 Okay, so welcome everyone to the eighth and previous to last lecture of this course. Actually, I will start with content from the previous lecture that I didn't have time to explain, which was experiments on. So I go back slightly on, on the close to equilibrium conditions. And this is a bit of a summary. So, and when this is the fact that it's a patient relation when you have a system for example a colloid in equilibrium, and you weekly perturb it, for example, applying a sinusoidal weak force on the colloid. The response in its under this week perturbation and the fluctuations without the perturbation are related by this quantity. And as I shown in this concrete also in this review that they put in the bottom. Well, however, when, for example, you are in a bacterial bath, so the original system is not an equilibrium system. And when you perturb it weekly, the response is not related to the fluctuations in the way it's given by the participants. So this is broke. And this has been shown, for example in my choreography experiments with biological systems for example in in an active myosin cortex you put a colloid, you measure the correlation function. You do the freedom form. And then you do the same but you have the response, you get the response function from these trajectories, and you will find that this, which has also been done. Okay, this is from the 2007 this was done, for instance with a sinusoidal force in the, in the group of Christopher Smith, this is a paper in science, you will find that out of equilibrium. The biological systems are in general out of equilibrium, and the response of the fluctuation do not obey for this mission. As you show in the figure on the right, you see that the correlation function is with times omega is on top of the response. This is really a clear experimental biological manifestation of non equilibrium, which has been shown in different systems, not only in myosin gels like this one, but also leaving all sites. This is an experiment done in France I believe, a few years ago. Also in another site you have a non equilibrium state so that there are various energy consumption but by for example molecular motors that are moving the gel. There are several responsible response experiments here, and drag a colleague and measure the response and compare with the fluctuations, and see that, again, this spectra spectrum of the correlations and, and the response are not on top of each other. And there is a manifestation of activity. In particular these two nice papers, they, they also used this equality from Harada sasa to compute the entire production in the surrounding meaning because the entire production that you get from Harada sasa equality is the one of the non equilibrium system. So in this situation will be, you have a particle in a, in an active gel. So the active gel is, is spending energy, and you'd like to understand how much energy dissipated in this in maintaining the system alive. So here, they, they get for example, you see in the figure in the bottom right, 10 to the two KVT, so you can this relate to to how many motors we have in the gel, etc. We did this also with the head bundles. These are very tiny cells in the, in the year of the bullfrog, and we get also similar sounds up to thousands of KVT per second. The interesting model is in red blood cells. So, you know very well, but it's a red blood cell and you may not know so well it's that red blood cells fluctuate and the membrane is shaking all the time. This is called membrane flickering. And it was thought for many many years that this flickering is an equilibrium process. This was because of thermal fluctuations near the red blood cell that is shaking this, the memory. So if you have a beautiful experiment, like pick a red blood cell with colloids and perturb one of the colloids, apply a force, as you see on the top, top left in my slides, and measure the fluctuations without this periodic force, and the response to the small And you can see in the figure in the bottom that when the red blood cell is alive, for this response to not match, and there is a signature of life and entering production here. So, it is a result of this being applied in many biodegradable systems and that it's really informative and it can help to quantify energetics of the matter. So, okay, if there's any question, please let me know. If not, I'll go ahead and start with another topic, which is somehow related, but it's irreversibility and dissipation. So now, what I'll try to explain is what happens if you are not close to equilibrium but you are arbitrarily far from equilibrium. Sometimes I cannot explain what was going on fully but I can give you some celebrated and useful results about this topic. Okay, so I will use my own hundred and not so if you don't understand anything, please ask. So, this is the question, what can we say about small systems arbitrarily far away from equilibrium. There are some paradigmatic months which I went through my course. One is, for instance, you have a particle in a potential and you modulate the potential as a function of time. Like, for instance, you have a periodic potential with as a period L, and you are changing the phase as a function of time like this. So this will be the potential time zero, this will be the potential at a later time. This way you're doing work and the trajectories are also reversible so dissipation. We know since since a long time as a signature in irreversibility. Another way, it's having a non equilibrium stationary state so the potential is fixed, but you have external force pushing the party, you can have a ring, you, you are pushing the particle, or you can have more complicated like for example, you have two thermal bats in contact, and one degree of freedom for example the X degree of freedom of the particle is at the cold temperature and the Y is at the hot temperature. This you can build in an experiment and this is also an equilibrium state. All right. So, there are quantities that are key in stationary states, one is for instance, the net velocity so you, you, you make many simulations or you look at many experiments at X of t at the position of a particle, and you, you measure the average slope, this will be the velocity, which will reflect on also