 And can you see my pointer? The pointer, we need to see your face. Yes, do you see the pointer? OK, excellent. Very good. Thank you. So thank you very much to the organizers for the opportunity to give this talk. I really would have liked, in fact, I intended to be in person in Trieste, but unfortunately, at the last minute, I was not able to travel. So this is work on a quantum theory of the triboelectric effect in collaboration with Professor Robert Lipsky, who's an expert on the theory of open quantum systems. So this is an exercise in the theory of quantum systems, which I suspect many of you will not be very familiar with. So the calculations will be flashed very quickly. I'll try to emphasize the conceptual points. And how is this different, for instance? This is very different from what Professor Marx discussed yesterday for reasons that I will try to make clear. So this is not a practical tribo generator for reasons that might be obvious to some of you. But it's the one that I'll consider because it's conceptually clean and allows to do some analytical calculations. So here there's a material, capital A, and there's a cylinder of that material robbing against the hollow cylinder capital B. And if you choose the materials correctly, you can produce an EMF, an electromotive force, that can drive a current to an external circuit. And this already is a little bit peculiar. So we learned in university the expression for the EMF from the Farting Maxwell law. But even though there may be magnetic fields here caused by the current, there certainly is not a apparently time-varying magnetic flux, average magnetic flux through the circuit as drawn. But there's also another important point, which is that unlike, say, in a Faraday inductor, if you reverse the motion of the cylinder, if you reverse the angular velocity omega, you will not reverse the EMF, evidently, right? The sign of the EMF depends on the choice of materials. So this is already very strongly suggests that this is a irreversible process. The generation of the EMF by the triple-electric effect is an irreversible process, which is my main point. So what is an EMF, an electro-motive force? So this is a concept introduced by Volta a long time ago. And he originally introduced it in order to describe whatever unknown mechanism was responsible for the separation of opposite charges against their long big attraction, whether it separated and kept separated. My point is that what I would emphasize is that the EMF comes from an active non-conservative force. So it's a force that can do a positive amount of work over charges that have gone through a closed path. So it cannot be derived from any potential function, not from a mechanical potential, not from a thermodynamic potential. And this is analogous to hydraulic pumping. So you can say a pump can move water up against gravity. So it builds up a gravitational potential. And then you can measure the strength of the pumping in terms of how much gravitational potential and the water gains, but that gravitational potential is not the pumping. And the point is that the generation of a EMF is an out-of-equilibrium process. And in the context of tribal electricity, there's an old school of thought originally in the Soviet Union that emphasized this very strongly that this kind of effects, tribal electric effects should be understood as out-of-equilibrium processes. And recently this has been taken up, for instance, by these researchers at UCLA. So what would be our attempt to write a non-equilibrium theory of the tribal electric effect? So this is inspired by, so my own background is in quantum field theory in high-end physics originally. There's a story about a Seldovich, a very famous Soviet theoretical physicist. I got this from a top that he gave some time ago, that Seldovich knew something at the time that not basically few other theoretical physicists knew, which is how a wind, the wind can raise away from the surface of the ocean. And I'll do, not only for historical reasons, but also because it allows me to capture some of the conceptual points in a very simple picture. I'll do a sort of a toy derivation of this in a quantum theory, which is a little bit strange, but this is what Seldovich was originally thinking when he did his work in the 70s on what is now called in high-end physics super-radians. So imagine there's a layer of air blowing with some velocity capital V and a layer of water. So you take it to be at rest, but the air can excite a wave in the interface between the air and the water and little b will be the phase velocity of the waves. So the angular velocity omega has little b times the wave number k. And whatever is happening in the interaction between the air and the water must respect momentum conservation, which is just a consequence of the translational interference of the system. And for simplicity, so here I'll take the kinetic energy of the wind to be the momentum of the wind squared divided by twice the total mass of the layer of air that's giving you the wind. And I can derive this use, time derivative of this using the chain rule, then I use the momentum conservation relation. And then a very funny, weird step is that I quantize the wave. So I say that the momentum that is being gained by the wave is equal