 I think we will start, but slowly by which is meant that anybody who is supposed to show up by 11.15 should still come. So, we start with this attenuation, everybody should sign. Now, I will mostly use the board, but maybe towards the end if I want to show some pictures I may also use the projector. So, I am going to where I should start by thanking the organizers for asking me to come in lecture here. It is a pleasure and so, mostly I will be discussing in this lectures self organized criticality and models of self organized criticality in particular sand pile models. So, but before we get into this models why are we discussing sand piles rather than something else in this college for complex systems. So, I think it is useful to study things like sand piles models because they provide a very good starting point for understanding complex systems. They are not particularly complex, these are rather so, this is called motivation. So, these are simple models by simple is meant the following. If you want to define what is the model you can easily do it and will be well understood by an eighth grade student. Like you know we will do this later, but you can see that definition of the problem is moderately straight forward. It is much more complicated for most other problems you know if you want to I do not know quantum field theory something or the other then it is hard to do. We will mostly manage without using too much of statistical physics even we will develop statistical mechanics along the way, but we will not need all the ideas one uses in statistical physics greens functions many complicated stuff which I have forgotten. So, we will develop them as it comes along. So, they are simple models, but sometimes simple models are not so, simple to analyze and we will check that actually even the analysis techniques are mostly quite straight forward and they are they will not use any mathematics which you do not already know by BSE and linear vector spaces, diagonalization of matrices, perhaps that is about the maximum we will use some amount of actually not too much else. So, the analysis is also simple and straight forward this word island sophisticated. So, there are things you can do sometimes you can do them in a very easy way sometimes you can do them in a very sophisticated way and we will use the easy way. So, we take that way involves using a large number of complicated symbols and mathematical structures and we will typically avoid it. So, it turns out that one can go to some distance using these simple methods you cannot go all the way and one actually ends up having to study a little bit more of the more complicated ideas and we will get a feeling for what these are and we will use them it is much easier to understand the complicated concept in a simple setting than in the full generality. So, we will not study the most general application of an idea we will start with a simple example and see how one can go along with it. So, of course, you know you can find large number of simple problems that is not enough we actually want to understand some problems of real interest and how do we get there and these problems are stepping stones for more complicated problems. You will study later in statistical physics equilibrium statistical physics non equilibrium statistical physics might even do some other stuff like pattern formation. In equilibrium I write physics, but I should have written statistical physics it is hard to distinguish because the boundaries are not so clear lot of stuff people study in field theory comes in the same field or you know some such thing thermodynamics comes there. So, we will leave it like that ok. So, and I guess the last point is that we will actually end up having some understanding of the real applications. So, there are applications it is not applications to real we will get to them you know you will see that at some level they are very easily understood these are applied sometimes the applications are not so, immediate and obvious and it is a interesting to figure out that yes one can use this simple ideas in this much more complicated problem which I did not think about at all ok. So, that is sort of a preamble to the introduction the real introduction is something like this that what do we want to understand in this subject in general in self organized criticality. And so, the answer is that there are many natural systems which we want to understand these natural systems show some common behavior and that is I want to first give an example of the kinds of systems we hope to understand using the kinds of things we will actually study. So, when you use the word fractal forms in nature ok. So, some of you would be familiar with this idea of fractals, but those of you who are not let me still introduce it here. So, quite often there are many natural forms which we see every day around us mountains rivers trees which have non typical shape and Mendelbrot realized or pointed out that these shapes can be understood quite well by some general idea of called which he called fractals he defined the terminology. And they are not well understood described by conventional objects which are not fractals like spheres and cubes and things like this ok. So, he gave the example of coastline. So, coastlines are you know the every island has some shape like this or there sometimes there are countries with boundary between them. So, there are country A and country B and there is a boundary he actually sides the case that you know the boundary between Spain and Portugal was listed as some number of kilometers in one in the Spanish atlas and the same boundary in Portuguese atlas was 30 percent longer ok. How can that happen? What does it you know is the same boundary? So, how can the length of the boundary be so different in different atlases? So, what was realized was that the different people were using different ways to measure the boundary. So, if I give you a boundary like this which is sort of jagged boundary. The way to do it is to take some divider and you know mark out how many times you have to go to go from here to here that is how we measure the length of a boundary. But, now if you use a length segment which is say a rope which is 10 meters you will get some number, but if you use a rope which is 1 meter you will get a different number because you know the same boundary which is here in 10 meters it looks like this, but if I go with 1 meter then I get a longer boundary. So, the length of the boundary depends on the divider spacing you used to measure it and that is not what is supposed to happen with lines have a length and there is a metric and what not. So, what he said was the this line is not really a line it is an object whose dimension is between 1 and 2. So, this sounds very strange in the beginning. So, he said that you should try to take the set and cover it with coins. So, suppose I have some set like this and I want to find the area of this set. So, what I do is I take some standard coin and cover it fully with coins such that no part of the stuff is left uncovered and the minimum number of coins needed provides me with the measure of the area that is the definition. Now, the coin in 1 d becomes a divider. So, you see the we have just extended the notion of a divider to measure areas as well if you want to measure volumes I guess the coins will become three dimensional balls and. So, you take the volume and cover it with balls and see how many balls you need that gives you the three dimensional volume of the set. If you cover it with two dimensional balls then it gives you the two dimensional volume of the set. If you cover it with one dimensional balls it gives you the one dimensional volume and then he said that if you take a line and then you try to find its one dimensional volume that will be this length. But, if you find it is two dimensional volume well that depends on how many d psi need what is the area of disk required to cover it now, but then you take the limit of this disk size goes to 0. Volume is equal to number of balls multiplied by epsilon to the power d epsilon is size of disk this is sometimes called the diameter or it can be the radius does not matter d is the dimension of the disk ok. And so, but of course, this depends on epsilon and then you have to take the limit epsilon goes to 0 to get this you know it should not depend on the size of the disk too much. So, now, if I take a line and measure it is two dimensional area it will go to 0 because very little volume