 The reason this one is one slides is because I wanted to show some pictures. In any case before we start it will be useful to ask if you have any questions on the whatever previous material there are no questions. So, let us start with this. So, in the last several lectures I have introduced a fairly large number of the order of 10 or so different models of self organized criticality. And all of these are simple models and these are supposed to be models of these complex systems after all this school is about complex systems. And so you know the title this is the title and it sounds a little bit of an oxymoron or self contradiction or whatever you like. And that is a standard objection raised against such models it says these models are too simple. And that is sort of supposed to be a dismissive comment about them. And the first part of this presentation is sort of a justification for these approach to study these models. It needs a little bit of justification because the people who complain are not stupid people they are actually professors in some place or the other or some such thing. And so you have to take the objection seriously you have to understand it and you have to also realize that the rejoinder is also a serious argument it is not a random comment you know you just say no no no this is like this. You think about it yourself you might be even convinced that these arguments are actually right. You have to decide for yourself some people will think no no no these are not simple enough or some people will say these are too simple they are not useful enough or they are useful. And the judgment can vary a little bit about this you know but anyway let me go on it works anywhere ok. So, this is sort of the general preamble it says that the phenomena we study in physics is very large distances are from 10 to the power minus 35 meters to 10 to the power 20 meters 27 meters and time scales go from 10 to the power minus 43 to billions of years and that is a huge range. And lot of people study these problems at the ends of the range. So, they study either the some things which are very very small you know at plank length or they study things are very very big which is the size of the universe. In particular amongst Indian students there is a very great fascination with studying either the things which are very small or very big and everything else it does not sound to be sufficiently interesting ok. So, you know astronomy or cosmology or particle physics these are the only subjects they want to study. And this one is a sort of advertisement for the subject and so I am justifying the fact that there is interest outside the ultra big and ultra small ok. So, there are many interesting not understood phenomena at human scales the kinds of questions you encounter in everyday life one should look at them one should be surprised at them one should try to understand why they happen and not only try to understand why something at plank length scale happens you know I do not even know how it happens. This is sort of not exactly the point, but you know I actually remember it very well because Abdul Shalam when he had visited Tata institute he once told us this story. So, he said you know he was from Jhang which is sort of near Lahore small city near Lahore and he said there was a professor of a college in Lahore and he was telling students you know there are all these fundamental forces all of us know about gravity. Now, there is also electricity in Lahore and the weak and strong interactions only occur in CERN in Europe. Now, so it is sort of of course you know Shalam knows that the things occur everywhere, but it is a question of what you find interesting and what is relevant at any given time and so many well let us go on. So, my aim here is to illustrate how simple models can help understand complex phenomena. You are welcome to interrupt at any time. So, this is a complex phenomena crack patterns in mud cuts region and photo is taken from the I think the resolution is not great, but I should have been more careful in you know transcribing one file from another place to another place, but anyway you get the picture. This is the catchment area of Ganga river and major tributaries and you see this river networks and they have interesting structure and can we understand that we have discussed it a little bit, but this is sort of more colloquial presentation. So, catchment areas of river basins can we understand the why the river network looks the way it does these are sand dunes. So, this is a quotation from Feynman lectures in physics volume 1 lecture 1. So, if there is a long preamble he says that if I were to ask I think all of you have seen this one, but I am reminding you he tells the story of you are supposed to pick out one sentence from the whole of physics and encapsulate the basic understanding which is the only thing which will be conveyed to future generations then what is it that is the equation and his answer is this one it says all things are made of atoms little particles that move around in perpetual motion. Please note that he does not say h psi equal to e psi he has not picked the general relativity equation either you know or mu nu equal to mu nu or whatever. So, I showed this in TIFR and some of my particle physics friends were incensed, but since it was said by Feynman they had nothing to say they were quiet, but they said we do not agree we do not agree with Feynman you are welcome to disagree, but you know he was sort of not so unwise a person and he thought about it then this is what he came up with all right. So, from the atoms you can make the bigger structures molecules that is like chemistry macromolecules like DNA living cells animals societies of animals and these are structures at bigger scale and people study all these things. So, but when they study these different subjects they usually study in different departments. So, you know the atoms are perhaps studied in physics departments molecules in chemistry you know there is chemistry and then there is biology and psychology is the mental processes in individuals and sociology even you lot of people get together that is the subject of sociology and ecology is interacting lots of many different societies different species of animals and at each level of organization new principles emerge which are you would not even get an idea that they are there if you are just studying atoms you have no meaning the word life has no meaning in itself because atoms are just you know like that there is nothing much to discuss, but life has lot of interesting properties and people like to understand it and it is different then there is this thing called love and it is not I am not making a joke I am making a point the point is this I guess it is very hard to imagine bacteria as loving each other there is no notion of love in bacteria, but if you go to animals like birds then you can kind of agree that birds love their mates you know they form bonds they stay with their mates for ever for their lifetime and if one of them is separated the other one seems to be somewhat sad from behavior you can see the behavior. So, you can see that love does emerge in animals, but not in bacteria it is a higher function which emerges only at a later stage of evolution ok and then so if you look at parrots or some birds like this they perhaps they love, but then they do not have a notion of culture they do not write papers in PRL for sure ok no so this culture phenomena which their culture is the same you know they behave in some way, but the behavior is the same now as it was two thousand years ago maybe ten thousand years ago and so not too much