 realizations of heat engines with colloidal particles and optical tweezers. Just before starting, let me just do something which will be do a small digression on locality balance because I got several questions about it. You are seeing my whiteboard, right? Yes, we can see your whiteboard. OK, just a five-minute discussion about it's an interlude on locality balance, which I assume for many models. But you can get a proof for land-driven systems. So take a land-driven system such as, for example, a simple one will be the following. We have x dot t equals mu f. This is a one-dimensional system. This force should have non-conservative parts. It's not only a potential, but there is no time-dependent force. OK, so there's just heat. There is 2d psi t. And remember that mu and d are related by instant relation. This is the mobility. This is the diffusion coefficient. And we have this. OK, so my claim was on the other day that p, probability of a trajectory given the initial value divided by the probability. I'm assuming non-equilibrium state, so I don't put p plus of x t plus. Given x 0 plus, this is the time-reversed trajectory, will be, in this case, probability of the trajectory given the initial state divided by the probability of the time-reversed trajectory given x at 90. This is in the forward trajectory, value t at 90. So I was saying that this is equal to e to the minus beta, the heat along the trajectory x t. And in fact, one can, from this equation, prove this equality. And I will just sketch briefly the proof, but I recommend you to go through it and ask me if you have any questions. But this is a sketch. So for land-driven systems, one can compute the path probabilities, the probability for a full trajectory. You can take a look. For example, there are very good books on path integrals. One is by Shulman. I believe it's called Path Integrals. And this type of calculation is done very, very well. It's very well explained. So the basic idea, I'm not going to be extremely rigorous. I'm going to be more physics style is the following. The probability for a trajectory times dx t. And this is what I understand by dx 0 dx 1 up to dx t. These are all the values that the trajectory takes along its way. It's equal. This is a change of variable to the probability for a trajectory of the noise times the differential of all the noises. This will be, again, I'm not being very rigorous. This will be dx i 0 up to dx i t. This type of change of variable tells you I could calculate the probability for a trajectory if I know the probability of the noise. And the probability of the noise is known because this is white noise, Gaussian white noise. So every psi has Gaussian distribution. And they are independent. So if you use this relation, what you can get is that p of x t for the trajectory is proportional to the exponential of minus. And this will be noise squared because it's Gaussians. So this will be 0 to t of x dot s. This is the derivative of time s minus mu f x s squared d by the amplitude of the noise, which is 4d. And then there is a ds. This you can find in many books. I'm not going to spend too much time on this. If you want to prove, I can also show you papers where this is proved very easy. So this is the probability for a full trajectory. It depends on the individual velocities and the forces applied to the trajectory. And you can compute easily from here the probability for the time reversal. The time reversal is suffering the same force. So I'm just time reversing. So what changes in the trajectory is the velocity. So instead of x dot, we have minus x dot minus mu x s squared divided by 4d ds. The only thing I change is x dot by minus x dot because I'm reversing the time. So from here, you can do the ratio p of the trajectory divided by probability of the time reverse. And you can easily convince yourself that here what you get is, OK, the first one will be x squared plus mu squared f squared minus, with a minus plus x dot mu f. Here, there will be also squares, which will cancel when we do this ratio here. And the only thing that will survive, and this you can see very easily, the only thing that survives is the cross term. So the only thing that survives is the integral of mu f xs times x dot s ds divided by 4d. OK, when you do the square, it's the square of the first, the square of the second, twice the first time in a second. There's a 2 divided by 4d. So it's 1 divided by 2d. And here is 1 divided by 2d. There's some, and you just get this, OK, from 0 to t. So many things have simplified. You just get a cross term. And then you realize in strato-nomitz that this is exponential. So I'm now going to use this. So d is mu kb t. So this is beta. Remember, beta is 1 over kb t. Beta, and then it's integral 0 to t of f xs strato-nomitz d xs. And this is minus the hit. OK, this is, plus is minus the hit. So as I show here, exponential of minus beta q of xt. So I have proved for a range of insistence, the local heat condition, as long as the hit is defined as a secimoto. So remember, when we had these doubts, why strato-nomitz, why hit? All here, I have used standard calculus. I didn't use the calculus here. So it makes sense to call this quantity here, the hit. So this is minus integral 0 to t f xs strato-nomitz d xs. All right. So this makes sense with eta-nomitz. This is just a regression. Take a look at home. Think about it. Check path integrals, and you will be more comfortable. All right. OK, just a short regression. I want to go back to the main part of my lecture, which is engines. So as I