is required to cover everything on the line it is three dimensional volume will be 0, but it is one dimensional volume will be finite. If you take a plane and you measure it is volume three dimensional volume will be 0, but what is the one dimensional volume of the plane that is the length of the line required to cover it fully it will be infinite in this limit ok. So, for each set there is a value of d such that if you go to smaller disks smaller dimensional disks then the volume becomes sorry if you there is a disk such that if you go to smaller size disk the volume goes to 0 if you go to smaller d then it becomes infinite and that value of d is the dimension of the space. This statement we made is clearly applicable and it works if you work in a one dimensional line whether it is like this or like this you will get a d equal to 1 if you work with two dimensional surfaces it will give you two if you work with three dimensional surfaces it will give you three ok. So, that is the definition of dimension let us write it down d is equal to volume d is equal to infinite for d less than d star is the lowest part visible to everybody no ok we will go to the other side here this one second yes there was some question ok. So, this gives you mean that definition of d and now I will just give you very quickly one example of a space where you get a known integer value of d that is called a fractal dimension. So, a space d is not. So, this example was actually known earlier it was done by Sierpinski first he said take an equilateral triangle and I am considering all the points inside the triangle ok and for the sake of completeness we will say that all the points on the boundary of the triangle will also be included in my set all the points on the boundary and inside is my set and outside is not there ok, but now what I do is I take the mid points of lines and join them up and remove all the points in the inverted triangle inside, but keep the points on the boundary. So, now my set consists of all the points which are on these 3 triangles with the boundary included, but not the points inside and then I repeat this construction I do this once again and again and again and again I keep this ok. So, in the end I get some set of points left because all the points on this boundary are never going to be removed all the points on this boundary are never removed and so on. So, he said what is the dimension of this set of points which remains and so well it is easy to check that each time I remove one generation of triangles this is the first generation of triangle is this one next generation there are 3 triangles next generation there are 9 smaller triangles and so on each time I reduce area at generation n goes like 3 by 4 to the power n because it becomes lesser with bigger n and tends to 0. So, for large n we are discussing the case n equal to infinity the area is actually 0. So, now I decide of this is this stuff is 0 area. So, I cover it with lines. So, you can check that if you take this line it has a length l, but set of these lines together smaller triangles will have a length which is sorry let us take this triangle it has all the points on this triangle boundary of this triangle will have some length l this triangle and this triangle together will have a length which is 3 l by 2 because each one is smaller by 2, but there are 3 of them next time will be 9 l by 4 and so on times l. So, the length of the boundary increases as we go to bigger n length of boundary goes as 3 by 2 to the power n of the n th generation boundary and so the length is infinite and the area is 0. So, it must have a dimension in between the two and so then he said that well let us say that I cover it with actually with balls of size epsilon and you cover number of balls of size epsilon is n of epsilon yes this problem and now n suppose I take this number of balls of size epsilon try to cover the minimum number of balls and now I decrease the size of balls to epsilon by 2. So, this will say that the number of you know I just decrease the size of each ball. So, that it becomes smaller by affected 2 then I need 3 times as many balls and so this says that n of epsilon goes as some power of epsilon which is log 3 by 2. So, if n epsilon goes like epsilon to the power minus d we must have 2 to the power d equal to 3 or d is equal to log 3 by log 2. So, that is the simplest example of a fractal whose dimension is between 1 and 2 yes sir. So, now coastlines are there people have gone and looked at various coastlines and use this method to determine the effective fractal dimension of a coastline and yes sir I it is not visible sorry firstly I can get does this work ok. So, then people found that different coastlines have approximately the same fractal dimension you can go to Norway or Sweden or you can go to Indonesia and you can look at the boundaries of coastlines and they look they have roughly the same fractal dimension up to numerical error which is you know which is necessarily there in field data. So, then it be there is one thing one can do is to have a lot of data you know from satellites or whatever and measure these fractal dimensions of various objects and there is a different thing you can do is to try to have a theory which will explain why the fractal dimension of coastlines is the same or nearly the same ok. So, our aim would be to go beyond sort of the description in terms of such numbers co-exponents of dimensions and try to see where they come from can be provided understanding the reason why the coastlines have a dimension 1.2 whatever ok. So, just to make contact with the next lecture. So, sorry we are discussing coastlines and so we said they have a fractal dimension t which lies between 1 and 2 and we try to understand why it is this way, but I can try to study some other physical systems. So, let us do mountain landscapes. So, for mountains if I draw a picture I am not very good at drawing pictures, but let me make a attempt I can draw something like the many people will agree including children 5 years old that this year it looks like a mountain you know it is a bad representation, but it is possible that this is a picture of a mountain 1 dimensional right now I drew a 1 dimensional picture, but I can also do this in 1 2 plus 1 dimensions. So, I can draw a real landscape, but then I can draw also a picture like this and people will say no no no this is not a mountain this is not the picture of a mountain. So, what is the difference between these two graphs for a mathematician is just a function f of x which is a positive function for us can you distinguish this picture from that picture of course, you can distinguish, but what is the characterization to be used to distinguish between these two pictures and the characterization people have found useful is something which is called measure of roughness of the stuff this stuff is much more pointy and rough than this one that is the difference that is why people can recognize oh sorry I should have drawn the third picture they said this is this does not look like a mountain either it is too smooth may be some prayer is or something, but it is not a mountain. So, if the surface is too rough it does not look like a mountain if it is too smooth it does not look like a mountain. So, it has to have some degree of some special range of roughness. So, how do you characterize this roughness that is the question and. So, the answer which turns out to work which is not immediately obvious. So, I will use characterizing mountains. So, there is a function h of x y which is the height above the sea level of the mountain at point x y and. So, no if you go to different points you get different values of h. So, I would like to study h of x y let us call it h h of x y minus h of x plus delta x y plus delta y and calculate its magnitude and calculate its average. What average are we talking about this is an average over the landscape you take a point x y fix a value of delta x delta y. So, that is another point at some distance away and then you move the initial point x y and see what is the average value of this thing. This thing will now be a function of delta r which is the distance you moved yes it is the height above the sea level. So, this thing I can average over x and y and also over angles of delta x delta y. So, this is now this called delta h which is the function of r which is the which is the variance variance of h separation r. So, how does delta h of r depend on r that is the question. So, without