 So, hello, everyone. End of the day, everyone is tired. I will try to be a little bit entertaining. So I am from Aberdeen. And actually this talk came after a good number of very interesting talks with rise a little bit of discussion. Particularly Andrea Stork the Overday, Andrea Karaniya Stork izgleda nekaj izgledajših reakcija. To je nekaj, da se počutite, z vsej hrabrištih fizicistov. Všeč nekaj izgledajši, nekaj izgledajši nekaj izgledajši biologij. Vse sem fizicist, tako da, da. Zelo je vse. V kandu, kaj imam počutiti, je tudi tudi, nekaj izgledajši, ki so evoči počke. Sveč, je nisem prijevno v svojo izgledanje več da je vse več, in je odpovojova na svojo vse vzostavljenje vsojoj, čudovite vsoje, in nekaj nekaj, nekaj nekaj, nekaj nekaj, nekaj nekaj, nekaj nekaj, nekaj, nekaj nekaj, In jaz je vsi obstahen odčivati, that this is in a very similar situation of what you encounter when you try to describe via the dynamic of fluids in more traditional physical set ups. Čest jah je, ki je to pričočen, tako je to svote, kdjeli zelo bilo nepoten, ki počtučijo tkaj semetri in počtuče okončne vršenje naši tem, ker je objevično objevaj nožnjih zeltikov. Evnoj bolje nedev ojunju. Neljere s vršenjem koncertem Piranic, ešte legcrivi so nisak všim počku, pričo to prične silni stren si. A zelo, biologija je zelo vzpešnjena v svojo vzpešnje, da je nekaj nače, da so našli, da taj požah čečuje vzpešnje večasne uročnejo, ne zelo vzpešnje konstantru, nekaj, da ne je to zelo, nekaj je bilo zelo, da ustavljamo inštačne zvrste informacije z data, a to je vzpešnje, kako se poživajte, ki je to izgledaj. To je ongojnja vrkla, tako je nekaj vrkla. Sreč smo nekaj izgledaj. Na različne, zelo, da se poživajte. V ideju je poživajte močnje. Sreč smo poživali o poživajte močnje. Jeste svoje dve izgledaj. Staljnje vrkla, šepe, vrkla in embryogenes in zebrafiskih embryo. In vse čo je zelo v abadju zelo, in nekaj občan, nekaj razladu vsak des travail, vse četko otvorega, Pravno je, da zelo, da ini nekaj interaktiv, vse del nekaj, analizovana in interasja in v tom, da je ono tako spokojovo in nešto evojenju aj in neč vzreti na vse. Tudi sarr, na povrstvenjem, ki da bi izgledali skoljiom nespečenje z vrvič plenty, če prvna su povrstvenje in ki se se pravati do trupov, Do they know where we have to go back? And how can you understand if they know where is their nest? But it could also be a problem that when you study the migration of large animal groups, are they going by a precise direction in mind, which can be dictated by a lot of different factors some individuals knowing where to go, magnetic field for migrating birds, configuration of the terrain, and so on and so forth, Or, they are just picking a direction, somehow a random, they find in internal consensus, there is nothing outside their group telling where to go, and then they go for a stroll, like they won't have some leisure. Like it's probably the case of the births of Irene and Andrea and many other people. Another example where this kind of question is very relevant, I've been discussing with Kizhejer and his collaborators in Dundee. They study cellular migration embryogenesis and the primitive straight formation in many embryo, in particular they study the cheek embryo, involves large scale highly coordinated flows, which involves 10 to the 5 of more cells. They believe, and there are some indication that this is driven by some kind of external field, chemotactic field, mechanotactic effects, so on and so forth. But the question, this is an example, let's see if it goes. This is some imaging from a paper of Dundee group, and you can see large migration flow here, cells on the boundary of the embryo. And this flow basically creates the primitive streak, which basically underlie the symmetry and the cordal spine of many, many, basically all vertebrates. So the question is, how can you tell in all of these cases if your collective motion is spontaneous or not, or there is something which is driving it? One, of course, can look at the time series of the orientation, so I can look where the group is going. And we know that if, I mean, this is basically what Offer was doing, that if you look at the direction where you're pointing, that the angle which defines the direction in which you are moving into dimension for simplicity, if you're not directed by any external field, it's going to perform a random walk, a persistent random walk with a certain length. And so one can think that in the other case, otherwise your animals, your particle know where they go, and so you expect them to fluctuate less. And indeed, here there are two cases, they are taken out of the simple Vyček model, we have seen it many times. So you can see immediately, these are time series of the angle, how the angle changes in time. So here I have checking the quantity for 100,000 time steps a lot. There are two different cases. I can tell you one came from a directed motion where there is an external direction, which came on top of interaction between your particle, and the other it's a totally free case where the particle self-organized without any external. So you can easily, I mean, each of you can tell, OK, this will be the free case and this will be the one with an external imposition where basically you don't turn a lot. But then suppose that you don't have access on such a long time series, you start to look at the thing on ten time, shorter time scale. This is the same time series, but just cut much earlier. You can start to doubt which is the correct one. Then you go 100 time shorter, you wonder which is the directed and which is spontaneous. Not totally sure. You can, of course, zoom, and maybe you can get some idea. You can use a little bit more refined techniques like auto correlation function of your time series, which I'm showing here. But you see basically that when you go to short time scale, it becomes not totally trivial to distinguish between these two cases. And there is even a worse scenario where I'm not giving you two cases where I tell you this is spontaneous and this is directed. I just have one set of data and I have to decide if it's