 Let me start by thanking the organizers for the invitation and for organizing this conference. So I want to talk about something that has not been talked about in this conference till now as far as I could see. And don't be surprised by the title, maybe there is nothing surprising, I will be surprised at the end if you are not surprised, but okay, I am surprised all the time anyways. So I will talk about synchronization. So it's a field that has, I mean, the non-linear dynamical physicists have been studying synchronization over the years in many different contexts. So what I decided to do was, okay, so we have been working on this synchronization for the last couple of years with Stefano and Arkady Pikovsky from Potsdam and Alessandro Kampa from Roomba. So what I decided to do was to give kind of an overview of the field and some results that I obtained recently with Alessandro. So as they say, I will try to uncover a little then cover a lot just to be okay. So what do I mean by synchronization? So this one doesn't seem okay. So let me start with some historical note. Apparently the first observation of synchronization was done by Christian Huggins as long time ago as in 1665. This was the time when he had just invented pendulum clocks and he was ill and he was lying on the bed and he didn't have Facebook and iPad and so on to play with. So he was playing with the pendula. So he had suspended two pendula from some wooden frame and what he noted, he wrote this letter to his father. He says it's quite worth noting that when we suspended two blocks at two clocks from two hooks embedded in the same wooden beam, the motions of each pendulum in opposite swings were so much in agreement that they never receded the least bit from each other. So basically what he did was he said these two pendula independently and at some point he found that the two were oscillating in phase. And this was the beginning of or this was the first observation, first documented observation of synchronization. So what do I mean by synchronization is basically just adjustment of rhythms of self sustained oscillator. So these are oscillators, they can oscillate on their own, you don't need to do anything and then you put them in contact or in interaction with one another and then they do something. For example, they might oscillate in phase as shown in the first picture here or they may oscillate out of phase, but they may also may not oscillate at all in synchrony and this is the and it turns out that the synchronization it is a rather widespread phenomenon that is there in you know in cells and organisms. In fact, in our heart the cardiac cells send thousands of pulses and then they all have to be synchronized in one way or another so that the heart can beat you know in rhythm. And it is seen in this cardiac pacemaker cells, this flashing fireflies, rhythmic applause in theater concert halls, I mean this has been demonstrated in a real experiment that after nice you know performance in a concert when people applaud the performer, so there is a particular rhythm in which the applause is observed. And also in electrical power distribution networks, it is a very important thing to have phase synchronization among the electrical power. So the path to synchrony, if you think of the experiment that I talked about on the first slide, see the path to synchrony is first of all what you need is you need some self-sustained oscillators, several of them not just two, but several of them. And secondly, there is to be some interaction between them which has to be weak because if they are strongly coupled then okay, there is no point in talking about 10 oscillators. I can as well talk about one. So there has to be some weak interaction and there has to be several self-sustained oscillators to be put in contact with each other. And the first analytically tractable model that produced exact results, I will come to what do I mean by exact result, it was proposed by Kuramoto in 1984 and it came to be known as Kuramoto model. There were many previous attempts by several applied mathematicians, but this was the one that was really successful and this is the model that I am going to describe right now. So what is the Kuramoto model? So what you do is that you take N oscillators, self-sustained oscillators and you put them in contact with each other namely that you make them globally coupled, that everyone is interacting with everyone else and each of these oscillators has got its own frequency of oscillation. That is to say that if you think of the Hagen's experiment, there were several pendulah and each has a natural frequency of oscillation. So to mimic that here what you do is you take many of these oscillators and each one has its own frequency of oscillation and each one is interacting with everyone else. And then so what you do is these are limit cycle oscillators. So the dynamical degree of freedom of the oscillator is just its phase theta i which lies between 0 and 2 pi obviously and here what I have done is I have plotted the or I have indicated the phases of N of these oscillators on a circle. So this is some initial distribution of the phases of the oscillators. And then this is the dynamics basically. So you ask how does the phase change as a function of time? So forget about the second term for the moment d theta i dt equal to omega i, this is the equation of motion of the i-th oscillator. This omega is the intrinsic frequency of oscillation of the oscillator and it is a random variable. It varies from one oscillator to another while the second term is to model the global coupling between the oscillators. So for example if theta j is equal to theta i you see that this term is 0 and so this will mean that the j-th and the i-th oscillators are in synchrony. So this k is the coupling constant and this 1 by n factor is included just to make this term of order 1 in the limit n going to infinity. Because if I do not have this 1 by n factor this term will be infinite in the limit n going to infinity. So the second term is the interaction and the first term is the intrinsic frequency and you see that there is a competition between the two terms. But the first term