a net velocity for the entire production. So this will be called the rate of entry production, I call it sigma sometimes. Until now, and the stochastic thermodynamics, there was not a very clear understanding on how the sigma relates to irreversibility of trajectories. And the best insights were from Pligo gene on Sager, etc., which developed the Leand response theory. So they showed close to equilibrium, you can write the entry production as a sum of forces times fluxes. Okay, for example, if you have a force could be a gradient of concentration, this will generate a particle current. The concentration gradient will be always positive. In this case, when you have one flux, when you have many fluxes, it will be a combination, you will have this type of relation. Sorry, I have a call here. So, close to equilibrium, fluxes and forces are related linearly. So you can say that F is on Sager coefficient times the flux. And then this makes that the entry production becomes a bit linear form in the in the fluxes and this makes that the entry production can be shown to be positive because of this on Sager reciprocity relation. Okay, however, this is from the previous century. So what do we do in the 21st century with thermodynamics? How can we relate irreversibility and dissipation quantitatively in non-equivalent systems? We know something about this because I already introduced something in the course. And we know that entry production, sorry. But these two casting entry production associated to a single trajectory can be written like this. Both one constant and logarithm of the probability for a trajectory divided probability in the time reverse process of the time reverse trajectory. I explained to you that when you average this over many trajectories, you get a cool back libel divergence between the probability to see a trajectory and the probability to see the time reversal in the time reverse process. This from information theory, we know it's always positive and that, okay, it is zero if and only if the two measures are the same. So if and only if the process and the time reversal have the same probability for all the trajectories. This only happens in one case, which is in equilibrium. If you have an equilibrium system, these two distributions are both probabilities are equal for all trajectories. For example, in Brownian-Monson in a potential, you can show this. What does this say for irreversibility and dissipation? It says, you take this and divide by kB. And this is a quantity of this physical. So this can be the heat over the temperature. It can be the sum of the heat fluxes to different paths to the temperature could be the work minus the free energy, depending on the system you will have a different expression, different quantity. It is physics. However, on the right side, there is only statistics. This is just taking trajectories and calculating probabilities. This is a very big insight because it is relating thermodynamics with just data analysis, just data, just information. And this is very useful for experiments because, for instance, you can go to a biological system, know very little about it, get one variable and measure the reversibility for one variable and then with this you will get an information about the irreversible. Okay, so this is, as I said, a connection between thermodynamics and irreversibility. So, a very big insight was, okay, the result I gave you now was general, was for all types of stochastic process, but what we can do is to do a particular case in which we will assume the system is isolated. So, as I said, until now I was telling you I have a system that has trajectories and it's in a thermal path. Now I will say, imagine, if this thermal bath is at a fixed temperature, imagine if now I can see all the molecules of the system and all the molecules of the bath. This super system, it's an isolated system. It's very via Hamilton. And I can think about the following, the probability for an entire trajectory of the system and bath, imagine I can measure all of them, because this is a determinant system is deterministic. This is equal to the probability to see one snapshot. So if you know in a Hamiltonian system, you know the initial state, you know the final state and you know all the trajectory. So the probability for an entire trajectory system and bath variables position and momentum is the same as the priority of all the molecules of the bath positions and momentum at one snapshot in time. This is like making one photograph of all the degrees of freedom of the system. Okay, this, that I'm explaining is a very beautiful paper, it's called irreversibility the face space perspective by Kawaii, Parano and Mandembrock in PRM 2007. This was a paper finished before my, I started my PhD thesis after this paper, which is very inspirational for me. So this is the priority to have the positions and the, so this is the priority to see a given a snapshot of all the bath and system. What happens in the time reversal, so the priority for a time reverse trajectory will be related to the probability of a snapshot in the backward dynamics so this will be the position of all the molecules. And now minus t minus the momentum, be careful here because when you time reversal trajectory, if you see, for example, a bank of fish moving, when you they move backwards, they are reversing the velocity, they change the velocities. So this is the forward process of trajectory. And in the backward, we flip the momentum and we go like this. Okay, we are comparing the same time in the drive. So we are doing a snapshot. For example, your process from time zero to time 10, we do a snapshot of time three in the forward and a time seven in the background. This is what we do. So this paper they show that this is the only information you need. Well, it's not the only because it's all the molecules or all the momentum position at the, at the snapshot. This is the only thing you need to get dissipates. You can prove that you have the same information in my snapshot of all the molecules that in the full trajectory of the system. But you can reveal the entire production from a single snapshot that has all the information to this national has the information of bath and system. Now, and what you can also show