to the momentum of a quantum of wave h, h, k multiplied by the rate which I'm making quantum of wave, which is f. So the rate at which the wave is gaining momentum is that same after h where omega, omega is vk. And then I see that the condition that the wind be blowing faster in the phase velocity of the wave is equivalent to saying that while respecting momentum conservation, energy is coming out of the wind more quickly than it is going into the wave. So, and this is in fact, if you know there's a lot of confusion in the literature, just as in fact basically the correct condition for when the wind raises a wave, a particular wavelength. So the wind speed has to be greater than the phase velocity of that wave. Notice that this would be, this inequality would mean that this is impossible as a conservative process, but of course this is not a conservative process because there's viscosity, particularly it was very important there's viscosity in the air. So in fact, this condition that the wind be losing energy faster than the wave is gaining it while respecting the momentum conservation means that there is power available to keep the air. And by what some people would call the basic principle of irreversible processes or non-equilibrium thermodynamics if a process can occur while respecting the relevant symmetry of conservation laws and producing entropy, then it will occur. This is very similar in fact, it's essentially the same as the reasoning behind what some of you might have heard of or the land-out criterion for the critical velocity of a superfluid. In fact, I think this is where Sambal which got the idea. It also can be translated into a theory for the generation of sonic booms and other shockwaves in hydrodynamics, Scherenkov radiation, also electromagnetic non-contact friction which I won't have anything to say here, but it's a very interesting subject. And we, some time ago, the same authors, my collaborator, Robert Lipsky and I did an analysis of this from the point of view of open quantum systems. But the story is that Sambal used this argument. So he never set his waves in the published papers but I got this from Tim Thorne and this is what he was thinking about to predict that the rotating black hole should radiate. This was the first prediction that a black hole could radiate. And this later motivated the work of Hawking that showed that a black hole had all the properties of a heap of that. But anyway, what is our key original result? So it's known that you get this kind of phenomenon which is in high new physics, it's called super radiance. This is not a super radiance that is popular in quantum optics, which is a different phenomenon. But it only applies to a boson, a sonic field. And it's known that you cannot have super radiance of fermionic fields. But our main result is that if you have two different baths for fermions, in this case, they'll be electrons. If you have two different baths and they're moving with respect to each other, this kind of reasoning gives you that they can be a pumping of current from one bath to the other. So okay, this I'll have to flash very quickly but of course if people are interested in the details I'd be happy to discuss this. So this is a calculation in the formalism of open quantum systems. What is the system? The system will be the electrons on the surfaces. So going back to this minimal travel or electric generator that I considered before there's a surface represented by a little A that's attached to the bulk material capital A and there's a surface that will be attached to the bulk material capital B. The electrons on the surfaces will be a system and they'll be coupled to external baths of electrons in the boats in capital A and capital B. So this is an exercise in a quantum field theory of irreversible quantum field theory. So this is in second quantization. So I consider the modes. So this would be the Hamiltonian for the modes. Little X would be either little A or little B. So those are the surfaces, capital X is capital A or capital B, so those are the boats. Okay, so again, I'm just flashing this very quickly but the point is that now there'll be motion. There'll be motion between one bath and the other bath because of the sliding. And this is introduced in this formalism. This has been done for a cylinder. The cylindrical symmetry makes it much cleaner to treat this analytically. But at the end of the day, nothing really depends on this and can be easily translated to linear sliding between the materials. M is the magnetic quantum number and omega is the angular velocity. And the other thing, so the surfaces are coupled to the baths. So the other thing that's very important here is that they're coupling between surface and bath. And this G represents some tonally amplitude for electrons to tunnel from the surface to either one of the baths. And I'm not making any attempts to calculate these that this would be difficult. This would require much more expertise in materials science than what I have. But I want to say some concrete things that can be deduced from this description without having any detailed knowledge of this G, so this couplings. All right, so we have electrons on both surfaces. We have electrons on both baths and we have coupling by tunneling between baths