cultural evolution occurs even in these animals which are somewhat sophisticated forms of animals. So, if you are studying cultural phenomena you have to go beyond biology you will not be able if you think of all animals or you think of human being just as animals it will not you will not be able to make much progress towards the cultural evolution you need totally different ideas and the words which people discuss these things in terms of are new words which are not used in biology much in rest of biology much ok. So, these things are called emergent properties all these life love and culture these are all emergent properties ok. So, now we want to understand these emergent properties and how do they come out that is called self organization because nobody is putting them in they somehow come out by themselves how did the birds get into being from bacteria I am not very sure somehow it happened ok. So, that is called self organization ok and we want to understand this emergence in self organization as a sort of generic phenomena and how to do it. So, it is not immediately clear that if you want to understand cultural phenomena you will be able to do it from sand pile models, but I am saying in general one expects that statistical physics deals with systems with many degrees of freedom which interact with each other and. So, if we have some experience with understanding how interacting systems is interact with each other behave then maybe we have some edge over other people who do not have that experience inside you know they have not dealt with these topics before ok. So, you may or you may not like the sand pile model, but this fact that statistical physics is hopefully useful in understanding sand should not be so much under dispute ok. Then if you do not like this particular stuff you can always modify and study something else you know modify the model make a change here make a change there ok. So, those are details. So, it says may help in understanding this of course, somebody will say that this is just a hope you have no proof that this actually happens ok. In fact, you cannot give me a good example where this happens and. So, I gave some examples and they say this is not good enough this is not good enough ok. So, this again is a contentious issue you can think about it and you may come up with an answer which is not always yes, but at least my belief is that we should try I think we may it may help other people have tried other things some of them work some of them did not work, but you know I am trying at the very least ok very good. So, this is the importance of simple models. So, I start with this quote from Einstein it says make things as simple as possible, but not simpler I think I mentioned it earlier that I do not think I agree with this point. Because if you have a complex system you are not able to keep it as simple as you cannot you have to simplify extra otherwise you will not make any progress. So, there is a definition of what is a complex system. So, it says this is my definition I cooked up it says cannot be described in terms of few variables. If you need to describe it in terms of 10 variables I think it is complex because my head kind of spins and I cannot really understand 10 variables at a time ok, but 2 or 3 variables I think I can understand. So, if you take a box of gas in equilibrium it can be specified in terms of 2 or 3 numbers this is a nitrogen gas at room temperature pressure the volume is so much you tell me these many three things composition and the pressure and temperature and then I am more or less I know everything about it is fully specified the behavior of the system is fairly well understood or specified at least I do not know about understood, but at least for specification of the system I only need to give you 2 variables. So, it is simple, but if you take the same box of gas and stir it up a little bit and I want to describe the same box of gas then I have to describe it using the here the velocity is this much here the velocity is this much here the velocity is this much in this kind of description and that is much harder and requires many more variables you have to give velocity as a field density as a field temperature as a field everywhere ok, then the system becomes complex ok. So, this so then it so we are going to try to discuss complex systems, but we want to deliberately ignore many details of the system and simplify further and this one I put in the red it says ignoring unimportant details is the core of understanding ok. So, I would like to explain it from a story I was trying to tell one friend of mine he was a film director and I was telling oh yeah we are studying models of rivers and he said how can you make a model of a river every river is different from every other Mahanadi river is different from Bramputra. So, I said yeah you know he grew up near Mahanadi and he was very fond of the river in some way he has emotional attachment to the river, but I was saying oh, but it is a river it is just a river. So, so suppose you are a medical doctor and a patient comes to you I have a stomach ache. So, you say oh ok what did you eat yesterday then you take these two pills go home you will be ok you do not need to know what color of shirt he was wearing yesterday did he have a fight with his wife or some such thing all these things are there they perhaps even affect the digestion system sometimes in some way right it is possible you know if you have anxiety it causes stomach problems it is known, but at the first approximation I do not have to worry about all this most medical doctors will not ask about how many fights did you have with your wife no and just to treat a stomach ache right. So, we are taking that position and the fact that the medical doctor knows that I should ask what did you eat yesterday and not ask what color of shirt you are wearing yesterday or even bother about the color of shirt you are wearing today does your name start with an S or with an O or whatever this is a understanding of the problem the problem is a stomach ache and the name has nothing to do with it right. So, I do not ask about the name. So, those details I am ignoring and that shows some core of understanding. So, we are trying to keep some basic phenomena which are core to understanding and then I try to answer the question at hand of course, you know if I ignore a crucial detail if I only ask that did you visit the temple yesterday or not and if you did not visit then that is the reason for the stomach ache may be there will be a problem. So, if you do not identify the core issues correctly then of course, you are not making a right model you are not describing it right, but if you are ignoring other things which we call irrelevant then maybe we are doing it right. So, I do not make any apologies for ignoring several details in the kinds of things we discuss ok. So, this is illustrated with some examples. So, this is fragmentation process I think we did not discuss it very vaguely, but I thought I will do it in some more detail here. So, fragmentation is a very important engineering phenomena from because lot of processes which occur in industry like mining you have to you know pull out lot of stuff you have to break lot of rock milling involve breakage of large material of matter rock and into smaller pieces and lot of energy spent in doing this and if you can sort of reduce the amount of energy spent then you are making great help in our industrial processes. So, in mining we want to actually break sometimes we do not want to break you know like airplanes we do not the wings to come off when I am travelling at least right. So, strength