Mark Newman - The Physics of Complex Systems - 02/10/18

[Music] hello thank you very much tip for that nice introduction yep that was a picture of man we're still together we still play music what I'm not doing physics today I'm going to talk about physics okay so I'm going to talk about the physics of complex system so as Tim mentioned I'm a professor in the physics department and also in the center of the study of complex systems which is unit and organization here at University of Michigan it's actually in this building the high-rise part of this building next door and it's sort of an interdisciplinary organization that brings together people working on different areas of science to collaborate on interdisciplinary things complex systems so you might might well be wondering what is this complex systems thing that sounds like kind of a broad term could describe a lot of things a lot of things are complex systems but when we're talking about science complex systems has a technical meaning so let's talk about what that is what is a complex system complex system means first of all they're systems that are made up of many interacting parts so examples might be atoms in a solid or a liquid or against many atoms in the air in this room molecules and they're all moving around and banging into each other and interacting other examples cells in a biological organism many of or interact create a whole organism people in a population of society animals in animal populations computers on the internet cars going down the road trees in a forest companies in a market these are all examples of complex systems of many interacting parts but it's not just that it's not just that they have interaction parts it's also that in order to qualify to be a complex system something interesting has to happen when they interact right if they you have lots of different parts and all they do is you know what they would do on their own then that's not very interesting if I have a big bucket of marbles and I tip my bucket of marbles out in the ground then I have a pile of marbles right not very interesting so complex systems are systems with many drinking parts that show what we call emergent behaviors which is that their behavior is somehow that they show some new behavior that you could only get because the parts are interacting so the the whole is more than just the sum of its parts I'll give you some examples in a moment this concept was famously discussed in very influential paper in the 1970s by the Nobel laureate Phil Anderson called more is different that sort of play on words you know have some people sometimes say less is more you know if you're writing sometimes you should use fewer words and it says more if you're playing music sometimes you should use fewer notes so some people say less is more Phil Anderson says more is different if you have more parts of your system when you put them together then something different happens so let's look at some examples a classic example is a condensed matter system like a solid you have a bunch of atoms and you put them together and they form a solid so they're interacting you have many parts they're interacting and they do something new they have emergent behaviors a classic emergent behavior of a solid is solidity itself right this thing is solid right there's no sense so collections of molecules or atoms could be a solid or a liquid or a gas but those terms are completely meaningless if you only have one atom molecule right there's no sense in which one atom is a solid or a liquid of gas we can only talk about something being a solid once you have many of them right so that's an emergent property of a collection of atoms now traditionally in physics we don't think of condensed matter as being a complex system but that's really just a question of history physicists means studying condensed matter for a lot longer than we've been studying complex systems so condensed metaphysics is its own thing with a long history that goes back for decades hundreds of years really so so we think of it as being a separate thing but really scientifically speaking it is what we call the complex system that is a great example of that many interacting parts giving rise to interesting new behaviors when you put them together a fluid like a gas or a liquid is another example fluid flow right the water flowing in a stream or something like that something you only have once you have many interacting molecules the navier-stokes equations that describe fluid flow are fundamentally a result of having many interacting molecules an ecosystem right an ecosystem is a good example of a complex system it's many interacting organisms an ecosystem and an example of an emergent behavior an ecosystem is evolution right so evolution as a result of natural selection is fundamentally something that happens when you have many competing organisms if you're going to have survival of the fittest in order to be the fittest there has to be somebody else to be fitter than if you only have one organism then there's no sense in which you're the fittest and then evolution simply doesn't happen it only happens because you're trying to out-compete the other organisms around you so evolution is fundamentally a thing that only occurs because you have interacting parts a market like a stock market is many interacting traders buying and selling from each other an example of an emergent behavior there is price setting in the market how do prices get set in a market well somebody off something for sale and somebody else says okay I'm willing to buy it at that price if nobody's willing to buy it at that price then you better reduce your price until you find the price where somebody is willing to buy it from you right so the setting of prices is fundamentally some of a negotiation between the buyers and sellers if you only had one person who would they buy and sell from right that only be one person you wouldn't have any prices there would be no meaning for prices traffic on roads cars on roads that's a good example of a complex system there are many cars on the road and they're interacting with each other if the one in front of you goes slower you have to go slower or to and so forth so an example of an emergent behavior in traffic would be a traffic jam right you can't have a traffic jam with one car doesn't work you gotta have more than one car in order to have a traffic jam so it's fundamentally something which only exists when you have many parts interacting okay so you get the idea this is what complex systems are in the technical sense systems many interacting paths that show these emergent behaviors so as a physicist the way we study these things is well there's various ways you can study them but I'm a theoretical physicist so the way I study them it's a very typical thing for physicists to do is I build models of these systems which means mathematical models or computer models and we'll see some examples in a moment so in a portent first point that we need to talk about here is that the models that we build of these complex systems there's a whole range of models that you can typically build from the simple to the complex so