 So, this is the, we have done that this is given as energy, so fluctuations in energy when you plot that low and behold it is a very good Gaussian, amazingly good Gaussian, another surprise of nature and then the width of that fluctuation is the special heat and that is the spring constant in my, so why it becomes Gaussian, there are two reasons of that. And I will give first the mathematical then the physical, but both can be considered mathematical, but before I go into that look at this functional form, why? Now remember e average is the value at the minima and that is a macroscopic quantity that the average energy of the system, so system must be minimum with respect to the average energy. So, I know first derivative will go to 0, so second derivative will be this quantity which is the sigma square which is the specific heat. So, this is the reason why this one of the explanation is that the total energy of the system or the fluctuation in total energy is Gaussian. There is a question asked I think yesterday or the day before yesterday and this is the answer to that question. Second important explanation that comes from a very profound theory of probability that I described I think in my third lecture and in my book also I described that quite a bit that is called central limit theorem. Central limit theorem says that if your total observable let us say X, capital X is a sum of a large number of small parts and if there is a weak correlation, correlation is allowed, but the weak correlation between them then the sum is Gaussian. This is called central limit theorem which is an exceedingly important part of statistical mechanics and I think I have discussed it couple of times in my book and I think in my third lecture we discussed it and this is what Rajorshi keeps complaining that how many times we teach the students, the students forget, but this you should not forget. There is a strong law, there is a weak law and I told you mathematicians are not given. This was also the person who worked on it theory of errors and all these things goes back to Gauss. Gauss I think had one of the proofs long, long time ago. I think Gauss was after Shakespeare, I think somewhere 1600 around that time or little later than 1600. I used to know all these things but forgetting. So, then central limit theorem says if I have something X which is sum over Xn, n is large number and then Xn are weakly correlated, then the Px an amazing theorem, just amazing theorem that this is the central limit theorem perhaps one of the most and I told you mathematicians are not given to give big names. We chemists and physicists we just mutilate and insult the English language but they have a very rarely one thing like fundamental theorem or algebra, whole algebra is based on that theorem, whole complex analysis. This is the whole much of probability theory is based on this central limit theorem. The central limit theorem says that this is Gaussian. One of the ways you can do that which is take up any book of probability, they will do in terms of binomial theorem. Toss a coin or take a dice then you will find the sum will become. Go on adding 3, 6 if you are throwing a die and then you will find that is Gaussian. Now all of you who do molecular dynamics or molecular simulation use this at very early stage of your simulation. Now tell me where? Not quite Baba, you sample from a Gaussian distribution. How do you create the Gaussian distribution? You take random numbers because I seed, call random numbers, add the random numbers. You know if you do not use a package like our days we used to, there was a function in photon called RAND but we used to write our own code usually because we are always pseudo random number and the not Gaussian correlation, we used to do it ourselves by doing you can play around little bit and make it much better. And sometime we used to assemble a language and to make it more uncorrelated. However one important thing if you do it yourself you call random number, call it 22 times, add them up, you find the sum of the Gaussian. All it takes is 12 so I don't need to go to infinity. Strictly speaking it has to be large but in real world just 12 gives it Gaussian. This is one of the reason many many things in observed properties in nature are Gaussian. Okay so then here is a table actually by a student made it today very quickly from the book. But here energy at constant volume, then CV response function. So these are the thermodynamic quantities. So there are actually three things one should have add here. One is the perturbation, one is the fluctuation and one is the response function. I probably have done perturbation, response function, then microscopic interpretation. This is as I told you it is a profound result of statistical mechanics. Perhaps one of the most I won't say the most but one of the most profound result of statistical mechanics and this is amazing that it tells you what is special heat. Then CP is that of enthalpy. Then volume and number density give you isothermal compressibility and you have to work in different ensembles to get that. NPT will give you volume fluctuation and gran canonical will give you number fluctuation. This listen carefully. This comes from NPT isothermal isobaric. This comes from gran canonical ensemble. You can calculate the fluctuation that give you both, give you isothermal compressibility and they give you exactly same result. You can do the simulation here in one ensemble, another ensemble. You will find