spontaneous or not. And remember also that the kind of diffusion in a group of N individuals, which perforce spontaneous collective motion, decreases with the number of individuals in the group. So the kind of noise becomes, in a sense, smaller and smaller, the kind of meandering and the persistent length growth. So short time scale is really difficult to tell by difference. There is an alternative way, which came from statistical physics and this is what I'm going to discuss here. And it's again based on the study of correlation function between fluctuations. So the main claim, which I'm trying to convince you, is that the nature of correlation between fluctuations equal time correlation. So here I'm talking about just looking at how a fluctuation in velocity or in the density at a certain point in your system correlates at the same time with another fluctuation at a distance r. Now, the nature of this correlation is radically different between the case of spontaneous collective motion and the case of collective motion, which is directed. In principle, if you have a system large enough and you are careful enough, you can be able to detect this kind of difference without accessing to a very long time series, basically looking at a few frames of your system. And this claim, which could turn out to be a powerful method to analyze data and to distinguish between the two cases, spontaneous and driven collective motion. Ok, now the story gets a little bit technical. I will try to keep some kind of not too complicated level due to the hour. The idea, you have already seen this kind of idea. When you, I start with spontaneous symmetry breaking. So there is nothing in the world which tells the group where they have to go. They just choose themselves. They talk one with each other locally. So they talk with their neighbors. They try, for instance, like birds to align their velocities and then they pick up a random one direction. And this is what physicists call spontaneous symmetry breaking of a continual symmetry. I don't know why there is written transition there. Sorry, mistake. That's spontaneous symmetry breaking. So you have already seen this drawing in many of this talk. Basically what happens is that since every direction is equivalent, once your birds, your particles pick up a direction, it doesn't cost basically nothing to change a little bit. And when there is a little bit of a fluctuation which brings you aside, nothing tells you this is wrong. Since you don't know where you have to go, you're just going around in some direction. This fact turns in the fact that the system is neutral to this mass situation. So you basically have this extremely long range correlation which emerge in your system, which you've seen in a lot of this talk. I mean, this is something basically you cannot escape if you have spontaneous collective motion. And this is true in a basically old system you can think of where you have spontaneous symmetry breaking, this continual symmetry and where there is nothing to tell you where to go. Another peculiarity, which I'm going to use in this method I'm proposing, is that in these systems the relative position of particles of neighbors, of beasts, of cells actually is coupled to the fluctuations. If they move, so if there is a fluctuation locally, it can make to sell diverge or to sell converge. So there is a coupling between the velocity and the density. So it turns out that this coupling makes also density fluctuation became long range. You can wonder why I saw interesting density fluctuation but the point is that I think density fluctuation are easier to measure. To measure velocity fluctuations, to measure velocity in the system, you need a little bit of tracking, maybe not a very long trajectory, but you need to identify for instance where the bird you have at time t goes after some milliseconds or some time anyhow. And this is not totally trivial why it is much easier to just find where in each snapshot of your experiment where particles are without knowing where they are going to be after a little bit. So in a sense density are very easy. Of course this theory has been worked out more than 20 years ago now. I am going to give a brief overlook but first the consequences you have already seen these kind of experimental data which came from Irena Giardina group in Rome. This basically tells one of the consequences I told you is that the correlation between velocity fluctuations for instance in a flock of birds either this spontaneous motion should be scale free but the correlation length is basically as large as the system size so it grows with the system size there is not a finite correlation length. And this is a proof that this kind of universe size applies. To bird there is a word of cautious that I want always to add since I mean their paper is clear but there has been some misinterpretations sometimes this has nothing to do with the bird being near a critical point where there is an order to disorder transition. The bird actually are highly ordered in this flock the fluctuation from order are extremely small but this is just a consequence of this spontaneous symmetry breaking which applies to the entire phase of collective motion so you don't need to be close to a transition. You can be terribly ordered with very small fluctuation and you still will see this kind of orientation fluctuation. Then there is something that is not speed scale free part but this is not pertaining to the stock but it doesn't have to do with order-disorder critical point in the sense of phase transition. Ok Velocity correlation now this consequence also means I mean this kind of property also means that if you go to Fourier space and this is something you already saw with Andrea Cavani's talk this for a number of reasons is more practical it's also a little bit technical but basically makes extremely easy to compute correlation function in Fourier space so basically what you have to do is just this operation you have to take here rj all the position of your particles in a certain snapshot you test the q represent wave numbers ok it's a vector in Fourier