 OK, shall we start again? Today's second speaker is Professor Henry Park at T.S. Please welcome him. Thanks for the introduction. And I know all your tangents. I learned a lot about stochastic somodynamics from Professor Chesson Lee last week. I think there are some new faces here. So maybe didn't learn. He taught you from the very basics to the very recent hot topics like TUR and speed limit. My talk is also about one example, using the stochastic thermodynamics to resolve the very longstanding issue for controversial problems raised in the conventional thermodynamics. Basically, I'm going to talk about the heat engine, even though the title looks horrible. It's very long and on saga or whatever. But it's a very simple problem. You know, the heat engine. What do you want to do with the heat engine? Usually, you want to increase the performance, make a better performance. What is better? Better is power. You want to increase the power. Or so you want to increase the efficiency of the engine. So you have to, you want to increase both of the power and efficiency. But is it possible to do that? Is there any kind of trade-off relation? If you increase the power, then you have to sacrifice your efficiency or not. Those kind of questions. So I'm trying to answer to this question in this talk. So starting from the very basic heat engine, you can find in the freshman textbook. You have an engine, a two-terminal lead, you have a two-terminal two reservoirs connected to the engine. Hot reservoir and cold reservoir. And hot reservoir will release some energy in the form of heat to the engine. And you can use part of this energy to the useful work. And remaining energy is going to be dumped into the cold reservoir. That's the basic thing. What is going on in this? Cyclic engine or the steady-state engine or whatever heat engine. Analytics is obvious. This is energy conservation. Coming in should be same as coming out. Interesting thing is thermodynamics. Because we know all the thermodynamics second law must be satisfied. So you have to capture total entry production. But engine is in a cyclic state or the steady-state. Engine does not have any contribution to the entropy increase. Only the two reservoirs. But these reservoirs lose some energy. And this is in equilibrium always. So entropy change can be written in terms of Claus's form, which was found very long time ago by Clausius. So this reservoir loses some entropy. But this guy, because you have an energy increase, increase of total entropy. So if you add up all these, they must be positive. This is second law. And one important thing is efficiency. Efficiency is something like how much you can do with some input energy. So W divided by the input energy. Put these things into here. These two occasions, usually these two occasions you can easily find. Efficiency has an upper bound, which is called Carnot efficiency given by the 1 minus temperature difference. This is all in the freshman textbook in general physics. It's a little bit more useful to rewrite this equation in terms of Carnot efficiency and enterprise production and the work. So I get rid of these all heat energy terms. Then you can easily find you can rewrite whole thing in this way. By looking at this, then you can easily see how to get to the Carnot efficiency. So this is usually for the cyclic engine. For the steady state engine, instead of this guy, you can use the enterprise production rate and work rate. This work rate is basically power. OK. In order to reach the Carnot efficiency, obviously this is 0 if eta goes to eta. See? In order to make that, what is the requirement for that? It's almost only s star equals 0. If s star equals 0, then this is 0. This is usually not OK. Enterprise production rate is 0, means usually we are thinking about the quasi-static process, which we learn as a freshman. But this power, w dot, doesn't have to be 0. If it is 0, then sometimes it can cause more problems because 0 divided by 0. So basically, some of the acetylene does not prohibit finite power engine with Carnot efficiency. So you can have a finite power in principle. It's not banned by the thermo acetylene. This engine with Carnot efficiency and finite power people usually call this a dream engine because people thought if you have a Carnot efficiency, then somehow you are thinking you cannot have the power. Then that's a useless engine because you have no power. But even if you have a maximum efficiency but with a finite power, then it's going to be a very useful engine. So this is a kind of dream. All the recent debates started from this paper by Benetti Saito and Cassati in Pierre about 11 years ago. They really calculate the efficiency and power in the frame of linear irreversible thermodynamics. Looks, sounds terrible, but it's basically linear response theory. It's something like you have again a hot or cold reservoir and you have a channel. So then the energy can flow from hot to cold. But if you also add a potential gradient like a chemical potential gradient, then particles also move. If particles move, then you can say you have some kind of work coming out because you move something from here to there. So you have a particle flux and you can have a free heat flux in the linear response regime. It must be linearly depend on this external force. External people call this thermodynamic force. We have some coefficient like that. OK? So of course, there is no gradient and there is no current. This is equilibrium situation. But if there is a gradient, then there is a current. One of the amazing discovery by Onsaga, the so-called Onsaga reciprocal symmetry, is these off-damer elements are symmetric. They are same each other. Onsaga got the Nobel Prize in chemistry by this work. This is, I'm not going to derive the whole these things here, but this Onsaga reciprocal symmetry is based on the time reversal symmetry in microscopic dynamics, if you follow his derivation. However, Kasimir, the later, he found when you apply the magnetic field for the electronic system, the magnetic field obviously breaks the time reversal symmetry. So if you break the time