and surfaces, that's what it says. And then we can write down kinetic equations for the population numbers of the electronic modes of the surfaces. As usual in statistical mechanics, there's KMS, Kugel, Martin, Schringer conditions that relate the pumping rate, which has a little up arrow to the damping rate, which has a little down arrow for electron coming from the bath or going into the bath. And the thing is mathematically, if you want, the key point here is that this motion, the relative motion of the baths introduces a Doppler shift in the frequency that appears here. So normally in equilibrium, the damping rate is always greater than the pumping rate, which is this exponential is always less than one. But if this M-capital omega introduced by the motion is large enough then that can change the sign of the argument of the exponent to make the pumping greater than the damping. In statistical mechanics, this is called population inversion. It's also behind, for instance, the operation of a laser. And the mu's that you see here are the electrochemical potential of the bulk materials, capital A and capital B. All right, and then we, so this, we can get current of electrons moving from one bath to another bath by other surfaces. But we'll assume that the system quickly reaches a steady state in which the populations, the population numbers of the surface states are not varying in the right conditions. So the thing is, so we arrive at that conclusion that their motion induces currents in both directions. So there'll be electrons that are moving from A to B but also electrons that are moving from B to A. And these are the expressions for those currents in terms of the pumping and damping rates. The net current flow between the two materials is of course the sum of those two currents. And the sign whether there is, so if the two materials are equal, these two currents will be equal and they will cancel each other out. But if there's any small similarity between the materials, one current can predominate over the other current. And again, I don't have time to explain this in detail but it is very important that it turns out that this process is self-limiting. But which I mean that if one current predominates over the other, so that you have a net transfer of electrons say from A to B, then the electrochemical potential of B, for the electrons in B is increasing, right, it's being charged up. And this charging up, so this change in the electrochemical potential discourages the current that is giving the charging and encourages the current that would give the discharging. So the process is self-limiting, it doesn't run away. Okay, so the side of the net current will depend on the relative magnitudes of this quantities. If you look at them, basically the strength of these is correlated to how much each of the material holds onto a certain electron. So this is consistent with what has generally been seen experimentally, that there is, at least roughly there is a triboelectric series, it's not perfect, which wasn't what I was talking about last time, I'll say more about it in a moment. But there's more or less a triboelectric series that when you rob one or two or the other, electrons will go to the material that has usually the higher work function. The series is experimentally seemed to be more or less correlated with work function, which has been seen in experiments also recently. Another very interesting thing that infused me a great deal as an undergraduate student, say you think of a Van de Graaff generator, the Van de Graaff generator has the same contact on the bottom and on the top, their metal brushes rubbing against the rubber belt, but electrons get off on the bottom and on on the top, so the current flows in opposite directions. And what is different is that the voltage is different, so that shifts the electrical potentials. Okay, so I'm running out of time, this is important because I think also Professor Marx emphasized this last time, that people used to think, talk about just this triboelectric series, but we know that tribo electrification is not entirely a material process. We know, and this is consistent with this description because I made no attempt to calculate this G's, this couplings, but of course the surface of the solid is a very complicated landscape and the energy levels vary in a complicated way over the landscape for the surface electrons and there's an exponential dependence of the tunneling rates on those energy levels. So it makes sense that over the landscape of roughness, there'll be regions where current flows one way and regions where it flows the other way. So the triboelectric series emerges only on the average over this landscape, but also the point that triboelectricity also depends on the strain of the material, of course the strain will also affect the surface electron levels. Okay, so I'm running out of time, let me just say one thing that we can immediately get from this theory is that there's an upper bound to how much voltage you can build up with this effect and it depends only on the linear sliding velocities written here BS, so I go back to my cylindrical system, I can write sliding velocity as a radius of the cylinder times the angular velocity omega. And this already is also interesting because