 this one. When you click on that, it is closed. It's a better sound there when you're in front of that. Sorry, how much work do you have? But now it's, uh, but why do you have seven chats and I only have one? I only have four. Sorry, I only have four. I don't have any chat here. Seven. Ah, when you click on it, when you click on it to read it, the notification will appear. So that's why you have no computer. Ah, the old messages are, you mean? No, they are not the number. Yeah, but I only have one message. I only have one. When you click share screen, so you're sharing your presentation so we can't see any other window that's open in your computer. Yeah, yeah, yeah, I know what I mean is that I don't see the chat. If somebody, I only have one message. Yes, I see. And three, at least seven. Well, anyway, if somebody wants to ask something, please use the microphone or use the chat, but somebody. We want to see, we want to find the minimal work to change a memory from a state to another state. Okay. And so we have a powerful tool, which is this one, the energy of the energy. Remember, the energy of the energy is the average energy of the memory minus t, the entropy of the memory, the shadow entropy. So we are going to apply this. And this is the minimal work to go from m to m prime. And m is a random, is the random, the probabilistic state of the memory and m prime is the final problem. Yesterday, I said that I will do the proof of this, but I want to do the proof just to because there is a conceptual issue that tells you why a memory is what, what are the requirements for it. This is non-equilibrium. This is to connect to non-equilibrium states. We will see why a memory is non-equilibrium. But you can imagine in the, in the memory something that can be in two states. So the curriculum, if the two states are, have the same energy, the curriculum would be to be one half, one half, but our purpose is to, the purpose of a memory is to be either in zero, in one, is this one? Yes. I think it's the imperfect that, no, the premise that I kicked out of. Yes. You stopped sharing screen for some reason. No, no, no. It goes out of the soup. I get out of the, of the meeting. Okay. I don't know why. Okay. So the, today I would like to do this proof at least in a very, not, not rigorous way, but to see one of the problems of this. This is non-equilibrium states. So one would think if you, if you have a system out of equilibrium, the system immediately tries to relax to equilibrium. So one can, one can say, is this true? This is the minimal work or I can, I have to do very fast transformation. Let me, let me show you how, how this formula, how this formula works. Remember, this is the free energy, the non-equilibrium free energy. Suppose that I, I, I start in one state. This is my system. The blue curve represents the state, the probabilistic state, and the black curve is the potential. So you remember that in equilibrium, the, the, and rho, rho is the, the probability density and H is the potential of Hamiltonian. It's a Hamiltonian. You know that if a system is in equilibrium, if rho is exponential of minus, it's not here. If, if rho is the state, this is the definition of equilibrium in statistical mechanics. There's a temperature T. This is the normalization constant and beta is the inverse of the temperature. So if I have a potential like that, the, the equilibrium distribution is like the blue curve. This one, the wrong work now. And the free one, but if this keep me out, keep me out. Suppose that you are in non-equilibrium system, in non-equilibrium state. So this is the potential, no? And in this potential, the equilibrium state is that the particle is the probability, is this one here for the probability is higher in the minima, in this case in the minimum of the potential. So this is the equilibrium state. And this is non-equilibrium. In this diagram, here is non-equilibrium and here is equilibrium. So if you just let the system relax, the system will start to, here the system could be a bunch of particles or a single particle and then you repeat the experiment and you plot the histogram and this is the histogram of it. So you can think of the two possibilities. So you start in non-equilibrium state and then the system relaxes. In a relaxed, a relaxation is an irreversible process. So you dissipate heat and well, you're not necessarily dissipating heat. That's the entropy of the universe increases. How it's irreversible. So you cannot, you cannot invert. If you want to go from here to here, you have to do work. Because from here to here is your work. And here, from here to here, you don't do any work, but you don't extract work. You don't do, it's only, there is only heat and entropy production. Okay. And now think of this transformation and doing the following protocol. Now you manipulate the potential and you do the following. Immediately, instantaneously, you change your potential to this value. Look that this is the inverse of this. So if I, if I manage to create this potential, the system is immediately in equilibrium. So I do this and now quasi statically, I go from here to here. If I do this, the entropy production is zero and the work done to do this is precisely the difference of free energy. And this is the minimal work. So this is the optimal process. This one, if I want to do the minimal work or extract the maximum work. And, and this is the process that achieves. Remember this important inequality. This is minimal. So it is the work is always bigger than this. The minimal work is achieved when I have this protocol. This protocol is not, not always impossible. This protocol. But if it is possible, I can extract the maximum amount of and I connect a non-equilibrium state with an equilibrium state. This is important because when you have non-equilibrium, you have non-equilibrium. It seems that you can, that there is an unavoidable potential production. Here is not the case, but why? Well, you can say, well, this is a cheap trick. It's just, okay, you start in non-equilibrium, that immediately you go to an equilibrium state. So you start here, but immediately at time t equal zero, you go to equilibrium and now you apply, you apply equilibrium. So this tells you, okay, you are cheating because you cannot apply this formula in non-equilibrium because what you are doing is to tune everything to turn the system in equilibrium immediately. And this is true. This is true for this naive picture. And you can say, well, for instance, if instead of doing this immediately at time t equal zero, you wait a little bit, there will be a relaxation. So it seems that this is impossible to achieve unless you immediately go to equilibrium. And you will say, ah, so I claim that this is very useful for non-equilibrium states, but you can say, well, it's non-equilibrium, but it's okay. Well, you are partly right. This is true that there is a case when this is useful, even if you don't instantaneously change to equilibrium. And it's precisely in memories. It's precisely, and this occurs when there are a huge separation of time scales. Suppose that the relaxation occurs in a time scale, which is very large, centuries or the age of the