of materials cement you do not want the buildings to fall or cement to develop cracks and you have to understand the fragmentation process how it occurs in order to avoid it occurring. Then there are these things like earthquakes where this I do not hope to change the earth usually you know in the cement I hope to make a new cement where the thing will break less often. In the case of earthquakes I do not really have some such hope I am not going to engineer the new earth which does not have quakes. The only thing I can do is to sort of mitigate the effect of predict the earthquake or some such thing. Then there are these things environmental studies you have to have study the erosion of rocks polar ice melting and again here you have to these things happen that is known we are not even trying to control them just now, but we are hoping to predict how much of it will occur in the next 10 years or some such thing. So, even trying to estimate the rate at which these processes occur is all that I hope to get sometimes in some of these complex problems. So, in particular in fragmentation process there has been a lot of work and usually the work involves in saying something like this that there is a big rock it is undergoes some protocol of breaking you hit it at various places then it becomes smaller rock then you hit the smaller pieces in some way they become smaller rocks. And eventually you get a distribution of pebbles of various sizes some are very tiny dust some are very big and can we understand the distribution of sizes of fragments it specializes the problem into some sub problem, but that is the one which we are trying to understand and it is expected or hoped that this fragment size distribution may be independent of material details you know whether you are breaking a rock or sulphur or an igneous rock or something some other thing it may be the kind of distribution of sizes of fragments you get does not depend on the material you are processing and at least for a class of processing protocols not for every possible thing, but for a class of such protocols. I think all these caveats have to be remembered in a real application and it is no use saying your theory is no good because it correct does not correctly tell me this because maybe it tells some other things which are useful and it is ok it is not telling me this, but you know that is not the only question which you need ask, but you should be aware of the questions we are trying to answer if they no question we are trying to answer then you have not thought enough about the question ok. So, here this was sort of our my start in this term. So, there was a paper in geoscience about glaciers and 2014 November and they said that curving glaciers says self organized critical systems. So, I described this to you last time I think, but the picture was not there. So, this is a sheet of ice and it cracks and cracks fall into the sea and this is what it says here. So, over the next century one of the largest contributions to several rise will come from ice sheets and glaciers curving into the ocean factors controlling the rapid and non-linear variables in curving fluxes are poorly understood. Here we analyze globally distributed data said something something we find that curving events introduces by brittle fracture of glacier ice are governed by the same power load distribution as in the sand pile model. So, you know since it said the sand pile model I tried to look at the paper a little bit closer and so, this is a inspired by the work of extra metal and breaking up of glacier ice. So, we take a square and so, this is a like this it is a square grid of so, it is discretized space and cracks are initiated on the left end and move right with constant velocity random transverse displacement and finite splitting polarity. So, crack starts on one end and it propagates to the right, but as it propagates to the right it also diffuses in the transverse direction sometimes it goes up and sometimes it goes down in our case we just assume it goes up and down with equal probability. And sometimes it can split one crack can become two cracks with some probability which right now we will see is independent of the position and then if two cracks come together they just merge and they go on like this. And so, there is a single parameter lambda here which is the split probability which determines the evolution yes. So, if the two cracks come they merge and move as a single crack this model was studied by other people before as a model of aggregation desegregation. So, what people had studied in the past was actually exactly this model, but they never produced this picture what they were doing was studying a one dimensional line in which there are particles which move around like the Takayasu model of sorts particle can move around they can diffuse and they can merge with some rates specified, but it was thought of as a one dimensional process evolving in time. So, the kinds of things they looked at was only the mass distribution of sizes or some such thing, but they did not look at the shape of cracks because the shape of cracks is a this is in the time history that is a non not a reasonable not an obvious variable in the problem where you think of time evolution of a some one dimensional model of aggregation ok. So, we wanted to look it as a process which produces. So, this cracks form and they develop and they go away and the wake behind they leave behind a set of fragments and we are looking at the distribution of sizes of these fragments and this is the picture of a simulation result with some rates and you know some details are there and you see this is the kind of structure which you get in this model. This comes in the steady state and even if you start with one crack it will break into two cracks they will separate from each other with time if then the two will become four and so on. So, eventually you get a unique steady state if you start with too many cracks the number of they will merge and they will get to a same density. So, the steady state is unique it depends only on lambda and then you can ask what is the steady state distribution of these cracks in the steady state and these are the pictures from real glaciers. So, you see there are crack patterns these cracks it. So, the model may not be perfect you know, but it is catching some basic phenomena which is there it is not fully unreasonable no, no, no even has to be sure of the clean made and so this is what it is and it was not easy to get these pictures because we do not get these journals in physics libraries usually, but anyway managed to get them just lambda which are the split probability. The steady state distribution of fragment sizes this is a little bit of a technical detail the rates of breaking and merging there is a rate to break and the rate to merge is one, but the ratio defines some chemical potential and there is a detailed balance corresponding to non interacting Hamiltonian. H steady state is equal to minus mu summation n i, n i is the occupation number variable at bond i you have to check this point, but once you check this point then it is clear what is the steady state distribution it is an independent occupation of different sites, but again the independent distribution of different sites will sound like a rather trivial distribution, but buried inside it is somewhere the distribution of fragment sizes which is less obvious. So, this is the statement here each fragment is a region bounded by two bias directed walks and these two so maybe I should go back to that picture again two bias random walks. So, this walk goes like this, this walk goes like this when