imagine the example of traffic on a road that we had a moment ago traffic on a road well it's a complicated thing the road is you know maybe it's windy and it probably has several lanes on it and it goes up and down over hills and maybe it's got bumps in its potholes and it has intersections with other roads and then there's a lot of cars on the road and all the cars are different you know there Fords and their Chevy's and it's probably some trucks in there and maybe there's some motorcycles and and then all the drivers on the road are different there are some drivers who are really aggressive and there are some who are more cautious so for so there's a lot of complex stuff going on in there if I as physicists we're making a model of this I would probably just ignore all of that stuff and I just say oh oh we'll just like make the road completely flat we'll make it one lane going in one direction no intersections totally straight all the cars are the same all the drivers are the same right so this is not realistic right but nonetheless we can learn a lot by making a model like this so there's a sort of spectrum of models that go from the simple to the complex what I was talking about was simple models over here on the left the simple models are unrealistic and that means in practice that they probably don't make quantitative predictions if you want to know how long it's going to take you to drive to work this model is not going to tell you that but because they're so simple we can often understand really clearly and in detail how they work which means we get a lot of understanding from studying these models at the other end of the scale so that's sort of that's that's like the spherical cow right you've heard that you know the legendary story about the mathematician who says consider a spherical cow in a vacuum in zero gravity not a realistic model of a cow but maybe it's a good starting point though complex models are more like an actual cow over here and complex models are like now I'm going to make a model of the traffic on the road that has all the potholes and has all the intersections and has the winding road it has six lanes in each direction and has Fords and Chevys and has aggressive drivers and more cautious drivers and so forth I could put all of that in the model and if I do that then I can make a model that has much more predictive power I can maybe actually use this model to tell me how long my morning commute is going to be but what it doesn't usually do is give me an awful lot of understanding of what's really going on in the system because frankly if I build a model that has everything that the real world has in it then it's just as complex as the real world I might as well just go and look at the real world the model is just copy of the real thing so it's just as hard to understand as the real system that I'm studying so both of these things are useful and there are people that do both of these things so simple models on the left here that means models that are unrealistic in a sense but give us good understanding complex models means good predictive power but don't give us a lot of understanding and there's a whole spectrum in between as well so most of what I do is over here on the left hand side as you'll see simple models but I'll talk a bit about some of these complex ones as well it's definitely people working on this one so we have this whole spectrum of these things from simple to the complex an interesting thing as a physicist working in this area something that I've had to get used to so physicists are perhaps alone in the world of science in that they have models that do both of these things simultaneously physicists physics is really remarkable in this way that we have models that are both simple and easy to understand and give accurate predictions so like Newton's laws for example which tell you how things move around Newton's laws really simple to write down just a couple of equations and yet you can predict the motions of planets absolutely perfectly you can predict where Jupiter is going to be a thousand years from now and you will be right you will be exactly right too many decimal places right so in physics we've got used to having these models which have this amazing predictive power and yet they're really simple things that you can write down in just a few lines right it's important to understand that this is not normal this is not how most of science works in basically every other branch of science now in biology and economics in the social sciences you don't get this you don't get simple models that also have great predictive power you have this choice you have to have either simple gives you understanding or complex gives you predictive power but you have to choose one or the other you have to decide where you're going to be on the scale so this is something that I've had to get used to working in this field is we no longer have what we are used to in physics these very simple models with good particular power this was famously dubbed the unreasonable effectiveness of mathematics by the great physicist Eugene Wigner saying you know it's really quite surprising that mathematical models have this amazing ability to predict stuff in physics right in no other area does this happen on the face of it we wouldn't expect this to be true and yet it is and it's a bit of a mystery why it works so well okay so let me give you an example of the kind of models that I'm talking about so simplified models not realistic so I'm going to go back to the example of the traffic again and show you a famous model of traffic on a road called the Nagel schellenberg model is invented in the 1990s 25 years ago by kind a goal and Michael Trachtenberg two German physicists and it works like this so you have a road here I've drawn it here this is what the road looked like so it's a one lane road going in one direction so there's one lane of traffic and it's only going one way and you break it up into boxes like that drawn there and the boxes can either each box can either have a car in it or not have a car in it so the red ones have cars in and the empty ones are empty and what happens in this model is just each car just hops one hop to the right like that and then they do it again and then they do it again they just keep on going like that so that's cars driving down the road okay so obviously this is a highly unrealistic model right there are no wheels on these cars there are no drivers inside these guys they're just red boxes and moreover they're sort of jumping in this totally unrealistic fashion they're all going at the same speed down the road right very much a simplified model of traffic on a road and yet we can actually learn things from this model so this kind of model is called a cellular automaton there's a whole class of model cell you saw Merton is any model where you've got boxes and there are some states of the boxes and there's some rules about how the states change so this is an example of a cell it's an automaton model clearly very unrealistic and yet we can learn quite a lot from this model so here's a little simulation I built of an eagle second boat model there's my road down