knowing anything about mountains can you say something about this function delta h well I know that at r equal to 0 it must be 0 and when r increases it should increase. So, how far does it how fast can it increase yes sir sorry separation is the separation between the base points on the x y you know x y points on the surface of a sphere separation is the distance between the points this point and this point the positions at the base of the mountain. Angle is the you know this one point is here one point is here and then I take the point here I keep the modulus are the same and I vary the position and calculate the average. This is by define this is the I mean definition of some quantity I can measure experimentally given a mountain range no because I am averaging over angle. So, the modulus of delta x plus delta y is held fixed and the angle is vary it is uniformly just now. So, it is only now it is only a function of delta r. So, what is this function delta h of r we said it must be monotonic in r. That is not a mathematical fact no I can certainly cook up functions which do not give you monotonic dependence on r like that, but the mountains we know of this will be lying between 0 and what the maximum height difference can occur between two points at distance r be order r because you know there is a slope and the slope cannot be too big for a real mountain range. So, this function is bounded by this function 0 and r. So, I my guess is delta h goes like r to the power alpha alpha lies between 0 and 1 this is not a proof as you see it is sort of a plausibility argument we are building up. So, then once I have this kind of stuff I can go to the mountain field data and measure the value of alpha for mountains in India and in Europe and in Latin America. And it turns out that quite often the value of delta is roughly the same and. So, then we need a physical theory to explain this let me just say that you know when you try to apply this ideas to real world sometimes it does not work. So, there are these things called the there are mountains which look like this which are called full mountains and there are mountains which look like this and what theory applies to one type of mountains does not apply to the other type of mountain. So, maybe we should not be very glib and say that oh I have a theory of how the mountains are formed and you know here is this theory and here is the value of alpha. You should sort of look out and check that whatever assumptions you have made in your theory are actually working in the real system. Sometimes they work, but sometimes they work only for one type of mountains and not for everything. So, there is a possibility that the theory you are using is too simple, but that does not stop us from trying to make a theory and then check if it works in the real world. So, we will say only this much for mountain landscape I have introduced an additional idea which is called correlation functions. So, the way we discuss these kinds of random shapes and matter is to discuss how the height at one side is correlated with the height at a different side and the measure of this correlation was this I could have taken the square which is easier perhaps you know it is the same thing. So, I take the square of the difference and measure the variance and delta h squared is the variance. So, we quantify the stuff by correlation functions and we say when you look at this correlation function it has a power law dependence. The power law dependence holds for all r which lie between 1 meter to 500 kilometers may be 1000 kilometers. So, there is a big range, but it is not infinite the power law will not work if you put r equal to 10 to the power 7 kilometers because the earth is not so big. So, there is always a upper cut off there is always a lower cut off to all these power laws, but in a very big range of the order of 10 to the power 5 decades or whatever the power law seems to work and. So, then we say that this system is characterized by these powers and if we understand a theory which will explain these powers then we have made some progress towards understanding the real problem. So, that is the way we will approach the problem, but let us make a list longer here. So, distribution of galaxies in the universe distribution of mass. So, when we look at the pictures of the stars from the telescopes when finds pictures like this there are some galaxies here then after some separation there are galaxies and. So, there are galaxies there are clusters of galaxies and the distance between clusters of galaxies is bigger than the typical distance within a cluster right and then there are clusters and clusters of clusters of clusters of galaxies and how do we characterize this structure. The answer is we say that here the answer is that mass within a radius r how does it depend on r. So, I stated my observation site go to distance r and see all the mass inside and make r bigger and bigger and bigger how does it grow. So, does people know how this mass increases with r in the universe when r is measured in mega per se for some such thing I do not know the distance between galaxies of the order of distance between galaxies any gases 0.5 0.5 it is kind of low yes somebody else if this was a what is called homogeneous universe then the mass would increase as radius cubed yeah, but you know. So, you take points in empty space with uniform density then for large r you will still go as r cubed, but it turns out that over a big range this power is more like 1.7 much less than 3 and so do we want to understand why is there is clustering in the universe and if there is then you know can we understand where this power comes from that is the kind of question we would like to answer. I am not going to say that in this course of lectures we will actually answer this question, but you know I am just trying to motivate the overall problem and then you can make your favorite question about the problem you want to study and you try to see if we can get some understanding of that kind of a question. But I will add actually just I think 2 or 3 more points to this. So, sometimes what people do so all these objects which we discussed so far they were actually kind of geometrical objects by themselves and we were trying to quantify the structure of these geometrical objects and these we quantified in terms of some dimension which we measured in some way you know actually each of the measurement was slightly different if you would notice. You know we use the same kind of words we are trying to quantify the structure in this case we measured it actually we use the same definition, but use the word correlation functions or use the word how much mass is within a radius r or how does the number of balls required to cover the space vary with epsilon. So, the other definitions were slightly different, but may be the basic idea was somewhat similar, but what if the object is not really geometrical object at all I do not know I like stock market prices. So, the price of the one particular stock it varies with time. So, what is the dimension of that object actually it makes no sense, but I can think of the price as a function of time for me that is a natural representation and the price does this and now whatever definition I use for the mountains I can use for the prices also, because this is a mountain landscape as far as I am concerned. So, it is very important to realize that the key point here is that one of the dimensions became time that was not really very geometrical to begin with, but with our representation it is possible to think of this as a mountain range and you can calculate the fractal dimension or the roughness of this mountain range and that will characterize something about the prices. If I can characterize the prices very well maybe I can use that idea to tell what will happen to these prices in the future for some such thing. Yes. For example, for this plot, if you take a time interval that is for example one point there is connected to the following point, maybe a week after, if you take an interval that is less than this for example one day, then you would have a more fractal. No, so this is what our idea was to use this idea here that the variance of the price difference will increase with the interval and how does it increase with the interval and that I will use to quantify the fluctuations. In some other words this will be called the you take the time series and you calculate the correlation in the time series and you calculate