the two sides merge then that is the end of the segment. The reason the walks have now become biased is because if you look at the left end then it slightly to move to the right if I look at a fragment and I look at the left end whenever a split occurs then I am moving to the right because I keep the left end of my fragment which is the right end of the walker. So, I move to the right more often to the left. So, these become biased walkers. So, many results are known. So, the scaling distribution for these kinds of stuff is sorry. So, this is called staircase polygons area under a brownian bridge and so on has been yes, but there is a probability to break into two. When you break into two I keep the right end which makes it biased. So, then you get a distribution like that and probability goes like you can show this result I will not go into details. There is a function g that is called some area function and it is a special function but has been worked out by area which also is interesting because area you know what did we have to do with either sand piles or with glaciers. No, it comes up in some diffusion problem he was doing some optics problem and he come up with this area distribution in this optics problem. And so, you know the mathematics is the same you get the same area distribution even in this new problem. I do not know how many of you have heard the name airing everybody has heard in some book or the other name occurs. I say he was studying diffraction stuff. So, segment this summary of fragmentation modeling we discussed a simple model of fragmentation of propagating branching merging cracks. Here the relative frequencies of different fragments ships can be exactly determined and there is some qualitative agreement is observations of glaciers. So, I was you know I was trying to argue with this know. So, I was asking these people who make pictures of glaciers they said no no no you know it is very hard to estimate the size of a glacier because glacier has a these cracks which you can see from top from satellites you can take photographs that is easy, but they also have a depth which is much harder to monitor. So, what matters in the end is you know whether it the volume of the fragment not the area of the fragment and the volume is a much more complicated variable. So, we cannot really predict very well which is true. So, you know you realize there was a model there was some interesting stuff this model has some interesting properties which let me say here. So, I will ask that there is this shape what is the probability that this particular shape occurs somewhere in my fragmented sheet I can determine this number and then there is this other shape which also occurs with some probability which I can determine again, but then I can ask that what is the probability that this occurs and this also occurs. So, it turns out very interestingly. So, I can say that something here and something here starting here this cluster is of type C this cluster is of type C prime is there a correlation between these two events of the shapes are the shapes of the cracks correlated and interestingly the answer is that the shapes are totally uncorrelated unless they are touching which is very strange because of course, in the actual problem there is a huge correlation they share a common boundary those shapes are not independent, but the shapes are nearly independent unless they are actually touching there is a proof of course, which we will not give just now you can look up the paper. So, this is the second problem I want to discuss river networks. So, rivers have been credals of human civilization and very important determinant of our environment. Nowadays there is lot of pollution and that causes lot of problems to people even now. So, of course, there are more spectacular rivers this is the Amazon river with all its tributaries and the colors are just distance from the last endpoint. So, this is a nice picture and that was in previous picture was also nice, but is there some common features in these rivers. Can we predict the structure of a network only the geological structure if I give you know if I do not tell you anything then of course, it is very hard to predict the structure of the river network, but suppose I tell you I know South America there is this kind of rock here, there is this kind of rock here, there is this kind of elevation here, this kind of elevation here then can I predict the river network. The answer is that this is also not very good because the network evolves in time and rivers actually are known to evolve substantially within 1000 years timescale while the geological processes are perhaps little bit on a longer timescale. So, the geological structure does not uniquely determine the river network. So, we modify the question are there some general properties of such networks independent of details of geography same for different river basins. And so, this is some phenomenology rivers have been studied by geologist and hydrologist for a long time they are also called there are also other drainage supply networks blood circulation network in animals which are similar in structure. And most of the maps show only major rivers or the big rivers for a hydrologist rivers are space filling trees because from every point there has to be an outward flow. And network may consist of thousands of rivers. So, this is the stellar ranking of rivers in a network this was also designed by hydrologist starting river has rank 1 when 2 rivers of same rank join the resulting stream has rank r plus 1. If a river joins to a river of lower rank then it rank does not change, but if 2 rivers of same rank join then the joined river has a rank which is 1 higher that is the definition. So, here all these n leaf nodes have rank 1 these 2 if join 1 to form river of rank 2 2 joins to 1 it is still 2, but these here these 2 and 2 join that becomes 3 and 3 joins to 2 remains 3 3 3, but it goes on in this case the lowest river is only rank 3. So, there are filmological laws which is Horton's law number of streams of rank r divided by number of streams of rank r plus 1 is nearly a constant. And Hex law it says length of longest river upstream river is area of catchment to the power a, but a is not 0.5 this is length this is area you might expect that this power is half, but it is not half it is 0.6 is clearly different from half. And can we understand these points can we derive these laws from more basic physical principles that is the kind of question we like to ask. And so, this is the Scheidegger model which I have already described actually, but let me repeat because there is no harm. It says discretize square grid uniformly sloping landscape uniform annual rainfall no loss of water by evaporation seepage all rain water goes to the sea. At each grid point the direction of the outflow is down left or down right with equal probability. So, flowing rivers flowing water from small rivulet that joined to form bigger streams and bigger streams and we ignore the bifurcation of rivers. So, you know the from each point there is only one exit direction and this is a typical picture. So, Scheidegger river network is a directed random spanning tree on a lattice. And the catchment basins of segments are staircase polygons back to the same issue many average properties of large networks are easy to calculate. So, one can show that if you pick a site at random fractional number of sites with outflow f goes like f to the power minus 4 by 3 for large l fractional number of sites with longest upstream path of length l goes like this power. So, you go to a site is a what is the probability that if I keep on going