 Today's lecture will be, this is the seventh lecture, remember today we have two lectures and tomorrow is the last one. Today will be about close to equilibrium conditions so systems that are driven out of equilibrium, but the force that is driving the systems out of equilibrium is weak and there are very interesting results for this situation. Please remember that what I have been explaining was general, was for any force, any degree of non-equilibrium, and today I will focus on stochastic thermodynamics of systems weakly perturbed out of equilibrium or weakly perturbed out of a non-equilibrium steady state. I will also think about a system in a steady state in which we apply a tiny, tiny dependent force, right? Okay, so before starting, I have divided my lecture in two parts, the first will be a lot of math and theory and the second will be experiments about these results. Before starting I'd like to request if you have questions, if it's not a problem for you to switch on your camera this way, if it's not a problem of course, this way I can see who is making the questions and it makes this course much more human, all right? Fine, so I'll go now to share the whiteboard because I'll do most of my calculations this way. So, okay, so today lecture seven, I will start with a very well-known result in stochastic thermodynamics, which is the so-called Harada-Sasai quality. Okay, so this will be my first result I will discuss the Harada-Sasai quality. This is an equality that connects dissipation or end-reproduction with how much a system does violate the fluctuation dissipation relation. Okay, this is a very nice result. And, okay, this is found, for example, in papers by Harada and Sasa, in PRL, this was the first contribution was in PRL, Physical Revive Letters 2005. The first one is PRE in 2006. I highly recommend you to vote the PRE. To be honest, I've gone through both and I understood much better this one. However, this one was the first, which they introduced this result. And also a fantastic work also by Harada only, okay, this is also Harada and Sasa, and a fantastic work by Harada, which is in PRE in 2009. Okay. This is Takahiro Harada and Shinichi Sasai, if you want more information. All right, so I will show an equality for a group of systems, which is non-governed dynamics, but this is coming generalized to more situations and recently there has been also a generalization to field theories. Okay, so I start with a non-governed system, which is the following. The system is over-damped, there is no mass, or the mass is very small, there is friction from the bath, there is an external force, okay, a force which could be conservative or non-conservative. Then there is a perturbation, which we will call this one, epsilon, Fp. This is a time-dependent force that is applied in the system. Epsilon is a parameter that we would say it's small. And finally, there is the white noise for the semi-mass. Okay, remember, this is a zero-mean and autocorrelation, given by a two-gamma committee. Okay, I'm not being very, okay, remember, this is a delta function. All right, so what is the path probability? The probability for a trajectory in this model. The path probability, which I usually call it like this, BXT, or a trajectory, any trajectory. X zero up to XT. Okay, I'm assuming time is continuous. What is the path probability? This is my question. So, I already told you about this, because in the end, here, the stochasticity comes from the noise. So the noise is a sequence of Gaussian random numbers that come one at a time. And this implies that we can write the conditional path probability, like this BXT, given X zero is the initial state. This is the probability of the full path given the initial state. Okay, I hope you're fine. BXT, this is equal to the oxide times a normalization constant times exponential of all this, which is the following. This minus, sorry, I made a mistake here. This you can find it in the book of Schulman on path integral, you can also calculate yourself. I just give you the result here, integral zero to T of this minus this minus this squared. This is coming from the fact that this is Gauss. So this will be gamma X dot S minus F X, that is, minus epsilon FPS to the power two, and then DS. Okay, this is the probability for a trajectory in terms of, okay, remember, this is a way, this is in reality a limit because we are thinking about continuous time is the limit when, when the delta T between every two observations is zero of the X zero up to the XT, and this is the same with noise. So this is the XI zero, the XI. Okay, but these are just the difference of what it's the most important thing here is this fact here. Okay. So, having said this, so this is the probability for a single trajectory. Please, please go. I say that S denotes trajectory. Sorry. No, okay, as is a time instant. So I'm integrating is a dummy variable. I'm integrating as from zero to T. Okay. This is an internal time variable. So I have a trajectory. Okay. Okay. Any other question. The question about the Jacobian. Yes, Jacobian comes out of the race within this too. And it also comes here. Okay, but it is not. It's something that is independent on on the trajectory itself, but if you want to be extremely rigorous. But for what I'm going to do next, the Jacobian is not playing any role. This is the main point. But yet, you just take into account the Jacobian exactly here. It's a normalization factor that depends on the system parameters, for example, on KBT on gamma, and it comes also from the Jacobian. You should put here the determinant of the Jacobian in transforming