is you can apply the chain rule for the cool back library. This means that knowing one variable as less information than knowing to. Okay, so if here you don't see all the bath molecules, all the momentum, etc. You only see one degree of freedom. When you get this an inequality, because this full a cool back library is greater or equal than the cool back library with one degree of free. So in general, when you do an experiment when you take data from a from a lab, you won't have all the, all the molecules. And you have to estimate the interaction from one snapshot, you will have this, and this will be a lower bound. This is something we very important in stochastic thermodynamics, most of the results we have in stochastic thermodynamics are bounds. So we doing the qualities. We say, there is an interaction of at least this quantity. Okay, you can. Yes, excuse me. I didn't get one point. So, before you said that we're dealing with Hamiltonian systems. And we're considering QMP as as project as labels for the trajectory of the whole system so bath plus system. So, how do we, why do we treat them as probabilities I don't see that because if we're following the entire system shouldn't be deterministic. Yeah, actually, okay. Here I wasn't clear enough because this. Okay, what they prove in this in this paper is, these are the system variables, not the bath variables. Okay. Okay, so they still have probabilities because they are, they are described by Bosnian issues into the minus beta times the energy. Okay, so here actually I was, I was not so clear but this should be all the values of the system of the system. Okay. Okay, thank you. Okay, I was mistaken. So you have system variables. If you, if you can measure all of them. Then you can track the reaction fully, you measure only few of them, you can only get lower bounce and this is the, the, the most common scenario in, in experiments I mean it's to cast the thermodynamics in general. So, going back to stochastic systems so let me, let me just say, and the general formula which was in terms of priorities of trajectories. Now I'm not saying it's isolated we just have the system, which is internal bath and the system has fluctuations. When you apply this equality to, for example, to the stochastic process imagine you have a process in two dimensions which has currents. Then you have a probability, joint priority for the trajectory in X and in Y so you have two degrees of freedom with follow trajectories. This will be a quantity that of course will be greater equal than doing the same thing only with X. So an important issue is, is this X informative at all. So it could be that even having one variable tells you nothing could be and here is a trivial, well, this is a here's an example. So when you have this model x dot equals to noise and why don't equals to cause X plus mega t plus noise. So, this is a 2D system that has a current, because this is a time dependent driving. And you see that the degree of freedom makes is a brown emotion. So it's equilibrium, whereas the degree of freedom y is non-equilibrium. When you compute the callback line of trajectories for X, you can see it is purely reversible when you only look at the X degree of freedom. If you look at why it is positive because it is irreversible in Y. So of course this inequalities work, but it can mean. So you have to take a degree of freedom that gives you zero entry production. So you have to be careful and take this in a realistic way in the sense that maybe you find a variable that contains no information about entry production. So these are the equilibrium variables. This is what happens actually what I was telling you before about the bath in the system. The bath is an equilibrium. So these variables do not contribute to the entry class. That's why I told you at the end, you don't have to take into account in the final result. Okay. So if you call screen and you measure the reversibility with equilibrium variables that will give you a signal and you will say the entry production is going to see them. But that doesn't give you anything new from stochastic thermodynamics. So you need to track variables that are non-equilibrium variables. And in particular there are some variables that are better than others, but this is a very long topic of research. Okay, so let me continue. And the next thing that I will try to explain is how can you estimate dissipation from single trajectories. As I said, well, you can look for example at one paper we did in 2012 in PLE with Juan Parrondo. The setup will be as follows. You have a physical system, this physical system is dissipating heat, the entry production and out of it you just get data. So you don't really know what the physical system, what the equations are, but you just get the data and you analyze the data. Okay, for people of you who are fans of data science, maybe you would like this problem. Okay, so you get the data and I'd like to know from the data what can I say about the introduction of the system without knowing much. Okay. Okay, one thing you can do is to infer the model and then measure the introduction model, but back then in 2010, we were not using any machine learning, nothing, just we wanted to, from pure stochastic thermodynamics principles, say something about the introduction of the system that produces this data. So a simple example is a Markov process. Markov process is a process which is jumping between different states, and we will assume it's nonically at steady state. So we will get, for instance, this type of trajectory and one can think of the entry production per unit time. So before you can also think about, and this we did a lot of research on this is interaction per jump. So every time there's a job, there is an entry production, and I will only read jumps in my trajectory. Some of the results I will show you are for this quantity, which is the interaction per step. So every time you are here you don't count it you just see this, then it happened this, then it happened this and so. So we don't take into account the waiting times, but the waiting times recently has been shown, they also contribute to entry products. Okay, so here is an exact calculation of the entry production for a Markov chain. Imagine a Markov chain