the middle there and I thought one car the car just hops from box to box like that and I also put periodic boundary conditions on my car so if it goes off that side it comes back on that side like that just keeps going round the end of Main Street is just the beginning again I learnt about periodic boundary conditions as a child by playing the video game asteroids which is a 1970s era arcade game where you fly this little spaceship and if you go off one side of the screen then you just come back on the other side of the screen okay so the very natural thing get off one side come back on the other side okay so that's what it looks like if I have one car in my system here I have eight cars so I ran my simulation again and my cars drive longer and they're all going at the same speed because they all hop at the same time right so I've got my eight cars they go off the side the screen there they come back on the screen there and this keep on going back okay good so far all right now what we're going to do is we're going to fill it up with cars we're going to put a whole lot of cars on the road and see what happens all of these cars okay now we're going to let the simulation run and see what happens whoa something weird happens they're going backwards okay why are they going that way not that way anymore so first of all they really are still driving to the right they're not driving to the left they look like they're driving to left but they're actually driving it right second of all this is a real phenomenon that you actually see in real road traffic so let's talk about what's happening here if you're a condensed matter physicist I know we have a couple of condensed matter physicists in the audience then you already know about this phenomenon what we have here is we're above half filling and the holes are moving in the opposite direction to the particles if you're familiar with that language that's all that's happening here if you're not familiar with that language then look at let's look at this in a bit more detail so what I'm going to do is I'm going to look at a slightly more sophisticated simulation a traffic on the road it's still gonna be a one lane road it's still going to be going only one way but it's going to have a bit more of a sophisticated representation of the cars so I have to admit I didn't write this program this is this simulation is done using a software package called net logo which is a standard software package that people use for simulations in complex systems it's free you can download it from the internet you can play around with if you want to it does all sorts of simulations and all sorts of things here it's doing a simulation of a one lane road with cars on it so here is my road along here and as a bunch of cars starting out on this road and I'm just going to let the simulation run and we'll see what happens okay so my cars are driving for the right again and you've got periodic boundary conditions so when they go off the right they come back on the left and there's a traffic jam right here in my cars and you see the traffic jam is moving to the left so it came back on again it's moving to the left even though the cars are moving to the right the traffic jam is moving to the left because the cars are coming up at the back of the traffic jam and joining the back of the traffic jam then other cars are leaving from the front and the net result is that it moves backwards in the opposite direction so the way the traffic is moving so that's actually what you are seeing on the previous slide with the Nagus record book model when looked like things were moving backwards actually the cars were still moving forward but the traffic jams were moving backwards so even though the Nagel strickenberg model is a really simple model it's actually getting at some real behavior in real traffic systems here so this is the thing that actually happens you may have experienced this yourself sometimes you'll be driving down the highway and you'll come to a traffic jam you'll come to a stop at the back of a bunch of cars you'll be sitting around there for half an hour in your car idling forward very slowly eventually you get to the front of the chap again and you move up again and suddenly you're driving just fine and it's like what happened there was no reason for this traffic jam right is this in the middle of nowhere there's this traffic jam on the highway there's no there wasn't an accident there wasn't a blockage there wasn't anything there you just sort of stopped for half an hour for no reason at all what happened what happened is there probably was an accident or a blockage or something and it was probably five hours ago ten miles up the high all right and it created a traffic jam and then this happens people come along and join the back of the traffic jam and other people leave the front and the net result is that the whole traffic jam moves backwards up the highway and by the time you encountered it it was ten miles up the highway from where of the accident it happens so you don't see anything but it's still sitting there this traffic jam sort of slowly propagating backwards up the highway these things can go for miles Hey so this is an example of a really simple model but nonetheless even though it's obviously unrealistic gives you some insight into a real phenomenon that can happen in one of these complex systems you can certainly make these simulations more complex here's another net logo simulation now of traffic in a grid pant a grid plan town with cars going that way and cars going this way and then they've got little traffic lights there you see there's the red and the green and they're switching backwards and forwards and so you can make these things much more complicated this is still pretty unrealistic right it's still not a realistic simulation of real traffic but but you can certainly make these things more detailed you can buy big software packages that are designed specifically for detailed simulations of traffic if you're a an urban planner and you're working for a city and you're going to change the traffic patterns or build new roads or something like that then these days they always do this kind of stuff before they'll do any actual changes to the roads they'll simulate it and they make these very detailed micro simulations where you'll have all of the streets and how many lanes they have traffic in each direction and all of the intersections and all the traffic lights and the road signs and then you'll have lots of cars on there moving around you do this really detailed simulation so now we're at the other end of the scale now we're at the complex model end of the scale where it's difficult to say whether I got much understanding from this model but it does have good predictive power this is the kind of model that can actually tell me how long my morning commute is going to take so people use these kind of simulations very seriously for doing whee urban planning and city design and so forth okay very good so there's one example the kind of thing I'm talking about now one thing that's true of all these models that I've talked about so far is that they are deterministic models