it's velocity dependent to something that the old Soviet school emphasized a great deal and which is difficult, it may be possible to accommodate in a theory of tribo electrification as a reversible process. And to close. Okay, all right, so just this, so if you put in the numbers, you get that the maximum voltage that you'll be able to build up just by a pure sliding will be of order 10 to the minus five volts, but as everybody knows, once you've accumulated the charge, you can mechanically separate the layer and multiply the voltage by how much you multiply the separation. And in the van de Graaf generator, this goes from maximum scale to meter scale. So you get 10 to the five volts, which is the right of magnitude. And also something that's interesting and nice, which I think is not trivial. If you look at this recent experiments, you see that the maximum reported density of charge are approximately one minus the other, which you would expect in this theory. Okay, so just let me close the point is we're trying to describe electricity as a new universal process. And there's some predictions that this model can make even without any detailed modeling of the surface electronic states. More precise comparison to experiment, of course, would require more precise control over sliding and over the surface separation, which is something that as far as I know, it's a little bit difficult to get an experiment because of course, stick slip and things like that. But let me close and if people have questions, I'd be happy to take them. Thank you very much. And here. In your expression for the current, you had a parameter gamma, capital gamma. What was that on slide eight? It was just, okay, yes. This is, this gammas are just some, so this, you see this gammas, the region, the first gammas have two, lowercase letter and uppercase letter. Lowercase letter is which surface? Uppercase letters is which bulk? And then there's, so it's a coupling of the surface electrons to the bulk electrons. The bulk electrons are considered as a bath, which with a simple baths with a temperature and an electrochemical potential. There's a pumping rate to the rate at which, right, the bath would pump the surface states in a damping rate to the rate at which they would go into the bath, related to each other by a gammas condition. And just for simplicity of notation here, the capital gamma is this sum. So it's a sum of pumping rate and damping rate for the coupling of surface A to bath A, plus the same thing, but for the coupling of surface A to bath B, and same thing with B, just a notation relation. But it's all in terms of this damping and pumping rates for the coupling of the system to the baths. In your theory, you are considering fermionic baths to couple with. In principle, there are also vibrations, so the phonons, how about them? Absolutely. And in fact, the phonons are bosons. So the phonons already have the normal super radians that people have known, at least where I come from in a point of view, people have known about for a long time. In fact, originally, the way that this project started was that I was trying to convince my, so I told my collaborator who was an expert on open systems about this solubility super radians which tried to do some things with it. And the original idea was whether we could say something about ordinary friction in terms of the super radians of soft phonons, maybe not normal phonons, but really phonons and porous surfaces and so on. So that kind of, when my collaborator realized that we could do something with electrons that had not been, so even the idea that you could get this analog of super radians with electrons as long as you had two baths and that this could have something to do, tribe electricity came up with kind of led that by the wayside, but this will definitely be there. And in fact, I suspect that there might be things to do in the microfysics of dry friction from that, from the production of soft phonons. Maybe just opportunity, let me just say something very quickly, this process, so this current, this opposing currents, they always dissipate kinetic energy, they always are producing heat, even if there is no significant net charging. So this also could have something to do, it could be a contribution to pure dry friction, which is something also that- Okay, we're running a little short of time. I have just one doubt about your initial Zildovic story. I thought the dispersion of water waves was not linear like you drew it. Right, absolutely, but here in this argument, this was not assumed in any case, so this V is not constant, this V will depend on the K. And in fact, in the case of waves, I've looked into this a little bit, it makes sense because the dispersion relation for surface gravity waves, as people call them, is that the velocity, little v, is an increasing function of a wavelength, there's a decreasing function of K. So it takes a greater wind speed to raise away with a longer wavelength, which is of course what you see, so small winds raise short wavelengths and you need stronger winds to raise stronger waves. But yes, you should not read this as a constant, this can be a function of K, which is definitely in the case of waves. Thank you very much. I think we'll now thank Alejandro Jenkins very much.