space you just have to take a different q typically small approaching the inverse of the system size and compute this sum where you take the exponential of this complex number which is the product between the q and the position of the births you take this sum over all the births then you take the model of square of this number reason what is simple you take an average if you can over a few different snapshot and you have basically your Fourier transformer correlation function ok and this is something which is rather simple to compute now these Fourier transform correlation function is some peculiarity like it's different in different direction if you look in different direction in Fourier space this is terribly complicated to measure even numerical simulation and I don't think it's simply accessible in experiments but what you can do you can average overall direction this improves your statistics and basically leaves you with the essential behavior of this object which is it is diverging with a certain exponent sigma as you go to smaller and smaller wavelength and I'm going to show you in a moment what it's doing and actually under some hypothesis this tonerantive theory tells you also which is the value of this exponent which is diverging there is a precise number for the criteria I'm going to show you it's of interest for physicists but it's not essential to discriminate between directed and spontaneous collective motion ok and then other consequence you have already seen today in your talk on bacteria is that this system are extremely sensitive I mean they have extremely large fluctuation in the local density ok and that's it this has been actually measured for instance this kind of things in a totally different system which is a system of epithelial cells we have measured actually this epithelial cells let's see ok when they express this protein here and you have seen this in the talk of Christina Marchetti also they basically have a transition to collective motion which apparently is spontaneous since when you go to measure these structure factor introduced before in density as a small q you exactly find this kind of divergence ok this is not extremely extended data which is due to the fact that we could for the moment only measure these on relatively small system of a few thousand cells but now we are trying to extend these but in principle this thing should go growing and diverging in zero as long as you increase the size of your system which means you are exploring smaller frequencies ok and also these other property which is connected to this one which is that fluctuations are extremely larger it means that if you go and measure for different sizes for the box in your system of particle and the fluctuation of this average these fluctuation are extremely large are larger than the square root of the number of particle which is what you expect in system which behave better which don't do spontaneous collective motion ok and again this is experimental data in this cell epithelial cell system and this dashed line is the theory ok so basically you have this kind of properties in this system this is spontaneous collective motion ok, I spare some details now we go to see what happens when the collective motion is not spontaneous but on top of this particle we choose a direction where is a preferred direction in space and to do this I put this kind of field H to exemplify I choose a V-check model and I put an external field of amplitude H in a certain direction, constant which tells on my particle as this direction is slightly better than the others ok and so basically I am pulling in a sense I am telling go that way before I anticipate the result of the theory then I don't think I will have the time to enter into the details but the results are the following what I told you so far about this long range correlation this divergence etc and actually I am introducing a cutoff in my system so if I look in real space I introduce a cutoff length which depends on my field above which this cutoff length depends on one over my field to a certain power certain power can be computed but this is again not so important and this telling me that my correlation are long range up to this length and then poof, they die and then the system realize the situation of the system realize that they are not spontaneous but there is some kind of overall direction the larger is the field so the stronger I am pulling my system the earlier this cutoff happens ok, that's video so this is something very clear if you go to Fourier space it's even simpler since a certain point when my q became small enough my divergence is cutoff and instead of growing as the wave number q goes to zero my system is going to go to a constant value ok, so there is a length scale or the inverse of a length scale Fourier space after which all that I told you doesn't imply anymore and my system behave decently and actually one can work out the expression a leading order in q and in the external field amplitude h for this structure factor ok, so before I had a divergence like one over q to some exponent this is just a pre-factor which depends on all the details of my system and not going to enter too much into that but now there is this correction which is linear in the field which tells me basically that when q became very very small and h is much larger than q this thing doesn't diverge anymore but it stops at a certain limit one over h ok I don't how much time do I have? 40, so ok maybe I can just give a flavor of how this thing is computed so this is a idodynamic equation of toner into theory so the idea as I told you is very similar to what you do in in Navias when you describe a fluid with Navias stocks basically find the important field of your problem which are connected basically to symmetry and conservation law so in this case you have the density since your particle are conserved at least on the time scales of the problem so the number of particle you have doesn't change and then you have the local average velocity since the essence of what your system is doing is breaking the invariance of all the