sideways. Okay. Other questions. Okay. Sorry about that. I can't follow the chat. I don't know why, but I'll follow the chat. Okay. I have the time now. Okay. Yes. Good. So now the question comes on. Given this, this system is larger and dynamics. What is the response? Response function. Response. And I will calculate the response function for the velocity. Okay. First of all, I must say that I'm following very much this PRP paper that I'm saying. And in particular within this PRP paper, there is one section, which is appendix B. So when you read papers, don't forget appendices are very important, especially if there are new results. And you want to look at the mathematical proofs in appendix B, which is called quick derivation of the equality. Okay, so I'm following, I'm trying to follow the shortest path for the quick derivation of the equality. Okay, what is the response? The response function is defined as follows. So what we will do is to compute what is the average velocity in the perturbed system, which is this. So the average velocity will be x, x dot t for value epsilon in the perturbation will be nothing but okay is the average of x dot. So the average of x dot, you can write it as this is the path definition of the path probability for a path. Okay, this is the way we write explicitly the average of x dot. What is the response function? The response function can be found by doing the following by making the functional derivative of x dot t of epsilon with respect to the external force, so the perturbation which is this. Okay, so I will calculate what is the functional derivative of this object, which is x dot, with respect at time t with respect to a perturbation at time s and be very conscious that you can have a perturbation at time s, which is before time t, and this will impact on what happens later at time t. Okay, that's why I'm putting here two times in this. When you do this, you can figure out and it's not so complicated to show that this is equal just you can show it from here, because you can do just the derivative with respect to this. This is an exponential, so when you take a derivative of an exponential, and then you say epsilon f is small, we will then say epsilon is small, you just get from the derivative, just the argument in the exponential. So, all you know what you get when you do this derivative is just this x dot at time t, okay, is this. This is the first product with what comes here without the epsilon, so it's just x dot s minus f xs. Okay, this is just this derivative, this functional derivative. And then we should be careful on what does this average mean, the average is done in the ensemble in which we have epsilon, okay, this is the derivative how much the velocity changes the average value changes when we are setting this value epsilon. So this, that's what we don't like this. Okay, this is the first calculus. So once we know this, we can define the response function as follows the response function at t minus s is excuse me. Yes. So, it's starting with. Yes, I'm using some notes, okay, because here to, in order to have this nice form and that this does not depend on the trajectory. It is more convenient to use the notes. Yeah, and I can, I can integrate this equation and start the notes. Perfect. I can use two different types of integration rule, I am using some notes just because for the sake of calculations. It's going to be much easier because I would just use standard rules of calculus in in migration. Okay, so I'm using some notes. Okay, thanks. Okay, so what is the response function, the response function is just the limit when epsilon goes to zero of this quantity. Okay, how much the velocity is changing when you have a very, very tiny and response. So we will do a limit when epsilon goes to zero of the delta x dot t with respect to an epsilon f p s. This is what I want to calculate when epsilon goes to see, you can figure out that this is very, very evident that way. When epsilon goes to zero, what you can get from here is just the same thing, but here at zero. So we will do this average, when there is no force. So this means that we will have beta over two gamma of x dot t. This is product x dot s minus f x s. Okay, this is in the zero. Okay, so without any right now, I can write this in the following way I will use now. And then the so-called velocity autocorrelation function CV t minus s, which will be the following quantity, it will be the stochastic velocity at time t minus the new velocity times the stochastic velocity at time s minus the new velocity. And this is in a system zero. Okay, for absolutely what I'm just saying what is this V is the average V bar is nothing but the x t at zero. All right. So, when you write this, this becomes x dot t x dot s. This is the first term, which is what it's is appearing actually up here. So, this is appearing here is that x dot s is here. All this doesn't know it. So that's why I'm not putting anything was let me put a circle just to be you want to be very picky. We just put a circle here. Okay, first thing is this, and then there is x dot team with the x dot s with the. And then there is V with V. So x dot average is V. So there will be minus V square minus V square and plus V square. So what you get at the end is minus the average square. And this is in a system. So you see, from this equation, I can continue up here. I'm using this equation. And what I get is that x dot t x dot s is the correlation function of the velocity plus V square. So it becomes beta divided by two. And then the square plus C t minus s. And then we have minus beta divided by two gamma of the