universe. Then this relaxation would never occur. Suppose that this relaxation occurs in a scale of centuries. Then you can do this process. You need to be very fast. And this is the idea. The idea is that, and this happens in a memory. In a memory, if you think of a memory or at least the typical memories, there are huge separation of time scales. And this is what I'm going to tell you now. But remember that with a huge separation of time scales, this formula, that the work is the difference of free energy, can be applied even without the need of such an artificial problem. So now let me go to information devices or information devices. So what is memory? What is the memory? The memory is something, it's a physical system that can adopt different states. In the case of a bit, it's very easy. It can adopt two states. And the states are somehow equivalent. And in DNA is the same. What DNA can, each space can adopt four possibilities. The machinery of the DNA works with any of the four. So they are equivalent somehow. And in a computer, each bit can be zero one, but they are somehow equivalent in the sense that they can be manipulated. And the second property, so they must be equivalent. And the second property is that they must have a long lifetime. Because in DNA, it's very rigid. You cannot, mutations are rare. I mean, in the computer, we don't want that the bits are flipping in a short time. We need a very long lifetime. And here you see the separation of time scales. The typical model is a two-way potential. This could be the register of a memory storing one bit of information here. And you see that, okay, let me finish. The system can be here or here. And in this barrier, it's much larger than KT. You know that the jumps are very rare. The typical time to jump is exponential. The energy by KT, this can be 10 to 20 years. So you say that the memory system has a very long lifetime. That means you can be in the same space for a long time. You can save the data for a long, long time. These are passive memories. Remember that CPUs are the active memories. This is the one in the current. This is passive memories. And the passive memory is that this could be the magnetization. I'm not an expert in computer technology, but this can be magnetization or whatever. It could be for DNA as well. This could be for DNA you have four wells. But this is one bit. And this must be very big. I mean, much bigger than KT. It's enough to be maybe 20 times KT. And then the time to go from here to here is an exponential of 20 times something. It depends on the time. In any case, you can make this very, very large. If you have a hard drive, you have to wait years to corrupt the information stored. Well, sometimes you don't have to. It depends on the temperature as well. Sorry, do we have to put in the work in order to transition from one state to another? Yeah. This is just the memory. We store here, but of course, to overwrite the memory, you need to have work. We need to do something like that, for instance, or even to bias one of them. This type of systems, eventually, it will equilibrate. It will reach a equilibrium. If you wait centuries, the system, here, there are two times scales. One is the time scale within the well and the other time scale is this one. In a memory, there is a huge separation between the two, separation of orders of money. If you wait long enough, then the system will reach equilibrium. What is equilibrium in this case? In this case, equilibrium means that the probability to be here is one half and here is one. Something like that. But of course, if it is a memory, you can manipulate this. For instance, you can bias this a little bit, not so much. Still, the particle remains here and here. If you are in this state and you bias the two wells, the particles don't jump. They don't. They cannot jump. So the state remains one half, one half. But now the equilibrium, if you compute this state, this is not one half, one half. The equilibrium probabilities are here is in equilibrium. If you wait centuries, the particle will be more likely here than here. Here you see what is a memory first. A memory is something that has these two equivalent states and they have a long lifetime. And second and more importantly, that in fact, when you manipulate a memory, you are driving the system out of equilibrium. Sometimes you want, for instance, if I write a zero and maybe I come here, I come from here, from this state and this one half, one half. But I can do whatever I like and I hear the system is out of equilibrium. I can even force the system to be here or I can measure and find the system here. And then my probability density will be located in this well. And then this is also out of equilibrium. So here in a memory, this is a generic memory with many states. Each state is populated with some probability. And this probability can be out of equilibrium. Here the probability, the equilibrium probabilities one half. And here, sorry, here the equilibrium probabilities higher in the right or in the left. And still the state is this one. So this is a a particular non-equilibrium. It's a particular system, a system with this separation of time scales. And it's a particular case of non-equilibrium states. Non-equilibrium, we think of non-equilibrium as something that relaxes to equilibrium. These systems are out of equilibrium, but they cannot relax. I mean, they can, they need to wait centuries to see the relaxation. So the relaxation is so slow that you can consider that the state is constant. And in these cases, the formula, this formula that the work is always bigger than Delta F, the one that we applied yesterday, this works even for non-equilibrium states. So we have seen what is a memory and that this formula is actually achieved, this, the quality is achievable in most cases. Actually, in the non-equilibrium, and when you have like the non-symmetric potentials, you have said that the equilibrium increased at a low time or no longer time needed. I mean, you spend a lot of time to achieve activity also in the non-symmetric potential. So you are saying like, for example, if the particle is trapped in the left side of the potential, it will remain in the side. No, I'm not changing to the other. No, eventually, if you wait for centuries, it will start to change. But with different rates, because it's not even a rate. With equilibrium, we have a rate of probability, depending on the height of the sensor, maybe. Yeah, yeah. For instance, here, the equilibrium, this is not equilibrium. Why? Because the equilibrium is higher here, right here. This is the state and this is the point. Yeah, it's not equilibrium. This is not equilibrium. Yeah. Actually, equilibrium, equilibrium is only when this, this equation is at the, what you have in a memory is always non-equilibrium, because equilibrium is not interesting. You want to store some information and this information is proper. In most of the cases, these keys are zero, one, because you have a well-defined, well-defined state of the memory. Here, when I consider, you see in computer science, you only look at deterministic calculations and deterministic, so your memory is a state and here we