said, this will be more experimental. So let me just go through it. So this lecture, we discuss engines and the paradigmatic engine is the Carnot cycle. This was the beginning of thermodynamics. Carnot is considered the father of thermodynamics. He was a naval engineer, and he proposed a cycle that is, say, the perfect cycle, because it achieves the maximum efficiency during his master thesis. So this was a totally revolutionary work during his master. Give me a second, because I think there is a question. OK, one second. Is there a question? No, OK, sorry. I'll go back to my. All right, if there's a question, please ask it out loud. So Sadiq Arnault is the father of thermodynamics. He was considering a system interacting with two thermal baths. This is what he considers his work in his master thesis an engine. He's not saying what is the system. The system is a gas. It's a gas or in this water. He's saying a system interacting with two thermal baths. And the cycle he was thinking about is what I showed on the left. It is called the Carnot cycle. And it's composed of two hydrothermal processes connected by two adiabatics. So in the hydrothermal, you put the system in contact with a hot, first with a cold bath, and you compress it. Then adiabatically, you change the temperature of the gas, increasing its pressure. And then you compress it. Sorry, you expand it when the gas is hot. And then you connect the two isothermal adiabatics. So for Carnot adiabatic was something very ideal. Adiabatic means there is no heat exchange between the system and the baths. And this is in practice impossible, as I will show you later. You cannot isolate perfectly any system. But this was his theoretical idea. And if you think about this cycle, what you get in a cycle, the entropy production also only contains entropy production in the baths because in a classical microscopic system, the entropy, the system entropy is a state function. So delta is thought, what I show on the left is minus the heat into the hot bath divided by its temperature, minus the heat in the cold bath divided by the temperature. So if you do this cycle very slowly, it means it's reversible. So there is no total entropy production. It is zero. This means the equation I put on the right that the ratio between the heat fluxes is related to the ratio between the temperatures. And now Carnot wanted to know the efficiency of a cycle of his cycle, which means what is the work output that you can get out of the heat intake from the hot bath? So you are giving fuel to the system. You are absorbing heat from a hot bath and how much of this heat is transferred into work. This is what we call efficiency. So if you use the equation on the right, top right, in the definition of the efficiency, you show that the efficiency equals one minus Tc over T8. So it is independent on the details of the energy. It depends on your details of the bath. This is called the Carnot efficiency. And you can show, it was shown by a Clausius from the second law. You can do the same thing I show here, but put an inequality greater or equal than zero. And you will get that the efficiency of any cycle is smaller or equal than the Carnot efficiency. This is a very remarkable result. It's a consequence of the second law. And it's all the engines and thermal machines in the world are less efficient than the Carnot efficiency. In particular, there's something also important. It's the power. What is the power of a car or of a machine? It's the work you extract divided by the time to do a cycle. And it is important that Carnot efficiency is obtained as zero power. So you do the cycle reversibly. It means that infinite time. So the work divided by infinite time is zero. So then in the next years, there was a development on what is the efficiency at maximum power. So we want to build engines and machines that are as most powerful as possible. So it is important to know how efficient they are in the regime of maximum power. And this, you can see in different power planks, they fit very well to the so-called Nomekov-Kursomalov efficiency, which is smaller than Carnot. And it's given by this formula I put below. Okay, it depends on the square root of the ratio between temperatures. All right, so now I will switch gears and discuss small engines. So colloidal heat engines. By the way, here there's a review paper we wrote in 2017. So I recommend you to take a look. So what is a colloidal heat engine? This is a heat engine. So you get heat from a hot bath. You transform it partially into a cold bath in the heat to a cold bath and partially into work. We move a wheel this way. A colloidal heat engine is an engine that has the size of a molecule. So of course it will be the amount of energy that gives in a cycle is of the order of KBT. So it's 10 to the minus 23 joules. This is really nothing. But what is important from here is that this energy is comparable to the fluctuation to the thermal energy of a single molecule of the bath. That's why these engines are very difficult to understand because they are strongly affected by fluctuations. That's why we need squastic thermodynamics at this point. All right. The first law is simple for a macroscopic engine. The change of the energy in a cycle is zero. It's a state function. Therefore the work extracted is equal to the sum of the heat. Remember, from the hot bath we get heat. So QH is positive. To the cold bath we dissipate heat. So QC is negative. This means that we can only get work if there is an imbalance between the heat fluxes on the two baths. So we