you have a few states, and you wonder what is the probability. Okay, this will be the Kulba-Kleider probability for full trajectory. And probability for a time reverse trajectory. What is the probability for trajectory in a Markov chain? It's simple, it's the product of initial and then transition probabilities. This is in the forward sequence, and this is in the backward sequence. So when you continue to do this calculation, okay, I'm interested in the entry production per unit time, so I have to divide by n, which is the number of data, and take the limit when n goes to infinity. This is what what is coming actually from from. Okay, this will be total entropy, but now I'm going to try to do the entry production per unit time or per step, which is this divided by time, when time was infinite. So I try to calculate this, and now I realize that I have X2, it's 1, X1, X2, Xn, there will be X3, X2, X3, so there is always the pairs are coming, and there is AB and BA in the denominator. So you can write all this sum. You can write it in one part that is equal to zero, you can show very easily, and n minus one steps that contain p of X2 given X1 divided by p X1 and given X2. So what, when you have a Markov chain, the irreversibility, which is what I'm showing on the right, it depends only on transition probabilities. So you just need to know probabilities of jumps, which is this. Okay. All in all, all this complicated formula became just this for which you can show that it is just depending on the probability of seeing first A and then B. That's it. If you, if you have a trajectory of jumps, and you just compute the histogram or how many times we see jumps one to two, two to three and three to one, and all combinations. This, this callback library is just comparing all combinations of, of transition properties. That's it. That's the only information needed to know the entry production in a Markov process. Just jump probabilities. Okay, so with a single degree of freedom, if this is my full process, so I just have a Markov chain, and I know there is no, this is all the description of the system. But you can compute this very easily this B to which is irreversibility at with two with words of lens to, and this is very nicely being equal to the entry production is yet blue and the black. You can also go at higher orders and measure the priorities of triplets. And this is equal to sigma priority of x one x to three divided by priority of x three x to this one. And this is equally good when you have a Markov chain. And of course, when, when there is a lot of irreversibility. This starts to have a statistics lack of statistical problems is very difficult to sample all these triplets when, when you have, you are very reversing. This PR 10 years ago or 11 already. What is more interesting me for you is what happens when there's hidden information. So, imagine this is your mark of chain and now, let's say for example you cannot see, you cannot resolve these two states. This is equivalent of building. The other line dynamics, which is Markovian, and this is the observed dynamics, which is no Markovian. This is called often a hidden market chain. Here, the problem is more difficult. And what turns out is that you need more than statistics of jumps to capture the entry production of the regional system. So that, for example, if, if the new data is fourth order Markov, you need to go up to fourth words of length four, which is intuitive, but we can prove this by doing this analysis as was before, mathematically. And when, when you have a hidden Markov chain, you will compute this, this these sub case, which are irreversibilities of words of length k and see that it increases up to a level. And when you saturate this, this growth, and this will tell you the degree of Markovianity of your system in a way, but in principle, this is a bit of brute force so you have to do a lot of histograms and it's complicated. But and we can still reveal interesting things. And this is what I'm trying to explain you in the last minutes. So for example, okay, this is ready to this problem. Okay, I don't know if there is volume. Imagine there's a car and it's trying to start the engine and you can only see the position of the car. So the car is fluctuating it's going back and forth. But you know that there's dissipation because there is friction of the wheels with the with the ground know, however, there is this bush that doesn't let you see all the information of the process. So in particular, you know there is heat. There is a fluctuating trajectory. And you're missing something that is okay there's current in the wheels. So the wheels are doing currents, but you cannot see because you can only have access to personal information. This is a typical situation in biology. Of course, this is just a bit of a joke. Just a graphical illustration but in biology this happens a lot. You can see a molecular motor that is dissipating, you don't see the dissipation, but sometimes the motor is not moving at a very fast speed so it's difficult to guess if this is active or not. In particular, this is a very nice model you can do is a flashing ratchet. It's very difficult that is jumping. There's a potential. So it is diffusing here and like this and so on, and this potential is switched on and off at the rate. So there is there is a resetting of the potential from from a piecewise like here to flat, which is a diffusion. When you have this, I'm sorry, because it's tilted if you have it straight, this will generate a current in this direction. But you can apply an external force to this to the system and compensate the current of the ratchet with an external force and reach a situation where there is no net motion in X, but still there are loops happening and still there is dissipation. And actually you can see what this is from long time ago, but you can do this trajectories and they are very simple. The first thing of this model is it's very illustrative so you see that at the stall force, there is no current and the reversibility in words of lens to be zero. So you go equally likely left to right and right to left. However, there is Android production so how can we see Android production in a time series that has no currents. Android production model