the power spectrum of the correlations and measure the correlations. So, we have got around to the idea that there are these systems like the mountains where they show correlations in things like the height of the mountain over distance scale of meter to hundreds of kilometers. So, what is giving rise to these correlations that is the next question one can ask. So, the answer is that well we have studied correlation functions in other contexts before and you know the knowledge of this other context should be useful to understand this problem. So, the so in this problem with distribution of mass in the universe is like density fluctuations of the matter in the universe, but I have studied density fluctuations of matter in things like gases when I look at the density fluctuations of gases. So, it is much more smaller system, but you know that all the distances are smaller. So, what what finds is that the correlations are roughly of the size of atoms and if you go to distance bigger than 10 atomic sizes or so the correlations die away the correlations typically die away exponentially with distance. So, that once you are 10 atom spacing away they are very small correlations. So, so I calculate rho of r 0 rho of r 0 plus and I calculate these correlations in a liquid in equilibrium and so what does this do well it has this kind of a structure I guess I should draw something more reasonable like this. It has this kind of a structure very close the next atom cannot get there there is repulsion. So, the correlation function is small then there is the first shell second shell and after some distance it becomes very small then you change the temperature the same thing happens it is very hard to produce correlations which are more than 10 atom spacing wide you change the liquid it will not help much. So, what people did what they realized that oh what if you go to the point where the liquid and gas systems become very similar then you get correlations which are long-ranged. So, so if there is this p t phase diagram for liquids like this. So, there is some liquid and there is some gas and there is some solid if you are at any of these points in this phase diagram the correlations are typically exponentially decreasing with distance. And the correlation length is order of 10 times the molecular spacing molecular size. So, no way you can go to a distance of 100 kilometers and still see the correlations. The only place where the correlations become big is actually this point critical point which is the end point of the line separating liquid and gas phases. And there people found you know that you just take the liquid and cool it down at the critical point it develops it becomes cloudy by which is meant that the scattering of light becomes very strong which is that the fluctuations of density are of the order of wavelength of light which is 10000 angstroms. So, the correlation length has become 10000 times the atom spacing you can do this better, but you know that the basic argument is that one gets one gets long-range correlations only the length near phase transitions in equilibrium systems who has written of course it is wrong. So, can you give me an example of an equilibrium system where you see long-range correlations I did not get you solid solid yeah ok solid can you explain how a solid has long-range correlations very good no I understand. So, the argument is that in a solid you have a periodic structure and so the positions are having long-range correlations that is correct, but this long-range correlation can be expressed as a you know just as a one point correlation function has a value and the two point correlation function if you subtract the one point correlation function product then the correlation length is actually small and finite I will write this down as an equation because it is an important point. So, rho r, rho r plus r 0 r 0 plus r expectation value as a function of r has long-range correlations this is correct, but then what I do is that I calculate rho r 0 rho r 0 plus delta r minus rho r 0 rho r 0 r rho r 0 plus delta r this is a bracket. So, this is called the connected correlation function and the connected correlation function even for solids is exponentially decreasing. So, we got to do better than this the question is can you give an example of a system which is in equilibrium which has long-range correlations. So, let us go away which is non-critical away from criticality can you have power low variation of away from phase transition points away from phase transitions can there be long-range correlations and the answer is actually yes the answer is that all materials have phonons which have you know. So, if you study the problem in some detail which I will not do here on board we can show that the there are modes with low wave number which are long wavelength modes which persist for long time and they give rise to power load tails in all correlation functions at least the time dependent correlation function. So, if there is a density mode here and then I was the density at this point in a solid because there is a sound wave which has gone to the other end and come back from the other end there will be a peak in the density at the spacing dependent on the little size of the crystal which is macroscopic. So, there will be time dependent correlation functions with very long range. So, of course, all solids have all liquids have phonons I think. So, every material that I can think of has these kinds of long-range correlations. So, why am I writing all these firstly I wrote it because everybody else seems to write this I read it in some place and I just copied it without thought, but when you think for it for just a little while you realize that no, no, no you can get long-range correlations even in equilibrium systems without too much trouble. I am not going into strange systems like x, y model in two dimensions where there are power law tails even in range in an entire phase. We are talking of ordinary liquids and solids and everything else and everything has these power law tails. Let us take some other example. So, if you take a ferromagnet in which there is of course, a magnetization, but there are these spin waves and the transverse magnetization fluctuations are very slowly decreasing and the fluctuation you know. So, there is a power law tail in the transverse correlation function of the connected part of the spin-spin correlation function. So, you can actually find power law tails in equilibrium systems it is not so hard, but these power law tails we kind of understand you know I read it in Kittel's book I forgot it after a while, but actually I kind of know. So, there is no big deal you are just making first about something which is rather trivial and well understood. So, I think that is the point the point is that you can get power law tails even in equilibrium systems, but they are usually very simple power law like 1 by r squared d k for some correlation function 1 by r 4 d k for some correlation function. If you want to produce a correlation function which decays with the power 1.7 that is very hard. So, that is what we say we say that in equilibrium systems long range different color is useful non trivial. So, everything else every other objection you will raise will be put as trivial and then we will not worry about it is as simple as this though the things which we understand we call it trivial the things which you do not understand we call non trivial and all the examples of correlation functions which are long range which are known fall in the trivial category most. I do not want to be awfully technical, but I am actually being somewhat careful. So, very good. So, we have discussed the fact that in equilibrium systems it is very hard to find long range correlations away from phase transition points. But we saw that of course, we are trying to discuss mountain landscapes and those landscapes are showing long range correlations, but the temperature in the mountains was not held constant throughout the billion years of history. In fact, if you study these correlation functions in even at critical point in the liquid you have to be very careful that the temperature is the same at the critical point to 1 part in 10 to the power 4 or 5 otherwise you do not see the long range correlation. The fluctuations of a milli degree will kill the long range take, but for the mountains they were going on for billion years that temperature was changing by 30, 50 degree may be even 100 degree centigrade at