need two temperatures to extract work from a thermal machine. This is very, very important. All right. So I will discuss different proposals and experimentalizations of engines which I will introduce you now. So one is the first was designed in Germany by the Beijing Group. The second that I know very well that's a Mexican experiment done to emulate a steam engine. There is another experiment in India where they achieved a bacterial heat engine. And then there's a work done by us in Spain, back then I was in Spain on building a carnoid engine on the microscopic scale. Okay. Let me first start with the stealing engine. Stealing engine, it's one of the most famous engines in engineering. It's just the four stroke cycle in which you have a cylinder and you compress it when it's cold. Then you heat it up in the step two to three on the left. So you are hitting the gas or the substance, the working substance. And then you let the system expand at the whole temperature. So this gives you a lot of work out of it. And then you again cool down the working substance. This is you see on the right, very clear. On the left, they could, this was the experimental design of the Beijing Group. They could implement this in the following way. So the working substance is a microscopic particle of melamine, three microns of diameter. The cylinder or the piston is an optical trap which we can open and close, compressing or expanding the one particle gas and we can heat the surrounding fluid which is water with a heating laser of 1,455 nanometer wavelength, right? Following this, you can repeat this cycle many times and measure the work in the same way I explained you in my lectures. So here there is only conservative work. If there's a potential that changes in time, you don't need to care about non-conservative forces. So it's just partially with respect to Kappa di Kappa. Only we are changing the stiffness of the trap which is the strength of a harmonic well, okay? So the black line you see on the top right is the work accumulated over many cycles. You see you extract work doing this cycle many times and if you do it reverse, you do work on the system as expected. In the insert, you see that there are two sub cycles. In the first part, you are compressing the particle well, not that we're not compressing the value, you're not compressing the effective volume of the particles, you do work and in the second part of the cycle, you are expanding an effective work. Of course, what you can do now with this experiment is to measure distributions of work in the cycle and measure its average which will relate it to the power or its fluctuations. For example, here is the average work. So this is the average work as a function of the duration of the cycle. The slower you do, the more negative is the work. That's the red curve. So that means that... Sorry, I can do all the patience. Sorry, give me a second. I think I can annotate. Okay, I don't know how to do it. Fine. So the average work is decreasing with the cycle time. It means we extract more work when we go slow but the power, if you do this red curve divided by the time, the power has an optimal value. So it is not maximum at slow driving but at intermediate. There is an optimum cycle time to extract power. Interestingly, there was a theory to explain this dependency of the work on time given by Sekimoto and Sasa. And they could fit very nicely. There is a one over tau correction to the free energy here. And finally, you can also take this data, the average work and the average heat and measure an efficiency. So this will be the average stochastic work done on the system and the average of the stochastic heat intake from the hot bath. What they see is that when you go slower you are more efficient but you are always below, you cannot. So the current efficiency is above this red line here. All right, the second experiment I will discuss is a colloidal steam engine. This was done in Mexico. So the idea is to emulate what is a steam engine in an old-fashioned train. And what happens is here in this experiment they have a particle which has magnetite and when it gets closer and closer to the center of the trap it increases the temperature of the fluid and creates a micro-splosion, a cavitation bubble that pushes the particle out of the focus. And then the particle comes back to the focus as I will show you. So here is a video. This is what happens with a particle of one micron and this is what happens with a particle of three microns. So you see with one micron, it's a small particle but the cycle time is fast and with a three micron particle the cycle time is slower but you displace the particle at a bigger distance. This they could measure with, they could analyze the data and see that the small particles you see is the figure in the bottom. They do cycles faster but they move for a smaller distance. There's less work down the particle and the bigger ones are doing also similar shapes but they go slower. The nice thing of this experiment is that the power is 10 to the eight times larger than the steel engine. So it's really, really powerful this compared to the other realization but there is some problem with this remarks that this motion introduces streaming in the surrounding fluid. So the bath is changing and that this you cannot describe this with a non-German equation and you need more complicated physics. That's why in the very end in this paper they didn't use the plastic thermodynamics because this is much more complicated to analyze. All right. Okay, so this was one