which means that their behavior is sort of completely written in stone the rules are very clear if I start the system in a particular state I know what it's going to do I put one car on my road and this is going to drive down the road and keep going round and round right doesn't matter how many times I run that program the car is going to do the same thing over and over again it's its terminus existe means it's completely predictable what it's going to do a lot of stuff that I work on is stuff where it's randomized or stochastic processes things that has some random element in it and this leads to the the methods that we call Monte Carlo methods which I'll talk about a bit now so in physics there are some things which really are random genuinely random so an example would be the radioactive decay of some radioactive isotope so you have some radioactive atom if you sit around wait for long enough it's going to decay it'll miss an alpha or a beta particle or a gamma ray or whatever some combination of those things that's a truly random process if you look at this atom you cannot predict when it's going to decay you might have some idea how long it'll take on average that's the half-life of the atom but you can't say exactly when it's going to be a might be a bit earlier it might be a bit later you cannot predict it and it's known in fact that there is no experiment you can do no Theory you can concoct that will predict exactly when this particular atom will decay it's a truly random unpredictable thing in fact so random that people use the decays of radioactive atoms to generate random numbers for example to create super secret codes that no one can break however there are also processes in physics that look random to us but they're not actually technically random so an example would be Brownian motion so Brownian motion is the jiggling of party in a gas if you have a little particle like a dust particle or something in floating in the air and you're looking down a microscope you'll see that it's jiggling around why is it digging around because it's being hit on all sides by the molecules of the air it gets hit by a molecule and jiggles it that way it gets hit by another molecule it jiggles it that way and you can see this under a microscope it's moving around somehow that's technically not actually a random process if you knew the positions of all of the molecules in the gas and how fast they were going and in what direction then in principle you could predict all of the collisions and you completely exactly had what it was going to pay of course we can't do that we can't measure all those positions and velocities and even if we could there are so many of them there's no way we could predict them all that's not a calculation we can actually do so as far as we're concerned when we look at this particle moving around it looks like it's random we can't predict it okay it turns out that you can do a pretty good job of saying how this system will behave just by assuming that it actually is random right if we can't predict it then it might as well be random that's roughly what we mean by random random means something that we can't predict so so this is the kind of thing I'm talking about an example of this an everyday example of this would be rolling a dice rolling a dice is a perfectly Newtonian thing right in principle we can calculate the equations of motion stuff but it's so unpredictable that it might as well be random and that's why we use it for generating random numbers so let me give you an example of the kind of models that I'm talking about these Monte Carlo methods here's a simple question so if I have a dice and I roll it how many times will I have to roll it before I get a 1 also that question is pretty simple 6 times on average I any particular time it might be more or less than that I get a 1 the first time I roll it or it might have to roll it 50 times before I get a 1 but on average I know it's gonna take six times because there are six numbers on the dice it's gonna take six rolls before I get a one alright that's obvious okay so let me ask more complicated question how many times do I have to roll dice before I get each of the numbers exactly once don't know that's a harder question hmm okay well so I can answer this by doing a Monte Carlo simulation what I'm going to do is I'm going to write a computer program and just to prove to you that this is not a very complicated thing to do here is the actual computer program that I wrote right even if you don't know how to program a computer you can understand this program pretty much well I can explain what this is doing this line here s equals one two three four five six it creates a set so as said is a thing like in mathematics it's a collection of objects or numbers so that says S is a set that contains the numbers one two three four five six which are the numbers on the dice okay and then this bit which starts with while is a loop the computer goes around this bit many times in each time round it generates a number n which is in the range 1 to 6 so that's what that line there is doing it's generating a number a random number between 1 and 6 so it's imitating the roll of a dice it's not actually rolling a nice it's just imitating it and then s dot discard that line there that says throw out that number from the set so if I roll a 1 throw the 1 out from the set so I'm just going to see how long it takes me to throw all of the 6 numbers out of the set if it's already been thrown out earlier then nothing will happen but if there's a one still in the set throw it out ok and then just go around that loop over and over again rolling a number and throwing the number out of set and you wait until all of the numbers are gone from the set and then you stop and you print out how many times you rolled there's really very simple programs just seven or eight lines there ah and I run this program here's me running it it's called dice dot py and I ran it printed out 14 so it's saying oh it took me 14 rolls of the dice to roll each number at least once but it's a random process right it's not the same every time so if I run the program again here's me running it again a second time it was 18 then it was 11 that was 16 it was 19 there was 8 all right so different number every time because it's a random process well what I can do is I can make a bar chart a histogram that shows how many times I had to roll the dice in order to get each number exactly once so horizontal axis how many times I had to roll the dice vertical axis is how many of these experiments I had to roll it that many times notice that the horizontal axis starts at six I have to roll the dice at least six times to get the numbers one to six right so that's the minimum it Peaks here around what 11 or 12 or something so the most common number is 11 or 12 but there's a long tail out here which says sometimes I have to roll the dice 30 40 50 times before I see all six numbers sometimes I just get unlucky so now here's the thing it turns out in fact that this particular problem has an exact answer you can calculate this is not that difficult to do so the first time I roll the dice I'm definitely going