reversal symmetry, then Onsaga symmetry doesn't have to be satisfied in general. However, he found interesting symmetry. If you change the magnetic field upside down, then you have this kind of symmetry. So this B field is up for example. This B field is down. So these two equalities, these coefficients are for the different system. One is magnetic field up, one is magnetic field down. Whatever he found interesting symmetry, this symmetry, Onsaga symmetry is broken for a given system. Using knowing all these things, they try to found from this linear response theory, they try to find the efficiency and power and all kinds of stuff. And I'm not going to tell you the derivation, derivation is rather simple, but this is the result. This is a normalized efficiency, normalized against the current efficiency. Here is so-called asymmetric factor, the symmetry between these two of them on an element. And forget about this red dashed line, I don't know how to get rid of these things because I got these things from their paper. This blue line is the maximum efficiency you can get for a system with some S, with a system with some H symmetry in general. And they found this flat line, along this flat line, obviously this is one, so this is current efficiency, and this flat line and the production rate is zero. But more importantly they found along this line, power is given by this, as a function of S. So what this means? When S equal to one, this line, this is usual microscopic term of symmetry, then this is the maximum point you can have, so current efficiency can be reached, but unfortunately on S equal to one, the power is zero. That's why we could not find anything with current efficiency with finite power before. But their suggestion is if you break the time with a symmetry, then you can go to this symmetric factor to not equal to one, then this is maximum efficiency, still current efficiency, and the production rate is zero, but work is finite. So there is a way to get the dream engine. This publication triggers many, many further studies to find really the dream engine is possible or not. How can you make the dream engine explicitly? So they do the same calculations, for example, so-called two terminal transport problem, which is temperature difference and chemical potential difference, electron transport in the classical limit, the magnetic field region. Then following this theory, on the Saga Symmetry theory, this on Saga Symmetry possibly broken there. But however, explain calculation show this on Saga Symmetry is not broken. This on Saga Symmetry result does not disallow the symmetric coefficient. They only say these two guys are same. So for example, if coefficients is even function of B, then these two guys must be the same. So unfortunately the could not find the asymmetric on Saga coefficient, where you can find the dream engine like that. So they went for a little more complicated system, so-called three terminal transport. We have two terminal, but you put the one terminal in between so you can control another, whatever parameter system parameters and try to find whether, and where the magnetic field is all there, then on Saga Symmetry does not have to be conserved. So they really found on Saga Symmetry is broken, is L is the matrix of this coefficient. So in this three terminal cases, L is three by three matrix. They found this. Wow, it's great. However, in this linear universal formulaic formulation, anthropocernical zero is not allowed in their calculation. So this is group, this is back. So again, you cannot find the dream engine at all. Right after that, inspired by the symmetric uncertainty relation, which you learn lasted for, I think more than two hours from Professor Jason Lee, this is based on the stock test of thermodynamics. So people try to calculate, is there any kind of similar trade-off relations between the power and efficiency? Indeed, they succeeded. Yeah, you see that this is only by one year later after T-wall was reported. They found so-called power efficiency trade-off relation is expressed in this way. You see left side is power, and right side is this, and theta by some kind of system, detail dependent constant. You see that if you go to eta goes, eta c, then right-hand side becomes zero. That means W dot cannot be positive finite. So they concluded probably there is no dream engine, not in the linear response regime and whatever, in general framework. But they calculate based on some kind of Hamiltonian systems. So it is not real general, general all different type of non-equilibrium systems. So later on, the same side, same side part, using the his uncertainty relation and derive this kind of trade-off relation. It's the same thing, but only the coefficient is a little bit different. This coefficient is basically work fluctuation rate. If this is not diverging, usually not diverging, then this is finite. So again, you cannot have a dream engine. However, I think you learned last week, this T-wall, standard T-wall only satisfied only for the overdamped systems. But in the same year a little later, another group, Decent and Sasser, derive much simpler looking trade-off relation, which is given by this. The car is also some kind of constant depending on the details of system. Using not T-wall, but another kind of trade-off relation is so-called entropic bound on currents. So when Eta was at the sea, he sees also you cannot have a finite positive power. This system is basically can be applied to the, this result can be applied to the under-damped, general under-damped and over-damped long-term dynamics, even with magnetic field. So story must be closed because they found even magnetic field, you cannot have a dream engine. But the problem is, then what is this, this guy's result about Benentie and Saito and Kasati. Magnetic field is on, then you can have unsurpassable coefficient can be possibly asymmetric. And some people found in the complicated structure you have really, Gonzaga symmetry is broken. Then what is the, so basically two results are not consistent to each other. So what is the problem? So this is the question you