right the energy cost just comes at the edge here where we have spins which are misaligned with one another if it's only the neighbors that interact and all the spins want to point in the same way there's some energy cost if you have a kink so to speak where the spins are misaligned but it's just some constant energy cost at any non-zero temperature there's going to be some probability per unit length of having kinks so when the system gets very long there's going to be some number of kinks per unit length which is non-zero and that means the spins on average are going to be as likely to be up as down because the kink doesn't care how big the droplet of spins is that turns over it only costs energy at the edge so for tau greater than zero kinks have a positive density per unit length and that means no longer in shorter in two dimensions it's a different story because in two dimensions at low temperature let's say most of the spins line up in the same direction then there's a droplet of spins that point the opposite way surrounded by spins that all point up so in that case the energy cost of that droplet of flipped spins is proportional to the total length of its boundary of the droplet and so there will be at low temperature some Boltzmann suppression of droplets which is exponential of minus perimeter divided by tau and so if the temperature is low large droplets are going to be very rare if the droplet gets bigger and bigger its perimeter grows, its energy cost grows and it becomes very disfavored to occur as a fluctuation so that means the fluctuations are highly suppressed at low temperature for the two-dimensional magnet but not in one dimension so in two dimensions we can get or three we can get a spontaneously magnetized low temperature phase in one dimension we can't so how do you actually compute this stuff while there's some exactly solvable models where you can find p and q there's a kind of a general approach which works somewhat well that Wilson and Fisher originally suggested which I'll tell you about only because there's sort of a moral which is if you're a physicist and you're trying to solve a hard problem you're kind of lost unless there's a small perimeter if there's a small perimeter then you can say alright I can start out with the approximation in which that parameter is zero and then maybe I can try to systematically calculate the effects of that parameter deviating a little bit from zero so now we know because I just told you that Landau theory works in four dimensions so for a small perimeter to try to get a handle on things Wilson suggested the deviation of the spatial dimensionality from four so Fisher Wilson in the 70's say consider spatial dimensionality d I'm going to call it four minus epsilon for which I apologize it's not the same epsilon but everybody calls it epsilon everybody calls that epsilon this doesn't mean the deviation of the temperature from critical temperature it just means the deviation of spatial dimension from four so when epsilon is equal to zero that's a case we can understand we can use Landau theory that's one right that's d equals three but what they did is they worked out a way of systematically expanding quantities like p and q in powers of epsilon in a power series expansion so we can expand because we're expanding about something we understand you can always do that we understand Landau theory expand beta all the things we'd like to compare with experiment as a power series in epsilon and then after carrying that out to some order try setting epsilon equals one because we'd really like to know what happens in three dimensions and there's no guarantee this will work depends on how rapidly convergent this expansion is but in fact it works pretty well and you can understand this point three two for example so we can expand beta by carrying out this expansion to a few orders and setting epsilon equal to one so if you don't know what to do look for a small parameter and it might not be obvious how to find one so of course you probably are wondering what it means to do statistical physics in 3.99 dimensions it's a good question so what they really do is they write down a bunch of integrals and so they have a formula that depends on dimension and really those formulas only make sense when the dimension is an integer so then you have to tell a story which is in the end I think not completely convincing about how well it actually makes sense for every integer dimension and if you think of it as some analytic function maybe you can argue if you know something else about the properties of that function that it has a unique analytic continuation away from the integer so we can define a function of d that really depends continuously on d and is defined off the integer values really they just looked at some formal expression which had d's in it and said suppose those d's are real numbers instead of integers without any a priori justification well actually you can treat it as a complex number because you're doing a power series as an analytic as an analytic function of d which has singularities in the complex d-plane and all that fun stuff but it works, it doesn't work great but it works okay, you can do a lot better than land-out theory and then something else that if you continue in physics you'll learn about because this subject of critical phenomena and scaling is really central in modern physics and classical quantum statistical physics and particle physics too so this is one last remark the great idea about land-out theory was it was universal it didn't depend on the microscopic details so the scaling theory