to get one of the numbers I don't know which one I'll get but I'll get one of the numbers and I'll strike that one out of the set right so suppose I roll a 1 on the first time so I remove the 1 from the set now I've just got to get a 2 3 4 5 or 6 and then I'm done right so I roll the dice again what's the chance that I'm going to get one of those 5 remaining numbers well that's five out of six right so how many times are going to have to roll the dice to get that well you just flip that probability upside down 6 over 5 is the expected number of times I'm going to have to roll the dice and the next time it's 6 over 4 and then it's 6 over 3 6 over 2 6 over 1 so the total expected number times I'm gonna have to roll the dice is 6 divided by K summed over K from 1 to 6 that mathematic expression here and it's not hard show that's equaled 147 over 10 or 14 point 7 so it takes on average exactly 14 point 7 rolls of the dice to get all the numbers ones sometimes more sometimes less but that's gonna be the average ok so here's an interesting thing we have a random process it's truly a random process but it also has an exact mathematical answer well that's not so surprising there's lots of things in physics that are like that but back in the 1940s and aluminous collaborations that the Los Alamos lab realized that you could use this process in Reverse this is the clever trick said ok sure if I have a random process I can do a mathematical calculation to calculate the answer to that but he said well let's use this in Reverse we have some mathematical calculation we want to do but we don't know how to do it if I can find a random process whose answer is the same as the thing I'm trying to calculate then I can just do that random process instead I don't have to do my mathematical calculation let's find a random process that gives the same answer as the thing I'm trying to do that's what Monte Carlo methods do it's find a random process that gives you the answer to your question so let me give you a simple example of that I'm going to calculate the value of pi right so that's definitely something that has a real answer pi is a different thing right there's no randomness about that is a definite number well might not know all the digits it goes on forever but there definitely are some digits there okay but I'm going to calculate by doing a random thing so here I have a circle and a circle is inside a square and the square is 2 by 2 could be 2 feet or 2 inches or it doesn't matter two units by two units okay two unit by two units square and a circle inside it so the circle is also 2 by 2 it touches the walls on all sides so that means its diameter is 2 so that means it's radius is 1 so the circle of radius 1 now you know that the area of a circle is PI R squared but R the radius in this case is 1 so that's actually just PI so the area of this circle is pi so if I could measure the area of the circle that would be pi ok so I'm going to do that I'm going to measure the area of the circle I'm gonna do it in perhaps a slightly funny way what I'm going to do is I'm going to take my circle and my square around it and we stayed on the wall and then I'm gonna throw a bunch of darts at it ok so I'm a really bad darts player and that means that my darts are just going to land completely randomly anywhere on this piece of paper all right so some of them will land inside the circle and some of them not gonna land inside the circle what fraction of them are going to land inside the circle well let's call that F the fraction of the land inside the circle is just equal to the area of the circle divided by the area of the square if the circle occupies half of the space then half of the dots again land in a circle right so the fraction of darts that land the circle is equal to here in the surface of a layer square but we know the area of the circle is pi we just throat that down and the area of the square is easy it's 2 by 2 so that's 4 so it's just pi divided by 4 so the fraction of darts that land inside circles four so all I have to do is calculate the fraction of dulcet line inside a circle multiply it by 4 and I get PI that's my whole calculation so of course I don't actually do this with actual darts because that would be dangerous and slow what I do is I do it in simulation on the computer I simulate darts being thrown at this circle so here here's my program but does it is again really simple I'm gonna N equals a million says that's how many dots I'm gonna do I'm gonna do a million dollar so a great thing about computers is a really fast there's no problem doing a million darts right so I'm just going to go around this loop here a million times each time I generate a random x-coordinate and a y-coordinate that's the point the dart hits the dartboard at and they ask is that point inside the circle that's what this line here is doing if it is count a hit and then at the very end I take the number of hits I divide it by the total number of darts that gives me on my fraction F I multiply that by 4 that gives me PI I printed out so again really simple program run my program it's called PI dot py I run it and it says 3.1 408 5 - that's not bad the actual value of pi is 3.14159 dot dot dot right so I've got the first three digits right the 3.14 I go right that's not bad it's not perfect why not because it's a random process you know sometimes a few more darts are going to fall outside the circle sometimes a few less it's not going to be exactly correct but I haven't done badly I've got it to like one part in a thousand they're running it again it gives me three point one four one 904 so still not perfect they're still pretty good not the same because it's a random process it's slightly different every time and then I got 3.14 games people one for blue 3.1 399 this time okay but they're all basically the same monster 3.14 it's telling me this now supply so I've calculated something that has an exact answer by doing a random process okay so there's an example of a Monte Carlo method this particular calculation calculating the value of pi by throwing random things at a piece of paper has actually mean people have known how to do this for a long time this this goes back to way before the computer apparently doing this basically this experiment exactly the same thing there's a variant on this experiment called bufang Snoodle experiment which involves throwing needles at a piece of paper when apparently this was a popular intellectual pastime for people in the late 19th century calculating pie by throwing needles the piece of paper finally it was a thing you did before you had video games there's a particularly famous example of this an experiment done in 1901 by Mario Lazzarini in which he dropped more than 3000 needles on a piece of paper and used the positions the hits and misses to calculate value of pi' accurate to 6 decimal places which would be really impressive if it weren't for the fact that it seems almost certain that he faked his results later on in the 90s a statistician came along and he actually analyzed Missouri knees numbers from his experiment and he showed that the fluctuations in the numbers were not nearly big enough