various times and that mountain is still there. So, it was not under a very precisely controlled conditions and you still got this power law correlations. How were they generated? That is the question and the answer is that in non equilibrium systems it is much easier to generate long range power law correlations and that is what we will try to discuss. This is what happens in equilibrium. In equilibrium systems it is kind of hard to generate long range correlations you have to be near the critical point. We will try to study other systems which are not in equilibrium where it is much easier to generate long range correlations. So, very good I am doing comfortable in time just once here in steady state. So, we put in this part in that steady state because if I say non equilibrium system then everything is non equilibrium you know it is very hard to say anything reasonable about it. But let us consider a sub class of non equilibrium systems which at long time go into something which to us looks pretty steady that is called a steady state that is as close to the equilibrium state of that we studied earlier. And in this non equilibrium steady states one can still find long range correlations and so these systems which show long range correlations are called critical systems. So, such systems they are called self organized systems. So, there are two words here one is called self organized and one is critical. So, this word it means long range correlations and what is self organized. So, self organized was a word which is. So, this word self organized critical was introduced by Bakhtang and Wiesenfeld in 1987. And they proposed a theory which is called self organized criticality theory of self organized criticality which is what we will roughly discuss. But the word self organized is much older it was introduced by Haken and Prigogen and they were thinking of things like living cells. And they said that these systems you know they are just a collection of molecules of various types. And they come together and they organize themselves such that there is a cell wall and there is a nucleus and they mitochondria and I do not know what else. And they perform all kinds of nice functions they can take food they can move around and so on. So, wait how does this happen how does a random collection of molecules get into yes please some question yes sorry which book ah reference I have not actually prepared I think nowadays life has become very easy you go to Google and Google scholar this is what all of you do. So, do not you know do not pretend that you do not do it try type this type self organized criticality Bakht and that thing will show up on the top. So, if you find some difficulty in finding suitable reference then you ask me I will provide you with reference send me an email sometimes I do not remember the full reference fully, but I will find it I will send you the reference yes sir. No I did not say that I said the opposite I said when I say critical I mean long range how does it imply no these are just words. So, we just call them because in equilibrium systems it was found that if you want to find long range you they are they were called critical and they were at phase transition point here there is no phase transition, but they still power loss and we still call them critical. They are critical only in the sense that if you make some disturbance it propagates to a long range ok and so critical just means long range for me it is unfortunate word, but maybe we do not have much choice now because you know in so many years have gone past people have got used to this word and changing words does not help too much. Now, I will give you an example there used to be a thing called atomic energy and of course, everybody realize that it was not atomic energy it was nuclear energy, but for some 30 40 years people continued to call it atomic energy because they felt that the name could not be changed and later they actually managed to change it and they call it nuclear energy and it does not cause any real problem. So, maybe someday people will find that critically not a right word and you can change the word to something else more politically correct, but you know I do not think it will be a big issue it just a change of name once you realize that the idea for criticality is not something critical in the sense that if you change a parameter this phenomena will go away ok. It is critical only in the sense that it if you do something here its effect is felt long at a long distance then everything is quite clear and there is no problem and you can use the same word or you can use a different word it does not matter ok. So, now the word self organized. So, as I said pregojin and haken discuss living systems or non living systems like lasers you know laser is a big tube you shine some light from outside and sometimes it produces a coherent radiation sometimes it does not produce a coherent radiation. So, they said that under some conditions the system organizes itself to produce coherent radiation or in a similar analogy under some situations the molecules will organize themselves into a living cell. So, the self organization was some subsystems with very small parts which organize themselves to produce a structure which has some function and it has long range it has some function. So, self organized criticality in combines these two ideas we have to start with lot of small systems which interact with each other in some way they produce a new system which is the steady state which has long range correlations ok. So, then ok. So, before we go into complicated philosophical discussion about self organization let us give a simple example of self organized critical system which is what pairbark gave in the beginning. So, he said that suppose you take just go to the beach and work with dry sand beach is not a good place to work with dry sand, but anyway let us take the case it is pretty dry quite often and then he said that you produce you just drop sand from top and it forms a cone like this that is called a sand pile all of us have seen it before. Now, if you introduce more sand then you form a bigger pile ok and he said that we it was noticed every when it was done before people noticed that there is a particular angle which is called the angle of repose which is what the surface of the pile makes with the horizontal and this angle is very reproducible you can just do drop any kind of sand and this angle is more or less the same for same sand if you take a different sand the angle is kind of different maybe I do not know what is the value of theta for normal sand outside in Italian beaches no clue nobody louder louder just say something it cannot be too wrong 16 16 I would say more like 45 that seems plausible all the sand which I know about the angle is around 40 to 45 degrees wet sand you can make actually very long columns and you can see nice pictures on the Google also no no no it is very interesting you can make wet sand can produce very tall structures the angle can be close to 90 degrees even ok. So, what is this angle of repose? So, it the statement is that if you produce a pile within the angle which is lower than the angle of repose then that pile is kind of stable and you touch it a little bit it will not do anything stay there and if you make a pile which is bigger than the angle of repose like this you can actually produce these piles by tapping suitably and being very careful, but these piles are kind of unstable because if you touch it everything will drop down and you know there will be a bigger balance lot of sand will go the angle will decrease to the value which is 45 degrees. So, he said this is a very simple example if the angle is less than the angle of repose the act of adding sand will increase the angle because you add it here and it will form some you know that average angle will increase, but if the angle is bigger than the angle of repose there is an instability mechanism which takes care of it and reduces the value to a lower value and so it goes down and so it is under the natural dynamics the system is attracted to this particular value which is the angle of repose and so the system organizes itself to go to the critical point between the stable and unstable phases. So, less angle is stable a bigger angle is unstable and it at the boundary of this. So, in some other literature it is called the edge of chaos. So, the idea is very simple and very attractive and intuitively obvious like I think this argument you