also has that property so the scaling functions apply to a big class of microscopic models which all scale the same way but not everything scales the same way the universality classes to a certain extent the microscopic physics doesn't matter you can consider lots of models which have the same universal properties if you're in a certain universality class then you have the same value of p and q in this scaling onsets and therefore all the predictions that follow from the scaling hypothesis but not everything's in the same class depends on the symmetry of the model so for example a uniaxial magnet which has the order parameter goes to minus order parameter symmetry and a rotationally invariant magnet which has a symmetry in which the order parameter is actually a vector and it gets rotated by some matrix like the rotation matrix and that leaves the dynamics of the system unchanged because no matter how you rotate the magnet it looks like the same magnet these have different symmetries and they're in different universality classes well you could already infer that from what I said about the lower critical dimension which is different in the case with xi goes to minus xi symmetry and continuous rotational symmetry but there's still universality in the sense that you can consider any magnet model you like with some kind of local interaction among the spins as long as it's rotationally invariant it'll be in the same universality class and the predictions will apply to it okay so that's it for critical phenomena so we got a half hour today and next week we have to decide what to do I'm going to do a little kinetic theory there's a lot of other stuff in the book a lot of it's fun much of it is applications of the principles that we've been learning up till now but I think from a fundamental point of view kinetic theory is probably the most important thing in the rest of the book and in particular I want to say something though I won't get to it today about a topic that we haven't discussed very much and that except in the context of critical phenomena that is the consequences of deviation from ideal behavior in gases and other fluids the effects of molecular collisions in other words on the behavior of a substance this is chapter 14 in Bell and Cromer and basically what kinetic theory means is while it means understanding macroscopic behavior from the point of view of microscopic dynamics which in a sense is what we've been doing all along and so what I'll do is first of all I'm sorry about this but I'll talk about the classical ideal gas a little bit more but then I want to say something about diffusion which is a consequence of non-ideal behavior that is from the fact that particles collide they're not really non-interacting and the reason is that diffusion is really a ubiquitous phenomenon in all kinds of physics settings it's the key to understanding heat conduction which is really diffusion of energy electrical conduction which is diffusion of charge of viscosity in fluids is diffusion of momentum mixing of materials like if we allow two different species of gas to mix with one another that's a diffusive process but understand all these things where you should know something about diffusion and that arises from the fact that gases aren't really ideal the molecules collide but first let me like I said consider a classical ideal behavior again from a somewhat more point of view somewhat different point of view than before I want to revisit the concept of classical equal partition so in general if we consider computing the internal energy of a system Boltzmann ensemble we say ok the mean value of the energy is a sum overall states of energy weighted by Boltzmann factor and then we divide by the partition function to normalize it so suppose for example I consider an ideal gas I have n particles in three dimensions not four, three I know how to do the calculation for four dimensions also well that's part of it also diffusion is not ideal because we consider it a particle collision but without particle collision how do we even make sense of the pressure well I want to make a distinction between the collisions of the particles with the walls of the container and the collisions of the particles with one another so in particular I guess we'll talk about this next Tuesday that's the ideal gas model and so the non-ideal the important property of the non-ideal gas is that if I have to follow the trajectory of a single particle in a large chamber it has a mean free path it changes the direction of its velocity on some characteristic time scale it does that because of the collisions with the other particles so if you put a number on the back of a particular molecule and followed what he was doing he would make some kind of random walk because of collisions with other particles and that's the aspect of non-ideal behavior that we'll want to discuss to understand these phenomena so in particular if I add some energy locally in the system so some of the gas is a little bit warmer how long does it take that heat to spread out well it's going to be delayed because of the collisions with other molecules so it's going to spread out gradually and that's the origin of heat conductivity that the fast molecules have to make a kind of random walk in order to travel a long way okay oh boy that was my closest call yet like you said this is a good opportunity I'm just stalling I guess I don't think so because I'm pretty damn likely to trip over it now okay or even before okay so if I consider an ideal gas I can classically what is it it's just a bunch of particles and each one has a position