are asking. Two-terminal heat engine, because three-terminal, we know that your entropy, zero entropy production rate cannot be reached. Still we are looking at two-terminal engine with broken unsurpassed symmetry. So is it impossible? That's the one possibility, but maybe it's possible. So the remanding can be really there because their calculation is also only applied to the discontinuum dynamics with magnetic field. So you may have something else. Or this is still possible, broken unsurpassed symmetry possible, but the remanding is not realized due to some other reasons. So that's the question you are trying to answer in our paper and I'm trying to answer to you today. Answer is of course this. So the conclusion is unfortunately you cannot have a dream engine, whatever you do. But there is some other ways. So we are only talking about the magnetic field here. Okay, let's go. And so to show that we are going into the exactly solvable model, which is something is nonlinear, you cannot solve in general. So you go to the linear Brownian particle engine, so-called, is under that system V is the velocity vector. And this is linear potential, harmonic potential case, whatever, stiffness matrix in general. You can have a non-conservative force like a torque inside of the system and magnetic field, Lorentz force. And the friction term is gamma in the friction matrix and psi is noise, noise can be noise for each component of the loss vector. You can think of each component of the loss vector as a velocity for the given particle. Well, v1 is, you can think of about vx, v2 is vui, but also you can think about v1 is first particle velocity and v2 is second particle velocity. But anyway, they are in a different temperature. You know, to make a two terminal engine, you have to have a temperature gradient. Okay, this is a simple system because even though this is linear, it's difficult to solve. So we go to the most simple system, two particle system or the two-dimensional system. So you have X1 and X2, one is T1 and one is T2, one particle is here, the other particle is here, and these particles are only doing the one-dimensional motion and they are all inside of this harmonic potential or harmonically interacting, whatever. So I'm thinking about this kind of system. So concretely, K is given by this. Non-conservative force is asymmetric of the element, it's generating a torque. And between these, of course, low-range force and this patient is just taking a simple identity matrix. You will solve this. Well, I'm not gonna show you the derivation, but you need the stochastic thermodynamics tools or technique to calculate the heat and all that in this calculation. So unfortunately, you learn all these stochastic thermodynamics tools last week, so you can do that by yourself. This is exactly so, Bobu. And you know, this guy, the Onsaga-Kashimi symmetry must be satisfied. Well, unfortunately, this is even function only. So Onsaga symmetry is restored even with the magnetic field as I advertised in the previous page. So no dremensin at all. Then one of the, my poster got an idea, go to the three-dimension. But I know that at the beginning, if you go to the three-dimension, you're gonna have a three-dimensional, three-term problem, then again, you're gonna go back to the same problem as in the linear irreversible thermodynamics framework. However, there is a way to escape out of that problem. That problem, we take three-dimension, or a stiffness coefficient here, same, and this is torque. So basically one, two, three-dimension, torque is around this way, a magnetic field in general. But main thing is we take this gamma only two-dimensional. The one guy is zero. And I'll take this temperature matrix, T1 is zero. What that means, what that means is this. This second particle is here, first third particle here, but first particle is outside of a reservoir because first particle is not connected to this because this is zero and T1 is zero, it is basically deterministic. So we have two terminals still and three-particle system. Is it clear? No? No, you have a third terminal that I just cut. Cut link to the third terminal means third reservoir. System is one, two, three-particle, okay? So this means this first particle, system is composed of three-particle, but this guy is connected to the low temperature, this guy is connected to high temperature, but this guy is outside of two reservoirs, okay? Then terminal means reservoirs, connection, okay? So this is the situation. We try to solve this thing analytically. It's horrible calculation. It will be three-by-three, exactly solvable. We succeeded in that and I'll show the one coefficient. For general B is too much complicated, so we take the BX equals zero, then we calculate L12 and L21. This is often an element of one side of the matrix. It looks horrible and this inside of this, here the C is also given by that, but if you look at, really carefully, then this C is all even function in B, so this C is all even function. Only thing is, oops, only thing is you have this guy and that guy. You see that this is just a minus sign in each other. This is all the function, so it is also where what, Casimir symmetry is conserved here, but also what symmetry is broken. This guy is different from that guy because of this BZ component, okay? So we found two-terminal engine and on-service symmetry is broken. Then what do you expect? Deneity, Saito, and Kasati theory in the linear irreversible thermodynamics, you can have a dream engine. You can have a finite power with current efficiency. On the other hand, then what about Dechen, the Sasa? He said, his derivation is also very simple. He showed the under-damped along the way, even with magnetic field, power should be given by this, so dream engine is impossible. So these two theories are contradicted to each other. What is the problem? It's a very simple problem. The stability of the steady state. Even though the K here is positive definite, torque force is there. It is still okay, we torque force solely, but you have a Lorentz force here. They combine each other, sometimes system can get out