longest upstream how far can I go and that is the power law with that power and can we understand these laws and hope in the all these laws previously discussed are obeyed with a equal to 2 by 3 in this model. So, summary of this discussion simple model of river networks is the Scheidegger model in this model catchment basins are staircase polygons and we can determine the relative frequencies of different shapes. And then this is the model may not be very realistic, but does help understand the origin of the observed regularities ok. So, when the initially the hydrologist pose the model there was a big mystery why is there this hex law why is there this photons law we have no clue at least I think at that time they did not have a very clear idea why this happens, but once you have a model like this then you understand the origin of the law in somewhat more basic level. And if it turns out that you know the real network is not exactly like this one Scheidegger. So, then I just say that oh well you know the in the real network like in the Ganga rivers they slope down then it goes like that then it goes like that may be the flow down is not everywhere in the same direction. So, I will make a more detailed model in which the flow there will be a more complicated profile of the slope of the land and that will give me a better picture or better agreement with observed data. So, right now that is the end of this one sorry I do not know how to operate this one then oh there is a mouse and then then we go to this one and full screen F 5 this did not work control F 5 ah control L ok. Thank you ok. So, this is a different problem which I thought I will discuss. So, it is modelling proportionate growth. So, this work is done with large number of people and I am only citing the main collaborators starts in 2002 which is long time ago. This was a student from Lausanne who spent 6 months in TFR then S.P. Singha and M.Chandrath, Vipsadu, Rahul Nandekar ok. So, why is this problem in what is the problem and why is it interesting this is what I will explain to you. So, the my interest in this problem comes from different perspectives there is some biological interest in the problem because it describes growth of animals there is also some physics interest which is pattern formation you know. So, you study different kinds of patterns and why do some patterns develop and there is some interesting mathematics which occurs discrete harmonic functions in tropical polynomials and I will try to give a general idea of this, but may be not great discussion. So, growth in biological systems different body parts in animals grow at roughly the same rate this is well known phenomena shown in this picture and I would say that this is very surprising and interesting and unexpected and can we understand why how this happens. Proportionate growth is typical in animal kingdom it is less obvious in plants and trees by looking at a trees height the small trees do not look the same as big they substantial differences by looking at the tree you can tell how tall it is looking at a picture of a tree sometimes quite often you can tell the how big it is ok. But the reason this problem is interesting for me is that this problem is much easier than the development of an animal from a single cell there has been a lot of study of development you know there is a whole subject called development biology, but the problem they study is precisely this you know how does a single egg become a chicken and that involves very complicated stuff you know you have to it is called morphogenesis which is generation of shapes and some part there is a spherical shell which has to become some part has to become the head some part has to become the tail and then feathers have to come and all these kinds of stuff is very complicated and it is not well understood whatever the biology is clean. So, I would say that let us consider a simpler problem you have a baby chicken all the body organs are formed and they just make a bigger chicken ok how does this happen ok and so let me explain the point the point is this. So, I have a hand there are these fingers baby has a hand which is smaller and they grow how come this finger grows at the same rate as this finger because they are not talking to each other. And as the system evolves the different parts grow independently then usually you expect a noise to grow or become bigger with time ok, but it seems that it does not seem to even at 60 years of age the you know some cells from the end keep on dying and they are replaced by new cells and so on and so forth. But the proportional proportion of the lengths of the fingers remains constant for most people which requires some amount of control something else is controlling the fact that the fingers remain of the same size it cannot be totally independent growth that control is hard to visualize clearly that is what we are kind of looking at how come what decides that these things grow at the same rate ok. So, also so the more we developed some models, but all these models are they have been lot of models of growth in which have been studied in physics in the past, but these models are different from all of them and so we wanted to understand them a bit better. So, this I point I have already explained proportionate growth requires regulation and communication between different parts otherwise you will not be able to ensure nearly proportionate growth ok. There is also another interesting point which is very surprising it says same food becomes different tissues in different parts of the body you eat some food it is broken up into some components all those chemical proteins whatever they go to the skin they become skin cells they go into the liver they become liver cells. So, same food depending on where it goes it becomes different cells can I make a model which does this and the standard biological answer they if you talk to biologist they say we understand growth there is nothing to understand they say that it occurs because of different chemical agents which achieve this. These are called growth factors inhibitors hormones there is a hormone growth factor called G 18 which determines the growth of the cells and there is an inhibitor and there is a hormone called this which controls growth that once they say this then the whole problem is explained there is nothing to explain further and that is like if you go to a murder mystery and you say the knife did it it is true the knife did it you know we are not denying the fact that the knife did it. But there is more we are looking for some structure beyond just the chemical agent which is doing the work, but the biological answer usually only worries about the agent chemical agent once they identify the agent their problem is solved they do not look further. So, this is the question can we find a simpler physical mathematical model that achieves this function this three parts of the function or two parts and ignore chemical detail we do not worry about all the different proteins and the DNAs which are there which actually do all this stuff. So, this is growth phenomena in physics these are different pictures of growth processes which have been studied in the past. So, this one is a copper leaves in zinc. So, you have some zinc copper sulphate solution and you put in an electrode and pass current then the copper goes to the electrode and it forms these kinds of leaf like structures. So, this is called diffusion limited aggregation and there is lot of theory which deals with diffusion limited aggregation it is well studied for last 30 years. This is the different phenomena it is Epsom salt crystals grown from