will try to do is it will try to make every oscillator run at its own frequency because omega i is changing from one oscillator to the other. So the first term will try to make everyone oscillate at its own will while the second term what it will do is it will try to bring about a global synchrony. It will try to make all the oscillators run in synchrony. K is yes as I said k is the coupling constant and this omega i these are quenched random variables like these are random variables which are not changing in time because every oscillator has its own frequency and that is something that does not change in time it is just constant in time. So there is a corresponding frequency distribution g omega and the question is what happens if I start with some initial distribution like this and let the dynamics go for some time. So of course we will be interested in the limit of very large number of oscillators not just 2 or 5 and the steady state is what you observe after you have waited for long enough that you see some time independent behavior. So here what I have done is I am plotting some quantity do not worry so much of the definition but it just says what is the fraction of oscillators that are phase locked. So this is a measure of the synchrony in the system and what you observe is the following that if k is 0 for example if k is 0 then this term is you know this term is not there then every oscillator can oscillate at its own frequency there is no synchrony at all and that is why this r fraction of phase locked oscillators is precisely 0 and then what you do is you crank up the coupling constant little by little and at some point you see that the interaction this term is strong enough to overcome the desynchronizing effect of the first term and what you see is that the fraction of phase locked oscillators it increases continuously and then like eventually when k becomes very very large every oscillator is phase locked. So this r is like an order parameter in this system. So what you see is that when k is very high then you have a synchronized phase when k is low you have an incoherent phase or unsynchronized phase whichever way you want to correlate and if the frequency distribution is unimodal that is to say it is not bimodal or something it just have one peak then this transition is a continuous transition it is not quite a phase transition I mean to be very precise it is more a bifurcation transition rather than a phase transition because it is not a thermodynamic system anyway. So this was something that Kuramoto observed and he could analytically compute the critical value of this coupling at which this phase this bifurcation takes place. So this is roughly the picture I wanted to show a video probably it comes or I do not know how it came or no it did not came okay forget it okay I had a video that I got off from the YouTube but anyway. So this is the picture that you have this equation of motion d theta i d t equal to this and this what I have done is I have plotted the phase diagram as a function of k. So when k is smaller than kc you have r equal to 0 and then the fraction of phase locked oscillators is 0 while when k is large it is non-zero. Now this model has also been studied later in in in presence of some reason it is not so this model it as such it involves just a quench random variable but it does not involve any noise. So this model has been later studied by allowing for fluctuations of this frequencies in time. You say that okay there are some intrinsic fluctuations in the frequencies as a function of time. So what people did or more specifically what Sakaguchi did was he included a Gaussian white noise in the equation some motion a Gaussian white noise which is characterized by having a mean which is 0 and some temperature some some parameter that that specifies the strength of the noise and what he found is that the phase diagram that you see up here that gets slightly that gets like modified. So now I have one more parameter namely the temperature. So this is the t equal to 0 line this is this one and as you increase temperature as you might expect that what will happen is that you know if you keep on increasing temperature it will be harder and harder to synchronize the system. So that is why you see that the critical value of the coupling to synchronize the system it shifts to the right it increases as you increase temperature and this one was also solved analytically by Sakaguchi. So this Kuramoto model as I said it is very very popular and it has been studied a lot you know what the last in fact last year in Dresden there was a conference to celebrate the 60 years of the Kuramoto model. So it is really the Ising model or the Ising model of synchronization it has been very studied very well studied in several contexts and but then still there are many unresolved issues. So let us look at you know why it is so you know challenging to study or you know what are the unresolved issues for example till now you know a model as simple as Kuramoto model you know nobody knows the steady state distribution of phases. If you ask okay what say I start from some initial condition and I let the dynamics run for some time and then the system will settle into either a synchronized or an unsynchronized space. So in the synchronized space no one knows till date what is the steady state distribution of phases you know relaxation to steady state and finite size effects these also have not been studied and the obstacles are first of all it is a dynamical system. So there is no thermodynamics there is no entropy to maximize no free energy to minimize so that is you have to really look at the dynamical equations of motion and secondly this dynamic it is a non-linear differential equation. So exact solution is difficult to obtain. So I just want to spend maybe 2-3 minutes you know like relating this Kuramoto model to HMF model this has been there has been some studies by us showing that this Kuramoto model corresponds to a particular limit of the HMF model. So I am back to our old friend namely HMF model. So in the HMF model you do not have oscillators what you have is we have some particles which are going around in the