rest, which is x dot t. And then we have minus F. Good question. Why is the average velocity dependent off time. Yeah, good question. But you see, when you put here epsilon to zero. What you have is a system that has no, it's a stationary state. What's happening here is that you have a force that depends on X. So it's like a potential doesn't change in time. The particle which is doing like this, but then you are averaging with the distribution of this, this potential. So if I average over many trajectories. This won't depend on time. So I'm doing this stationary average of the velocity. It will depend on whether this is just conservative or not. If you have a particle in a ring, and you do X dot X dot T over many trajectories, you will have a net velocity. If you are only in a potential if this is just partial view of partial X of you, this average velocity will be seen. That's this question. Okay, I mean, I still am not clear, but I think I can try it. Okay, okay. Okay. Anyway, okay, so this is the average of a, over many, many, many infinite trajectories in a time T of a system that has no time dependent. So it's not. There is not here as an independent driving is either an equilibrium system or a non-equivalent steady state. This is very important that this process for epsilon equal to zero. I just remarked this for epsilon equal to zero. The system is either an equilibrium system, equilibrium dynamics or non-equivalent steady state. Okay, this will depend on the nature of it. If F is just the derivative of potential, so F is just minus partial X of you, we will have an equilibrium dynamics. If F is the derivative of a potential plus an external force, we will have an, the amplitude system is a non-equivalent steady state. Okay. This is for epsilon equal to zero for epsilon positive, we have a given system so we have either an equilibrium system that is driven or a non-equivalent steady state with driving. Either equilibrium or non-equivalent steady states, steady state plus drive. Here, X dot of T depends on time. So this partial T is non-zero. Here, because we are in equilibrium or non-equivalent steady state, the net velocity equal to zero. Okay. So this satisfies a little bit the question. Let me advance because otherwise I will not be able to explain the key result. So, okay, so we continue with this. Just be aware that here there was a gamma that was killed in this part. Okay, in the second part it's surviving. Very nice. So this is the response function at T minus S. And now what I will do is I will sum the response function at T minus S plus the response function at S minus T. Okay, what is the response at time T if I apply a pulse at S and plus the opposite. So if you take this equation and sum T minus S and the case of S minus T, you get first of all the following beta average velocity square plus C T minus S. Okay, here I'm using the fact that the velocity velocity autocorrelation function is even an entire reversal. So C of T equals C minus T. So C of T minus S equals C of S minus T. That's why there are two factors here. Right. This is one. And then the second part has TS and ST. So we have minus beta over two gamma. X dot T, ST, okay. STATONOID F says plus X dot S STATONOID F XT. Okay, so this is what we get and both are in the unperturbed system. This is the average of X dot times F. It's not X dot. So this is something else very important. Okay, so now what I'll do is I will do the limit when S goes to T from below. Okay, if I do this, I get something like two are zero. And because S going to T minus or S going to T plus S going to T basically makes that T is equal to S. So we are evaluating the response at time zero becomes beta B square plus C zero the autocorrelation function of the velocity. These are correlation factors of velocity, very important. Minus and then T equals to S on the right is just beta over gamma X dot T STATONOID F XT. Okay, at the same time. And then this is at C. Okay, so I don't need this part. And this is the main question. So in reality I could stop here and tell you this is a fantastic equation which relates a dissipation with breaking off the partitioning situation. But if you see this, this equation, you won't see it very intuitive. So just keep this in mind. And let me just introduce something quite relevant that is the Fourier transformers. So, and I think there was a there was a question about it. So let me try to explain now. Okay, so this is our main equation and I'll try. I don't know if I can copy this. I will make a try if I can copy this equation. I manage. It's always quite complicated to learn a new technique every day. Edit. Now we cannot make it. This is all so fun. Okay, so I take this and I copy. No, I cannot copy. Okay, so I go by hand. The result was the following. The result was to R0. The response factor of the velocity time zero and equals beta. You have square plus C0. Not correct. C0 minus beta over gamma of X dot t. Okay, in the system. This is in the system. This is in the system. This is the equilibrium correlations correlations in the system. This is the response when you weekly perturbed the system, and this is an average in them perturbed system. All right, so now I introduce the Fourier transforms in this in the same way. In the paper, Fourier transforms. So I'll first define, for example, the Fourier transform for the autocorrelation function, which is a, I'll define it as follows in this convention DT of 2 pi minus infinity to plus infinity. DT of Ct. E i omega t. This is in the way they define Fourier transforms in this. Okay, just be aware of one thing. That is that, and this