very nice proposal and another one that I really like a lot is the so-called bacterial heat engine. This was realized in India in the lab of Ajizud. It is really a beautiful experiment where they realized an analogous experiment to the one I showed you before of the stealing cycle. So again, there are isothermal compressions and expansions and isochoric heating and cooling which is done in the same way. You heat the water or you let the water relax to its clean and temperature but now the difference is we are doing this cycle in this bath which is filled of bacteria. So the bacteria are, you can think about it like self-propelled particles that give very strong kicks to the colloidal particle. Their motion is non-negligible because the bacteria are as big as the colloidal particle. They are one micron length whereas the bath molecules of water they are very, very tiny. It's nanometers. So when you run this engine, you can also get movies. Very nice movies. You have the colloid in the center and the bacteria are a bit blurring on the background moving all around. Then you heat up and you see the bacteria going a bit crazy and pushing the particle in all directions. Okay, the consequence of this interaction between the bacteria and the particle is one of the key consequences is that the distributions of the position of the particle are non-gaussian. You can see this in the histograms on the left or top the red histograms it's much more evident than the blue histograms because the water is hotter. Moreover, the dynamics of the particle shows super diffusion at short times. So it is not just describing a super simple and given equation but you have to add more complicated stuff like color, noise, et cetera. So this is also a consequence of what is going on. All right. The interesting thing here is, well, the main point. In the previous slide, the distribution is not Gaussian, right? The black. Yeah, the black line is Gaussian because it's in log scale. Hmm. So Gaussian is e to the minus x square but if you take the log of e to the minus x square becomes minus x square. Okay. So in log scale, the Gaussian is a parabola. Okay, no, okay. So actually... I asked this before because the vertical lines, the histogram is not symmetric with respect to the zero. Yeah, well, this is also experimental errors because you cannot run this cycle forever. You can only run it a few times, maybe 100 times but it is really, really unlikely that the tails are symmetric. This is within experimental errors. Okay, thank you. Anesthetical errors, okay? Yeah. That's a good question actually. Excuse me. Yes. What is the presentation of those bacteria under the diagram, those blue and red representation of bacteria? I cannot understand. Why they are blue and red? Yeah. Or what the direction of those bacteria? Okay, this is just an artistic representation of what is going on. This is the first thing. They are not for bacteria, they are many more because this is a cultural medium that they do. So it's not that they are for this. There are lots and lots of them. The color means that there is half of the cycle, the bacteria are in a fluid that is hot. There is a 313 Kelvin. So this means around, I don't know, maybe 40 degrees Celsius. So only temperature, there's no difference between these four diagrams of bacteria, just temperature. So in the top left and top right, the bath is hot. So there's high temperature and bacteria are moving much more frenetic than when they are in the half, lower half of the cycle, where the water is at room temperature. Okay, so we are doing the same cycle as before as here in the top left, but now the only thing that changes is the bath, instead of being water, is water with bacteria. I see. That's why they put the bacteria red and blue. It means the surrounding fluid, it's hot or cold. They're cold or hot bacteria, okay? Thanks. You're welcome. All right, so and also the matter of fact, this degree box square in the middle, it's the diagram, the, it's like pressure volume diagram for the passive engine in the similar conditions. So having bacteria makes you that the effective volume of the particle increases. So it's like moving a bigger guts and also the effective temperature is increased. So it's like, it has many effects, changing temperature, volume, pressure, everything. So the net effect in the work is that you extract maximum work with the help of the bacteria. So if you see the top figure, this is similar to what I show you in the stilling. So compression expansion, compression expansion. And after one, two, three, four cycles, you have, look at the Y axis, you have 0.4 KVTs. Look at now the second plot in the bottom, one, two, three, four cycles and you are in minus six KVTs. So you extract much more work, which is kind of expected because the bacteria are pushing the particle uphill in the potential. So you are, you have an extra source of energy here. Indeed, you can do this for different temperatures. So you can heat the water. So you have a lower temperature for the water, at least the room temperature and a higher temperature, which can be anything. You can put at 40 degrees at 50 or 60. And here they somehow introduce an effective temperature. So you can take these histograms and feed it to Gaussan even though it is not a Gaussan and get out of it an effective temperature. And they see that they can reach effective temperatures of 1,000 Kelvin, which is crazy, no? Of course, this is the meaning of effective temperature. It doesn't mean it's a real temperature. I will go into this later. So if