and a velocity so in the classical case I can completely characterize it by the position in three-dimensional space of each of n particles that's not enough by itself I also need to know their velocities or momentum but if I know the position and the momentum of each one of n particles I know everything classically right now suppose we're talking about one particle as we discussed many times if we have a particle in a box we can sum over all of its orbitals when we sum over orbitals I can replace that by a factor of volume divided by 2 pi h bar cubed if it's a particle in three dimensions and then I have an integral over momentum sometimes I've written it as an integral over the wave number but here I've written h bar times the wave number to get momentum and then I divided by h bar in the pre-factor so you can think of this volume as arising from integrating over the position of the particle so it's really summing over all of the states for a single particle is doing a three-dimensional integral over its position and its momentum and dividing by 2 pi h bar cubed for a particle in three dimensions since h bar is really h over 2 pi if you like we could call that h and that's a nice formula it's kind of easy to remember it means that we can think of each quantum state which is what we're counting here as occupying a volume in this six-dimensional phase space which is Planck's constant h cubed but if we want to do classical statistical mechanics Maxwell and Boltzmann and those guys they did all kinds of classical mechanics they didn't know anything about h bar in fact all we're going to need to know to understand what those guys did sorry they were all guys is that summing over states is the same thing as integrating over the six-dimensional phase space for each particle times some constant and the constant's going to drop out of the things that we compute so counting states to a classical physicist means integrating over position and momentum for each particle up to some normalization constant and it doesn't really matter how I choose it so suppose in particular I'm talking about an ideal gas in which the energy is just the kinetic energy there are no interactions so no potential energy and if the particles all have mass m the kinetic energy is the sum actually it doesn't for what I'm going to say it doesn't matter whether they all have different masses but is the sum of momentum of the first particle square divided by 2m up to the nth particle square divided by 2m and so when I compute the expectation value of the energy well I actually get three n terms each one of these you can think of as three terms the x squared, y squared and z squared components of momentum so I get three n terms and they all look the same because when I, well maybe I shouldn't maybe I'm doing too many things at once this is the integral now over phase space for the first particle second particle etc up to the nth particle and the numerator and then I have e e to the minus e over tau and the nominator I have almost the same thing except no factor of e so that's what I've got but this is just the sum of three n terms these three n terms so suppose I pick any one of them like the x component of the momentum of the first particle squared then all the integrals divide out between numerator and denominator right because I don't have you know the e just multiplies e to the minus x component of p1 squared over 2m and the integral over everything else is the same in the numerator and the denominator and I have all together three n such terms so we have three n and then times an integral over one particular momentum of 1 over 2m p squared e to the minus p squared over 2m times tau divided by an integral dp e to the minus p squared over 2m tau so just by multiplying and dividing by tau right because I got a p in the numerator and the denominator after rescaling the integration variable shouldn't be three n equals three n times and this is equal to u because I have three n such terms three n such terms rather and so that just becomes tau times this dimensionless integral so this is three n tau and then I have the integral of dx x squared e to the minus x squared where now x squared is p squared over 2m tau divided by integral dx e to the minus x squared and that's this dimensionless number which is one half the momentum and hence x is integrated from minus infinity to infinity in numerator and denominator so that's just three halves and tau so as you already knew the average energy per particle in the classical ideal gas is three halves tau but now let's look at this from a somewhat more general point of view we wrote an energy function which was a sum of squares it was just some quadratic function it could have been any quadratic function and we would have had a similar derivation namely what we found is that for each one of these terms the contribution to the internal energy was one half of the temperature because every quadratic quantity appearing in the energy gave a contribution which looked exactly like this just a factor of tau times this ratio of dimensionless integrals which is one half so if the energy is equal to sum of squares of variables that means that u is equal to one half tau for each term suppose we had many harmonic oscillators instead of many free particles for example then I would write the energy as the sum over a label for the oscillator one over two of the mass of oscillator i momentum squared plus one half mass times frequency squared x i squared well each one of these appears quadratically each momentum in each position and so I could do exactly the same analysis