can random transformations. Okay, so as you said, non-equilibrium is not interesting to store information, right? Yeah. Okay, one can prove using the non-equilibrium free energy that the non-equilibrium free energy is related with the Shannon entropy of the memory. So in this, to prove this formula, we assume that the system, remember that there is a huge time scale separation. There is the time scale of the relaxation within one of these regions. These are the regions of the two wells, but now this is an example with four regions. So there is a time scale, which is fast, it's very fast within each region, and there is a super slow time scale, which is the jumps between regions. These are the information states. These are my bits, and it's very likely to jump, but it's very easy to, I mean, it's very fast here. So we can assume, and this is a good assumption, that the system is in equilibrium locally, but out of equilibrium between regions, and this is the peculiar non-equilibrium states. They are locally in equilibrium, but out of equilibrium, if you consider the probability, and in this case, the non-equilibrium free energy system. I don't want to give you, this is a, you can prove this, those of you who are, who know very well the statistical mechanics, you can prove this formula. This is not difficult. You just, you have to use the definition, the definition of free energy is this one. This is the Hamiltonian, minus g, the entropy, the Shannon entropy of the state, and this is the Hamiltonian. The state is non-equilibrium. If this is, if this is equilibrium, this is k-p-loxid, as in statistical mechanics, you can prove this formula, and this is a property of each state, of each of these regions. If the memory is symmetric, in fact, the only important term here is the entropy, and we can get this formula, which is that the minimal work to go from m prime to m, sorry, to go from m to m prime is the difference of entropy in the, in the memory. And this is, this was expected, but there is all this treatment to do it, and this work can be achieved in memories because this separation of time is k. This is important. And now you can apply this, for instance. Let's do the second one. Suppose you have a memory which can be in any state, it's completely random. So it has a lot of entropy, Shannon entropy. So this is your starting, this is from m, this is to go from m to m prime. So you have a lot of, you have a lot of entropy, and you want to overwrite and write everything zero, zero, zero, zero, which has zero entropy. So this is zero. So the work to do this operation to go from a completely random memory to a zero, zero, zero, zero memory is just the entropy of this. This is Landauer's principle. You remember Landauer was, that the Landauer principle tells you that to erase a bit, but remember what is erased a bit is to overwrite a bit. So you have a bit which is zero, one. Remember the, you are in zero or one with probability one half, and then in the restart to zero, you erase the bit by forcing the system to go to zero. So this is, this has an entropy log two, and this has an entropy zero. So this is log two, and this is zero. So the minimal work is KT above. Sorry, I have problems that- There is a T here. Ah, okay. A KT. A KT is here. Well, it depends on the unit, but it says, there is at least a temperature. And sorry, in the first, in that formula, in the box, what was FM? Sorry. I didn't explain detail because I want you just, okay, if you want to see the details of this, we assume that the system is locally in equilibrium. So you can define a partition function for each state, and then FM is just a free energy associated with this state. So, because I know that not everybody is familiar with the crystal mechanics, I skip these days. What I want to show you is that Landau's principle follows, remember that yesterday I had this historical introduction, so we presented the, the still, the Maxwell beam on the CR engine, the Landauer principle and the Bennett solution. So what we are trying to do yesterday, we saw the CR engine with ebromos and so on, which uses mutual information. Today, we are looking at, for instance, Landauer's principle and the Bennett solution by using mutual information, which will appear now. There is a, I would buy information. Oh, no, that's your, okay. You can have the other way around, and this is interesting, you can have a memory without all zeros. This is very low entropy, so this is zero, and you can disorder the memory. So you will have, this is positive, and the work is minus this, which means that I extract work. But I can go, if I have a very, a very older memory, all zeros, and I disorder it, I can extract work. There is a very interesting paper by Yarsinski and Mandel, you know, that is in the, in the paper that I gave you, this is the citation, it's a, they have a tape because the memory could be, you did, when you write, this is the memory as well, right from any, so you have a tape with zeros and ones, it can be magnetic or can be whatever, and the tape, it's all zeros, zeros, zeros, and there is a machine, which actually is like a chemical machine. It's a kinetic model, like the ones that Barato showed. I don't know if in this school you will have Markov chains, okay, it's a very simple kinetic model. Kinetic means that it's like reactions, like A goes to B, B goes to C, so it's a Markov chain. So it is, the machine is fed by a tape with zeros, zeros, zeros, zeros, the machine disorders the tape and extracts work, and work, this is a information machine. So you can have the two, now, now the, the endowment's principle is more general, and it can go in the two directions, so you can go, if you order, if you order a disorder memory, you have to dissipate it, but if you will do the other way around, if you go from the tape with zeros, or a bit with zeros, you disorder it, you can extract work. These are called information reservoirs, and there is a lot of literature on this. And it's information as a fuel as well. Information is a fuel, essentially because information, but by information we, we mean something with low entropy. Something with low entropy is a fuel. And the C9G, for example, it uses information to extract work. Well, this is different. This is a feedback state. Okay. We will go back now to that. This is different. If the C9G, you have a statistical system, you measure and you use information in the, in the Aldo, Aldo-Cami, depending on the information system. Here, in the information reservoirs, you have a memory. Only a memory and the, which is order, and then you disorder it and you extract it. Because it's a chemical thermal part, by disordering it. And everything is, is encoded in this free energy, okay? So free energy is the important. The information, as a fuel, you see now, I mean, for example, some correlations. Yeah, although it's harder to, for instance, since it's smaller, it uses just the fact that it is low entropy per week. It doesn't use correlation. Actually, this is the kind of weak host of them. You can have something which is, you can have a tape with probability of being zero equals zero. I mean, probability of each bit equal to one half, which is