you have questions, be patient. And, okay, another important result is that out of all the work you get, most of it comes from the non-Gaussan tales of the distributors. So they can separate the work events that come from the central or common events, typical events, and the work cycles that come from red events. And these are the ones who dominate in the work extracted from the bacteria. All right, so this is a very beautiful experiment and I highly recommend you to go through to the paper. Next, I will discuss the Carnot engine, which is part of our work. This was a collaboration that we did in Spain where I was a teaching student slash postdoc, it was in the middle. And I would like to highlight the name of the first scientist which is Ignacio Martinez. He has been my collaborator for many years and we spend many times, many hours in the lab together working. And I'd like also to highlight one thing that is, sorry, how this collaboration started. So this collaboration started in a summer school. You see, it has been 10 years now, 2011, almost 10 years. And I'm telling you this because this is the best thing of schools which is interacting with other people, meeting students who are interested in the same topics. And in my case, it was really good that I met Ignacio in this nyaki, we call it in this school and this collaboration started there. Okay, so just to let you know, if you have a chance to make another school in real life, please do it. All right, so what did we wanted to do in this collaboration, what's the following? We were thinking about making a Carnot cycle. So the previous experiment I showed you, they had all the ingredients, almost all the ingredients to do the Carnot cycle, except one, which is designing an antibiotic process. And a second problem of this experiment is that the control of the temperature is very slow. So you can, the way it was being done was we have the particle, that's a distribution, we heat the water and then the fluctuations of the particle increase. So it looks like the particle is in a bigger trap, right? This has two problems. The first, it is slow, it requires some time to heat the particle this way. And second, there is a limited temperature. So if you heat the bath above the vaporization temperature of the water, the water is gone. So you can heat up to 100 Celsius. On the other side, we thought about a trick to avoid these problems and the trick was the following. The trick is instead of heating the entire bath, we heat only the particle. How can we do this? Well, at the level of the distribution, we can do this by shaking the particle with an invisible hand, right? Because what is happening in the top process? The process in the top, what happens at the level of the particle is the fluctuations increase. So what if now we only tune the fractions of the particle and keep the molecules in the bath at the same temperature? In principle, with a nice invisible hand, we could do this fast and we could do this with no temperature limits. This was introduced in this PRE 2013 and I will now explain how we did this. The setup is simple, well, simple in a way because it is a colloidal particle. It's one micron diameter. It is trapped in an optical tweezer. It is in water. And then on top of that, we put two electrodes. These are, I mean, you combine in a supermarket these electrodes we are using, but what you cannot buy in the supermarket is the generator, electric generator that we put at the two ends of the electrodes. So what we do is we apply a field, an electric field that is very strong. So the particle is in water, but fields an electrical force in the X axis. And the particular peculiarity of this force is that it is not an AC or a DC field, but this is a Gaussian white noise field. So we give a field whose intensity is Gaussian white noise. So we are generating random numbers in the computer. We plug into a generator and the generator gives a random field on the particle. This is called the white noise technique. So when you have this, the dynamics of the particle becomes as follows. So it is like a line given equation with a 10 dependent potential as before, linear potential. And there are two noises now. The noise xi is the thermal noise, which is the water around. And the noise eta is an electrostatic noise. In reality, it's an external force, but it is like a noise because it's random. It's one. It has zero mean as well. It is Gaussian. So we are generating noise and numbers. And its amplitude is nothing to do with temperature. It's an amplitude sigma square noise, which is whatever. It is independent of the thermal bath. That's why he put the last question. And this is what we get out of the equilibrium fluctuations of the particle. So we can trap the particle and get the autocorrelation function of X and also of the noise. The noise is, the autocorrelation function is flat in the Fourier space. So it means it's white noise. The colloid, when it's in the trap, without the noise, it has a Lorentzian power spectrum. It means the motion comes from a Langian equation just like this. This is briefly. When you add the noise, what happens is you put more energy in all the frequencies. So what it means that it's the same as if we had a particle in a hotter bath. This is the spectrum you get when you just hit a bath near a particle. So it is mimicking. And you can design this as an effective model. It is mimicking this equation, which is a Langian equation with an effective temperature, which is the temperature of the water plus something that is proportional