so you saw that the answers the answers here they fluctuate you don't get the same answer every time and what what he found was that lazareva these answers didn't fluctuate enough didn't fluctuate as much as you would expect them to based on the number of needles he dropped and based on this he concluded that you know there was a one in a million chance or something that he actually have gotten the results as good as he did so seems extremely likely that actually he liked his results it's kind of sad ok so let me give you a couple of examples of of calculations which we can do of random systems and Monte Carlo style calculations we can do with random systems complex systems type things the first example I want to give you is something called diffusion limited aggregation which is a growth process a very simple model it was invented by a professor here at the university of michigan lense and a professor in the physics department back in the 1980s he invented this beautifully simple model so what it is is you have particles in this model are represented by these blue dots in this picture and so a particle just sort of wanders around kind of like that Brownian motion that we talked about before it's sort of moving around at random like this and if it meets another particle then it sticks to it just sticks there and then it doesn't move after that just sticks away so what you have is you have some blob of particles like this in the middle and you have another particle that's coming on the top there and it's going to wander around until it sticks somewhere on the outside of the blob and at that point it stops after that it never moves again it just stays wherever it's stuck on the outside and then another particle comes along and wanders around and stick somewhere else and you keep on doing this that's it that's the whole model you ask well what happens if I do this you might think that what would happen would be well you just get some big blob you just sort of grow slowly out would particles stick here but it will stick here particle stick here they just dig at random points and so you just get some big brown blob growing in the middle of the screen amazingly that is not what happened something much more interesting than that happens so I wrote a program to do this one it's a little bit more complicated than the other programs we saw though it's not that complicated is like 30 or 40 lines or something like that not going to go through it I'm just going to show you what it does you start off with one particle in the middle and you start throwing in these particles and they stick wherever they want to and you get this so to be clear here I'm just showing where the particle stick I'm not showing them wandering around before they stick because that would take too long and I want this to you know not take too long so I'm just showing where they stick and you get this beautiful pattern Oh also the colors the colors are there just to make it look nice they don't mean anything right so you get this beautiful pattern you get these sort of filamentary things where it sort of grows out in these fronds from from the starting point in the middle what's going on here well what's going on here is that for any particular particle that comes on is much easier for it to stick at the end of one of these frogs here or here here than it is to stick somewhere here right down in the middle in order to get here in the middle it would have to navigate its way down this fjord here to stick somewhere down here and in doing that it would have to be very careful not to bump into either the sides of the fjord because if it ever does then it's just going to stick there right so it's really unlikely that it would just happen to go all the way down this fjord get right into the middle and then stick somewhere that's just not going to happen it's much more likely that it would stick somewhere around the outside much easier to do that so that's what it does it has a much higher probability of sticking on the outside here than it does in the middle so you end up getting these sort of beautiful filamentary things that grow outwards so this process very simple though it is is thought to be akin to growth process that we see in biological systems for instance in corals which grow by taking nutrients and minerals out of the water but around them but that's much easier for those nutrients and minerals to stick to the ends of the fronds than it is to get into the middle of the growing coral and so they tend to end up with these filamentary shapes like this now these pictures here that they look like real corals but actually they're not real corals these pictures were themselves created by doing a more sophisticated version of the simulation that I just showed you with growing things on a computer these were done by a Dutch physicist called Yap Condor in the early 2000s and they look really like looks like some real corals that I've seen and so this is is believed to be a good sort of general model for how these things grow so again instead of giving us some understanding really simple model but giving us some understanding of how you would get these filamentary structures that you see in the growth of corals all right the next example I wanted to talk about is one that I've worked on in my own work here at the University of Michigan and this is based on something called the percolation model so percolation again is a really simple model so so the way it works is you have a bunch of points or nodes like this arranged on a grid rows and rows of them and you just throw connections down between them just put a link between them in those two and those two and you just keep on doing this and you throw lots more connections down you're just throwing them down at random anywhere whoops come back just throw these connections down random so you get something like this all connected together so that's it that's the whole model you just take these dots and you connect them together at random why is this an interesting thing well what's interesting about it is the way it behaves depending on how many of these links you throw down or another way of saying it that is what's the probability that two nodes are connected together high probability then there's lots of links if there's low probability then this very few looks so if there's only low probability they being connected only very few links then you have something like this over here on the left there's just a few connections lying around but basically it's all still separate stuff it hasn't been connected up very much on the other hand if you throw down a lot of links like this over here then it all gets connected up I can get from anywhere to anywhere else now in this maze because everything sort of got connected together I can get from here to here so there's a path that goes all the way across the lattice from one side to the other for instance here there isn't over here over here there's no way I can get from one side that's the other because it hasn't connected up and in between those two extremes there's the magic point where it first connects up when you first find that you have some path all the