a disorder tape. In principle, if you look, this is another thing that when you compute Shannon, if you don't look at correlations, suppose that you have a tape which is, suppose that you have a tape which is zero, one, zero, one, zero, one, et cetera. And you just, of course, it's not disordered. But if you look only at the frequency of zero, so once you get the probability to be zero, the probability to be one is one half. So you will see the entropy is scale up. But if it's not, it's actually there to be zero. Well, maybe it's scale up because of the first thing. But at the entropy of the whole chain, so this is entropy per bit. If you are naive, and this is, but the entropy per bit is almost zero. Okay. So now we can address the problem of looking at the physical nature of the demo, because the demo is a memory. So we needed this analysis of memories to incorporate the physical nature of the demo. So now I'm going to, this is essentially Bennett's solution of the Maxwell Demon of the steel arranging. And the idea is to consider the steel arranging and the demo as physical systems. And the demo measures the position. Remember that we have the steel arranging and the demo measures the steel arranging. So the first problem is if we need some work to measure the quantum. So we need a model of measurement. These are classical measurements, no quantum measurements. So there is no random, well, there is some randomness, but there is no there is no collapse of the way quantum. It's a classical measurement. But still we need a model of a classical measurement. We, this model, we call it the ideal classical measurement. And we have a system which we want to measure with some states X and the observer. The observer will be a memory and it has a state M. And of course, what is the measurement? The measurement is that M becomes correlated with the state of the system. So think of the steel arranging, this will be the steel arranging, X is left, right. And this will be the demo. So M will be left, right. And after the measurement, they become correlated. Actually, if the measurement is error free, X will be equal to M, right. But I put M prime because the M changes in the measurement initially can be whatever. And here is correlated. And think of that. This is what the exercise you did yesterday. If the measurement is error free, then M prime would be equal to X. If it has an error, but M prime would be equal to X with a probability of minus epsilon or it would be the opposite to the probability of epsilon, whatever, whatever model you can. And of course, this can be more complicated than the steel arranging. This can be, this is the outcome of the measurement. We will assume that it's discrete because it's easier, but it can be even continued. You can have a pointer, a measurement device, and M can be X. We will assume also that initially the two are uncoupled. So this is a typical assumption that they are uncorrelated and uncoupled. So there is no any interaction. And after the measurement, they become correlated. But this is the idea of a classical measurement. The system does not affect it by the measurement. This is impossible with quantum mechanics, but in classical mechanics we can assume that the system is not affected. And we will assume also that after the measurement M and X are correlated, that they are uncoupled in this physical sense. I mean, they don't have an interaction energy. Of course, to correlate to systems, you need to switch on some interaction. Remember, we are trying to do a physical analysis of this. So there will be a Hamiltonian. And this means that the Hamiltonian, before and after the measurement, the Hamiltonian, there is no interaction between the two systems. And in the middle, there will be some interaction. Okay, if you have this assumption, M X represents the state of system unobserved before the measurement. So the joint probability is factorized because they are uncorrelated. And after the measurement, I would have a correlation. But X would remain the same. So the form of the probability after the measurement is like that. X is the same as here. It didn't change. And M changes according to the conditional probability. If this is error free, this conditional probability will be a delta function. If it has some error, it will be like the one you saw yesterday. For instance, if this is left right, it will be left left, it will be one minus epsilon and right left epsilon. Exactly what we did yesterday. So yesterday, the difference is that yesterday M was just the outcome of the measurement. You didn't look at the physical nature of the observer. And now it's a physical system. So there will be a Hamiltonian. And there will be a Hamiltonian for the system and for the observer. And there will be a shadow range. Remember the non-activity energies, this one here. And now before the measurement, everything, the shadow range of the sanity when the particles are independent, the systems are independent, the Hamiltonian is also the sum of M, the Hamiltonian of the observer class. They're of the system because they're a couple. So everything is additive and the free energy is additive. So before the measurement, we have this condition. And after the measurement, this is the interesting part. They are the couple. So the Hamiltonian also is the sum of the system plus the observer. It doesn't matter if they are correlated or not because the energy, I have the energy of the system, the energy of them. But they are correlated. So the shadow entropy now is not added. So how can we convert the shadow entropy of two systems? How can we relate this with the individual entropies of each system using the other expression for the mutual information? This is amazing that the mutual information, you remember that there are two ways of, there are three, but here it's the right. So yesterday we analyzed a lot this, well, it was M yesterday because there was no need of separating before and after. Now I use it, right? But yesterday you used this one, you remember? So, well, no, we use this to calculate, but actually we use this one. How the uncertainty of my system reduces when I measure. Now we are using this one, which is how the correlation reduces the entropy of the, well, if you like, how the entropy of the, well, we can do, we can, this is the reduction on the uncertainty if I measure. And this is a different thing. This is a measure of the correlation. This is the entropy, the uncertainty of the X and M considered independent. And this is the uncertainty if you consider the correlation. Or in other terms, the entropy, the entropy of the two is equal to the entropy of each one separately minus the mutual information. Well, I can take this formula and see there is H with this Hamiltonian is the free energy, the other free energy must be. So this is very nice. This is the the mutual information tells you the contribution of correlations to the free energy of the whole thing. And appears, so it appeared yesterday appear as the reduction of uncertainty now appears as the contribution of correlations to the free energy. Now if you subtract before and after the free energy of the system cancels and the work to