is in principle complex, because this is real, and this is a complex number, but C of t is symmetric. So it is even with with respect to the reversal and this is cost omega t. Plus I sign omega t, which is all. So if you multiply, sorry, even by even. Yes, sorry. Okay, someone is calling. If you multiply even by even, this should be no zero and if you multiply even by all the integrals will be zero. So in the end, here we will just get something is DT of 2 pi. C of t. Okay, this means that city le omega is real. That's only a real path. Okay. This is the first thing. The second thing I want to say here is about the response function. So the response function the freedom form. I'll call it like this, the same way BT over 2 pi out of T. I omega t. So it is not to guarantee that the response function is real in general it has both real and imaginary parts. This is very important for you to realize. And when we were doing this limit here. When we are doing this limit, we were doing as going to T minus in reality. When we hear when we do the limit as going to T, we will have in reality, this is not very well explained, but this should be towards zero realities are zero plus plus or minus. If I am going to be very, very mathematical with this. Okay, so when you do this, and you have this. So, of course, now the R t minus s will be the inverse freedom form. So it will be the omega over 2 pi are put it on from omega e minus omega at t minus s. Okay, this minus infinity to plus infinity. So what we are interested in is are at zero plus plus are at zero minus, and this becomes, you can show very simply, this becomes the integral the omega to pi. Please realize that here t minus s t minus s, if we put the minus is equal to zero is the integral of the full is this integral with e to the zero which is one. So the R zero plus plus or minus becomes this times two, and this is the real part. Okay, when I put a prime is the real part of the Fourier transform of the response funds. Right, this is one thing. And another thing is from here you realize that C tilde, sorry, not C tilde, C of zero. The result will be this one, but now I'm going to write it in a more fancy way, because I'm going to use this free transform and see zero is nothing but the integral the omega to pi of C tilde omega. Okay, so, of course, this is, if I put here, omega, sorry, t to zero. It's not here but in the inverse free it's the same thing as here, you get this one. So, this means, this means that this part. This part here. All this part I will change it by this. And this part. By this. And I'm just missing one part for the quality which is this, this will be a X dot F. And it is not so difficult to show that X dot F. Nothing, but mind the heat dissipate in the system. Okay, in the amperter system. Of course the amperter system. The system is either equilibrium or non equilibrium state state if it's equilibrium. Here q dot is zero. There is no dissipation of heat. If it's only clear in the state it is not q dot is negative. Okay, so this is equal to this. It is not so, I mean it's not just one and I need to prove this because what I showed in my previous lecture was that this is equal to minus F X dot. Okay, but you can show that this is equal to X dot F. And I leave it to you as an exercise. This is a nice exercise. So, you see that in this equation we have R zero, which is response C is correlation and this is dissipation. So, when you put all of them together, you get this very beautiful equation, which is the form the heat dissipated in the amperter system is a pre factor which is the friction coefficient times v squared plus the integral from minus infinity to plus infinity of the omega by two pi of the correlation function of the velocity. So, we have omega minus two KBT, the real part of the velocity of the freedom of the response function of the velocity and in the aspect. Okay, so look at this equation it is really beautiful and really insightful, because it says the following. The equation of the amperter system is related to two things. First, the net velocity, this is like the drift, like gamma v square is when you have a particle in a fluid moving at a constant velocity, its dissipation will be gamma v square. This is the naive path. And this is a measure of the extent of the violation of the fluctuation dissipation relation. So here is an equality connecting dissipation, because this at the end in the system is the Android production rate in units of KB connecting with violation of the fluctuation dissipation relation. So why am I saying this? Well, because there is, and when we have q dot equal to zero, which will be equilibrium systems, this implies, okay, also the V bar is zero. And we have see the omega equals to KBT r prime omega. And this is the so called fluctuation dissipation relation. And it was proved by Kubo. Quite some years ago, you can find it. Okay. Another Japanese Japanese are very strong in this topic report. This in the year 1966 is a very important and classic result in statistical physics. Okay, so if you go to the paper of Kubo, you won't find it written like this, you will see here the imaginary part, but this is because he writes, and very many people write often the correlation function for for X instead of for B. So that's what it brings me to the conventional conventional formulation of the formulation of of the fluctuation dissipation relation. In fact, so what we are doing now is auto correlation function of velocity can be written as this for the transform of V, for the transform of V at minus omega. This you can show that this related also today. So we have