solution. So, you just have solution put in some Epsom salt cool it down then crystals grow and then you break the top and take them out and this is the kind of structures you see. This is the different growth process which also has been studied you know this is growth of snowflakes is sort of similar phenomena and this last one is called an invasion percolation cluster. So, you have some porous rock you push in steam and see where it goes and it goes into some shape like this and the geometry of these structures is also studied a lot. So, I looked quite hard honestly and I did not come up with many processes which were different all of these in some qualitative sense ok. So, in all these cases what happens is that the inner parts do not grow further all the growth occurs on the outside it just keeps on growing bigger and bigger and the proportionate growth is qualitatively different because when the percent becomes bigger the heart also has to become bigger by the semium fraction ok. So, none of these previously studies models is doing that and that is what we want to achieve ok. So, then we wanted so this was sort of one this was the part dealing with growth you know. So, proportionate growth is different from all these other growth that the keep that the bottom line. So, now this is spatio temporal patterns. So, understanding pattern formation is very important you know they are these turbulent structures in fluids no flakes continental drift they produce patterns some of which are very cute and you want to understand how they happen and some such thing. This is the large scale structure of the universe through the cosmic microwave background you see lot of interesting structure. This is Rayleigh-Bernard convection cells you have liquid heated from below and it produces convection cells which are roughly hexagonal in shape. Here they put in small bit of very tiny aluminum particles which go round, but when they go round they can you can see them and this is a long exposure. So, you know you can see the tracks of particles. So, the white color kind of tells you how the particles move in the convection cell and these are drying mud patterns I showed you earlier also you know it is a fragmentation pattern ok. Most of these are not very tractable analytically you know I can produce nice pictures, but can I produce a theory to produce these pictures that is the question ok. Yes, it is because of something we do not understand very well yeah that much is true. Suppose it was non Markovian with something, but it will still be not it will the insides will not grow only growth will only occur at the surface yeah it does not forget then what I still do not get the kind of patterns I am looking for I was trying to explain some physical problem I started with and putting in non Markovian character or Markovian character is not enough it may be needed, but it is not enough DLA like starting point will not work whatever you do because it will not produce proportionate growth it will not produce growth from inside it will only produce growth on the outside yeah it has been studied internal and external oh this one oh this one we have not come to the previous one this ones oh the DLA can have lot of variations I do not worry about that because for people have worked on it for 30 years I am sure there are several variations by now, but none of them produce it proportionate growth that much I am sure of it cannot because it is just the structure of the model is growth occurs on the outside I do not worry about the extra details ok you can make the worker non Markovian you can make them jump around from here to there it will not produce proportionate growth still whatever you do ok. So, the fact that simple dynamical rules can generate complex patterns is very well known it is not a new reason phenomena we are discovering for the first time. So, it is well known in the game of life you can have this simple rule there is a one dimensional line x i take value 0 and 1 at the next instant of time you take the two neighbors add them or two whatever you get is your number and you just keep on evolving by this deterministic rule the rule has been written down in one line then if you start with this initial condition with everything 0 with 1 1 which I denoted by a star then you produce a picture which looks somewhat complicated it actually produces a sierpinski gasket you can check or you can prove and you know. So, it produces a somewhat more complex pattern from us very simple word and so, for example, there is also this thing called Mendel growth set which people have seen no doubt it is also produced by simple equation z prime equal to z squared plus c and you see all kinds of complicated structures. So, however, in many of these cases you can produce complex structures, but you cannot describe them well ok. So, for Mendel growth set I only see pictures I think any more than pictures one has not seen much I think there is a proof somewhere that the fractal dimension is two or some such thing which I do not know whatever it means it does not tell me much about the shape of the Mendel growth set ok. So, we will describe to you in the remaining 45 minutes no 30 minutes 30 minutes a simple cellular automaton model which shows proportionate growth it shows complex and beautiful patterns it is exactly analytically tractable and we will even study what happens if there is a little bit of noise added in the problem. So, the model is very straight forward it is the sand pile model you take the sand pile start with everything with height 2 and add particle at the origin then there is some toppling topple then add again at the origin this toppling and the keep on toppling and after you have added 10 to the power 4 particles at the origin you look at what you have got and this is a colour plot of what you get by using 0 1 to 3 are denoted by red blue green yellow you get a picture then I add 200000 particles you get this picture and 400000 particles you get this picture and so it is easy to see that as you increase the number of particles added the pattern grows and it grows proportionately that is the visual proof of the result ok. You can start with other beginning conditions so if you start with everything 0 then you get this picture you can take some other background other lattices so this is something called Manhattan lattice sorry F lattice alternate columns occupied 1 0 1 0 stripe like initial condition and then you produce this kind of a picture. So, by varying the initial conditions varying the lattice you get different pictures different models but all of them seem proportionate grow ok. So, that is something we would like to understand ok yes yes of course yes no this was not recurrent you can start with everything 0 it does this. No in this case we are actually studying on an infinite lattice on an infinite lattice outside the region affected there are lot of transient the forbidden configurations so the notion of transient in recurrence is not useful because we never reach any recurrence all of these configurations are transient once they once you add more particles they disappear they do not come up again ok. So, model what is the model we use only two basic facts from biology one is that food is required for growth if you do not get food you do not grow and the second is that the cell division is a threshold process if there is a cell you give it some food it will not immediately break until it has enough size it will not break into daughter cells ok. So, we use just this threshold and then we in our model we say that once the nutrient at the site becomes at least 4 units then the cell breaks into 4 parts and