 Now, what is very important is that what if two events are correlated? Now, if two events, for example, I am tossing die, I die twice. Once I get five, another time I get three, are they correlated? They are independent, if the, my experiment is then two independent event will be, they are just probability of happening two is pa times pb. So, if it is in case of my, I am throwing a die, then one over six into one over six, that becomes one over 36, like if I have one and five, then I get one over 36 for that giant power distribution. This is very important because in real world, in many, many cases, these independence, these independence theorem you can say, independence is not valid. So, not at all affected by one, getting one three of dies is not important by the two, hence the probability as I said is multiplicative. Again consider two events a and b. Now, we want to know what happened in this case, they are not correlated at all. One outcome of second experiment does not depend my outcome of the first experiment. However, as I keep saying in the most of the case in the world, these two are indeed correlated. That is what is interesting. What is important when there is a correlation? You have a phase transition when there is a correlation. You know, we live in the world because things are correlated. If they are independent ideal gas, then we won't survive. So, now if we can ask the question given an event a, what is the probability of the occurrence b? This is the called the conditional probability that I give you an information. Now, if a has already occurred, then probability b depends that a has occurred. I give an example, I take a liquid molecule and around the liquid molecule there are 10 neighbors. I tell you that there is a 10 neighbors now. Now, next time about say 1 picosecond later or 100 femtosecond later, I ask you what are the probability the number of molecules will be 10 or 11 or 12. And you will know now it is within that number that will be either 10, 11 or 12. So, conditional probability says knowing an information of an experiment. When you know that, then before even you do the next experiment, you have an idea of what would be the outcome. So, this is the what is the presence of correlation or correlated events. This is a very common term we used in probability theory or physics or chemistry that the two events can be correlated by many different ways, but in the examples I am giving you, they are correlated by intermolecular interaction. So, the study of statistical mechanics. So, we have now talked about random variable. We have talked of sample space. We have talked when the two events are not correlated like throwing a dice or tossing coin. However, in real world, as I said, things are interesting only when they are correlated. And so, study of statistical mechanics is that is what Bojman tried to do all his life. This correlation between two particles in a binary system. Ideal gas does not have a correlation, but you can see when ideal gas has such rich predictive power. When you take binary thing into account, you again go very far, but you have to work much harder now. So, study of statistical mechanics is a study of different types of correlations between atoms and molecules, whether liquid or gas or solid, superconductivity or whatever you do. This is essentially studies of these correlations in different form of other. So, this we call ideal behavior when total absence of correlation of molecules that we call very important term non-interacting limit. So, this non-interacting limit is the limit that I said we will do later. As I already said that the correlation is essentially conditional probability. So, when you talk of conditional probability, the two events, one happening and what is the probability, the next experiment will have this outcome. That is essentially is the our beginning of the correlation. So, this is a quantitative measure. So, conditional probability provides a quantitative measure of such correlation. That is how we construct, for example, radial distribution function or other things. As I said this is one of the central quantity in many body systems whether colloids or liquids or your austral ripening all these things essentially depends on this kind of things. So, I have some problems for you to do that two unbiased eyes are rolled. Calculate the probability that you already did. What is the probability of the number? This is the simple thing. Now, I will give you an interesting result that the, what is the probability that numbers of the two dies are different. And you can see that if I sum of the two outcomes and plot the probability, of course, nothing can be below 2, nothing can be above 12. But this is interesting structure because the one that is here is the maximum way it can happen. 