way across the grid from one side to the other that's called the percolation transition so this is the anything where this transition from system where it's all separate to a system where it's all got connected up with this translation transition the middle of percolation transition so why are we interested in this well people studied this process for a lot of reasons but the reason I'm interested in it is because I'm interested in how diseases spread so imagine that each node in the lattice represents a person and there's some person over here representing red who has the flu okay and then this person could give the flu to some of their friends so their friends in this case means the people next to them on the grid but they're probably not going to give the flu to all of their friends because you know they might not talk to some of their friends this week or they might talk to them but they might only talk to him on the phone in which case you can't give them the food now for whatever reason there's some probability that you give flu to your friend some probability that you don't so I'm going to represent that by putting the lines in so the lines represent who I gave the flu to so this red dot here gave the flu to this person this person but not to this person down here okay and then those people can pass it on to their friends and so fourth so it just goes if follows along the lines and it's eventually going to get to all of these people here that's what we call a percolation cluster it's the clump of connected nodes that you can reach from wherever you started just by following the lines right so the disease spreads to the percolation cluster in which it starts and the probability of the lines the probability of connections in the percolation model is equivalent to the probability that a person transmits the flu to another person right so there is an exact mapping here there is an exact equivalence between this percolation model and the spread of disease the probability of connections between nodes in percolation model becomes the probability of spreading the disease between two people and a percolation cluster in the percolation model becomes an outbreak in the epidemiological sense it's the collection of people that got the disease in the fabric so now this business that we talked about on the previous transparency of this percolation transition becomes very important if you're in this situation over here where it's not all connected up it doesn't matter where the disease starts it's not going to go anywhere right it can't spread because this isn't all connected up yet if you're over here on the right and disease starts somewhere it's going to spread all through all throughout the world almost everybody's going to get the disease and right at this intermediate stage the point where it first connects up that's the transition between the two regimes one regime over here on the Left disease can't spread another regime over here on the right the disease can spread and there's a transition in between them right in the epidemiological language that's called the epidemic threshold but it's really exactly equivalent to the percolation threshold in the percolation model these two models are really exactly equivalent to one another so we can use what we know about this random model in physics to tell us about the spread of disease well fair enough this is not a very realistic representation of the actual population here we don't actually live on a grid right we're not actually friends with the people to the left and the right of us and four in front and behind more like more like we're in some network like this so the nodes represent the people and there are connections links representing who's friends with whom and then some of those links I've shown in bold here those are the ones where the disease is being spread long italic right so it's the same thing as before except that now the connection is not a regular grid it's some network of who's friends with whom and then the Z starts somewhere and it only spreads along the bold links and it goes wherever it's going to go and it's reached this cluster of people and those are the people who got disease so in axial thank the way we represent diseases we're taking the network of who's in contact with whom and then we're doing the percolation model on that network so instead of doing a little square grid we're doing it on the network so in order to understand how a disease spreads we need to know what the structure of this network is that connects people together so people have spent a lot of time working on that trying to determine the structure of these networks this is one of the things that I do in my research this is a picture from a study by my friend Valdis Krebs that was about the spread of TB tuberculosis people think of tuberculosis as being something that romantic heroines die of in Jane Austen novels but but it's actually become a serious health issue in the last 25 years or so because of the it's it's really bare biotic resistant strains of tuberculosis which are resistant to treatment and and so I've become a serious health problem so people are concerned about this so people look at networks like this which is the network of spread of tuberculosis so tuberculosis is spread by airborne infection you know somebody coughed on somebody else so you have to be in close physical proximity with someone else breathing the same air to pass TB so this would be the network of who's been in close physical proximity with whom and you can see that there's interesting structure in this network like there are these star-like structures you see around here these are what we call hubs in the network they're nodes that have a lot of connections to other people and you could see that that would be important from a disease spreading point of view if there's somebody who has a lot of contacts with other people and if that person got sick they could spread the disease to a lot of other people so that's an important thing to know about there are some people who have contact with you know ten thousand people a day and if you're the person who sits on the front desk is reception or the person who drives the bus or something like that you can have contact with a lot of people and then if you get sick that could be very bad here's some other examples of networks this is a network of contact between school children in a u.s. high school this is this is one of the networks from my own work this one's initially one this is a picture of the internet which is not on a network of contacts between people but but it is the network over which computer virus is spread so people who used the same kind of mathematics that we use to study the spread of human diseases also to study the spread of computer viruses in which case you want to know what the shape of the Internet is what the network there is that spreading disease here's an actual example of a disease spreading over a network this is real data from an actual outbreak of an actual disease SARS severe acute respiratory syndrome which is a sometimes fatal form of pneumonia they had a bad outbreak of it in Southeast Asia in 2002 2003 