give to 10 year old child roughly he will understand and by the same token it is possible to talk to graduate students and convince them of something which is not even true. So, you should check that everything I said is it actually working I said it, but you know is it true. So, then people made very careful checks with sand does it happen like this now there is an angle of repose such that if you are below then it is stable if you are above it is unstable and so on and so forth. And they found no if you work if you are very careful and you try to measure things well then there is a thing called angle of stability and there is a called angle of drainage. So, if you keep on adding sand typically the angle keeps on increasing to a big value and then at one point there is a big event and the angle suddenly reduces to a lower value which is called angle of drainage and the angle of drainage is lower than the angle of stability by 2 degrees or 1 degree. So, if they look about the same 45 47 you may you may not you know they are rough surface is slightly rough you cannot even tell easily, but suppose there are two angles suppose there is an angle of drainage which is less than the angle of stability then the system oscillates between these two stages you keep on have some sand on a table. So, there is a finite size table like this and I drop sand from here then it forms a pile like this. Then I keep on adding sand it will form a bigger pile and at some stage the height will go down and what you see is that when you measure the flux out it will behave like this some small stuff then there is a very big event then there is some stuff and there is a big event and there is some stuff and there is a big event. So, this behavior is called charge and fire behavior. So, for a long time you are charging charging charging and when a particular threshold is reached there is a fire and everything goes down and then there is a slow charge and build up and then there is a fire and charge and fire it is roughly periodic in time yes please. Flux out is the flux of you know because this is the finite table when I keep on adding sand the sand keeps on dropping out and I can measure how much is dropping out. So, this charge and fire behavior in other context is called stick slip behavior. So, you have this stuff here solid resting on another solid and I apply a force. So, if I apply too small a force it does not move and when I apply too big a force it moves with some acceleration actually, but if you apply just the force required for it to move the old theory said that there is a coefficient of friction mu right. If you have a force bigger than coefficient of friction mu then it moves it is less it does not move the name associated with this law is of Coulomb. So, it is a very famous guy and when you do not mess around with him severely, but anyway if you actually study the behavior in many solids the behavior near the threshold of force is a bit more complicated. It is called stick slip behavior for a while the thing moves and then it gets stuck and then you apply a little bit more force then it moves and get stuck again and then you need to apply less force to keep it going and, but then will get stuck again and so on and so forth. So, the real behavior is a little bit more complicated than carry catcher behavior one might make in some simple model and the fact that you know there is stick slip behavior when you move one solid against another is very well known to violin players and people like that. So, it is not an unexpanded phenomenon it is not something which you have to do very complicated experiments in IBM lab to realize it has been known for a long time that you get if you apply friction on one surface to another you get stick slip behavior sometimes which can be used for making music or some such thing. However, in other situations it is not always that you get stick slip behavior sometimes you may get this kind of a simpler behavior. So, what we will do is we will study the simpler systems first and we will introduce the complication made by all this periodic firing and stick slip behavior a little bit later. So, with this warning we are going ahead and describing what Pierre said happens in sand piles. So, he said sand pile on a finite table in steady state with slow driving. So, with slow driving means that you add one grain at a time. So, you have a table you have a pile and you keep on adding grains and add one grain at a time and then ask what happens. So, whatever happens is called the steady state of the system after long time the system will look like it is in some steady state and we are trying to describe what happens in this steady state and he said that again if you measure the out flux out flux will not be very nice well behaved function. So, sometimes when you add a grain it does not do anything it just sticks there and but sometimes there is some grain comes out sometimes may be three grains come out then after sometime some grain comes out. So, he said the out flux will have many times nothing happens sometimes some event occurs the size of the event which is measured let us say by the number of grains will go out can show a lot of variation it can be one grain or one thousand grains or one million grains coming out and these occur with different frequencies. So, this system has slow drive slow uniform drive it has what is called erectic burst like relaxation. So, you add one grain at a time, but for a long time nothing comes out then all of a sudden hundred grains come out then nothing comes out for a long time then time ten grains come out and so on and so forth. So, this is called burst like relaxation and then size of events has huge variation as we said with some probability few grains come out with significant probability a huge number of grains come out five thousand come out or five thousand or more and this probability can be measured in experiment. And then you find oh, but this is also probability that fifteen thousand come out that is also measurable quite big and so the size of events has a huge variation. So, what are the functions which describe the size of events? So, he said probability that flux out is equal to f goes like one upon f to the power x for one much less than f much less than f max is a function of the size of the system I will write L is the size of the system ok. So, there is a power law here and this power law you can measure and it is something like 1.2 or some such number and nobody has been able to calculate this exponent very well in theory, but in experiment you can measure it sometimes it comes out like that sometimes some people did the experiment they found this charge and fire kind of behavior. So, they do not see a power law. So, then they said that self organized criticality does not apply, but that is too strong a statement he says that for the experiment that I did that theory does not work they did not say that it does not work anywhere ok. So, I think if you make the claim that some theory like that theory of self organized criticality is that theory of the universe and works for everything then of course, there is a problem it does not work for everything, but at least it describes some kinds of systems quite well and one should leave it at that ok. So, then he said that here is a power law x is non-trivial, but it can be measured in some experiment or you know whatever and you can see what it is, but then he said that look this is very similar to the problem of earthquakes. So, in earthquakes what happens is there are these continents they are moving against each other some 3 centimeters per year or whatever the same rate at which your nails grow ok and then this once these big rocks are coming together they build up stress and this is occurring at a very steady rate because the continents are moving slowly and steadily, but once the stress becomes too much then there is some break up and there is some relaxation event which is called earthquake too much stress cannot be held. So, some breakup occurs and some thing which is called an earthquake which lasts for a very short time ok. So, the picture of earthquake intensity as a function of time you can just put a seismograph and let it run for years and you see a plot something like that one ok. For a long time it is a quiescent period then there is some earthquake tiny one and then long quiescent period there is another earthquake and so on and so forth. And the sizes of these earthquakes have been known for a long