2 is 1 plus 1, 12 is 6 plus 6, this. So, as you have, for example, 8, 8 can be 5 plus 3, 3 plus 5 plus 4 plus 4. So, this is 6 possible outcomes and you can get the answer. Another thing is that from a well suffered pack of 15 playing cards find the probability of drawing an ace king and equine. The order of draw is maintained. You know, you are doing that combinatorics and permutation. The order is not important. Now, this is again, when order is important, then of course, you have many more outcomes. When order is not important, when order is not maintained, when order is important, yes, you have less number, but order is not maintained. Every time you have lot more outcome. And next is, this is my favorite. I have talked about it there. And let us see a monkey typing a line from Shakespeare's Hamlet. He thinks it is like a whistle. Then you have 27 characters probability of writing. So, very interesting is that suddenly from this kind of rather in a class textbook war in all your high school math suddenly it graduated to a frontline research problem. This happens statistical mechanics. This is protein folding and most important paper of protein folding the Leventel paradox. So, this is the beauty of the probability theory that everywhere you have to construct the elementary model. And the basic idea again, that if you do, if the monkey has no, like it typed randomly, then it takes, how long it will take, it will have 10 to the power of some 33 attempts. On the other hand, if you, this 2 to the power of 78 is something like 27 to the power of 8. And then this 2 to the power of n. This is 10, somewhere that is 10 to the power of 33. Because I have done that, but anyway, you can do it yourself. And yeah, it is a little bit more complicated that would be block of the correct 8 letters. This is a little bit simpler. You know, this block of 8 letters is made simple. But if you do, there should be a blank here. We think it's like a weasel, then you will have 27, 28 including blank, 28 pieces. So, you have to have 27 to the power of 28. That many, that's your sample space. 27 to 28 and that is what I said 10 to the power of 33. Here it makes it a little bit simpler. There is a block of 8 letters. But whatever, so this is a leventhal paradox that leventhal post in terms of protein folding. Now, so this is the elementary probability theory. Now, I am going to do something extremely important and that is the central limit theorem. Now, as I told you that, you know, assumptions are not given to use this kind of terminologies that they have one thing they call fundamental theorem of fundamental theorem of algebra. Now, what is the fundamental theorem of algebra? Anybody remembers? You read it many, many times in your high school and your B.Sc. Fundamental, I told the mathematicians don't at all. Nothing is interesting for them. When they use a language like that, it's very important. What is the fundamental theorem of algebra? Guys, you should read up your algebra a little bit. No doubt people don't respect chemists. So, that is the theorem is that you have a polynomial of degree N. How many roots it has? N roots. And now, if this depending on the values of ABC you can have all real or you have complex conjugate in PS. Why it is so important? All your numerical work, whenever you are finding a solution we solve by method of roots. Even you are underlying the programs you are using. Fundamental theorem of algebra is being used. Now, this central limit theorem is an amazing theorem just an amazing theorem. It is the most fundamental theorem of not one of the most fundamental theorem of probability theory. Saying that it is very important chemistry. It is a sum of N number of, we have talked about random variables. Variables which can take random numbers within a sample space. N number of random variables, let me do, I am tossing the coin N number of times, the dice N number of times. Now, I define S as a sum of the random variables. I give you the average number. Then this central limit theorem comes from nowhere. It tells you the probability that the sum has value S is this thing. But it is magical theorem. I think in my book I have probably three times I have discussed central limit theorem. That is really magical. I have gone to many, many comprehensive, in physics I do not have to ask. They will always ask a static comprehensive or cumulative exam what is central limit theorem and explain central limit theorem. So, this is now why your energy of the system is Gaussian and this full width at half of energy distribution, what is the full width at half in energy distribution? Specific heat. This is delta E square is Cv, Kv d square Cv. Volume, which is sum of the volume of course, that is not fluctuating, but you have to consider the empty space. That is also then Gaussian and that also isothermal compressibility. And the way we used to do, we did not always trust the computer program to give us random variables. So, we used to, but we have to sample from Gaussian distribution. If you are simulating an arrangement dynamics, then your force has to be Gaussian distributed. And then we used to form this, from an ICD we form the Gaussian distribution, then we sample from that distribution. So, this is what we use everywhere. It is something we are routinely using in term dependence statistical mechanics. Equilibrium statistical mechanics or anywhere. So, many of the results, if you know the center limit theorem of statistical mechanics, they become trivially A. So, total energy of the system, then weakly correlated, they have to be weakly correlated among themselves. This is a very strong theorem actually. And so, this is the specific heat as I just described. In variably Gaussian with the standard deviation, then the other thing that you know, end-to-end distribution in a polymer chain and where the total end-to-end distribution is sum over sum over these things. And then this end-to-end is Gaussian distribution. And that again follows trivially. You do not have to do anything. No random