circle. So every particle has got its location given by theta i so this is the angle and the corresponding conjugated momentum. So the equations of motion as we know is d theta i dt is equal to p i and d p i dt is the force which is exactly the one that you see in the Kuramoto model namely sin of theta j minus theta i there is a global coupling. So this is the HMF model now suppose what you do is that you keep this system like immersed in a heat bath you know you put this system inside a liquid or something that can be approximated as a heat bath and this liquid is at a constant temperature T. Then what will happen is that this energy conserving dynamics will no longer hold. So you will have a Lanjavan dynamics namely what you will have is you will have a friction term coming from the interaction with the heat bath and there will be a fluctuation model by some Gaussian noise where this strength of this eta i t will be given by the temperature namely if I look at the two time correlation. So this is a model of course nobody said that the liquid will be injecting Gaussian fluctuation but this is a model delta i j let us say this is j and delta of T minus T prime. So this is an HMF model in the canonical ensemble HMF model in contact with a heat bath now what you do is that you include till now the system is still in equilibrium. So if I do not have the heat bath it is in micro canonical equilibrium if I put the system inside a heat bath it will eventually settle to canonical equilibrium but what now what you do is that for every particle you turn on an external torque which is this term omega i some torque which is a quench random variable that is for every particle you choose a torque from some distribution G of omega this is G of omega as a function of omega. So by doing that what you are trying to do is you are trying to inject energy into the system. So this is a quench random variable you inject energy into the system by hand and now what will happen is that the system will never relax to an equilibrium the detail balance is manifestly broken by this term because you are injecting energy into the system. So it will always go to a non-equilibrium stationary state a state in which there will be non-zero probability currents in the phase space namely that at a technical level you know if this term were not there the stationary distribution would have been e to the power minus beta H where beta is just 1 over temperature but if you include this term the system will never be able to relax to canonical equilibrium because the canonical equilibrium it does not solve the stationary state of the system. So what we have been doing is the following that this is some work that I did recently with like Alessandro Kampas. So you consider a system of coupled identical oscillators. So here what I said is that you have many okay by the way I forgot to mention the relation to the Kuramoto model sorry. So this is still HMF model in contact with the heat path and there is an external torque and now if you take the over damped limit that you take the limit where gamma is much larger than the inertia of the system. So then you reduce this second order differential equation to a first order differential equation which will be gamma d theta i dt is equal to omega i plus k by n this term sin of theta j minus theta i plus eta i t. So this is precisely the Kuramoto model in presence of a heat path if I put t equal to 0 then this is precisely the Kuramoto model okay up to some scaling of theta and so on temperature. So what we have been looking at is the following system that now you consider a system of oscillators which are identical that till now I have been talking about oscillators each one of them had a different frequency of oscillation but let us say they are identical and then they are synchronized they are all running at the same frequency by default right from the beginning. So the system is trivially synchronized at zero temperature there is no noise and the system is trivially synchronized and now what we want to do is we want to start the local anisotropy effects I will specify in a minute what do I mean by local anisotropy effects. So it is basically to model some defect or something that is intrinsic to every oscillator some defects in the system and to see how they enhance or destroy synchronization. And what we get in the end is very rich phase diagram namely we find the phase diagram in which you have either time independent or time periodic steady state solutions and you get both equilibrium and non-equilibrium phase transitions within the same framework and there is a re-entrant transition in the sense that if you remember Kuramoto model like in the Kuramoto model what was happening was that you have to crank you had to crank up the coupling constant for the system to be synchronized and then the system remains synchronized while here what happens is that you know this is some parameter that you tune. So the system is initially synchronized by default because I started with identical oscillator. So as you crank up the this parameter at some point it loses synchrony the system is no longer synchronized but like again it gains synchrony if I keep on increasing the parameter and this is what is known as re-entrant transition. So the system is the following that you have n globally coupled oscillators it is the same as in the Kuramoto model picture the only new addition that I have done is this inclusion of an on-site potential term. So this is the interaction this is the intrinsic frequency which I promised is the same for all and there is some noise and there is an intrinsic on-site potential that is acting on every oscillator and which is characterized by W the strength of the on-site potential is characterized by W. So here are the so the first thing to note is that the dynamics is not invariant under you know if you make a Galilean transformation to a co-moving frame you cannot scale out the effect of this frequency term. This frequency term will remain as such because of this on-site potential term it is a technical point not so important. So let us look at some general features of the dynamics. So the first thing is what