measure, sorry, I will shut up. Okay, this is the minimal work to measure is the increment. This difference, I don't speak too much, which is the increment of free energy in the apparatus because this is inevitable. Plus, and this is the interesting term plus the mutual information. So it appeared in a very different way. And of course, this is what we are going to compensate afterwards. You see, before we can extract the work API. And now I have to put the work API. And this is the solution of the of the maximum. Here you have the whole thing. We can have the sealer engine is a particular case of what we call feedback motor, the feedback motor, you have, you have some physical system. And you have an observer and the observer measures something and manipulates the system according to this message. And we call this feedback feedback. This is an engineering feedback means that you act on a system according to the message. So what is the the non nuclear free energy? In the beginning, I mean, they are no interaction, they are correlated. So the free energy study, the free energy of the system. After the measurement, the correlation appears here, of course, the memory, the observer, the demon changes its state and it appears here. What is the minimal work? The minimal work to do this operation is the difference these cancels and I get the difference of free energy in the measurement device or in the observer plus KT. You see that here I is always positive. So here to create correlations to measure you need to do some work. Now you have the feedback. And the feedback is what we studied yesterday. The feedback essentially is to erase, to destroy correlations. And we know from yesterday that we can take out from the bath this amount of work in the feedback. And finally, and this was Bennett also idea to complete the cycle, you see that here in the feedback, the memory of the demon, this doesn't change because the demon still remembers the outcome of the measurement. But now he has to forget. And there is this random eraser here. And then the demon must go back to its initial state, which is this one. And if you compute the minimal work to do this operation, this is minus delta okay. So this is the minimal work to complete all the steps in the silicon. And now you see yesterday we only had this part because we didn't care about the cost of the operation. Now we have the cost because there is a cost of measurement, there is a cost operation. And if you sum the three discusses with these discusses with this and the minimal work is zero, the minimal work is zero. Or if you like the cost, this is what you extract. But to complete the whole thing, you have to spend a work on measurement and a work operation. And when you are not doing that work, you are referring to a new correlation, right? Yeah, this is the state of my product. Yesterday we got the second law. No, we got that. Now we see that to extract KTI, to extract KTI, you have to remove correlation. And in fact, this is what happens when you do the feedback, which is moving the pistol, extracting and putting back away. You are essentially removing correlation between the demo and the analysis. Now this analysis also tells you something about the feedback. This is an interesting thing that I realized months ago. Yesterday we saw how you achieve the maximum extracted work by doing this reversible protocol and so on. But here there is another condition. To extract the material out of the work, you have to remove all the correlations between the demo and the analysis. The scale that is breaking. Yeah, I will do it. The only thing that I would say is that some books interpret Bennett's solution and they say, no, the cost is in the erasure. And there is no cost of measurement. This is not true. The cost is distributed among the two. Here you see that there is this delta X. So you can tune, depending on the type of memory you use, you can tune this as you like. So you can, for instance, you can make this zero by making delta X equal to KTI and then it will appear here. Here if you make delta X equal to minus KTI, this is zero and this is KTI. And this is, Bennett shows an example of that. So there is a zero measurement work and there is a KT log two in the Cedar Ending, KT log two work, which is the Dawes principle. And you can have different, you can have here zero and here, sorry, you can have this zero and here are a lot of, here on the work here. Now in the two steps, measurement and erase, you can have all type of combinations, depending on the, on the specific realization of the memory. And this is what it is, this formula here. And with that, we will finish and and just let's check the chat. Yes, there are two messages. Yes. Okay, good morning from Abuya, Nigeria, everyone. Sorry, I came late today. That's no problem. Okay. Hello. Hello. Okay. Hello. Thank you for greeting us all. Okay. So join us for a coffee break. Yes, of course. And now I see the chat. Yeah. Okay. Okay. So order, where is order memory? Here order disorder means a low entropy. So here, if you look at the Cedar Ending in this way, you are using an information. And it's for, this is the memory of the demon is an information reservoir. And by, by, by disordered in this reservoir, you are, you are extra. So this is the, this is equivalent to the model of the demo using information. So you, you see, you can look at the, at the Cedar Ending. Once you have the whole picture, you can look at it as completing the cycle by erasing, no? By, by turning all these ones to zero. And then, and then you have to spend a lot to permit and then you lose what you want in the feedback or just by letting this, by creating a memory, disordered in the memory. But for, for that you needed the memory in this state. So if you like the entropy of the universe, you are, you are saying that you are increasing the entropy of the memory and use this increase for extracting it. So once you see this, it's like the Cedar Ending is not so exciting, it's more boring. It's like entropy is constantly the universe. So you go, you disorder something, you extract work here, you extract heat, no? Or what? You extract work. Since it's a cycle, this is, this is work. And the work comes from the heat. So, so the entropy of the, of the back decreases, but it's compensated by an increase in the memory. But of course, this is not the cycle. This is the problem. This is not the cycle. You cannot, you cannot feed, you need to feed the machine with, with an order. So so this is to answer something who says, can measurement be regarded as kind of information eraser? This is kind, it's more than, well, this is more than an information eraser. This is the creation of information because you, here there are no information, everything is zero, there are no, here there is a lot of information, which is the information of the different states that the Selerianic has adopted in the, in the same cycles. So for me, measurement is, measurement is the correlation between two systems. And there is another question, when a measurement is done, the right distribution of the observable will become narrower. Yeah, this is what we saw yesterday. And information is also narrowed