 irreversible absorption and yesterday I kind of got close to the end of car parking and then I was rushing and got myself a little balled up. So let me just recapitulate what we learned about car parking. So what we have in mind here is we have cars of length one that permanently in any spot of length X greater than one. And again in the same spirit as what we did for dimer and camber parking the basic degree of freedom is the empty interval probability namely the probability that we have an empty interval of size X but we don't pay attention to what happens outside of X. And by using that degree of freedom it turns out that we can write local equations for the evolution of these empty interval probabilities. So in the case of car parking so let me call EXT the probability of an empty interval and then we were able to write down a rate equation for this empty interval probability so it's DE of XT dt. So there's minus X minus one EXT. So that's one term in the equation and so this corresponds to filling a parking spot whose length X is bigger than one. And so the number of ways or the number of places you can put a car of length one in a spot of length X is X minus one places. And then there is the effect of parking in a larger parking spot that impinges on the interval of length X. So there's a minus two because I can park at either end of this interval. The integral from X to X plus one EYT dy. I'm going to write this more neatly. So this corresponds to filling the interior of interval. Here is where I have a larger parking spot but I part of the car impinges on the empty interval of size X and so this is the last term and this corresponds to the situation where the parking spot is larger than one. And then there's a similar set of equations for the case where the parking spot is smaller than one. In that case all you have to do to write down the equations is just invert or reverse the roles of X and one and so you're trying to park not a car in a parking spot but you're trying to put a parking spot over a car. And so all you have to do is just reverse the order of X and one and so you just have this and all I did here was just wherever I saw an X and a one I just reversed their roles. And so what we did then is we did the same kind of trick of trying to solve these equations. We first focused on the on the top equation and we introduced this quantity EXT is equal to phi E to the minus X minus one T. Whoops, other way around minus sign. And if we plug this on Zotz into this equation then we end up with just a simple equation for phi and we can solve that equation and so what we found last time was phi as a function of T was equal to the following thing. It was equal to EXP minus two the integral from zero to T one minus E to the minus T prime over T prime DT prime. Okay, and now now that we have this in some sense we have the solution but what we want to extract from this is the coverage of the system. And so the coverage we can determine the following way. So the coverage row of T. So that's nothing more than one minus so that's like if I look at a little infinitesimal region of space is there a car there. That's one minus the probability that that infinitesimal region of space is empty. So that's one minus E of zero T. So it turns out though that to find E of zero T we can go back to our original equations and see that if I look at here if I look at x equals zero from you know since this is true for any x less than one let's look at x equals zero. So at x equals zero we have DE zero T by DT. So we just have minus E one T and then there's there's no contribution this because this integral goes over zero range so this is equal to minus E one T. So from this we can now integrate this and get E zero T. So we'll have E zero T minus E at of zero at T equals zero but the probability that an empty an interval is empty at T equals zero it's all one so that's just minus one and so that's equal to minus the integral of E one of T DT from zero up to T. But if we go back to here we see that from this equation E one T is the same as phi. So finally our coverage which is one minus E zero is just whoops yeah so this is minus sign. So one minus E zero is just the plus of this guy which is plus the integral of phi DT. So finally we have our result that row of T is equal to the integral from zero to T of phi of T DT. And so let me just write this out now. So this is the integral from zero to T DT DT I'm sorry phi of T prime DT prime so it's DT prime phi of T prime and so that's EXP minus two the integral from zero now to T double T prime one minus E to the minus T double prime T double prime DT double prime. So that is sort of a recapitulation of what I did at the end of yesterday's lecture. Now there's one interesting feature very interesting feature of this result which is we can ask how quickly is the final coverage approached. We saw in the case of dimer and camer deposition that the final state was approached exponentially quickly in time. Let's ask what is the behavior of the approach to the final state. So let's look at row of infinity minus row of T. So row of infinity will just be integral of this guy to infinity. So I take row infinity minus row of T that's just gonna be integral from T to infinity of this of this expression. So this is the integral T to infinity DT prime EXP minus two zero to T prime one minus E to the minus T double prime T double prime DT double prime. And now if we're interested in the asymptotic approach well let's so let's have T large so that means that T is large here which means that when I'm doing the integral DT prime I'm starting from a large value of T. I'm sorry large value of T prime so that means that inside of this integral T prime is large. And if you now stare at this integrand for large T or large T double prime this exponential can be ignored with respect to the one. So if we just were to ignore this then you would have E to the minus two over T double prime integral. When I integrate this guy I would get minus so it's asymptotically minus two log T prime because I just ignore this guy so I integrate one over T double prime that's log T that's gonna be just log of T prime so I have minus two log T prime but then I'm exponentiating it so when I exponentiate it this is nothing more than T prime to the minus two power and then I'm integrating it from T to infinity and so the whole thing is one over T. So in this case the approach to the final state is much slower and it's power law rather than exponential and so you know this is what we've all encountered if you're trying to park your car in downtown Rome or downtown New York there's very few parking spots and it takes a long time before it fills up fills up. Another interesting aspect of this which I'm not going to discuss but it's it's fun to think about which is like suppose we're talking about parking in a real city with no well-defined parking spaces like Rome or Triesta okay cars leave so there's also desorption of cars and one can ask what is the final state of the of this absorption desorption process so there will be like some final steady state density and it turns out that there is a nice analogy between this reversible car parking problem and granular compaction there are these very beautiful experiments that were done a long time ago of just taking sand and just tapping tapping tapping and looking how the density would increase as a function of time and it turns out that the density has an inverse logarithmic relaxation to the final jamming density so if you imagine doing car parking with a very high rate of cars trying to park and a very small rate of cars trying to leave the system this problem this process also compacts and it has the same features as a logarithmic inverse logarithmic compaction of granular sand and so it's it's a fun problem to think about okay so that sort of concludes everything I have to say about irreversible absorption in one dimension and what I