direction in space and picking one so there is a breaking of the symmetry in your system so this is the script should be the scripted by a vector which tells you where you are going so these are the slow field of your theory and then you basically write this kind of partial differential equation which includes ok this is can look a little bit curious but basically includes all the possible derivatives to the lowest order which are compatible with the symmetries of the problem ok basically so you put everything so if there is not a strong reason not to put a term in this equation you put the term ok and then you can look so actually these are all adaptive terms these are all diffusive viscous terms there is a noise source since you want to equation in your system there is a pressure which is here and there is this magic term which is already seen which is exactly the term which luckily breaks the continuous symmetry and tells this velocity field to be non-zero, to be a value here which is v0 which is non-zero ok and this is basically the original theory of tournament 2 and there is a pressure which is here and what we do is just to add external field there is nothing particularly fancy you add this external field and you can go on now you consider the fluctuation in your system so small changes with respect to the average velocity and the average density of your system so you just see what happens if you put small fluctuation in your system you work out horrible equation which came from the previous one but this can be done so you go to Fourier space which is a standard technique you linearize with equation so you just consider at the beginning very small situations and you discard everything which is higher and basically at this point becomes a problem of linear algebra even if a little bit annoying and this leads you compute the structure factor in Fourier space at least to leading order in the so for small wave number and for small external field now technically you have to integrate over this time frequency to get the equal time structure factor you do all your calculation and you get this result here this is a prefector which I hope is correct and then this is this behavior you saw there is this one over q to some power the prefector in your field which act as a cutoff so the only thing is that here you have an exponent 2 which is what you get out of linear theory remember I basically said I consider all small situations now the original equation if you go back and look at also no linearities and this becomes a little less trivial to take into account and you can use basically dynamic randomization group arguments which is what also Andrea Cavagna talk about over the day and there are basically number of arguments to conjecture that to a certain degree that this exponent from being a q square became a q z ok and this is what gives you this characteristic length scale I show you before in the results ok and so you get basically this kind of final result you also average in all the direction as I told you in your life and you get this final result now we are going to test so you can test in numerical simulation these kind of things to start with I took again the V-check model I showed you before with an external field and what do you see here there are a lot of curves here is my structure factor here I have this wave number I told you so I measure as smaller as smaller wave numbers these correlation function in Fourier and I have different value of this external field they are pretty small compared to interaction but they already do the job and you see if I don't have an external field these things diverges like 1 over q to the z which is this line and the black dots seems to diverge as soon as I put even a small very small value of h at a certain point this term here became small and this dominates and so tends to have my system converge so basically my structure factor converges to a finite number basically the message is if you look at this quantity in experimental data you should be able now this is a very extreme case where I have an extremely small external field but look I don't know at this purple curve which is still small it's 100 of the interaction between particles the effect of the external field but you see that the purple line very clearly turns and doesn't grow anymore while the black line which is spontaneous motion grows indefinitively so basically if you're able to measure up to these small frequency which means up to these large sizes you will see a clear difference in the fluctuation correlation and you can say ha directed motion then you can test the value of exponents by doing some rescaling and more details which are more for physics you can check that the normalized theory works better than the original linear theory before doing Rg you can actually check which is simple just taking the value of this structure factor for q going to zero that what is left exactly scales like 1 over h and this is actually verified very neatly in the structure factor you can look again at the giant number situation which also the general friction get cut off if you take a box which is large enough at certain point they stop fluctuating in an anonymous way the number of particles in the box became larger than this cut off length introduced by the external field you see that the fluctuation becomes normal so this is possibly another way of measuring this kind of directed motion but they are strongly correlated ok ok this is a small detour there are other predictions which came out of this theory which basically tells you how the order parameter is going to change when you apply an external field the external field promotes to a certain extent order and this is telling you this theory can also tell you how the other parameter changes thanks to the applied external field this is basically what defines the susceptibility in physics and just adding this slide the susceptibility is diverging in the thermodynamic limit and there are consequences of this spontaneous symmetry breaking and I am just adding since recently the Nis Bartolo group actually happened