does this on-site term try to do. So this on-site term what is this corresponding potential as a function of theta. So what it tries to do is that it has 2 minima at 0 and pi so what this on-site term on its own tries to do is to make theta's crowd around 0 or pi because these are the 2 points at which the potential has its minima. So this is the effect of the on-site term while the interaction term as in the Kuramoto model will try to make all the theta's come together. So this is the effect of the interaction term and thus on-site plus interaction will lead to some static arrangement of theta's around either 0 or pi and it will lead to a time independent steady state just this term and that term because this is an interaction that tries to make theta equal to theta j and this will be minimized if theta is either 0 or pi. While the frequency term what it tries to do this intrinsic frequency term what I call a drive here this the tendency of making the theta's come together and crowd around 0 and pi will be opposed by this omega term that will tend to drive the phases at a constant velocity and it will never lead to any time independent state it will lead to a time periodic state. So there is this competition between a time periodic state or a time independent steady state at long times and the question is what is the phase diagram like and of course if I increase you know this is more or less the picture at small temperature and if I increase the temperature there will be desynchronizing effects brought about by temperature. So the phase diagram it turns out is rather rich of course one has to scale out some parameters and you know reduce the number of relevant parameters. One trivial limit is when you put w equal to 0 when you put this onsite term equal to 0 and this is a drive equal to 0 this becomes just the HMF model for which we know that there is a transition between the synchronized or unsynchronized phase at temperature equal to t half. So let me just so just point out that like when omega is not equal to 0 when you have this term then the system will go to a non-equilibrium stationary state as opposed to a stationary state. So the phase diagram is the following yeah the question is what happens to the stationary state and do we see synchrony and transitions. So let me skip this one let me just show some you know the three parameters or the three axis of the phase diagram are like this capital omega and w and t. So these are the three axis. So let me take a look at some slices of this phase diagram. So this is what you see. So this is a temperature equal to 0.3 and the two parameters are w and omega c this is the common frequency of the oscillators and this is the strength of the onsite potential. So what you see is that of course if omega is very large then there is no hope to get a time independent steady state. It will always settle to a time periodic steady state while if like omega is small there is a there is a region where the system settles into a steady state and you might have synchrony that is r not equal to 0 or r not equal to 0 and there could be a first order phase transition in going from synchronized to unsynchronized phase or there could be a second order phase transition. So a first order phase transition will be signal by having a hysteresis loop as is seen here while a second order would mean that it will just go continuously from this to that. So what happens is that if the temperature is here somewhere sorry if the value of w is somewhere here then what happens is that you start with the system being in synchrony but as you increase the strength of this common frequency the system becomes desynchronized which is kind of counter intuitive that all the oscillators were synchronized to start with but yet they get desynchronized but eventually they become again synchronized but then they go to this rotating or time periodic solution. The same thing happens at t equal to 0.5 and if t is larger than 0.5 namely here then what happens is that you lose this time periodic solution you are always in a time independent stationary state and it turns out that this rich phase diagram namely that there are first order transition, second order transitions, time periodic state or time independent steady state. It turns out that everything can be solved for like analytically. I will not bore you with the analytics but just to say that what we do is that you go to the limit n going to infinity and you write down the Fokker-Planck equation which looks a bit complicated but yet it can be solved for exactly. So, this is the stationary state solution and once you know the stationary state once you know the stationary state solution then this different phases that I talked about namely you have either a steady state unsynchronized phase or a steady state synchronized phase or a time periodic phase they can be found out by either linearizing the Fokker-Planck equation or doing some self consistent analysis but everything is exact and like everything can be done analytically in the limit n going to infinity. So, let me just come to the summary. So, what I talked about is a system of globally coupled identical oscillators in presence of local anisotropy and the basic feature is that there are these oscillators everyone on its own is oscillating at the same frequency but then there is this term that accounts for some interaction between the oscillator and there is some global anisotropy and a noise and this system shows an amazingly rich phase diagram namely it shows every possible scenarios that you can think of more or less that you have either a time independent or a time periodic steady state and synchronized or unsynchronized phases and like everything can be solved for analytically. So, future directions for example, it would be interesting to study relax dynamics to steady state how long does it take for the system to relax to the steady state and dynamics in the steady state and effects of inertia like what happens if I play the same game with the HMF model that there is an anisotropy and there is a drive and what are the finite say is effects that one is to also judge your assist and thank you for your time.