want to now do is to turn to absorption in higher dimensions and it turns out that in higher dimensions there's only one thing known and as you're going to see it's partly because now the geometry of the problem becomes you know complicated and because of that one can do only looking at the asymptotic approach to the final state so let's look at absorption d greater than one and let's talk about the absolute simplest thing we can think of which would be absorption of spheres in a continuum d-dimensional space and so let me just do two dimensions because I at least I can draw it on the blackboard so we absorb you know circles on this plane here and whenever there's an open spot that can fit a coin we put another coin we keep doing this ad infinitum until there's until those remaining spots are at a point where there's it's not big enough to fit a coin and so we reach jamming and the question we might want to ask is how quickly do we reach jamming the actual jamming density is a hard problem that we don't you know you know there's no full solution of but at least we can determine how quickly jamming is approached and understand something about the dynamics of the process altogether so let's look asymptotically in a large time limit when you know the system is close to jamming and so you have to search for a while before you find a spot where you can put down another coin and so you know here is say one coin here is another coin and let me try and arrange this right so here's a third coin and so you know when you put down their coins the thing is that the next coin like you can't put a coin here because it's going to overlap with this guy so if this is of radius r there's a region of radius 2 r which might be called the exclusion zone where I cannot put a coin inside of here so similarly there's another exclusion zone here you can't put anybody here there's another exclusion zone here where you can't put anybody but there's a tiny little region here where if I put a coin here it just barely going to fit inside of the remaining spot so in the long time limit what determines the dynamics of the process is these tiny little exclusion zones where I can put like another particle and so the dynamics is basically turned by the density of these sort of allowed zones so let me call CLT the density loud zones and the index L here is to indicate that these allowed zones well first of all they're geometrically complicated they're typically like these sort of triangular con concave triangular regions but they have a characteristic size scale L you know like maybe it's not equilateral but you know there'll be a characteristic a length scale associated with this so density of allowed zones of scale and how do these zones disappear well if I put a coin here then that zone is disappeared so I can write down a rate equation for how the density of these exclusion zones changes with time and since they're decreasing well okay I'll put it on over here minus sign they're decreasing and how do they decrease well it's proportional to the density of exclusion zones of size L to begin with so let's see this is C of LT is equal to minus this but the probability that you put down a coin is proportional to the number of zones times the area of the zone because that's you know there's that's how many ways you can fit a coin inside of the zone so it's proportional to the area of the zone which scales like L squared so I'm going to put here squiggle to note that this is kind of an approximate identity and then I have to just sum up over all exclusion zones of any size I'm sorry I don't sum up this this is just this is just the yeah this is all there is to it so for a for a fixed size I already broke my first piece of chalk for a fixed size this is this is how the density of these allowed zones disappears and so this equation is easy to solve because it's just exponential and so I'm going to get from this C of LT scales is e to the minus L squared t so that's how these exclusion zones disappear second point is now we're going to ask well how does the coverage change so how does D row by D T how does that change well the coverage is only going up and it goes up every time a coin can land in an allowed exclusion zone and so how do how can I put a coin in a loud exclusion zone well so first of all I have to have an exclusion zone of size L I want to integrate over all sizes of exclusion zones and so this is going to be integral e to the minus L squared t let me just check one thing oh yeah so sorry once again I mean if I want to increase the density it's proportional to the exclusion zone again times the size of exclusion zone so that the part that the coin can actually land inside so there's going to be an L squared DL okay so how do we do this integral and here once again scaling is easiest way to estimate the integral because here we see the combination of variables L squared t so what I can do here is I can multiply by root t for this guy I can multiply by t over here divide by 1 over t to the 3 halves and so this now becomes an integral of say e to the minus z z z I'm sorry e to the minus let's call it z squared z squared d z times 1 over t to the 3 halves so I don't care about this integral because it's some number of order 1 but all the time dependence is contained here so this is telling me how row is increasing as a function of time and finally row infinity minus row of t which would be the integral of this quantity the scales is 1 over t to the 1 half so that is the main result and in fact the only result that's known about irreversible absorption in higher dimensions which is that the approach to the final state is power law in time and power law because again there is this continuum possible shapes that can accommodate additional coins and this result is true for two spatial dimensions and you can kind of see almost immediately that if you were to do the same calculation in higher dimensions so the only difference is that you would have here not L to the 2 but L to the power d there would be just d's floating around everywhere and so in in general dimension this is t to the minus d over 2 and so this is the main result for irreversible absorption of isotropic objects in higher dimensions and you know people have played lots of additional games of this you know anisotropic objects you know objects can move objects with different shapes but this is the main result did you have a question yeah yeah oh I'm sorry what am I doing here wait a second what am I doing just bear with me a second t to the 1 over d sorry yeah that's stupid okay all right so you know once again I'm kind of at the end of a particular topic so if there's any questions from whatever audience is still left I mean you know fire away and before I go on to something else so now I come to a topic of the dynamics of non-equilibrium spin systems this is a major part of this course because spin systems play a huge role in our understanding of many body systems and statistical mechanics in general and understanding the dynamics of a non-equilibrium spin system has been a very rich problem for which there's still many interesting open questions left so I'm going to talk about the dynamics of the ising model so the ising model is a very simple idealized model for ferromagnetism where you imagine that you have a lattice and each side of the lattice is a two-state spin they can either point up or down and there is an interaction energy that favors ferromagnetic order and this is embodied by taking the Hamiltonian minus j summation let me call this sigma i sigma j where the sum is over nearest neighbors i and j are nearest neighbors to each other and here's sigma i is equal to plus or minus one and so if sigma i equals sigma j then it distributes minus j to the energy if they're misaligned it contributes plus j so the interaction energy itself favors ordering of the spins and what competes with that is thermal agitation and so there's