of the stable steady state, which has been known, the, this kind of stability has been known only two or three years ago. People thought B field doesn't do nothing, so you have always a stable Boltzmann distribution. But as soon as you add the torque, B field is a combination of effect on the system, and system can be out of the stable non-nuclear steady state. Linear irreversible thermodynamic, they do not think about the stable steady state or unstable steady state. They just think about, okay, you had a single is a steady state, stable, then there it is going on, okay? So we checked the stability of this thing. In some cases, analytically some cases, too complicated, we go into the numerical test to find out the stability analysis, stability region. You see coming back to the first page of this guy, this is Eta, Eta C, WDadiq, the positive along this line, and of course in our system, as is once I estimate the broken, so we are somewhere here. So we wanna go there, then what do you expect? Where is the stable region? Immaculately, it's all unstable. And stable, on a single one, the stable region going up to reach the top plateau. No other reasons going in. So controversy between this resistance of dream engine of these old walls from the Onsaka symmetry and all that is not really a controversy. If we consider the stability condition, if stability condition is there, then there is extra constraint, then you're gonna have no dream engine is possible. How much time? Yeah, I'm good. This is my last speech. So this was the question you are asking at the beginning. And so is this. Two-tone heat engine with broken Onsaka symmetry can be realized. However, stability requirement make the dream engine realize, make dream engine not realize, impossible. That's our result. And this trade-off relation by power and efficiency by especially Deton and Sasa is working. It's really solid result, even with magnetic field, so impossible. But there is a way, we know, we work very long. You know, to break the Onsaka symmetry, which gonna basically prohibit the dream engine, you can use some other non-magnetic field. So using some other external force to break the time-reversed symmetry, then you may have a dream engine. So one example is velocity-dependent force. Broken, of course, it's a time-reversed symmetry. Broken, which can be realized by active reservoir, which is very popular, I guess maybe next speaker wanna talk about it. And it's a first cooling in the plasmus, cold damping, many where a phenomenological velocity dependent force is really present. So in there, if you use that kind of things, you can realize the dream engine possible. Also, Deton and Sasa's result about this power-efficient trade-off is only for the under-damped long-jabang dynamics with white noise. You can have non-mocovian noise. Basically, this active reservoir is also considered as a non-mocovian active reservoir, memory, or nobody will look at the discrete dynamics. We don't know whether there the dream engine is possible or not. That's the still open question. Thank you for your attention. Any questions? So how would you define the dream engine for discrete dynamics? Carbon efficiency or the maximum efficiency with finite power? But I mean, it would stay at the delta W over delta S state. There's no W dot and S dot. No, S dot can be zero. If S dot is zero, then you have a carbon efficiency. The double dot is finite. Discrete system? Yeah, yeah, yeah, that's right. How come I know? I don't know about the discrete system. We tried for the, you know, the quantum heat engine problem is basically because the state is discrete in the system. You can think about this here, the jumping system. Right. But that's an example. Example-wise, if you solve it, you cannot find a dream engine. However, there is no general proof like a Dacheng and Sasa for the continuous dynamics. That's the only thing I say. If you go to the new quantum system, coherence factor also change a lot so carbon efficiency can be overcome. So that's all different problems for the quantum case. Another question? So you've been only talking about the steady state autonomous heat engine. So is there still not possible for the periodic heat engine? Cyclic engine. Yeah. Cyclic engine is also can be included by in the Dacheng and Sasa formulation. So even in the cyclic engine, you know, at least if you follow the Dacheng and Sasa, you cannot have a dream engine. Maybe I missed it in the three particle system. How this temperature was zero in the particle one. Are we set to zero? For example. By a steady state or something? You know, you have particles interacting each other. Then these particles inside the one reservoir. That is T1 and T2. T1 and T2. And they are totally connected, these two particles. Yeah. And the particle in the middle. Middle can have T1 minus T2 kind of, I don't know. No, we can interact. But this guy does not inside of the reservoir. It's something like, it's a very simple form of heat transport in the long chain. One end particle is in the reservoir T1. The other end you have T2. But the other particle between is not in the reservoir. Temperature is zero of that product. We set the temperature is zero because it is outside. Temperature zero is nothing but there is no interaction between the, there is no interaction between the stock that's reservoir. That's all. Okay. Other questions? Only thing because, just one mention. Because this is life science workshop, I did nothing about the life here. I think Professor Shen already talked about a lot of life. But heat engine is, can be connected to the biological engine, you know, the many ways. Especially information engine. That guy just have talked about the last week. The virus is sensing something, taking information and going to the that direction type of information engine. But information engine genetically can be converted to the heat engine. One-ton mapping is possible. So if you learn something about the heat engine, then you can also learn about the information engine. Through that bridge, you can reach the life science. Okay. Okay. Let's thank Professor Park.