calculation. It follows from center limit theorem. And then many, many other cases. Now, in statistical mechanics, we have the phase space density and all the probabilistic description goes there. Pair correlation function as I described. Brownian motion I described. Protein folding, leventhal paradox. The monkey typing Shakespeare's sonnet. So, these are the things. They are all essentially the A of probability theory. So, to summarize this part of the probability theory is that it is something essential and you have to, you never know when you will, but you better develop a good understanding of the correlations and sample space and the probability that would really stand you in good way when you are doing these things. Any questions? No. Basic idea is to following. You have a random variable X. Let us say position. And you now do an experiment, find out what is the position. Now you do one more. So, whenever you think of random variables, you have to think what is the outcome. What are the values it can have? That define the sample space. Now, first I do the experiments, find all the sample space. Now, if I do an experiment, I get a value, say 10. Now I am repeating the going to the experiment. If it is not correlated, next value can be anywhere in the sample space. However, if they are correlated, then it will be a constraint on that. It will be near about that 10, something like that. Now, this is, anything else? Any other question? Yes. Sure. Very good question. This is just a very well-defined discrete experiment. Outcome is discrete. We are tossing, we are doing a dice. And that is the reason it is cosplay thing. But if your outcome is a continuous, within again a sample space, then you have no problem. Then you have your smooth thing that you are looking for. That is what is important. Even when you do computer simulations, we always tell students to look at individual values. What are your outcomes? And have many times, one or two results go out of bounds, then something is wrong. And ultimately many times the program becomes unstable. So, an idea of, for example, in a many-body simulation, energy has to be conserved. What do you mean energy has to be conserved? It is fluctuating, of course. But it has to be fluctuated within a given bound. If you do NPT simulation, then volume is fluctuating. But you need to know if it fluctuates too much, we discard it. Because then we say, my sample space is not what it should be. So, instead of blindly going and using a computer program, it is very important to realize what is going on inside, not inside the program, but in the problem. What are the kinds of things? Because one of my students now simulating the old problem I did long, long time ago, the diffusion, 30 years ago, diffusion in a triangular potential, which is a very interesting trapping incident. Once one professor from Cambridge, came to me and was very excited about that work. He heard of that work and he heard Bangalore. He came to study of chaos. And I was least interested. In my young age, I would do what I want to do. Anybody coming and telling me that let's do this problem, I would not take that. I should have done that. But that problem now, my student, I told him just simulate it. Find out, what happened, there is a trapping. It is not trapping in local, it is trapping in a trajectory space. So then it goes like this along a line. Then after some time it goes off. It becomes diffusive in a very long time. So how it goes from one trap trajectory to another trap trajectory. And how the diffusion sets scene. When you publish that we did some work but we didn't have this kind of computer power. So we could do only up to for example, elementary steps of 1 million or so. To do this you need to run many billions. So I am asking, I am telling this student, look into it carefully as a problem of mechanics. Now we will, anything else? 1 over root over n in all the power distribution that comes in that comes in of course from center limit theorem. If you do just the binary coin then 1 over root n comes in and so basic idea of course one gets from other than center limit theorem you want a physical insight 1 over root n. Is that what your question? So one of the thing is that you know the statistical mechanics that when we get delta E square and we divide delta E square by n and a system is stable if it goes is n going to infinity the relative fluctuation has to go to 0. So we get root over n by n 1 over root n that goes to 0 and that saves our day and we become okay with that. Now precisely 1 over root over n comes from in the tossing of coin that I believe comes from our application of starlings theorem okay but physical insight of that I have to think about it is a very good question probably I knew but now I do not remember it why it is 1 over root n why not 1 over why not n to the power of 1 third yeah that is different that is very different thing that is when you have instability or you have things going out of bound yeah you get always 1 over root n central limit theorem gives you 1 over root over n and many of the times we are happy with the central limit theorem giving 1 over root n is there anything deeper into that that is why 1 over root n why not a little different probably there is a very simple explanation of that but I have to think about it and I will get back to you about that right now I am more into probably getting into the next phase