that exactly the same. Magnetization is the external magnetic field. Magnetization is the response and this is the response function. Then I put an external electric field. This is the response polarization and this is the response function, the electric constant. So, all the time experimentally is measuring the electric constant without really knowing what the hell they are doing. But it is the theoreticians or responsibility to understand these things and that has happened. That's the way it has happened. Experimentally much of the time need to work extremely hard to get a result and they really have not much time for this theory and all these things and you will find there is a kind of disregard for us which is painful but that's the way it is. So, now there are certain very interesting result that I want to tell you and that you should know. Here is the specific heat. Ice. Then these are solids, ice, aluminium. Look at this. Now you expect solid to have less fluctuations and that specific will be the same, less. That indeed is less. Now look at the things, liquid. Look at that. Ethanol is so much bigger but water has almost twice this specific heat. Why? I come to that. Liquid nitrogen this one, benzene this, benzene so much bigger than water but much smaller. Mercury is really has much less specific heat. Hydrogen partly because there is an anomaly because they have the ortho para equilibrium. As a result of that they have a very large fluctuations. Helium has quantum effects in it which allows it to really explore much larger energy fluctuations but look at that. As you would expect this is less, this somewhat more and where this is little bit more. So now it is a standard questions we ask why has water specific heat is that of more than twice of benzene which is bigger than water and also bigger than ethanol. So this is a question I am now asking you. Explain why water has such large specific heat and that is very good for you otherwise you would die of fever. You know our body we are very lucky that water has large specific heat because 70% of our body is water. Otherwise your protein if temperature some comes outside your protein would become temperature of the body will rise and your protein will become unfolded and you die. Now you need to tell me why and how water has such a large specific heat. It is due to hydrogen bond. There is a bit more than that. This is a good beginning. Hydrogen bond certainly if hydrogen bond not there then water would not be liquid first of all. This is only at 18. Much heavier things are gas but water because hydrogen bonding. So it is hydrogen bonding but what next? Ethanol also has hydrogen bonding don't forget that. Ethanol has, ethanol forms three hydrogen bonding through oxygen one through hydrogen. What are forms? How many hydrogen bonds? Four. Four hydrogen bonding. Still answer is not there. Come on. That you are saying this is a semantics. You are just putting the same thing in words. We will not be able to waste too much time here. That the reason is that because of hydrogen bond there are lot of excitations in the system. There is a if you do the normal mode analysis that means you can freeze water or take the temperature out and then do a normal mode analysis means you calculate the density of states or you can even take water in little water temperature and do the velocity time correlation function and do a Fourier transform. That will give you the oscillations and as a peak you will find water as three pronounced peaks. One is about 50 centimeter inverse that's because of rattling but there is this peak at 200 centimeter inverse which is hydrogen bond excitation. Now tell me what is one KBT? Thermal energy how much in centimeter inverse? 300 Kelvin how much? 206 centimeter inverse. One KBT is 206 centimeter you might look surprised but that's what it is. So one KBT 300 Kelvin is 206 centimeter inverse and hydrogen bond excitation the network is vibrating like that that's a Raman active mode that is 200 centimeter inverse then there is a vibration which is about 600 centimeter inverse. So water because of the hydrogen bond it has all these collective excitations so when you raise the temperature then all these excitations get populated so before temperature goes up you need to populate them and as a result of that kind of extrinsic nature because the ethyl groups are hydrophobic so there is a local order but this kind of extensive network we call hydrogen bond network that is so peculiar of water that is not there. So this is a beautiful slide that I think I will give it to them to put it on the book. So now this is what I have been talking free energy expansion and the response functions because of the minimum or a quantity x is either pressure or y then this is 0 but then the expansion becomes that in free energy and these are the expansions in energy fluctuation is a specific heat volume fluctuation compressibility and if I have the fluctuations in the system is the magnetic susceptibility so what is important these free energy expansions who free energy second derivative are these response functions those are the same functions that give you that tell you how much a system going to react given the example of water it is essential for you and your body that water has large specific heat otherwise you die off and exactly that these quantities are the ones which are most important most important constitutive properties of a material they are the ones which measured in the very beginning