to measure in a in a system of self propelled colloids submitted to an external flows which plays a role of this external field H had some measure of this response of the system to the external field which are compatible with our theory which predicts initially some kind of linear increase of your order parameter and after a certain value of the external field which depends on the system sides a growth with the exponent one third so this is already an experiment verification of another aspect of this theory and the last thing that I want to show you I don't have an experiment as I told you and the nobius objection on what I show you so far is you showed us the Vitchek model the Vitchek model in a sense is a wonderful model but it's a little bit dull particle just orient one with each other there is no steric repulsion there is no I mean the model is fully compressible so if you want to set this to people interesting in to send migration the cells are not like Vitchek particle cells occupy a volume they actually the kind of epithelial cells also Christina discussed are at the confluent limit where the pecking fraction is one or even slightly larger than one so can you still detect density fluctuations and correlation of density in this kind of system so I basically use the same model you have already seen in a few of these talks to call it collision on Vitchek model essentially it's a Vitchek model where you also have steric repulsion and I added an external field so not only my particle likes to go in the direction where they get pushed and this is what does the spontaneous ordering but also they have a direction which is better than the other which is given by this angle theta h so I'm just playing exactly the same game but in a model which is a little bit richer and this model actually has been shown to be able if you tune the parameters smartly enough to reproduce very closely the data of the cellular migration experiment I showed you at the beginning of my talk so the full dots are this model tuned and these open dots is the experiment I showed you before so I want to put myself in with a model which is reasonably close to experiment in cellular migration and to see the same thing and basically this is what I got again here I have a structure factor here I have a case where my field is 0 so the motion is totally spontaneous and I have this divergence as far as I like to go and here I applied very small field here the smallest is compared to the same constant between different particles essentially so 10 to the minus 2 and 2.5 10 to the minus to very small values but again you can see that if you look at one of this curve the green one for instance you see a clear movement towards a constant value so this is a clear sign of directed motion this is a clear sign of spontaneous motion if you want you can also divide sorry this should be q not h you can also divide by q to this magic power z I show you in the case of spontaneous motion you are flat in the case of directed motion you decrease this depends on how you prefer to look at the data so the point is there is a qualitative difference between these two kind of situations collective motion and directed motion and I believe that it is possible to observe in if you have large enough system if you have large enough number of particles to observe in real experiments and it can be used to discriminate for wild beast cellular migration and fish and other groups between these two cases and the recipe is very simple first of all hope that your moving system is large enough otherwise if you have 10 particles that's not the idea for you then compute the structure factor as I told you and all you need to know is the position of each particle or of many of the particle if you miss a few doesn't matter since this is statistical measure so if you have a 10% loss of your particle you can still survive now check the behavior of this quantity of loku if it is diverging you are spontaneous if you are constant you are driven that's the basic idea by the way these also work all the result I told you also work if your external field only affects a fraction of your particle so if you have a 10% of informed particle in your system since they know where is the nest they know where they want to migrate all these result apply it doesn't matter as long as you have a finite fraction of your particle which they know where they want to go all I told you is true so it's quite general there is one word of advice this is criteria maybe I mean the point is it could be that your system is not large enough to observe this kind of saturation of a structure factor this tells you if you are directed if you see a saturation and you are moving you are clearly directed it could be that if you see something it is just since you are not looking at a large enough system so if you have a very, very tiny preference for the direction maybe you need a larger system to detect so we cannot exclude directed motion with this method in principle but we can confirm it if we see this kind of saturation and the thing is moving that is going to be directed motion there is no doubt in terms list, but not last what we would like to do next and they would like to do this with the group of Roberto Charbino in Milano is to try a controlled experiment of course one could apply to a case where you know that there is chemia texis but an idea is to have one of these cellular migration experiment on some micro graded substrate so basically the idea is to take the substrate in which these cells are crawling and scratch it with parallel lines and the cells should be encouraged by this way to follow the parallel lines so we are introducing a preferred direction in our system and we believe that in this case we should observe the signature of directed motion so this is what we would like if we get some findings in the next months and that is basically the conclusion so my talk I have to thank my group in Aberdeen the black people is the people who collaborated to this kind of project and these are my external collaborators the people in Milano and John Turner in Oregon