time to be described by something called Gutenberg Richter law which says that the frequency of an earthquake frequency of earthquake of magnitude of energy release e sorry burn ok. So, the phenomena which was sort of being described as a sand pile can also be used to describe something like earthquakes and at this level of description the things are working I think that is a great gain because before this time people knew the earthquake they knew that it occurs because of stress build up, but there was no reasonable way of explaining how this Gutenberg law follows from the fact that the stress relaxation gives rise to earthquakes ok. And now maybe we do not have a detailed theory to calculate the exponent y, but at least there is some more mechanism which will explain the occurrence of these power laws in this kind of model. So, I have to explain a little bit more. So, what you said was what happens in the sand pile? He said in the sand pile there is some sand surface when you add a grain then there is a he said that in a sand pile there is a local stability condition. It says the local slope is bigger than something then it is unstable and then a relaxation occurs. This is a threshold condition if the slope is less than something nothing happens it just stays there ok. So, if the stress if there is if you add and the slope becomes bigger then some transfer will occur, but this transfer may make the slope bigger at some next site where it goes and so there is a sequence of instability events which is called toppling events and so there is a this set of events is called an avalanche in the sand case and this avalanche can have different sizes sometimes means stop after 5 or 6 topplings. And in the earthquake once you have a stress build up then some local breaking event occurs of rock, but as a result the stress is transferred to the neighboring blocks and then they may become overstressed and they may also give rise to this stress hold relaxation. And so now we have a qualitative explanation of how the earthquakes occur. I have necessarily kept the discussion at a very qualitative level because it is under important to appreciate the qualitative point before getting lost in some technical calculation of details of exponents ok. Sometimes the would get lost for the trees. So, I am trying to emphasize the overall qualitative feature that this is the phenomena we are trying to explain this is what we can do even before I calculate the exponent x in great detail. I know that there is some exponent x which we need to calculate ok. And the fact that there is a power low tail to this distribution is rather easy to see ok. So, now what I like to do is to yes sir no the answer is short answer is no in my understanding that self organized criticality is a framework for explaining such phenomena ok. If you want to explain how the neurons fire in the brain which is one of the applications people make of such theories. Then you need to know something about neurons and you know something about the brain in which they are organized. The theory of senpiles which is built on you know this some square lattice with some grains how will it would how will it ever hope to tell about neurons. No no very good what is it good for it is good for providing an overview of some well view about the things. And then if you need to calculate behavior of neurons which you are welcome to do. Then you need to learn a little bit more than just the well view you need to get your hands dirty you need to figure out what happens in this particular system. And then you can do more, but the general theory does not necessarily give you the full answer to every question you need to ask ok ok. So, now let me first define the senpile model of Bakhtang and Wiesel first I think the key point in such in making models is that they should be nice and simple and they should not be too difficult and they should be sufficiently difficult and non trivial. So, this is what Bakhtang managed to do. So, he said let us work on a lattice like this ok. There is a square lattice for the moment and it has number of grains at each side. There is a side I and there is a number of grains is called z i and I guess z i is bigger than equal to 0 ok. And then he said that let there be an instability condition instability threshold say if z is bigger than z i if z i is bigger than z c then relaxation occurs. So, what he said was that you have a the picture is like this that you have a grain and you see put you can put three grains on top, but if you put four there is too much and then things topple as simple as that. So, if z is bigger than three in our case toppling occurs and one grain to each neighbor. So, in our case z c is equal to three. So, if z c is at least four then four particles will go out to the four neighbors and the side will have fewer particles now and its height is reduced it is stable. The neighbor yeah then it may become unstable then you apply the same rule again to the neighbor yeah. So, at the boundaries at the boundaries particles mainly. So, if you topple here there are four grains three go here and the fourth one goes out that is the simple model. So, what happens in this model that is the question now. Now, it is a more precisely formulated problem previously we were working with a very broad class of models which we did not have very precise rules, but now the rules are written down already nothing else will be added. So, what you do is you start with something. So, what will be added add grains at random grains one at a time at random relax. So, you just add a grain to one of these sites at random and then if the site is if the system is unstable you relax it if relaxing this one causes instability at some place relax that one and keep on doing it until you get a new stable configuration and then you add another grain and then you relax. So, it says repeat. So, you keep on adding grains keep on relaxing and then add more grain and relax then add more grain relax what you get is called the steady state of this pile and then the claim is that in the steady state of this pile you will see that behavior already drawn in the next graph and now maybe I am in a position to calculate the distribution of event sizes in this model. So, that is what we will try to do yes I do not know what is continuous time you add a grain and then you relax and then you wait for some time or continuously adding is the issue we did not address here we added one grain and you waited long enough for the disturbance to die out and then you added another grain. So, the way I imagine it that I add one grain per minute, but the relaxations finish within half a second. So, I add a grain relaxations occur sometimes they take 2 seconds instead of half a second, but it does not bother me I am going to wait for 1 minute and then I add another grain and then I add another grain after a minute after a minute. If you increase the spacing between these to 1 minute or 1.3 minutes nothing much will happen 1.3 to 7 minutes real variables instead of you know fractions does not matter. So, one can try to generalize the model in some way you know suppose I define the model originally with discrete time you can ask what will happen if I make the time evolution continuous I think that is a good exercise that can be done, but one should do it later. So, I read long ago there is a famous statement of chairman Mao he said you should walk before you can run. So, once we have understood how to solve this simple model then we can think of possible extensions to other cases, but not before. So, in this case let us be sure that we have understood the problem I will work one by one board it is an empty pile you had a grain becomes height becomes 1 then you had a grain it becomes 2 then you had a grain it becomes 3 then you had a grain becomes 4 and topples and then the height is 0 and then 1 2 3 4 topples and then 1 2 3 4 topples. So, that is the steady state in the steady state there will be a periodic variation of toppling with time. See all of you will understand what I have said and I have not written down any equation on the board. So, 1 by 1 board is trivial 2 by 2 board. So, on a 2 by 2 board I add grains I guess what are the stable configurations you can have at each side the height can be 0 1 2 3 the number of stable configurations can be at most 4 to the power 4. So, number or stable configurations sorry number of stable configurations is equal to 4 to the power 4 and then the system will be in one of these