this competition between thermal agitation and ferromagnetic order that gives rise to a phase transition now the ising model has played a paradigmatic role in our development of statistical mechanics and so maybe it's worthwhile to just spend five minutes giving a little bit of history to this because I you know will the students know like history of the ising model no maybe maybe not okay so let me give a little a little bit of history because it's very colorful and very interesting so 1920 Wilhelm Lenz who was a very respected you know hair doctor professor in Germany proposed to his PhD student Ernst easing and even though we pronounce it the ising model if you're speaking German you say easing so he proposed to his graduate student Ernst easing you know we really have to have a spin model for I'm paraphrasing a spin model for phase transition the most successful model for a phase transition was the Vanderwall's theory which turns out got Vanderwall's a Nobel Prize but it was a mean field a description of phase transition it was not microscopic in character it was just purely phenomenological but it described phase transition between a fluid phase and a gas phase listen yeah I would like to know why is the meaning of my nose in front of the ising model and the meaning of my nose so I say it again why is there what why is there why is the meaning of my nose in front of the ising model yeah so the minus yes for fair that's for ferromagnetic interaction so notice that if sigma i equals sigma j the product is one and it contributes minus and negative energy is low energy which means that it favors it at low temperature this does that answer your question does it I hope so anyways yeah so if it was a plus sign that would favor spins to be misaligned and that would correspond to the anti-ferromagnetic ising model anyway so you know lens told easing solve the ising model and so he developed it you know he's a transfer matrix approach to solve the ising model in one dimension and found no phase transition so it was a flop and in fact easing after this taught high school in Germany he actually suffered a lot during World War two but then he came to the United States and he taught like a junior colleges and high schools in the United States so he was basically out of physics altogether and discovered only very late in his life that he was actually very famous it's kind of a funny story but anyways so because there was no phase transition in one dimension you know then people said well there's no phase transition one dimension there's no phase transition any dimensions the model is not interesting but then in the 30s people like pyros looked at the model again and was convinced that there was a phase transition in two dimensions and higher and then I think it was in 1937 it was Cromer's vanier developed this duality relationship that showed that there had to be a phase transition and then in 1944 Lars Anzager actually solved the two-dimensional ising model exactly and it's a tour de force of mathematical physics if you want to be either inspired or depressed depending on your perspective you look at this paper and you say holy smokes this is amazing and in 1952 I guess it was TD Lee then found the magnetization of the two-dimensional ising model and apparently there was a story that in somewhere around 1950 Anzager had solved for the magnetization and never published it but he wrote somewhere in some conference proceedings not a conference but just at a conference he wrote on the blackboard I found the magnetization of the two-dimensional ising model wrote on the blackboard never published it but then TD Lee solved it and and wrote a paper on it and again if you want to be inspired or impressed or depressed you could read this paper and the thing that I find the most amazing is first of all the mathematics is just amazing but then he writes at the very end I'd like to thank Bell Labs for giving me a summer research internship to do this project over the summer you did this project over the summer or I forget if it was Bell Labs he thanked or whatever it was but you know he did he went for a summer project and solved the two-dimensional ising model anyways so after that you know people then have spent a lot of effort trying to solve the ising model in higher dimensions and it seems like this is never going to happen so if you really want to be super duper famous solve the ising model in three dimensions see where you get okay but anyways so the thing is that the equilibrium properties of the ising model are fairly well understood and roughly speaking the spatial dimension plays an important role in that in one dimension there's no phase transition in two and three dimensions there's a non mean field phase transition with non trivial critical point exponents in four dimensions and above mean feel mean field theory turns out to be asymptotically correct and then we can understand the properties of the ising model just by solving the mean field limit which is certainly much simpler so that is the equilibrium properties of the ising model and one could fill up an entire course of discussing how to solve it in one in one dimension two dimensions scaling approaches renormalization group and all of that but I'm not talking anything about equilibrium properties I'm looking at non equilibrium properties the dynamics of the ising model and it turns out that the basic problem statement is the following or the most interesting case is the following which is that when you say you were looking at non equilibrium properties you're imagining that you start with a system at one temperature and you change the temperature and you ask how the system responds and so if you start at low temperature you're starting with ferromagnetic order and you raise the temperature you'll just go to a disordered state that turns out not to be terribly interesting the more interesting situation is starting at a high temperature where the magnetization is zero and cooling it to where the magnetization is non-zero below the phase transition temperature and asking how does magnetic order emerge from the lack of order and there's there's three different types of situations one can consider one can start at high temperature and cool to the critical temperature one could start high temperature and cool to below the critical temperature or one starts at high temperature and cools to zero temperature and it turns out that if one's looking at the relaxation properties it doesn't matter whether you cool to below the critical temperature or zero temperature it's all the same dynamics so to make life simple let's just go from cooling from where we started at high temperature and cool to zero temperature and let's do it infinitesimately quick so we'll do it in you know instantaneous what's called a quench from infinite temperature to zero temperature another aspect of this is our our initial state could be anything about the critical temperature where the magnetization is equal to zero and again it's simplest to start at infinite temperature so there's absolutely no mag no ordering between the spins and suddenly quench to zero temperature so that is the basic phenomenon that we want to understand so the basic goal is understand in instantaneous quench to t equals zero quench from say from t equals infinity to t equals zero so at this stage it's worth showing the movie now so I have a very brief movie just takes 10 seconds show that shows a typical relaxation situation we can play it a few times so you you really see it so this is some so the blue and red or I mean the blue and yellow are two different phases of the icing model and we set the system going and by the way can everybody see this who's out there in the virtual world can someone tell us can someone chime in do you see it okay very good so again the blue and the yellow are meant to represent the two different phases spin up and spin down and we start with our