to the strength of the noise. So it means, this means that if we add a very strong noise, this will be like heating the particle very much. And if we add a weak noise, it will be like having equilibrium. All right. So here are some experimental results. So when you increase the field, the distribution broadens as expected, and you can use an effective equipartisan theorem to define a temperature like I shown in the bottom. This is the same as considering this model and saying this noise comes from a thermal bath. And what we see is that the noise comes only in the x-axis. In the y-axis, the particle is like in equilibrium with the thermal bath. So we see that along the x-axis, the temperature is very high, but along the y-axis, there's nothing. So we are going to do and to look at the fluctuations only along the x-axis. And a curiosity is that we can reach a maximum of 3,000 Kelvin, which is, well, in between the opposite temperature of the water and the temperature of the sun. So of course we could reach more, but then the experimental setup was not going to survive very much. Of course, this is the temperature of the particle. It's not that the wires and the water and the components of the setup are at this temperature. This is just an effective temperature. Please take it, not as a real temperature, but just as a strength of the fluctuations. Okay, again, something that I showed you the other day is that you can also do non-equilibrium processes like dragging the particle back and forth and test the fluctuation theorem by Krux. So we could do this and we show that the temperature you get from Krux is the same temperature that the ones we get from equilibrium. So this is a consistently check in a way. All right, so now it comes a bit more the essence of the cycle, which is the following. To do analogies, how do we do a isothermal compression in this setup? So an isothermal compression is easy. We just compress a gas at a given temperature. So this we can do by we take the particle in the trap and we just increase the intensity of the laser. So this makes the trap is stiffer and the volume of the particle is smaller. This is a way we do isothermal compressions. What about isochoric heatings like in the stilling engine? So this we can do also simply by as follows. We don't change the trap, but we add the noise. So if you don't change the trap, the volume is the same, but if you add the noise, the fluctuations increase. So it means the particle is hotter in a way. These are simple, these are things that I believe you can guess from what I was teaching to you in the previous week. But what is not so obvious is to obtain adiabaticity. So how do we design an adiabatic process? So following Karno, an adiabatic process means there is no heat, it's in the system and the bottom. But this is not possible. In practice, it's not possible because there are unavoidable heat exchanges between the colloid and the bottom. So the question is, can we isolate effectively a Brownian particle? This is what we call micro adiabaticity, right? So micro adiabaticity is an invention we did, which was the following. We design a process in which the stochastic heat vanishes on average. So if you do the process once that is heat, imagine KBT. If you do it another time, you will have minus KBT. And if you have a distribution, we look at the process that attains the distribution of the heat, which has zero mean, this was our goal. So adiabaticity on average, micro adiabaticity, we call it. So the theory says that in principle, this is possible to do because if you do the process very slowly, it means the total entropy is zero. Adiabaticity is the same thing as saying isoentropicity. So it means I should design a process in which the system entropy doesn't change on average. And here is the trick that I put there. It is to conserve system center. So if you design a process in which the available phase space is conserved in time, you will get an isoentropic process and if you do it slow, a kind of adiabatic process. We could do this experimentally. This is what I show on the right. So the red line is the total heat. The blue line is the heat in the X variable. So here is very important. Some of you were asking about under damp systems. When you change temperature, you cannot forget that there is a cavity of heat in X and a cavity of heat in V. You cannot describe this system without the mass. So you can use over damp approximation to describe the dynamics, but not the thermodynamics when temperature is changing. This is a key insight that we was discussing in two of the papers I told you and also in our papers. Very important. So here is the trick. So the assumption is we are close to equilibrium. So we are doing a very slow process. So when you have this, your distribution at any time is canonical. And the distribution here is quadratic in X and quadratic in V because you have, sorry, it's Gaussian in X and Gaussian in V because you have a parabolic potential in X and you have kinetic energy, which is quadratic in V. So this means that you can compute the average system entropy and from the assumption in the second line, you can get this very nice result that the system entropy in average is equal to the area under the lips. So under the distribution of the phase space. So it's the average of X square. This is the fluctuation next and times the average of V square. This, if you look at the figure in the bottom, we can see this like a shell or like an egg, no? If we design a process that our initial phase space density is the blue