then it goes to 4 neighbors that is the model ok. So, this is reduced to the well studied centile model this is repetition for you now non-negative integers at z i at site i there is a rule there is a relaxation rule rule for forming pattern is that add n particles at one site on a periodic background and relax. So, these are deterministic patterns there is no randomness in this next time you do this experiment you will get the same picture what we are looking for is this proportionate growth and characterizing the proportionate growth ok. So, this is centile model topling rules we can get rid of this one. So, key observations is that the diameter of the pattern you get will vary as root n that is clear because there was some background I kept on adding particles outside they have not changed all the particles inside have to be in the region which is affected by the topling, but the maximum height inside is fixed to be 4. So, the area affected should increase linearly with n and so the diameter should increase as root n the proportionate growth is observed, but it is not so easy to explain and then there is this extra result which was very surprising for us initially when we saw it. So, we looked at this picture and you look here and you look here and you look here if you look here and zoom up this region you find perfectly periodic structure ok and then when you grow to a bigger size the perfectly periodic structure just becomes bigger in size ok. So, the perfectly periodic structure in each way this is what we call a patch and the big picture looks like made up of large number of patches which are sewn together to form a guilt I guess that word is familiar to all of you ok. So, examples of periodic patterns in patches in some region you get this picture you zoom up it will be very big if you make a big lattice in some region you will get this picture perfectly periodic structures in some region you get this is and they are boundary of one region will be this picture adjacent to this picture with some boundary separating them and so on ok. So, so simpler so we wanted to characterize this pattern, but we found that the square lattice pattern is not so easy to characterize. So, we went to something called the F lattice which is very similar to the square lattice at each side you have 2 arrows in 2 arrows out and the rule is that when the height becomes more than 2 2 particles leave in the direction of outgoing arrows it is a directed sand pile as opposed to undirected sand pile and then you get this picture which is a bit easier to understand what is so easy about this firstly all the boundaries are straight lines and there are only 2 types of patches periodic patches one patch is where everything is height 1 and there is a second type of patch which is a alternate red and yellow which looks orange here ok. So, there are 2 types of patches that they are arranged in this crazy way and can we understand this one there are thin lines yeah these are called defect lines and they are there and the size of these lines is kind of small, but they are important as you grow the thing they move and that is what leads to growth that is a feature which we are not addressing just now we will ignore this transient lines in this first description. So, some characterization may be extended to other backgrounds exactly same pattern found is. So, this was a different lattice I showed you the Manhattan sorry I showed you the F lattice where at each side you had 2 arrows into a arrows out, but we constructed a different lattice called the Manhattan lattice where there is a whole row thing going up whole row thing going down up down up down and then left right left right left right it is a different set of arrows than the original one, but you grow it on this new lattice you get the same picture. So, even using different lattices you get the same pattern. So, the pattern does not depend on the lattice so much of course, it depends on the lattice, but not so much ok and so ok. So, this is the Manhattan lattice with density half this is with density 5 by 8 and they produce same picture and then we. So, we were showing these pictures to biology friends and they said you have these models, but you know this is not at all biological because in biology there is noise and your model has no noise deterministic cellular automaton models who cares we are not interested. So, we said we will put in noise. So, what we did was we put in 1 percent this pattern is grown on 0 1 0 1 checkerboard background, but we take 1 percent of the sites and they change their height from 1 to 0. So, it is on a random background noisy background and you still get the same picture overall structure of the picture remains the same you can see the details some complications and then you make 10 percent noise then the picture looks like that it has changed the pattern is not identical to the you know these two pictures are not identical, but at least the gross features remain the same there are 8 petals on the outside and then there is a new next layer of 16 petals and so on and so forth right. So, some robustness in the pattern is there even if the details of the patterns are not identical. So, this is the same picture, but now we said 1 can become 0, but 0 can also become 1 with some probability yes please. No, I am not claiming that all noise will be preserved the pattern, but at least for some classes of noise the pattern is preserved the pattern does not disappear as soon as you add the littlest bit of noise there is a degree of robustness it is not perfectly robust to all perturbations you can imagine some kind of noise which will kill the pattern that is not a problem, but at least there are some kinds of noise which are which are not killing the pattern I am interested in explaining that part. The part where the noise kills the pattern I do not care no that is less interesting for me ok. So, here what we did was we added 1 to 0 and then you see a slightly different picture what is difference between this picture and this picture here you have this pattern here where there is some density here and there is a different density here and these two patches have a boundary which is fairly sharp. In the new picture the boundary is not so sharp it has become fuzzy, but still there is a density difference between the different parts of the stuff. So, the pattern is there to that degree, but it does not have sharp boundaries anymore ok. So, this yes please. So, in the previous one what we did was we changed 1 to 0, but here we also change 0 to 1 ok. So, it is a slightly different type of noise and it has a slightly different effect. So, these simple rules give rise to patterns that are unexpectedly truly true to life. So, what I was doing was I was preparing a talk. So, I went to the downstairs garden and I took out a picture of a flower this is a real picture of a real flower and then we were trying to make a model with a picture like this from my senpai models which looks like this. So, if you take a background in which the background has some different shape then you get a picture like this and the size of these leaves can be adjusted by changing the density of the background or changing the periodicity of the background and stuff like this. So, I was adjusting the length of the petals to match this one we got it these are roughly the same, but in addition you get surprisingly this funny stuff which are like these ones and you get this corolla which is like this one and I swear it we did not put it in it just comes out by itself ok. So, the model is of course, very simple, but it seems to have some properties which occur in the real problem