in the distribution. No, but you are not right, which in the sequence, the disorder sequence contains more information than this. This is also a philosophical problem of information theory, that information, we think of information and we think of meaning of the meaning of a message. But in information theory, this was why information theory is so successful because Shannon gave up, I mean, Shannon removed all semiotic or all reference to the meaning of messages. He only cared about the statistics of properties of messages. So this, although it's random, this contains more information or it could contain more information than this one. This doesn't contain any information because once you know, I mean, once you know that if you like, you can think of compression of information in computer science. You can compress this. If this is 100 zeros, you can just say, well, this is 100 zeros. And everybody can reproduce this. And it's only two words, 100 zeros, three words. But to describe this, if it is really random, you can describe this. You have to say this is zero, one, one, zero. So this has more information than two. Okay. For the last part, well, I find that I want to do two things. One is to explain the main ideas, how this can be implemented in, how can this be useful for biology and chemistry and that. But before, because I think now is the time to discuss this question. I want to talk a bit about a more philosophical issue, which is what it's information in physics or in general, this information. And I have this question because what I told you is you have this review paper that I send it to you. But it is also based on a review that we wrote with Jordan Horowitz and Takahiro. Takahiro is one of the persons, the guys who really restarted on this business of the modernity of information. And Jordan and I have done a lot. So the three of us are like experts. But it was the interesting that we were writing the review. I can have experience of that. There were a lot of discrepancies on fundamental issues, of course, in the mathematics and everything we agreed with the mathematics. But when you started to think about the fundamental questions like what is important. We were, especially the conclusions, we were like writing. So we had two very different ways. I think they are complementary. But for them, for Takahiro and Jordan, this was like a revelation, this scheme. And in this scheme, you see everything that I've told you before of the long, long life states and so on. It's not so necessary here. Actually, I will show you that you can do this even with continuous time in the moment. So for them, they were the revelation that the motor information is a measurement of correlations. And that it appears as a term, a contribution to the free energy. So for them, the key point of information is correlation. You can measure correlations with a correlation function. But the mutual information is a nice way of measuring correlation because it appears in the free energy. So the free energy, the contribution of correlations to the free energy is the mutual information. So for them, information is just correlations that are created and are destroyed in the system, in systems, in bipartite systems, sorry, systems that are composed by two. And that's the end of the story. So correlation and creation and annihilation of correlation. And my idea of information is that this is not enough. This is not enough. You need also the large separation of time, because this is my view at this morning. Okay, so I think probably there are the two things are for me, if you don't have if you don't have large separation of time, it doesn't make sense to talk about information. But I'm not so sure. But for Jordan, you are from many other people, or you must be in this team, it's only whenever you have a correlation to systems that correlate the states and so on, you can adopt this information approach. You can interpret the system. I think you need the key point is this separation of time scales, which is in DNA, in memories, etc. Probably these two viewpoints are related also with the fact, with the distinction in information, which is the distinction between digital and analog information. This is more related with digital information when you have really the states which are discrete and you cannot jump between one and two. And this can be in continuous time or discrete time. And then the information involved in this type of view point is more analog information. And one problem that I've not seen, this is a mathematical philosophical problem, what are the advantages or disadvantages of analog information versus digital information. This is something that, well, it says that in digital information you can have error correction. I don't know if somebody has thought of a life, if life would be possible with analog information. Life uses digital information. Other information in biology is analog. But DNA is digital, and it is not so clear. For me, it's not clear. You can have an evolution with analog information. That was my point. I mean, I started learning this thing and always my question was, why don't we do things in continuous time or continuous variables? But when I worked at the lab, I thought, okay, you can only measure with a finite precision and only in a finite rate of acquisition rate. So maybe things that are continuous, a variable of time, maybe like an idealization. I think that it's not physically realized. Yeah, I know. You record in a very reliable way to record analog information. Now and now, they sell more vitals, but you can't. Whenever you have a mold, you can only measure with a finite precision. And also in a finite number of instances, you can measure at every time you want to. Well, for the mold, you also need you also need an activation. I mean, you need something with a long life time, because otherwise long, long with respect to the scale where you are using the device. So when do you need that something? How long do you need? You rely on the memory first. So we decided that at least the time of where you are manipulating this information. Very typical. Typical realization, right? Maybe. Yeah, no, but she's asking, when I say long lifetime, is how long? In technology sphere, the longest the better, the longer the better. In biology, I think you see in the time scale where the process is going on, but also I, well, this is. You can multiply, for example, 200 times more about the time of the process you want to measure. Because, for example, the time scales of tracking the process in biology is much larger than in biology. With respect to the process, with respect to the process. For instance, in this case that we are, would be in the, there is a time scale where you manipulate all these things, so the states of the memory must be much longer than this scale of time. And this, for instance, look at this, this is what we call an information reservoir or fuel. But if you wait, if this is a memory, magnetic memory, and you wait, or you heat it up, or you wait for one century, this will disorder by itself. This is an irreversible process, actually, when the memory, because you are disordering something and you are not