initial state with a random distribution and then we set the system going and so what you see is that there's a very complex you know coarsening process going on the point here is that if we're going if we're quenching to zero temperature the spins like to be aligned and because they like to be aligned that means that an interface where two you know where boundary between misaligned spins it the system doesn't like it so it's trying to minimize the length of the interface so the dynamics is basically minimizing the length of the interface as a system evolves and so as it evolves here the interface which is very ramified beginning gets shorter and shorter and shorter and eventually you get to the point that you reach the ground state so that is the type of phenomenology that we'd like to try and understand so it turns out that even in two dimensions there's many unusual anomalies associated with the dynamics that's fine thank you very much and so first of all before we understand what happens in two dimensions which actually is very difficult I'd like to look at simpler cases which will be the mean field limit and the one dimensional case and we'll see that in the mean field limit in the one dimensional case already here there's a lot of interesting mathematics to understand a lot of interesting phenomenology to understand as well okay so the basic thing is that we've got to then start by writing down the master equation for the probability distribution of a given configuration of spins so our basic variable is the probability of a configuration of spins so actually I'm Sigma so this is this is the probability of a spin configuration collectively I'm called Sigma at time t so as I move along here I'm going to start getting a bit sloppy with notation because it's always hard to write these curly brackets but you know for the first time and just being pedantic so this is meant to represent a state if I have n spins in the system I have two to the n possible spin states and so this is the probability of being a particular one of those two to the n possible spin states and what we want to compute if we if we could really do everything we want to compute this full probability distribution but what we're going to content ourselves with is understanding just the evolution of this probability distribution in state space so once again you know like I'm going to go back to the very first picture I think I drew on the very first day which is saying that you know we can represent our state space as these two to the n points and so here is a particular spin configuration s and I'm going to think about a very simple class of dynamics known as single spin flip dynamics where if you're in a particular spin configuration the way that you change is you pick a spin at random and you flip the spin and if you you know and depending and we're going to see what are the rules in which the spin flip is actually allowed but we're going to take a spin and so and we're going to flip one of them so let me just draw two points in the state space so I don't clutter up this drawing so here's another state where I can think of a spin prime and so the prime here is meant to denote that a spin at given site has changed its state and so there will be some rate w and I keep changing from sigma to s so this is sigma sigma the silver there'll be some hopping rate sigma to sigma prime that I pick up the I say the I spin so you know maybe I should even put an index here which I'm not going to do throughout the rest of this lecture because it's too gloppy a notation but I pick one spin the I spin and if I flip it then I'll go to a new state sigma prime where the I spin has been flipped and there'll be a transition rate w sigma going to sigma prime and similarly there is a hopping rate w sigma prime going to sigma and so in principle you know I have two to the n spins I have you know lots of arrows everywhere here but all I want to do is draw these first two arrows now you know the one thing about these hopping rates at the moment we have no idea what they are and the way that we try and determine what the what they are is by a condition known as the detailed balance condition and this is kind of an ad hoc you know an ad hoc assumption but it works really well and it seems like that's the best we can do and the idea here is the following which is that I have my state space and I know that in equilibrium the probability so p equilibrium is proportional to e to the minus beta the Hamiltonian and if my Hamiltonian is this thing over here then this is proportional to e to the so now the minus becomes a plus beta j summation s i s j and if I flip the spin then I'll have e to the minus beta the Hamiltonian but if spin i is flipped then here I'm going to get you know p equilibrium here is proportional to e to the minus beta the Hamiltonian which is proportional to e to the now minus beta j summation s oh and I keep using s I mean to use sigma sigma i sigma j sigma i sigma j so now I can so so these are the equilibrium probabilities so in principle if I want to be in equilibrium I want to ensure that these equilibrium probabilities exist and the way that one does that is first of all let me write down the master equation itself so deep dp by dt so again there's a gain term and there's a loss term and the gain term is because of hopping events where I go I'm at some other site and I hop to this particular site and so there's a summation so there's a sigma w sigma prime sigma prime sigma p sigma prime t and now again I'm not going to write down all the indices because it becomes a bit too tedious but what I mean by this and a sum over sigma prime so what I mean by sigma prime here that means that I take my configuration sigma I take one of the end spins in the system flip it that defines one of the states of sigma prime and I'm doing a transition rate from a state where one spin out of the end is flipped and I'm going to the state I would care about sigma and so this is a sum over lots of different states where you know n different n minus one other states where the one of the other n minus one spins is flipped and so the rate of flip going from sigma prime to sigma times a probability that's a total gain probability and then there's a loss because I can do the same kind of thing sigma prime w sigma so I started at state sigma I flip one of the spins with the rate sigma w sigma prime and I have the probability being in state sigma and again you know so I should have written here curly brackets with an eye here to note that it's the eye spin I should have summed over you know sigma prime I I hope that's understood so I don't have to write so much you know gloppy gloppy indices everywhere okay so the thing is that in equilibrium I mean this is all we know about the system is that in equilibrium p dp by dt should be zero and so that tells us that the left-hand side I mean the two terms here have to balance so again let's let's look at what's what are the meaning of these two terms here so this is the total influx rate into site I from everywhere and this is a total outflow rate from this site to everywhere in the system and so what we need is for those two rates they have to balance however this is a very non-local condition and it's not easy to deal with such a non-local condition and so the detailed balance condition is actually a much stronger condition and makes things much easier which is that you say that well I want the sum of the input and output rates to be equal the detailed balance condition is that I say instead of the sum of the rates are equal I say that every single rate across every single link is equal so it's a much stronger condition but it also simplifies things so the detailed balance condition is just w sigma sigma sigma prime p sigma t is equal to w sigma sigma prime sigma prime sigma p sigma prime t that's all so that means that this the total flow this way equals a total flow that way and this is