a return from effects in this way, minus i omega, return from minus at minus on. And this shows that this is imaginary unit i times minus i is one, and then omega squared, so it's omega squared x to the omega x to the minus omega. So this means that the auto correlation function of the velocity at omega is omega squared, the auto correlation function of x at home. Okay, this is a very simple result to prove another analogous result you can show is that the response function in the velocity is. This is what what just is behind all this is just i omega, the response function in x at omega. So this implies that the imaginary part of the velocity to creation function is omega times the real part. So this is the actual function in x. Okay. Yeah, I think I made a mistake here. I think, okay, this is the real part. And here, okay, here we are relating imaginary real, and there's the same analogous with real and imaginary. So all in all, if you want to write this one. You can find out what happens in kubo's paper kubo doesn't write it like this, who writes it for x and for x, the conventional formulation is like this, c x omega equals, instead of two kbt will be an omega squared to kbt divided by omega squared imaginary part of the response function of x. Okay, so you can also write her as as a, instead of in terms of response function of x of velocity with response function of x, the only difference will be that here, you will have to put x, and here, omega squared. This is the response function of x and imaginary part. Okay, so if you go to a literature, you will find both formulations and they are q. All right, so with this I am done with the result and I don't know if I have time to the other concept I wanted to explain today, which is also related to the question response. Okay, I'll try to go through and if I don't manage and I explain you more later in the second part of the lecture. Okay, this is a very also exciting topic, which has a very funny name, which is the following, is the Frenzy. The Frenzy is a concept introduced by different authors and, for instance, Christian Meijs has now a review of this topic in the physics reports. 2020. And there are also very good papers about this also called, okay, this is now the most accepted name but there are other names such as traffic. There's also activity, etc. Okay, and there are other relevant authors in this, in this topic, which, for example, Marco Bayesi. He has very, very beautiful papers in this topic. Why not as well. Was the PhD student of Mr. Meijs and Urna Basso as well as very nice papers. I did one paper with Pierre Paolo Vivo as well. So there is growing community working on this part. And, okay, let me try to explain what it is. So, I told you before that a priority for a trajectory has a form, and typically we can write the priority of any trajectory as follows. The priority will be so-called action will be a functional of the trajectory and P0 will be the probability of the same trajectory in another process, it will be a reference process. Okay. So, in other words, I will define AXT as the algorithm for the probability in a reference process to see a trajectory divided by the probability in the process of my interest to see this trajectory. All right. So, one particular case is when this one, when this one is the probability in a conjugated with the time reversal. If we do this, A will be the entry production, but this is more general what I'm going to explain. So, first of all, I can show you that one can, if you want to do an average of XT, X of time T, this will be nothing but, well, in the measure P, this will be, you can do it like this, average of all trajectories of X of time T probability of the trajectory. Here it could be something else, it could be a function, it could be anything else. And you can also write this as the average of XT with E to the minus A. Like the way you change measure with P, C, okay, XT. Then you can do the average of anything in one process. And using this A, you can do the average in the other process. In other words, in other words, A is just a change of measure. Now it comes, what is the so-called frenzy? So what I'll say is that I will split A into parts, one that is even under time reversal, and the other which is odd under time reversal. So, I will say that A, it will be D minus S divided by Q. Okay, S is the same thing I explained you before, which is the entropy production, and D is this new object that is called frenzy. But actually, instead of calling D, because I use D for diffusion coefficient, let me use another word, another symbol that I like more for this object, which is phi. Okay, phi is the frenzy, and S is the entropy. Okay, so what are the two things that characterize this? So the entropy, as we said, is a functional that is odd under time reversal. So you measure the entropy production of the time reversal trajectory, you get minus the entropy reaction of the original trajectory. However, so this is the entropy reaction. And what is the frenzy? Frenzy is the symmetric part of the action. Okay, I did everything with D, and now I have to trace back. The frenzy is as follows. When it applies to the trajectory, it takes the same value as when it's applied to the time reversal. This is called the frenzy. Okay, what is called frenzy? Well, I can really, the best is to look at examples, but it is because it's related to the number of jumps that are happening in a trajectory. So the entropy is related to the reversibility, but frenzy is related to how many jumps in any direction are happening in your trajectory. Okay, but this is easier to see with