getting all the work that you get if you do the process. Okay, so this is, I wanted to finish with this, but I think was, so in, in the last 10 minutes, I want to finish with something else. But this is the main, I wanted to finish with this, with this philosophical question. But, so, well, you can, I think they are complementary. So this can be useful at this, I think that the, for the fundamental role of information in physics, meaning this. And, and maybe to apply to biological systems or other systems, this is enough. Okay. So I let you choose team. And, and, and I will finish the course with some tool for those who like this one, but I like this one. But how can you use these ideas to apply for, how can they be applied to biological systems? But it's, it's, I will not, today's session is not very, I don't want to give you so much. This is something that I will do in the, when it's here, it is summarizing this slide. The theory that I want to just briefly show you is, is done by Jordan Horowitz and Massin. They are responsible in this paper in 2014. And they introduced the concept, well, the, the concept of information flows was introduced by Abad, Abadayan, and Armenian. And it was this Armin, Armin, Abadayan, and Gwente Mahler. And they introduced it much in 2009. But actually this paper is, this is more, well, this is, there are more sources in this paper. So, and the, the idea is to use, the idea is to repeat this scheme. The idea is to repeat this scheme, but in continuous time. So can you do something, for instance, suppose that now, instead of a system and server and, and, and do all these procedures, measurement, feedback, eraser, you do everything in continuous time and X and M are two stochastic processes. So you have two stochastic processes and form a single stochastic process when you process it together. And, and there will be a mutual information that can grow and can decrease and so on. So essentially, if you look at this, it's a bit like that. So you have two stochastic processes that they evolve, they create correlations, they destroy correlations. And the idea is to do this in continuous time. So I would be very, I, I, I will not mention very much in details, but this is the, okay, I will try. Okay, look at the, the definition of, of mutual information. Remember, we have different definitions, but one that is especially interesting is this one, for correlations. Now, now we are trying to follow this idea that information, that mutual information is, is a measure of correlations. And, and, and this is the best way of expressing this, and if the correlation, this is, remember, this is, this is the number of bits I need to describe X. This is the number of bits I need to describe Y. And this is the number of bits that I need to describe the two together. So the mutual information is the number of bits I save if I describe the two separate with respect to describe the two together. And from that, I can get that H. So the entropy, the mutual information you see. Now, now look at this, this is Charon entropy, but you can think of, of thermodynamic entropy as well. So what is the entropy? If the, the, when, when two systems are correlated, entropy should have it. The entropy of the, of the, of the, is less than the sum. And this less, is, is, is, that's the mutual information. So this is why the correlations reduce the entropy and increase the free energy. So the free energy is always my, so, now suppose that I have two stochastic processes. So now X, X and I depend on two. Sorry, professor. There is just one more question. Is the minimum cost of gaining one bit of information through measurements still K2, LN2? Well, this depends very much on the, actually, depends on the memory. It is in the formula that I showed before. It's not K2, not too necessary. It depends on the memory and so on. Okay. So not necessarily. You can look at what we did tomorrow, yesterday or today. Okay. Please continue. Continue. Now suppose you have two stochastic processes and well, let's multiply, if we multiply by K, this we can, we can, so we multiply by K here, the Goldman concept and this would be, this would be the entropy, the physical entropy. So now you can write this as a function of time and it's temperature. No, sorry, temperature. Well, Goldman concept is equal to one, so we have this formula, which is the entropy, how the entropy of the system evolves. And now we can take the derivative, the time derivative of this and the time derivative of this, that's that. And here, this is a very simple formula. It's just the time derivative of this. It tells you how, if you have two systems evolving in time, how the entropy of the, this is the real entropy, the entropy of the universe, how changes. It changes according to the, it's the contribution of, it's system, classic contribution of the combination. And you can convert this into free energy if you like. The work, the work is always bigger than the free, than the derivative of the free energy and the derivative of the free energy would be, the f would be like that. It would be the energy of x minus the energy of x, y, the total energy. Why with derivative, no? The derivative minus s time this entropy, so time s, x plus that, sorry, time temperature. Okay. And if I, if I have here the, the energy of x, the energy of y and the interaction energy, I can write this as the work is bigger than this, this energy, this entropy is the free energy of x, this is the free energy of y and now I have the correlations and the, the, the, this, this formula contains all the scheme that I, it was in my slide, contains everything. So for instance, suppose that the, what is the work to measure? Okay. In the work to measure, I integrate this over the measurement. The measurement, this, I have the, this integral is zero because at the beginning and at the end of this zero, this is zero because the system doesn't change. So I get just the change of free energy in the, in the observer plus the create the, the increment of, for the feedback I can do the same. This equation contains all the questions that we have before. And the, moreover, this can be applied to a continuous time. So if I, if I have two systems and they evolve, I can, I can just apply this formula to any interval. And this is the idea of information flows, information flows, they compute the, the, you can take two stochastic processes and you like and compute this and calculate the minimal work to evolve in this way. So you see that one can, these two equations are equivalent. So one can just work with the entropy instead of the, of the free energy. And this is what follow it and, and it's possible to do in the, in the, in this paper. Okay. But there is a last thing which is the following. So you can, you can now go to the, to this formula, the, the entropy production. So I'm sorry, the entropy here. And you can calculate the entropy production. So the, the entropy change in the universe. This is what we call the entropy production. When you have a system in Kota with thermal patterns and so on, the entropy of the universe changes as follows. It is the entropy of the system, which is in this case, s x t, s y t plus the change of