which have been kept which gives rise to these structures ok. So, then one should understand them better that is the statement. So, we were very ambitious we said oh we can also produce directed sand piles you know I like directed sand piles. So, this is a model in which once a topling occurs it are sense of particle up and down and right, but nothing to the left. So, then it produces a picture like this we said this is called the larva pattern because it you know this is the mouth where you put in add particles the food comes from there and it grows to the right ok. And then you look at the picture oh it seems there is a tail region there is a head region there is a thorax qualitatively. So, growing sand pile patterns give simple models showing complex patterns and proportionate growth. In some cases we can exactly characterize the limiting pattern only in a limited number of cases and it has robustness to the initial background and that is what there is, but I still have 10 minutes. So, these are the references there is a article in current science which is available on the internet you just type current science and you will get it free you do not have to subscribe. And there is a model in J stat sorry there is a paper in J stat which has appeared by now and then I was just want to expose a little bit to the mathematics involved in understanding this which we did not emphasize I looked at the pictures they are easier to explain than the mathematics. But there is this stuff called complex numbers and there are things called analytic functions complex numbers and they are very nice structure and you already had a whole course on this conformal field theory which deals with the analytic functions of complex numbers you know it depends on the fact that the complex analytic functions of complex numbers have a lot of structure ok. But suppose you have a problem which is defined only on integer points on a lattice then can we have any no soup the numbers are m plus i n where m and n are integers you can define a function on this set of points. But then can I use the theory of analytic functions for these functions. So, it turns out surprisingly it is not very obvious, but it is true that you can have something called discrete analytic functions the discrete analytic functions are the functions which satisfy this discrete Cauchy Riemann condition which is that you go to a site and you calculate the derivative of the function by taking a finite difference in this direction 1 2 3 1 z 1 minus z f z 1 minus f z 3 by z 1 minus z 3 the derivative of f. But if this derivative is equal to the derivative calculated from the x direction for all points then the function is said to satisfy discrete analytic discrete Cauchy Riemann conditions and it is said to be discrete analytic. So, these functions have nice properties firstly this all the functions satisfy discrete Laplace's equation, but you have all other structures like sums of discrete functions are also discrete analytic functions, but product of discrete analytic functions are not discrete analytic. So, you have to work harder, but once you have the discrete analytic function they can be used to explain the patterns once is in this problem. So, this is the explanation of that part could just written like that I do not want to explain this it says we find the coefficient of the linear term is too hard to explain I will not do it my time is limited. So, if you want to characterize the pattern then it turns out that it can be expressed in terms of this discrete analytic functions you know. So, for some other problem we found that the solution is a airy function what is an airy function is the solution of that differential equation. In our case the solution is expressed in terms of discrete analytic functions. So, it provides me because there is some instead of known about discrete analytic functions I can use the existing theory about this that is all I need to say. So, these conditions determine the pattern completely if you sort of say that my pattern is a discrete analytic function then in some sense you completely characterize the pattern now of course, you have to work hard it is not so trivial, but just like if I say that the there is conformal symmetry then I get lot of you know I say oh, but this exponent must be 1 by 8 this exponent must be 1 by 7 by 12 or whatever. Similarly, if you impose these kinds of conditions on the patterns then you can characterize them with somewhere ok. So, there is another thing called tropical mathematics 5 minutes only which I would like to explain. So, tropical mathematics is a very interesting branch of mathematics where what they said was that suppose I take a new definition I have real numbers a and b, but I define something called a plus b equal to maximum of a and b and I define a times b is equal to a plus b ok. These are some strange definitions of addition and multiplication, but if you use these definitions then the addition is commutative and this addition is distributive over multiplication and all the rules of algebra work ok. So, you can do all the mathematics with this new interpretation of plus and multiplication ok. So, that provides very interesting new structures for example, there is a famous let us see. So, standard properties of usual addition and multiplication continue to hold that if you add you know associative property you first add these numbers then you add this to that or you first add these two then add that one that will still give you the same answer. And there is a fundamental theorem of tropical algebra you can multiply numbers you can define polynomials and then there is a fundamental theorem of tropical algebra is that a tropical polynomial can be always written as a product of linear factors which is an extension of the famous fundamental theorem of algebra that every polynomial can be written as a product of factors ok. So, sometimes by realizing that oh, but I can think of the multiplication in this way that gives me new insight into the problem ok. So, anyway the mathematicians studied this tropical algebra actually lot of people in Brazil studied this in the beginning and some of the other people were making jokes about this is the kind of mathematics which people in the developing countries do that is the origin of the word tropical algebra is the kind of algebra with people in tropics study ok. Actually they were respectful and they were making a joke, but anyway this name was given by French mathematicians ok. So, in my case the statement is the piecewise linear convex function can be represented as a tropical polynomial I do not know what this means, but if you have a piecewise convex function you can write a tropical polynomial which will be the graph this piecewise constant function will be a graph of the tropical polynomial. And then if you take these growing sand piles and you draw the toppling function which is the number of topplings which occur at site x as a function of number of particles added this function in the large n limit seems to be a tropical polynomial. So, that is the reason why the tropical polynomials are presumably interesting for the study of sand piles. So, I think this is the most interesting physical application of tropical algebra because people have studied it in the past, but it was a very unmotivated exercise I think. And we are starting with the model we want to understand and this algebra just seems to be showing up which is useful for understanding and hence one studies it further. So, that is a different motivation then studying the problem from the purely mathematical angle. I think that is all there is. So, I will stop here and.