that I just did look at the terms one at a time divide out all the integrals in numerator and denominator that are the same and I'd be left with a one half tau for each variable that appears quadratically I get one half tau for each p i and one half tau for each x i contributing to u in the case of n harmonic oscillators if the sum goes from one to n maybe that's confusing because I don't use n to mean something else in a minute so let me just put it this way the internal energy classically if the energy is a sum of squares is equal to one half tau times a number of quadratic degrees of freedom so suppose for example we have an elastic solid so we have n atoms in three dimensions and that just behaves like three n harmonic oscillators describing the vibrations about equilibrium of the position of each one of n particles in three dimensions so we have three n x i's well how's I don't know three because we're in three dimensions and n atoms I'll just say three n times one half tau from positions plus three n times one half tau for momenta so all together three n tau correspondingly the heat capacity is three n that's just what we called before when we encountered it as a limit of divide theory the law of do long and petite it's just this classical equal partition principle again the principle that every quadratic freedom contributes one half tau to the total energy now we know quantumly that this applies only in a certain high temperature limit the so-called classical limit this classical equal partition is a temperature going to infinity limit of quantum theory and so in the case of a harmonic oscillator in particular we know the expectation value of the energy of a single harmonic oscillator as we've discussed a number of times by now is the level space saying h bar omega times e to the h bar omega over tau minus one which becomes just tau when tau goes to infinity that's two quadratic degrees of freedom each contributing one half tau the x and the p and it becomes something small at low temperature times e to the minus h bar omega over tau as tau goes to zero so we say that the degree of freedom freezes out and gives a much suppressed contribution to the total energy and thermal equilibrium when the temperature is small compared to h bar omega where omega is the frequency of the oscillator but the classical equal partition principle kicks in at high temperature high compared to that oscillator level spacing just one more quick remark which also recall something I said earlier so I can summarize this by saying when we speak of equal partition one half tau contributes to u for each accessible quadratic degree of freedom where accessible means not frozen out that's sufficiently high temperature compared to the characteristic energy scale that it behaves like a classical variable so if I consider a gas a molecular gas like a gas of dipolar molecules in addition to the center of mass motion of each molecule we can also consider vibrations which has some characteristic frequency omega and the rotations of the molecule around either one of two axes you can either rotate out of the plane of the board or in the plane of the board and in the case of the vibrational modes of course the spacing is just h bar omega where omega is the vibrational frequency of the molecule along its axis and in the case of rotation the energy is one over twice the moment of inertia of the molecule about the midpoint of the axis times angular momentum squared about either one of the two axes call that l1 squared and l2 squared these are also quadratic degrees of freedom so in this case since it's an oscillator when it's accessible that contributes another tau there's an X and a P in the harmonic oscillator when that degree of freedom is not frozen out and this is also another tau since there's an angular momentum squared about each one of the two axes two more quadratic degrees of freedom so for the internal energy from the center of mass motion we have three halves I guess I'll write it this way we have n tau times three halves that's because there are three quadratic degrees of freedom the three components of momentum and at high enough temperature plus another one for the two rotational degrees of freedom plus another one for the vibrational degree of freedom these freeze out at different temperatures in fact the rotational modes are present at lower temperature than the vibrational modes the vibrational modes freeze out at a higher temperature for typical molecules so what we'll actually see in the if we look at the heat capacity as a function of temperature divided by n actually at very low temperature we're in the quantum regime where the heat capacity goes to zero but then when we enter the classical regime it'll be three halves per particle but that's because the vibrational and rotational degrees of freedom are frozen out and then it will jump up to five halves because of vibrations at a sufficiently high temperature sorry rotations come in first typically because of rotations and at a still higher temperature where vibrations are not frozen out it goes up to seven halves but anyway the principle is one, oh boy that was a close one one half tau for each quadratic degree of freedom which is accessible meaning a sufficiently high temperature with some characteristic energy spacing in this case determined by h bar squared the typical value of angular momentum and the moment of inertia in this case determined by the vibrational circular frequency at sufficiently high temperature the degree of freedom becomes accessible and then contributes one half tau to the total energy per particle