true for every single link across the network and so it's called the detailed balance condition it's sort of like stated as an article of faith and I guess that's all it really is and then but once we have this we can actually work with it and mathematicalize it and start understanding the dynamics of the ising model under this detailed balance condition okay so let's just go a little further with this so that says it w sigma excuse me yeah why do you say that the detail balance is a lack of faith because it seems to me that it's the only way in which you can have a solution with time reversibility so well no I mean it's an act of faith because the true condition is just that I want the sum of the rates into any node and is equal to the outflow rate if I have if I can make a model where I could actually enforce that that would also satisfy the necessary equilibrium condition so you know detail balance is much stronger because you're saying that the sum of the rates on every single bond is equal does that address what you're worried about yeah more or less it's okay thank you okay all right so anyways so let's just look so the point is that with the detailed balance condition we can determine the ratio of the forward rate and the backward rate so this is nothing more than sigma prime t p sigma t but this is equal to so here we have p sigma prime so that's going to be p sigma prime so this is p of oh yeah so here is p sigma and here's p sigma prime so this is e to the minus beta j sigma i sigma j and then I have e to the plus beta j summation sigma i sigma i sigma i sigma j and so this condition allows us to determine the ratio of the hopping rates we don't determine the actual hopping rates themselves we only determine the ratio but that means that if I don't know the full rate there's an overall time scale which I can choose to be one or anything I want but all that really matters though is the ratio of the rates because that's going to determine like which states are favored and which states are disfavored by the dynamics itself okay so that's formally what we can do with this but now let's actually work out some details and write something down that we can actually work with and understand like what is the meaning of this because right now this looks like some formal mumbo jumbo and we don't know really what we're talking about until we can actually say like what is the probability that a given spin is going to flip in a given situation so here I'm going to use a very handy dandy trick that works when you have these ising variables that are plus or minus one so where's my trick yeah so I'm going to use the following trick so I have it's a following if I have e to the a sigma where sigma is equal to plus or minus one and in fact sigma could be zero as well so it could work for even a three-state system so this thing is equal to cos a sigma plus cinch sigma so because the cosh is an even function so cosh of sigma is equal to cosh of minus sigma so that takes care of the plus or minus one and if it's zero then cosh of zero is is is I guess let's just do one I want that yeah so let's just do plus or minus one so in that case here you can rub out the sigma and here since cinch is an odd function this is the same as cosh a plus sigma cinch a and so this I can write as cosh a one plus sigma tanch a and so I'm going to use this trick to now simplify what we have over here so the ratio of the rates w sigma going to sigma prime divided by w sigma prime going to sigma so that's equal to so it's cosh so this a here is so you know this is a sum over you know since I'm focusing on site I this sum is over the neighbors of I I hope you can see that and this is the sum j of the neighbors of I so the thing which plays a role of a is beta j the summation over all sites j of sigma j so the point here is that there's going to be a cosh a in both numerator denominator that cancels and then I'm just going to have what's left over so I'm going to have one plus and because of the the numerators with the minus sign so it's one minus sigma I tanch of a and what a is it is going to be beta j summation over all neighbors j of sigma j and similarly I have one plus sigma I tanch of beta j summation sigma j and now this object here so it's telling us like the local environment of the system so you can think of this is like the local field experienced by site I so I'm going to define this thing to be little h sub I the local field because of the of the neighboring spins so we can write this a little bit more simply as one minus sigma I hyperbolic catch of beta h I and then have one plus sigma I hyperbolic tanch of beta h I now we've now found the ratio of the transition rates but we don't know the rate itself but now just for convenience I'm going to say well let me choose the amplitude and such in the following simple way I'm going to find w sigma sigma prime is equal to one half one minus sigma I hyperbolic tanch of beta h I so I've chosen my amplitude this way so this is the basic quantity that I want to be dealing with and let's get a feeling for what is actually contained in here so again this is the probability that site that's the spin at site I will flip and let's look at the limit where beta is going to infinity which corresponds to zero temperature so in beta goes to infinity the hyperbolic tanch is either going to be plus one or minus one depending on the sign of the local field so if the spins in the neighborhood are aligned with you then you're going to get plus one for this quantity you'll get one minus you know so if the spins are aligned then you're going to get the product of yourself with your local with your local neighborhood is one and so you get one minus one is zero so you're going to get a flip rate of zero on the other hand if your local neighborhood is anti aligned with you then this product will be minus one you get one minus minus one you'll get one plus one is two divided by one half you'll get one and so the point is that this model corresponds to majority rule so after all this algebra we have a very simple way of describing the actual dynamics from the point of view of a simulation that if I'm in a local neighborhood in two dimensions and three of my neighbors are saying vote for Trump and one of my neighbors saying no vote for Bernie Sanders I'm going to vote for Trump because that's what the local majority is telling me to do so the only ambiguity here which we'll discuss momentarily is what happens if two people are saying vote Trump and two people are saying vote Sanders what I do in that case well we'll come to that momentarily but anyway so that is the starting point for now beginning to understand the dynamics of the Ising model and so what I want to do now is work out two specific examples one is the mean field limit which is you can solve for everything and it's very pedagogical and useful and then I want to turn to the case of one dimension which is really an unbelievably rich subject even though we know everything there is to know about the one dimensional Ising model okay I guess I want to keep this for later it's much better when these claws are dry than when they're wet so we have w sigma sigma prime is equal to one half one minus sigma I tansh beta hi okay so I want to solve this case first of all in the case of the complete graph which is a very simple way of implementing them the mean field limit so let's look at the complete graph solution so what is a complete graph it's a it's a bunch of sites and sites in which everybody is connected to everybody else so in the case of four sites you have these connections as well as the diagonal in both directions and so everybody is a neighbor of everybody else that means there's very good mixing everybody knows that everybody else is thinking or doing and so this is a realization of the mean field limit where you can replace like the individual spin by the average spin and this is a simple microscopic way of