then you add a grain it will go into some other one and I add it add at random what you get is a stochastic process with some transition rates it is a Markov process. So, we have a Markov process now this is a word which is a little bit beyond the BSc level I decided, but it is a name. So, if you know what is a Markov process you just have to give a definition which takes 3 lines the system is in some configuration there is a probability it will go to a different configuration right and then in that configuration there is a probability that it go to this one or to this one or to that one and in such a system there will be a long time steady state and we are looking at the properties of the steady state. In particular we can say that you have this long time stationary series what is the distribution of event sizes number of topplings then I will have to work with a system which is L by L and I will work out this toppling matrix and I will find the steady state I will find the distribution of event sizes and it will be a power law and I will know the power that is my aim in life ok. So, it turns out that so far people have not managed to get this far even for this simple model getting all these next steps and working out explicitly what is the steady state what is the distribution of configurations in the steady state what is the probability that exactly our things will drop out when you had a grain cannot be done in close form so far ok. So, then I become a little bit uncomfortable. So, let us give a short proof that whatever happens the answer is going to be a power law ok. So, give that proof the proof will occur in this much space and so you know you should stay awake and you will be able to follow the argument once you have decided that you want to follow it ok. So, let S be the number of topplings adding a grain in the steady state. So, S is a random variable on different times it takes different values S is a random variable. So, we are interested in finding what is the expectation value of S ok. So, so what happens I add a grain here and then it just sits there does not do anything then I add another grain it comes here and then I add another grain it comes here and it topples and but it topples this one and this grain also so if you take any particular grain and you look at its trajectory in time sometimes it sitting at some place then it moves to some site then it sits there for some time then it moves then stays for some time and it moves and so on. So, every grain has some trajectory on the system and eventually it leaves it is added at some time it moves around and it leaves at some time. So, how many steps does it take to leave let us first ask this question ok. So, I can do this I do not know the number of steps it leaves is greater than equal to constant times L because you know it can be added at a distance anywhere from the boundary it has to leave at the boundary in order to reach the boundary it has to take order L steps ok. So, let us say order L. So, but now if you start with some pile and add 1 million grains one after another then the you add up all the topplings made by each grain that is that number of steps each grain travels that will be just equal to the number of total number of topplings made in this whole experiment divided by 4 sorry multiplied by 4. Thus number of topplings felt by number of steps made by grains is equal to 4 times the number of topplings made in the experiment because in each toppling 4 grains make a movement right. So, the average number of topplings per added particle is equal to the average number of steps grain average number of steps taken by a grain right and the number of steps is bigger than equal to L. So, the average number of topplings per added particle will be of order L bigger than order L it is a trivial bound at some level ok, but what we have done is we have exchanged the time average and space average. I look at a grain at a time and see how many steps it moves do not look at one avalanche and what it happens in one avalanche look at one grain and what happens to a grain and on the average it has to move L steps before it leaves each of the grains is doing this I have added 1 million grains. So, how many topplings have happened 1 million times L times some constant. So, how many topplings per added particle order L because that million cancels out that is all that is the argument. So, now this average value of S increases with L at least linearly. So, now if L goes to infinity this must go to infinity. So, what is the distribution such that the mean diverges to infinity in the thermodynamic limit in the limit of large L it has to be a power low. If hence probe S is greater than equal to 1 upon S to the power plus epsilon for S to the power 2 otherwise it would not diverge. Let us write you know there is a low problem. So, let me write x with x less than 2. So, that is all the argument is very simple and straight forward it does not invoke any of the complicated rules of that toppling of the sand pile. So, it is actually valid for a very wide class of sand pile kind of models you add something somewhere there is a toppling it goes to the neighbor when some under some rule or the other. There is some toppling rule we are not telling you in great detail then it will go to the neighbor it will get out of the system. In all such cases the mean size of avalanche will have to diverge with L and hence there will be a power low and hence the system will be critical in the sense that the power there is a power low tail to the distribution function very good. So, with a slightly more careful argument one can actually show true stronger result expectation value of S is greater than equal to constant times L square and so it is time is 1 o clock let me take 2 minutes and I will finish this argument and then we will stop. So, what it says is that now I am I am giving a slightly more careful argument. So, it says you mark a grain in my experiment and at each toppling it can go, but where will it go. So, with equal probability all the four grains go in different directions they are all different, but which one we go where is decided at random in a permutation. So, I will go first grain goes here, second goes here, third goes here, fourth goes here or 1, 2, 4, 3 and so on all the different 24 combinations occur with equal probability this does not change the original model in any way because I can assign whatever rules for which grain goes where and the model does not change. So, now with my new rules each grain when it goes from this side it goes with equal probability to one of the four sides. So, each grain does a random walk in the space of configurations when it goes to a side it stays there for some time some length of time which I do not know, but I do not care for the length of time its weights and then it moves again. Once it moves again it moves with equal probability to one of the four sides and so on. So, now my problem reduces to this one. So, I will write average value of s is greater than equal to average number of steps a random worker to reach boundary when added at random. So, I have a lattice I put in one grain and this grain does a random walk and it leaves then I add on the grain it does a random walk and leaves. The only question is how many number of steps on the average before the grain leaves and the answer is it takes L squared steps order L squared because that L would have been if it when took a straight path which it usually does not manage to do. So, the number of steps it actually takes is order L squared this number can be exactly calculated if I am a little bit familiar with the theory of random walks ok. So, this I will not calculate this will be your first homework assignment basically I think it is a non trivial problem and you can look up some place or the other to get the answer internet is a great help no no no. So, what you should do is to just read up and either work out yourself or read up from some place what is the proof of this result that the average number convince yourself one way or the other that the average number of steps of worker takes is order L squared. If you work in a system of size L and you added random everywhere ok and this answer holds also in d dimensions where d is true nothing changes everything all the arguments given were robust and they go through with this change ok. So, we will stop here now. So, we will meet again at half past two ok. So, can we solve the one dimensional problem? Yes. Yes. Thank you.