circle and we change the dynamics in a way that the area inside the circle is conserved, we will reach adiabaticity because this system, sorry, these dynamics will conserve the area inside the lips, which is X square times V square average. Okay, that's a bit of the story. We design a process in which we conserve the phase space density, which was proposed in the papers in the bottom by Czequimoto and Boa and Celani as well, also from here from STP. And the main story is this one, we compress in the X direction by increasing the stiffness and we need to expand in the other direction in V and we do this by heating. So we need to compress at the same time heat, but this has to be done in a very, very precise way. And if you follow the theory in the middle, this system entropy goes like T square over kappa. So we must, in order to reach adiabaticity, we must make a process that T square over kappa doesn't change in time. So for example, if kappa is linear, T should be a square root. Or if T is linear, kappa should be linear with T, kappa should be a quadratic in T, T square over kappa. This is highly non-trivial, but this is something we could do with our experiment because we could change the noise intensity as we wanted and we could change the laser intensity as we wanted as well. Let me check the time. Okay, if you have any questions, please take a good moment to ask because now I'll show you the results from this experiment. But up to now I just, I'm just using an effective equilibrium model, okay? I'm saying I'm driving the engine very slowly. So locally at time T, the distribution is canonical. It's the only thing I'm saying. Okay, so if there's no more doubts, I'll show you, I'll present you the protocol we did. Here there are two baths, but there are effective baths because the cold bath is the real bath with water and the hot bath is when we add the noise, the fluctuations of the particle are like in a bath of 525 K. So these are the two baths we have. And just look now at the top figure. So we have four parts in the protocol and isothermal compression when the bath is water. Then we do a micro adiabatic compression. So we are doing two things here. We are heating and we are increasing the laser trap intensity. You see the green lines increasing. So I am increasing the stiffness. So I'm making the trap more and more and more stiff. And at the same time, I'm increasing the purple line, which means I'm increasing the temperature in time. Please realize here that the temperature is linear and the stiffness is quadratic. That's why we are doing T square over kappa constant. And then we do the reverse process like isothermal expansion and adiabatic expansion. So when we close this cycle, we can measure effective temperature and the stiffness. And this is what I show in the top right. These are experimental, all these experimental results. The lines is the theory from Carnot. And you see the different types of symbols. It means we do the cycle slow or fast. If you do it slow, we are very close to Carnot prediction. If we do it fast, we are out of equilibrium. So it is not perfectly matching Carnot's diagram, but this is respected as well. In the bottom, I show you other famous diagrams. It's the Clapeyron diagram, pressure versus volume, and Carnot diagram, which is temperature versus entropy. And the nice thing is we could realize here, you can see adiabaticity very clear because there are these isothermal, sorry, these adiabatics in which the entropy doesn't change. So there are two vertical lines in this diagram, which we can achieve experimentally. This is like, it was a very good achievement, I think in the field, because it was the first Carnot cycle, the design ever in an experiment. And you can find in Wikipedia this cycle is a very, it's really a classic result in thermodynamics. But experimentally, it's very challenging because of the fact of doing adiabatic processes. This was the key point. All right, once you have this, you can again quantify the work. On the top left, I show you examples of the work in a longer cycle. You see it is fluctuating. Every cycle gives you a different value. You can accumulate the work done over many cycles. This is what you can see in the top right. And the nice thing is you can now identify in the different parts of the cycle, where do you do work, which is in the compressions, look at the bottom left, and where do you extract work, which is in the expansions. Out of all this noise, you can average, you see that the trajectories are very noisy, but you average and you see that on the average, you get negative work. So you are extracting work from the engine. And on the bottom right, you see, we could do this cycle at different speeds. So when we do the cycle time very long, 200 milliseconds is the one in the bottom, is when we get the maximum extracted work. So work is more negative when you derive the cycle very slowly. This is all suspected from the theory. The theory is here. So on the top left, the blue curve is the work as a function of time. And you see it goes one over tau. You reach the best work extraction for when you go slowly. The heat does the opposite because the energy is not changing. So there is a first law. This is what I showed on the left. And on the right, it's more exciting in this result because I'm plotting the power first. So average work divided by cycle time. It has a maximum at intermediate cycle times. And the maximum is obtained at the Novikov-Kurson-Arbon efficiency. So this was a prediction from many years ago, which is for the