achieving this now it turns out that for the Ising model on the complete graph one should modify the Hamiltonian slightly so let me write what the Hamiltonian is this is going to be a summation minus well I'm going to choose this sort of the J the interaction strength I'll choose to be you know one but in fact one needs an additional factor of 1 over n summation I less than J so you count each pair only once sigma i sigma j and there's a minus sign out in front and the 1 over n is necessary so you get a extensive energy because there's roughly n squared pairs here and so for a system of n spin I think my microphone it's still working so to ensure that the energy is extensive we need to have a 1 over n normalization factor okay and so what I want to do now is to solve for the dynamics of the Ising model here so here I'm going to refer to my notes for a second so now I'm going to use w of i to denote the rate at which the ice spin in the system is flipping so again I'm being a little bit elastic with my notation but I hope by context it's clear what I'm talking about so w i is the rate at which the ice spin flips and so this is equal to one half one minus sigma i and then I have hyperbolic tanch of my local field which is now so there's beta over n summation J not equal to i of sigma J so this is the analog of what we had over here so now my local field is that everybody else in the system is yelling at me and so I have to add up over all these other spins and then the coupling strength was 1 over n and so approximately this I can write in the following way 1 minus sigma i hyperbolic tanch so if this sum went over all spins in the system so the average you know so this divided by 1 over n is just the average spin value in the system which is nothing more than the magnetization so with the correction of order 1 over n I can replace this summation by just the magnetization itself so this is tanch beta so that's my flip rate and now the thing that we want to compute you know now we're at the stage we're ready to do some computation which is let's ask ourselves what is the behavior of a given spin in the system so let's look at dynamics I don't know dynamics spin I so we thought we focus our attention on a single spin in the system we ask well how does it change and so the point is that sigma I how does it change well you know again the way that you should think about the dynamics being implemented here is we have these different rates of picking a spin and flipping it so we pick a spin at random and with this rate we're going to flip it so in order for sigma I to change and so if sigma I changes what happens is if it's spin up it's going to change a spin down or it's been down to change the spin up so the change in sigma is minus twice its initial value if it's plus one it goes to minus one so it changes minus two it was it was minus one it goes to plus one and this changes again minus its initial value times two so it changes it changes by minus sigma I with rate w IDT and it's equal to itself with the rate one minus w IDT so we can say the delta sigma over delta time so the change is minus two sigma I and the time is one over the rate and so I put time down here so this is one over the rate so this is one over w I and so we're going to get minus two sigma I w I so this I'm going to write here as sigma I dot but what we really want is the time rate of change of like something that's thermodynamically realistic which is or thermodynamically realizable which is the average spin so let me now define s I dot as the thermal average of sigma I dot so what we have to do is compute the thermal average of minus two sigma I w I thermal average of all of this so that's what we want to compute and now you know these computations in general turn out to be kind of pleasant and you know the algebra of the ising variables makes all this a very pleasant game so let's see what comes out of this so we have minus two sigma I but w I is one half one minus sigma I tanch beta m and then we want to take the average of all of this stuff so there's the first term there's two a half sigma I with a minus sign so the first term is the thermal average of sigma I so that's minus s I and then the second term so the two and a half cancel and then I have minus and minus becomes plus sigma I times sigma I but here's something very useful simplification which is that sigma I squared is one independent of whether sigma is plus one or minus one and square is one so then we're going to get plus so all that goes away tanch and now what's more useful to look at is the time rate of change of the magnetization so the magnetization dot is equal to the summation over all sites I divided by n of sigma I of s I dot and so we add up over all spins so here we're just going to get minus m and here we're just adding up n identical terms and dividing by n so nothing changes and so we're going to get minus m plus tanch beta m so this is the mean field equation for the evolution the ising well m dot is equal to minus m plus tanch beta m so in the end we have a very simple equation and now it's a it's fun to actually look at the consequences of this simple equation so it turns out that the right way of approaching this is before you attempt to do any analytics on this is one should graph you know it's it's a first-order differential equation it's what we call a dynamical system or one variable dynamical system and it's very useful to plot as a function of m m dot and see what happens so without this term here then we just have a line that looks like this and so without this term here corresponds to beta equals zero which is infinite temperature so infinite temperature the dynamical behavior is given by this and so if m is negative then m dot is positive and so this becomes a stable fixed point and so in the case of beta going to zero or in fact it turns out beta less than a critical temperature which will come to in a moment this is stable fixed point so without doing any work we already know that at high temperatures sufficiently high temperatures magnetization equals zero is a stable fixed point of the dynamics on the other hand for beta small enough I'm sorry yeah for beta big enough then you see that this hyperbolic tank starts off with a positive slope and so the positive slope near the origin is competing with the negative slope of this term here and one finds that for beta bigger than a beta c that this function is going to have a positive slope near the origin so it's a function that looks something like this and now this is an unstable fixed point because I'm just to the right of it I flow you know m dot is positive I flow away if I'm just to the left the m dot is negative I flow away and these become the stable fixed points and there's two stable fixed points that m equilibrium and okay so just I can't remember the notation to have my notes so just say minus plus m equilibrium and minus m equilibrium they're symmetrically located there are two solutions and so we don't pick out which of the two solutions is selected that actually depends on the initial condition so and and then the last part is what happens right at beta c and it turns out that right at beta c this so I guess I'll put it up here now so for beta equals beta c and in this case the critical temperature is equal to one and right at beta c then the slope of this guy so you see the beta is equal to one the slope of tanch near the origin is one that matches this and so there's zero slope at the origin so the function looks something like this so this is beta equals beta c equals one and so this is again a stable fixed point but you approach it not exponentially quickly because because the slope is zero and so there's a power law decay to the equilibrium state so again I'm plotting m dot as a function of m so the thing about doing a graphical analysis is that you can understand what's going to happen without doing any work and so the other feature is