Steve Bourget II from Duke University put together this picture showing the actual path of the disease through the population in that specific outbreak people have also done you know you can go to the other end of the scale and do the detailed simulations so I've been talking about these very simple models percolation and stuff like that there's there's a whole scale of these from the simple models to the complex like we talked about before so this for instance is a computer simulation of the spread of a computer virus done by this chap at UCSD and you can see this virus starts off sort of relatively slow spreading around the world and then you wait a little bit and suddenly it's everywhere people have worked on very detailed micro simulations of the spread of diseases so you know this this is sort of the disease equivalent of that traffic simulation I talked about where you put all the roads in and all the intersections and everything like that this is from a study called the epi sim study spearheaded by Steve Ewbank at Virginia Tech so this is a simulation of spread of disease in a specific city so the city they studied was Portland Oregon so it's an actual City a real place and they have all of the topography and all of the streets and all of the buildings and they have the cars going down the streets and then people in the cars and they have the people down to work and the kids go to school and all of that stuff is in there very detailed simulation of everything so now we're at the complete other end of the scale of like putting in the kitchen sink we do a really detailed simulation maybe it's not so useful for getting understanding but it's very useful for making predictions if you want to actually predict you know how many people are going to get the flu you know how long it's going to take how long it's going to be before they recover those kinds of things then you can use a detailed simulation like this to actually make quantitative predictions about how things are going to happen here's another example from Victoria klitz this group this is this is actually so somewhere in the middle this is an example of a model that's of halfway between this simple and complex what they were interested in here is the role of Airlines in spreading disease around the world so the primary vector by which diseases spread around the world is on Airlines airplanes are basically just little metal tubes full of germs hurtling through the air and and so they made this this simulation in which they had all of the airline routes the actual airline routes that the airlines fly from each airport to each other Airport and how many passengers are on each flight and all of that is in there in great detail what they didn't do is they didn't simulate what's going on inside the cities in great detail they could have done something like that study of Portland but they they've made a very simple model inside the cities where it was just sort of people randomly milling around and meeting each other and passing disease on so this is sort of somewhere in the middle this is a sort of semi complex model that sort of got some complexity to at the airlines but some simple modeling as well it's a combination of those two things and they could use this to predict for instance if there's an outbreak of the flu which cities it's going to get to first which ones it's going to be worst in how many people are going to get infected when it's going to get what date it's going to get to this city versus this other city and so forth okay I'm very nearly out of time I've just got a couple of minutes left I want to show you just one more quick example before I finish so all of these diseases that I've been talking about so far or what we call s-i are diseases that means diseases where you start off in the susceptible state you can catch the disease you get infected and then you recover that's the flowchart for an SI R disease there are other diseases for instance there are si RS diseases those are ones which dollars are set for you get infected you cover but then after a while you lose immunity and move back into the susceptible box again meaning you can catch the disease the second time that doesn't happen for instance with the flu once you've had a particular strain of the flu you can never catch it again you have lifetime immunity but there are other diseases where you lose your immunity and you move back over here again or there are more complicated things still so this can give rise to some interesting problem so missioning behaviors this circular si RS model this is actually very closely akin to something else the studying in physics called an excitable medium which is something where you can sort of trigger it and set it off and then and then it relaxes back into what we call a refractory state and then after it's been in refractory state for a while it can be triggered again and you can set it off again so this is used as for instance a model of of the tissue in the heart that causes your heart to beat for instance it's an excitable medium another famous example is a chemical reaction called the police officer Burtynsky reaction what this what happens when you do this is you sort of get waves of disease he imagine this is the landscape you got a wave of disease here behind it you're in this refractory state this is people who've recovered and they can't get the disease in front of it this is people who are susceptible than they can get in the seas so that means the disease tends to move downwards in this picture into the area occupied by the currently susceptible people leaving this refractory state behind it so as I say this is related to a bunch of other things including nonlinear chemical reactions like the volusia ultimate insky reaction so just going to leave you with this this is another simulation done using the net logo package of precisely this process so you can imagine these as waves of disease spreading across the landscape there are actual diseases where you can see these waves happening they're actually not usually human diseases because human diseases don't really spread across the landscape they mostly spread in airplanes but but but animal diseases show this kind of wave-like behavior for example I've seen movies of rabies spreading across the landscape and you get these waves of infection that spread across the landscape and it looks very like this kind of thing so this is actually a simulation of a chemical reaction the so called belousov Jabotinsky reaction but it's mathematically very closely related these si RS models of the spread of disease I'm just going to leave that running I'm out of time so I'm going to stop talking now I hope that I've convinced you that physics has some interesting things to say about a whole range of these complex systems from various different areas of science I'd be happy to answer questions if you have them but for now that's it I'll stop talking Thanks [Applause] [Music]

Channel: Michigan Channel

Views: 8,459

Rating: 4.8994975 out of 5

Keywords: +university of michigan, +physics, +newman, +complex systems

Id: 2L64AhoKamE

Length: 57min 36sec (3456 seconds)

Published: Tue Feb 13 2018