examples. Okay, so now, knowing this, what we can do is compute the probability for a trajectory divided by probability for a time reversal trajectory in a non-mechanical state. Okay, this is very easy because now we say this is e minus phi xd minus s xd divided by kb. And now divided by e of minus phi of the time reversal trajectory minus s of the time reversal trajectory divided by kb. First of all, phi of xd and phi of xd reverse is the same. So this goes away and you just get an e today. Okay, I think I was missing a factor two here. I'm missing a factor two. I think I mistake, I am mistaking the sign here. I'm mistaking the sign here with a plus. Okay, I think with a plus this becomes s xd divided by kb. So, and this brings back to the definition that I introduced in my course. And this is s top, okay, s xd is kb times the logarithm of the probability for a trajectory divided by the probability for time reversal. Okay, this is consistent with this. All right. However, what if now you make the product that you do probability for xd time. Yes. Are there results in the conventional, the same as Kramers-Kronig relations. I must say, I don't remember. I don't remember which were the Kramers-Kronig relations. Oh yes, they are. Who is being disposed to you? Okay. Good to know, but I don't, I didn't remember this. Good point. Thank you. All right, this time. So now if you make, instead of the ratio, you make the, if you multiply this time this, the under reflection cancels because this is equal to minus s xd. And the only thing that survives is the frenzy. So you just get minus 25 xd plus. Okay, so in other words, the frenzy, it's for a trajectory is related to something more weird, which is minus the logarithm of the square root of p xd times p xd bar. Of course, you see, this doesn't make any physical sense because you see that the distributions, path probabilities have units. So the only thing that makes sense is the change in frenzy. So if you do this, the change of this quantity, you do d. But as many quantities in, in, in thermodynamics, if you do d at t minus d at zero, what you do is basically you will have minus the logarithm of the square root of p xd, p xd bar, and then p x0, p x0 bar. Okay, so things that make sense are, make sense are changes in, in frenzy. Okay, so this is not so important as what I'm going to say next, which is that the fact that, okay, as I said before, the average of any functional of a trajectory, p xd, in the, any, with, with, with measure p, so we have a physical process, we want to know the average of the work, we can write it as the average of e to the minus a xd. So this is the action times the same functional in a reference process, whatever process we want. Okay, and this a is defined between p and pc. This implies that you can show this easily that if you have a weak perturbation, you will have the following that this will be a similar or equal to. So we'll develop this. Sorry, there's some noise in the background. We'll develop this for a small. So then this will become one minus a times omega zero. So this will be and omega at zero minus eight. So close to for x and small, this will be the first contribution to the response. And this you can show if you do, for instance, the response is in this way, the epsilon of omega xd at epsilon. And then this will be the limit when epsilon goes to zero of omega, evaluated a trajectory in epsilon minus omega xd in the amperter system, divided by action. This is the definition of the derivative of how much it changes an observable when we have a weak perturbation when absent. So from this equation in the middle, this is similar or equal than minus the functional. Times a xd. This is trajectories. In. So this means that we can write the response. We're studying this way as one half. Omega times s s prime epsilon in epsilon minus omega times five prime epsilon. So it means that there are two contributions. This is the derivative with respect to epsilon. There are two contributions. One is an entropic entropic term. And the second is a frenetic term. It's a term that is proportional to the frenzy. And this is very important. It was a very important insight of this theory that entropy and frenzy for the response of any observable are equally important. One is one half and the other is, it's a factor one. So for instance, if you take this example, omega goes to one. Okay, this is the most silly. Edgar. So I think that is a question so. Regarding the absence of the a should be proportional to absence of course you should have. Epsilon time a no in this formulas and not necessarily not necessarily because. Okay. Here you should have the A. The epsilon. Yes, yes, yes. Okay, that's the point that's the point. Sorry, I forgot that you're right. You're right. In the same way we were doing the derivatives function derivatives as here. Okay. Yeah. Okay, that's, yeah, that was a typo. Thank you. Very, very good point. Okay, that's why here that is s prime of some these are functional and very active perspectives. Omega equals to one, this is one valid functional is the most silly you can think of this implies so of course the derivative of one is zero. This implies this equation. Okay, this is the central equation from this theory of, of frenzy. This is the key result. So you put seat one, the derivative one is zero, and this one, but what you get out of it is the following. You get that the entropy minus the frenzy. You can show this in a quality very simply. Therefore, you have something like this in the systems get very equal than five. So it is like a tight bound to the entire production tighter