that even if you know the answer before you start then it's a lot easier to get the right answer by algebra whereas if you don't know the answer it's very easy to like get yourself balled up and not know what you're doing so now that we know the answer let's derive the answer I mean we might as well do it so let's see what do I want to say here yeah just one more thing about this equilibrium state here which is that well how do we find these equilibrium values so okay so for t less than tc or the same as beta bigger than beta c I have the equilibrium state defined by m dot is equal to zero so zero is equal to minus m plus tanh beta m and I can write the zero is equal to minus m and now I can just do a power series expansion so you get beta m minus one third beta m cube plus dot dot dot dot and so I have here one third beta m cube ignoring higher order terms is equal to m beta minus one and now that we know that the critical temperature is one then close to beta equals one we can forget about this gives me a higher order correction so just set beta equals one over here and so what we get is that m is or m equilibrium is approximately equal to the square root of three beta minus one and so this defines a critical point exponent because normally at the phase transition when one writes m scales as t minus tc to the power beta and so we infer right away that beta is equal to one half and this is the critical point exponent for the magnetization too many what oh yes oh boy so for those of you uninitiated in the audience we typically use beta for the inverse temperature but also beta is used as the critical point exponent so for the purposes of not confusing let me call this b and b but the point is that the magnetization is measured is normally defined with the exponent beta one over t one over square root of t I'm about I'm about to do that yeah yeah yeah okay that's fine all right so as I say we haven't we you know I haven't done anything I haven't really done any derivation yet so let's now derive the approach to the equilibrium state so for beta less than one which is above the critical temperature so the point is that we're in this regime here where there's no other fixed points and so in the long-time limit all we care about is what's happening very close to the origin and so we can write m dot is equal to minus m plus tanch beta m which I'm going to write as plus beta m and so this is equal to m beta minus one all right this is minus m one minus beta since beta is less than one this is a positive coefficient so we're just going to have exponential approach to the final state that is that m of t scales is e to the minus one minus beta t so there's a sort of a critical relaxation time that scales like long over one minus beta so if beta is small the relaxation time is order one and as beta approaches one then this relaxation time grows and grows and at the critical temperature it's infinite and something else happens there okay and similarly if beta is if beta is less than one then the point is that you asymptotically in a long-time limit we just have to look locally near each of these fixed points clearly there's a non-zero slope near each of these fixed points and so locally near each of the fixed points we're going to have an equation of the same type where instead of computing m dot where compute the deviation of m from its equilibrium value and we'll have an m minus m equilibrium here and we're going to get basically the same behavior and I'll leave that as exercise for someone who wants to just do it but let's look for beta equals one so for beta equals one then I'm going to have m dot is a proxy equal to so I have minus m plus beta m the first term in expanding the hyperbolic tension minus one third beta m cubed but if beta is equal to one then this term cancels this and I'm going to have m scales is minus one third m cubed and the solution to this is that m of t asymptotically scales as one over the square root of time so at the critical temperature there's very slow relaxation to the equilibrium state of zero magnetization whereas everywhere else there is exponential approach to the final state so I'm going to stop here for just a minute to see if anyone's going to ask a question because I kind of at the end now of a little of a story the mean field solution for the ising model so any questions from anybody do we maybe should use this so other people can hear you in full generality I think this this exponent should be t to minus I know beta over new z no I'm sorry so in general if you so this exponent the exponent of the relaxation of the magnetization should be when you do scaling t to minus beta over new z I guess yes yeah so I mean I haven't introduced any dynamical exponent yet you're correct I mean there is match this one over root t is related to a critical point x you know dynamical and in this case beta over new z would be yeah I mean I you know it's like I grew up in the time of all the fancy business of the critical exponents and so if you asked me this question 40 years ago or 45 years ago I knew the answer off top of my head but now okay but I think it matches in matches yes okay okay all right so fine so now we've we've solved the one-dimensional I mean the the the ising model in the complete graph limit and what I want to turn to now is the ising model in one dimension because it can be exactly solved and it's very beautiful and it has lots of interesting connections with other statistical mechanical models so now let's look at the 1d ising model and so the Hamiltonian of this system would be equal to minus j summation over i for example right this is sigma i sigma i plus one because I'm only doing a sum over nearest neighbors yeah ask away do you want to use a microphone yeah that's true yes so you started by different dynamics with the detailed balance right yes so if you define another type of dynamics would be I mean well so in fact this is this was like a frontier area some time ago because you know people were very frustrated that doing single spin flip dynamics the dynamics was very slow and if you wanted to like look at critical point properties you have to mega simulations you need supercomputers and all that and so various people have invented other algorithms where instead of flipping a single spin at a time you flip many spins at a time there's the Swenson-Wang algorithm there's the Wolf algorithm there's all kinds of things there that people have implemented and you want to check the detailed balance holds and so that's usually the hard part of these more fancy algorithms to verify the detailed balance actually works but yes there are many you know there's many kinds of dynamics you can invent I mean single spin flip is simple easy to appreciate but it's extremely slow and so that's the whole motivation for doing these other algorithms where multiple spins are flipped simultaneously thanks okay all right so what dimensionalizing model so we have this and usually it's simplest to think of this is as the spins on a periodic ring so that you know we just go all the way around once and I don't have to you know that's fine so it turns out for reasons of simplicity it's more convenient to write this as j over two summation over all i sigma i times sigma i minus one plus sigma i plus one oh boy let's try to make the penmanship better sigma i minus one sigma i plus one and the reason for doing that is that it makes a simpler story for writing down the hopping the flip rate so now let's look at w sub i the flip rate of the ith spin and so this is equal to one half one minus sigma i and then hyperbolic titch of beta oh I erased it I erased it so what was here there was a summation over all site neighbors j of sigma i minus one plus sigma i plus one and then I have a two so beta over two I hope that's right just bear with me a second here yeah okay so I already made a mistake right at the very beginning here so let's let me just yeah let me