efficiency at maximum power, which we get in this experiment as well. And if you look at the yellow, well, yellow green symbols on the right, you see that increasing the cycle time, here I'm plotting versus one over tau. So you should look at the plot from right to left. Increasing the cycle time, we increase the efficiency. And we can reach up to 0.92 plus minus 0.06. So it's very, very close. Okay, this is in Carnot units. It should be this times Carnot efficiency. So we get very, very close to Carnot efficiency with this device. Okay, moreover, up to here, I showed you average work divided by average heat. This is what is the efficiency here. So you do many cycles. You measure the heat distribution. You do the average of the heat over a cycle. And you divide by the average heat. Sorry, you do the average work in a cycle divided by the average heat in a cycle. But what if now you think about the efficiency in a single cycle? In a single cycle, the efficiency is different to the average because it's working one cycle divided by the heat in one cycle. Now imagine you have a cycle where there is very little heat. So KBT divided by zero is infinity. So it means you can define a fluctuating efficiency or stochastic efficiency given as the work in a cycle divided by the heat in the cycle. And that efficiency is not bounded by Carnot. And this is an efficiency that is, it has been introduced by Christian Mandembroke and Massimino Esposito, among others. It is, for example, the first reference I put in the bottom, Nature Communications. This is some so-called stochastic efficiency. So one can define distributions of efficiency. And the papers I show in the bottom, they introduced and they derived universal properties for this distribution of efficiency. And one of them is that if you look at few cycles, the distribution of the efficiency over few cycles can be above Carnot. They can be super Carnot efficiencies. And this we could measure in the experiment. You see that when the cycle time is small, it means that the lower part of the plot, this, what I'm plotting is a distribution when it's a bit abstract this plot, but if you want, I can explain further later. This distribution has events above the Carnot efficiency. And there were other theoretical results. One is that the tails of the distribution are power loss and others that the distribution can be bimodal. So they can be two peaks in the distribution of efficiency. So there are two, actually, there is a result that says that in some types of cycles, the Carnot efficiency, even though it's the average, it can be the most unlikely efficiency. For this, I highly recommend you to look at this Nature Communications paper. It's called the unlikely Carnot efficiency. So you can get average efficiency with this Carnot, but at the same time to be the least likely. This is a very, I mean, it's a theoretical result, but it's very fundamental and interesting. Okay, this is something more advanced. It's related to the large distribution functions. So from the distributions, you can compute distribution of efficiency, the large distribution function and find that there is a minimum. The minimum in a large distribution function means it's the most likely efficiency. And well, actually it depends how you do it. I think here it was the least likely efficiency the way we express it. And we also showed that there are two reversible efficiencies. So here we are averaging the entry production over trajectories with a given efficiency. We see that trajectories that have Carnot efficiency are reversible. They have zero entry production, but also there's a second reversible efficiency, which means that there is another efficiency, which is the one in this minimum at which the accuracy are reversible. So this is a very fundamental result that we could test. We don't know, we don't have a very intuitive meaning of what is this second efficiency, but it's something, it's an open question in the field. And okay, I finalized my lecture today by saying there has been other interesting setups such as, sorry, the one I'm showing on the left, which has the advantage that you can use the work to move particles. So here I'm telling you, I'm doing a Carnot, but the work I'm doing, I'm losing it because I'm not using this work to lift the weight. However, you can cleverly design a trap that is like here, like elliptical and use this type of cycles to move a particle in cycles to use, to make usage of the work in other words. This is called the Brownian curator. It's a classic paper in Pierre 2007. You can also look at it. And also there was this incredible paper in science where they designed the use of our white noise technique, but to do, sorry, to do an engine with a single atom. So we did this with, I call it, but this can be also realized with an atom. And here I should highlight the name of Obinaba who was a nice CTP diploma student and he was involved in this science paper which is really, really amazing. All right, and here are the conclusions. Mainly, Carnot cycles can be realized experimentally on the mesoscopic scale. And this brings to many interesting phenomena which come from fluctuations, such as surpassing the Carnot efficiency for a small number of non-linear cycles. And this is a very active area of research. There is lots of people working now on engines, both theory and experimentally. And I really encourage you to read and to get introduced to this amazing topic. So with this, I am done with what I wanted to explain. I'm up into questions. I'll stop at the beginning.