this is the end so my students are fond of cartoon so they are they say that I am trying to fluctuate because it is certain life must have some responsibility so they are not great cartoons but you know statistical mechanics such a difficult subject is good to have some cartoon then no cartoon so this is a picture of Einstein I really like so we now this is the chapter which you have to read and again the same kind of things that I have been talking here that how the same thing and the equations from there but there are lot of very nice things that I have discussed here and quite a detail because this is my favorite chapter so energy fluctuation specifically the derivation is given here is a little better than what is given here you can see the derivation in full glory sequence is little different from what I told you here then the fluctuation in other response functions full derivation is given here ok the full derivation is given here and this is both grand canonical and npt ensemble and then there are some very nice system size dependence on fluctuations that we just discussed yes no non-linear phase transition all free energy is minimum with respect to all thermodynamic properties whatever you choose as Einstein showed there are two independent variables so you have to be like he if you look Einstein's statistical physics there is somewhere around chapter I think Landau d6 the old second edition that I studied somewhere after 30 or 40 around that he had this Landau has this chapter of fluctuations and there he discusses this that Einstein chose selected two independent variables entropy and volume and as I told you paramount you also discuss pressure and temperature Landau because that is connection with the hydrodynamics and discussing very important experimental results which can also be through entropy and volume but it is getting very complicated pressure and temperature is a better variable the rail-abinoir spectra the three peaks that is explained by hydrodynamics ok so it is minimum only thing you have to make sure that you have the two independent variables do not take two dependent variables only near a phase transition only near a phase transition this spring constant just goes that's why it's like here so long I am here I am I am here I am stable but when I come here it is no longer stable that exactly happens in phase transition the system falls out of the cleaves and it forms out of the cleave this quantity just becomes like that flat do we call it softening very important term in physics softening of modes yes absolutely I don't know why you are confused it has to be minimum although it is not stable it has to be K H D that's why I am talking about spring constant just take a spring do that spring comes back right so you are applying a force spring getting elongated who determined I could have put that also that I give a force elongation is L that is omega square X so that has to be you know this is a good question but you have to think and make yourself believe because these are the fundamental and exact results that I told you in book I had and I wish I kept it that is really statistical very significant result from statistical mechanics explaining your high school thermodynamics and physics is coming from this chapter we don't question it but we should actually think about it because it is just too fundamental to ignore there is no point of doing science this is really beautiful so this is same thing I discussed this royal distributions and now going back to different ensemble micro canonical grand canonical ensemble and the maximum means now omega has to be maximum okay and then partition function has to I told you partition function is the quantity that is maximum and I explained in the morning why partition function is maximum partition function is maximum because e to the power minus e by k B T as I said or omega is that that particular state is selected which has maximum number of microscopic states and then this is what we discussed and this is what we discussed there is a question actually one of you asked why it is Gaussian and this is the reason why it is Gaussian we do it fundamental level and the distribution of specific heat and thermal conductivity all these things we discussed but I would strongly recommend you take a look into this book this is my book because I do not think there is any other book that has discussed this so beautifully except I think patria statistical physics but then he does not connected to this water specific heat and the kind of things we physical chemist or chemist believe that is not their cup of tea they will go to liquid metals and not interesting as I said mercury is specific heat is point how much point 2 solid specific heat so small except when they go to phase transition there is specific heat diverges order disorder transition order disorder transition aluminium and manganese and there goes order disorder transition but that because you have this multiple ways to arrange things with different energy so a binary alloy undergoes a spectacular phase transition anything else we will next class Monday we will start with the applications of the partition function and describing some way and getting some very very important results which are even today used in drug design or all kinds of thermal and I think packages from NIH and other people use what will be very I never knew that in drug design we use the circuit tetrodiquation which is derived with monotomic gas ideal monotomic gas