is is very counterintuitive, but we can do it by just looking at the three. The three is just the callback liner with information of triplets, for example of seeing one to and zero, and seeing zero to one that in the case there is no current. It reveals irreversibility enough to say that this is an active system under the central products. I mean a big improvement you see we increase by four or five orders of money to be three, and adding more data of longer words of more correlations gets you closer to the enterprise aspect. But of course, when you do the nine for example, the data becomes to discuss your three states, but words of nine become very unlikely to see some of them. So this is useful. And, okay, I just finished with giving you three reference. Okay, this is an application we did to experimental data with the bullfrog. This is also very nice paper, I highly recommend you to read because it discusses and symmetries and how to detect the arrow of time or the reversibility in correlation functions. The comparison function is even on the time reversal, but in this paper from the, I think it's the 86 or so. They said you have to look at correlations of extra time t, extra time t plus tau and extra time t plus two tau or three time. So this is three time correlations to reveal the reversibility. This is a was already an explaining this paper in the 86. Now we are doing a revival of some of the results in a different way. And also this, sorry for the mouse here. Very nice recent paper as well where they include also the effect of the waiting times in the trajectory. So what I show you until now was entry production per jump, but you can extend the theory to enter production per unit time by also looking at the irreversibility of this in the, in the waiting time, which, which is a highly retrieval as well. And I think this is, this is it from, from my side, I think it was much faster than what I expected. But I'm open to questions or doubts or anything. There's a question saying, so you are going to hire order decay to take into account. No more. Yes, exactly. Exactly. The system is not Markovian D3 is not equal to the two. And the more I can, I can share this with you. Very good question. Yeah, exactly. So when you are Markovian D2 already saturates the three is equal to the two there's no more information in three order correlation that into all the information is containing jumps. When you are Markovian, when you are not Markovian, you don't saturate it too. So you get more irreversibility when you look at three D3 and more of these you get D4. And it could be the covia situation where you never separate as well. This is the example I was giving now actually. Here, when, when you hear we are losing information because the, the full information is having X and Y. When we put what we are doing here is irreversibility only with X. The information we have is smaller and the resulting process is not second order of three or Markovian is even more. So we see the nine is almost saturating. This is, I think the fight on the nine so with a very simple projecting, projecting this X, Y to X, we already get a Markovian process with higher order. And this you can do also in the, sorry, also in this example. In this example, if you, if you group two states into one, the resulting dynamics. I'm not sure I will be terrible Markovian it's, it's not clear to me. So just this type of Markov chains and he said transformations generally do not just bring you to the next order of my community but brings you to higher order general. That's a good question. The questions. Any other question. Excuse me. I have a question on the first part of this today lecture. Yes. It's possible to say that the a functional you introduced the, when you talk about frenzy is somewhat related to the narrative theorems and the generator of. I must say, I don't know. There is a good paper I can, I can tell you about entry production as another invariant, which I can send you, but I don't think they look at it in this, in this way. So I must say I'm not sure. Okay, because the work calculation you showed us remember to me the, you know, the calculation you do in the field theory when you talk about another theorem. But I remember those so that the, the energy is the product generator of time translation. Okay, you know that there is a PRX paper, where they generalize to fill theories. So, probably you, you're right, but I'm not sure how, how they do it mathematically have to have to go into the details of the paper, but it's a good point. I can tell you a reference if you're interested. Okay, thanks. Thanks. I'd like. I'll send it in the chat after the question. Thank you. Okay, there's a question of look at this talk. Can you say then, can you connect your microphone and ask it. Yeah, yeah, of course. So, I was a bit confused or surprised by the fact that in the initial derivation of the Rada quality. We had a response function that was not zero even if the time of the perturbation was previous to the, to the time of the effect and usually due to you have not this, this phenomenon. So, yeah, but in the end when I take the limits I assume, I assume causality when you take the limit so you have to be careful on this. So in principle I derive it, I put it rough I put t minus s and they can be anything, but you must take into account with your t minus s, t is greater than s. And we use s minus t, s greater than t. That's why later in my lecture I put zero plus or zero minus. So it's, you have to be a bit careful on this, but, but you're right. So, in reality, there is no response from the future to the past. I recommend you to go to the paper because this is explaining more rigorously in the other proofs, the proof I give you was the fastest, the quickest, but not the most recovers. But this is a good point and this appears. Okay, so you take a different limit depending on the order of times. Exactly, exactly. That's why it appears are zero plus or minus and then the imaginary but so be careful with this. It's a detail but it's important. Okay, thank you. You're welcome. If not, we can also close this, we can discuss about the exam and this year's and promising. Yes, so I'm, I can create a room to discuss about that. So I'll stop recording now.