let me just start over again minus j sigma i summation sigma i minus one minus sigma i plus sigma i plus one so try again I'll get I'll get it right eventually minus minus j summation over all sites i sigma i sigma i minus one plus sigma i plus one so this is erase this start so scratch the last two minutes start over again so I'm going to write my Hamiltonian in this way it's just more convenient because again you know the definition is I saw I take sigma i summed over its neighbors so I'm explicitly writing the sum over neighbors is the left neighbor and the right neighbor so this is my Hamiltonian and so now the flip rate is equal to one half one minus sigma i and then I have hyperbolic titch beta summation over all neighbors so it's it's just I don't there's no sum anymore because it's just the neighbors it's a local field so it's going to be sigma i minus one plus sigma i plus one times j yes I am counting each pair twice it just there's just convenient okay so anyways and now comes I want to sort of take advantage the algebra of the Ising model so I'm going to now do the following thing which doesn't know violence which is I'm going to divide here by two and multiply here by two and now this variable the sum of the sigmas this variable is equal to plus or minus one or zero that's all that can happen and now I want to take advantage of the following fact that tanh tanh yeah so tanh a x is equal to a tanh x for a equals plus or minus one or zero because the tanh function is an odd function so if I change the sign of a inside that changes the overall side of the function and if a is zero then tanh of zero is zero and so I can put it here with no violence to the system so it turns out that we can now take this variable which is inside the hyperbolic tanh you know again let's maybe have my brackets look a little bit better so here's curly brackets on the very outside and then here we have hyperbolic tanh of all of this stuff here and so I can take this variable and put it outside of the tanh without any violence and all I have left back inside is hyperbolic tanh of 2 beta j so let me now define the quantity gamma which is hyperbolic tanh of 2 beta j and so I can rewrite this flip rate is equal to one half one minus sigma i it's gamma sigma i sigma i minus one plus sigma i plus one I miss wait a second there's one there's one more mistake here and that's because I erased my blackboard is there's there was a one half here originally and so there's a one half here originally okay so this is right yeah no no the one half because that's still inside the hyperbolic tanh so tanh 2 beta j is what I'm calling gamma oh oh yes so yeah this is why I needed my other blackboard so was this was this my rate let me just remind myself what it is yeah this this is the rate and so yes the one half when I bring this outside there's the one half here so okay everything is fine okay very good so the thing is that by by this trick at zero temp by this trick we can sort of get rid of all these spin arguments inside the hyperbolic tanh in principle this is a complicated thing to deal with because if you expand the hyperbolic tanh in a power series you've got all powers of sigmas you've got a there's a lot of combinatorics involved but when you do it this way you know you just have like very simple products of sigmas to deal with algebraically so let's now use this flip rate to compute like the evolution of the magnetization and you know so we'll follow the same program as we did in the complete graph limit and we'll learn something interesting hopefully we'll learn something interesting so once again we want to compute s i dot which is the thermal average of sigma i dot and so what we've seen from before is that you know when a spin flips and I it changes by minus twice itself with the rates w sub i so this is nothing more than minus twice w i sigma i thermal average so let's now compute this average and see what we get so the one half and the two cancels so there's a minus in front and then I have sigma i times 1 minus gamma over 2 sigma i sigma i minus 1 plus sigma i plus 1 and so the first term just gives me minus sigma i our thermal average is sigma i which is minus s i the second term is the minus and the minus cancel so we have plus gamma over 2 then I have sigma i sigma i which is 1 so we're going to have plus gamma over 2 sigma i minus and now we take the thermal average and we're going to get s i minus 1 plus s i plus 1 so this is the equation of motion for the spin and now if you look at it you realize if you stare at it with the right frame of mind that this is almost the same as one of the first equations I wrote on the first day which is it's just the one dimensional diffusion equation you see that involves hopping from i minus 1 to i i plus I mean from i minus 1 to i i plus 1 to i it's almost looking like a diffusion equation except for the fact that there's this coefficient gamma sitting out in front so aside from that it really is a diffusion equation and it turns out that this equation has a very simple solution which I don't remember off the top of my head but for the initial condition s i at t equals zero equals delta of i zero that is that I start my initial state where spin at the origin is up and all the other spins are equally likely to be up or down and so we might expect that if I'm at least at high temperature that this spin should disappear it should it should relax away and so for this initial condition the answer is the following s i of t is equal to the following thing it's equal to i yeah i sub i of gamma t e to the minus i gamma t so again this is the modified Bessel function of the second kind of order i and that's the answer so we have the exact answer for the spin value at a given site and the thing is that what we're interested in typically is a long time limit and as and so it turns out that in the long time limit this thing goes like e to the minus one minus gamma t so for time going to infinity so this is going to zero so in fact we learn more or less nothing all we know is that the spin value disappears and if gamma is not equal to one it decays exponentially quickly and if gamma is equal to one and if you go back to where i had to erase it but when beta goes to when the temperature goes to zero which is beta going to infinity we have hyperbolic tangent of infinity which is one so when the temperature is zero then magically this becomes one and so in fact this has a following answer one over the square root of t in the limit for gamma equals to one which is the same as the temperature equals zero so at zero temperature the relaxation is slow it's power law at finite temperature the relaxation is quick it's exponential but we don't learn very much because all we see is that like the spin goes away so uh and in fact exactly that's right so i mean the common you just made is there's no spontaneous magnetization in the one-dimensional easing model at any non-zero temperature so it's what we expect there should be no order at any finite temperature but the point is that we're missing an opportunity because there's a lot of interesting behavior still hidden in not the average value of the spin but the spin correlation functions so in fact the spin correlation functions are the elemental objects which was with which we should focus on to understand the dynamics of the one-dimensional easing model so i'm i think i'm just out of time uh so i guess maybe that's a reasonable point to stop but let me just sort of tantalize you with the next part of this which is that after doing all this work and computing the average value of the spin we learn nothing so how can we learn something and the way we learn something is by looking at the spin correlation functions and there we will see that there's interesting dynamical behavior and it's very beautiful and it's very helpful and it gives us a lot of insight so that's what i'll do starting in like four hours from now