than the second one. It's not only that the entire production is getting zero, but the entire production is also greater or equal than the frenzy divided. This is sort of a second. Okay, but let me go further because sorry Edgar. Yes. It will be twice the frenzy. Yes. Yes. I made a mistake here. Yes. Yes. Yes. Exactly. This is twice. Correct. This should be twice in the way I defined. All right. And let me just wrap up with this. This is probably one of this and thought as the most and the best application of the results, which is when you use as omega, you use a time integrated current, which will be called gay. This will be a time integrated, integrated current. It could be, for example, the position of a colloid could be a work up to time t source of time integrated current because you are summing the work over a different time intervals. Heat into reproduction, etc. It could be also your mark of process with four states and you count, for example, every time you jump from one to two, you count plus one. And when you jump this way, you count minus one minus one. This is also time integrated current. So you will have jumps when this is happening. And minus one when this is happening. And at the very end, if there are drifts, you will have this will be a time integrated car. You can apply the same quantity and the same relation sorry to the response of a current, and it will be written. So in the same way. So you will get this same equation, but now instead of omega, you will have any current for Markov Markovian system. The response of a current will have this term that goes with the the entropy and this term that depends on the furnace. If the current is the entropy you will have here a sort of an S square. So if, if J equals to S, this is a positive contribution, but this can be negative. And this is very important when you consider examples and one example that is very much. If you have this J, I'm going to simplify notation very much. You will have the J with respect to epsilon is J as prime divided by two minus G. Okay, so why this term is important. This is being discussed in the last years. Very nice papers one I can highly recommend you with examples. This is a version of one of us and mice, one of us who increased and mice. This is not in a very well known journal but it is very, very well written, we like it is J. I think it's from an Indian conference. J fees, come cities. In 2015, some very, very, very nice paper, I really like the examples, it's good for you to read it, because you will see the fancy can be easily measured in some modes. And, and then discuss for instance, the following system which is an active matter system. For instance, you have a system like this where there are some obstacles, and you have some swimmers that are trying to pass these obstacles and they have a drift. These swimmers are of course in an active, sorry, in a thermal bath, but the swimmers are managing self proper particles. This is paradigmatic model of active matter. This type of dynamics can be described with simple stochastic processes actually. And what happens here is you would say, okay, what happens when I increase, let's say the field. So imagine you put, for instance, a concentration gradient of food. And here it would be actually the opposite. So there will be concentration of food along x like this. So bacteria would like to go where there is more food. Then, and this will define a parameter which will be the slope of this, of this concentration gradient, and this will determine the net velocity of this, but you know, at least, if there is no this objects, these obstacles. I think that when I put more and more gradient of food, the bacteria will move faster, but in some experiments has been shown that, okay, first the net current of bacteria. When you increase epsilon increases. This is actually the effect of this part. You can, you can see, but then at some point when you. This will be another situation with higher value of epsilon or concentration gradient. It can happen that there are there are crowding events. So the bacteria are not helping each other and they, they are and they are obstacles to themselves. So it can happen that there are trapping effects here. Okay. The bacteria are passing. But when there are many, many people trying to go, there is here a trap. So they are trapping effects. These trapping effects can be captured by the simple models and you can really go to this paper and check an example. And this is a manifestation of of frenzy because here, there are a lot of jumps in all directions, which can be quantified with a with a quantity of this. And this generates that when action is large, you can have this type of response. So this is called. This region is called the region of negative. This is called the difference and response response, and it can be understood the best way by having this phonetic term in the, in the description without this. It is not possible. So this type of dynamics will have a monotonous increase. This would be without tendency. This will increase the current like this, but never show distance effect. So this is very, very important for active matter system system to the extended volume, etc. Okay. Again, let me just remark that here there was a field, so it's like a field pushing in the side. Okay. I don't know how much time they have, because I wanted to show you experimental results. Some slides. Maybe we can take a break of five minutes and then a good point now. Yeah, it's a good point. It's a good point. Let me know when do we stop the recording just to be sure. Okay. I stopped the recording now.