entropy in the reservoirs. This is essentially q divided by t. You have many reservoirs. This is a, how much work goes to the system? How much, how much heat goes to the reservoir? Okay. And, and this is bigger than zero by the second one. This is the second one. And here you can replace, here you can replace this one. And you can do a lot of things you can do when, when, so the important thing here is the sign of the, of the objects. So the information is, is minus. And this is, this is a plus. So you, you want to beat the second law. So you want to, to reduce as much as possible. For instance, if you want to pump heat or you want to motor, you want this to be negative. Of course this cannot be negative because of this. But thanks to, yeah, if, if I, if I have, if I have a total information, if I get this negative, so if I reduce the correlations, I can increase, I can do this negative as well. Okay. If I increase the correlations, this is positive. And then this is a minus. Then I need resources from the environment. So when I measure, I increase the correlations. I need resources from the environment. When I exploit the correlations, I can, I can make this negative. I can extract heat from the person. Okay. This is just a very brief idea of what they do in the paper. This is the first thing who uses equation and this one who analyze the energetics of systems. Now we don't need the separation of time script. We don't need anything. This is why it's just how the motor information and, and the entropy of the reservoir can be, you can use one resource. This is what, this is a resource. This is heat and so on, so you can extract. And this is another resource. So it is an exchange between one and the other. And just the last thing, I will show you this in the next slide. What they do is this in Markov chains. And they want to calculate this. They want to use this formula that the total entry production is sx plus si plus, sorry, minus i plus the entropy in the reservoirs. And this is bigger than zero. This is okay if you apply to the Maxwell demonetism like that. Define what you want to apply this in biology to autonomous systems where there is no engine forcing. And the system gets rich in stationary state. When the station, when, when the state is stationary, everything is zero here. You know that the stationary means that everything is zero. This is not zero because you can have closer. This is zero. So they managed to overcome this by defining information flows in a stationary state. This is a bit technical, but I want the estimation to, but you realize that there are different tools. And this is a very important tool for those who are working with biological systems in stationary states. This is a good preference. And the idea is that the mutual, the mutual information is zero. And you can split it in two terms, one positive, one negative, which is the flow of information that goes from system one to system two, system x to system one. If somebody is interested, I can explain details. And what it is more interesting, they can derive two second laws for system x and system y. So even though this is, you have a motor, for instance, or something like that, if you interpret the motor as following these lines, you can split the operation of the motor into a path, which is creating correlations, which can be interpreted as a measurement. You may have to continue studying. And some path, which is exploiting correlations, which can be considered as, can be interpreted as a tool. And moreover, you can calculate the efficiency of each of these two processes. So in a machine, that can be interpreted like that. You can really look at, I mean, assess the efficiency of each process separately. Okay. I'm sorry that I can't tell you, because this is a, we need two hours and so forth. The main idea is this. Okay. So I go back to, you see that this is, this is really this theory by four of the telephoto, it's only based on this path. So I will do this question. And thank you. Let's thank the professor. Are there any questions from the audience? And yes, let's check. Well, I will be here tomorrow. And so somebody wants to. What do you think are the main challenges? Well, it's for the wrong table. In this field, in information. For me, the main is to exploit this part to see how, how information devices appear in each person. For me, it's very important how we are building has, I mean, we humans are building information devices every day and increasing the capacity and building the information devices. It's not trivial. You are building a physical system that has this symmetry breaking and so on. So this is for me an interesting question. And biology. Yeah, I didn't talk about that. Many people working on information theory applied by sensing a new system. You work on endosomes, which is a kind of information processing from the membrane, the nucleus in itself. Of course, it has DNA, splicing DNA from RNA, all these parts. And before it is interesting, maybe not so much in the thermodynamics, but the primary information theory. Of course, then you have quantum information, the connection between quantum information. What about information generated by black holes? Are these results applicable to cosmological models? I would like to know about that, but I never understood the black holes. And I don't understand this. I know that the entropy of a black hole is spinning with the surface and so on and so forth. I'm more interested in cosmological approach. Yeah, I'm interested because here you see that it is a rhodicity breaking. Rhodicity breaking means when you have this long life time, it stays because you break her rhodicity in the system, the system cannot be explored. In the tool, the system cannot be explored. And rhodicity breaking, of course, in many important stages of cosmology, like nuclear genesis. You have nuclear genesis, you have symmetry breaking. Or when you create a planet, when there is a gravitational collapse, the planet could be here or here or here, so there is symmetry breaking as well. And all these ideas of the entropy of the universe increasing and so on is based on the fact that the system is revolving, which I think is completely nonsense. It's not revolving. Yes, of course. One last question from the chat. Okay, so what do you mean by the relation between information and a large time scales? Not necessarily. Of course, this is also very interesting information in complex systems. No, but it's just this thing that we mentioned before. Large time means that in a hard drive that you need that the zero sum ones don't flip spontaneously. You don't like time. And at DNA, you don't like to have mutations every time you turn your cells back. So you need real reliability. Reliability means long states that last for a long time. Okay, so let's thank Professor once again. Before we proceed further, I believe we have some announcements. Yeah, yeah, yeah. Personally, thanks for coming here. We had a very intensive course and I don't believe that all of us have said that we will not have more, but it's important to take a rest and see a bit. And all of us are waiting more from you in the future. Thank you very, very much.