 I apologize we had some technical problems and but now we are ready to start so welcome all to this day of tutorials or the workshops or workshops on stochastic thermodynamics and then we start with Massimiliano Esposito will give a review on inclusive stochastic thermodynamics so Massimiliano the floor is yours. Okay. Thank you very much. Sorry for all these problems. Can you hear me well? Can you just confirm that you hear me well? Yes. Yes, we can hear you well. Okay, and do you see my full screen or do you see also the videos on top of my presentation? I see the full screen and on the side I see the. Okay, then let me put it in the corner like this and hope that it's going to work like this. No, I don't see your screen, your video on the screen, on the slides. Okay, so you see my slides. Well, I can start. Okay, so I was asked to give a tutorial on inclusive stochastic thermodynamics, which means as I found out that inclusive refers to the fact that it's encompassing both the system and the bath. And so my way of understanding that is what are the microscopic foundations of stochastic thermodynamics when we look also at what happens into the bath and so my plan will be to try to show how we can recover these laws that originally were from phenomenological thermodynamics and I think it's fair to say now that within stochastic thermodynamics in the sense of the theory that one can build on top of a stochastic dynamics which can be seen as a kind of effective theory for the system itself where the bath is not explicitly present. And so the current theory of stochastic thermodynamics basically has achieved the goal of mimicking reproducing all the requirement from phenomenological thermodynamics mean namely splitting the energy of the system into two distribution, one that is called heat and another one work and the rational behind that splitting is the second law, which identifies in the change for the system entropy. And the reversible part that is this Sigma called the entropy production. And the reversible part that is better times the heat so in other words the heat is that part of the energy that ends up contributing to the entropy balance and which can be thought of as the one contributing to the entropy change into the reservoir. And the one contributing to the entropy change in the system and the one in the reservoir is this entropy production and what characterizes. Yeah, what what another key important key property is the fact that entropy the rate of entropy production is also positive. The system relaxes to equilibrium and at equilibrium, the rate of entropy production is zero. And also if I consider reversible transformation, the entropy production along those is approximately zero approximately is in this in the sense that it is a much lower order than the change of entropy and beta times the heat in the second Do you see my mouse when I show. Yes, yes, I can see your mouse mouse. Thank you. Now that I will move to trying to, okay, where does this come from from a more microscopic fundamental point of view, and namely, from a system reservoir picture, in terms of a quantum mechanical description of the full system Hamiltonian one. My talk will be into two parts. The first one is the one concerned with system reservoir setup. By this I mean that the system is typically putting contact with the reservoir and stays in contact with that reservoir. I will first derive exact identities in that setup. And then I will show you how one goes to the, how one derives so classic thermodynamics from it and how these results follow from so classic them and I follow from these exact identities. In the second part. I will treat the setup of repeated interaction where now it's more than the system is put in repeated contact with different systems that can play the role of the reservoir. So it's, it's kind of iterative. And we will first consider non autonomous setups and then mostly autonomous setup within scattering theory. So that's how I'm going to address this problem of the inclusive approach to specify system. And at the end I will comment on other possible approaches. So let's start with the system reservoir setup. In mind here. You don't see my titles, or do you see them. I see open one. You see it okay I don't see them but you see that's good. I'm going to consider an Hamiltonian of everything that's my full Hamiltonian system, which is made of a time dependent system Hamiltonian about the reservoir or I put E for environments but in words I'm going to often say reservoir sorry for that. The reservoir Hamiltonian or the environment Hamiltonian and how the system and the environment interact and the time dependence can be both in the system or in the interaction. So the dynamics of these entire system is simply ruled by the unitary evolution with the full Hamiltonian. So you can choose that's an important that's actually the only assumption of what this that these are the two only assumptions that I start from a factorized state between the system and the environment. And my environment or my reservoir is at equilibrium in a gift state but my system can be in any state. And now I will call the system dynamics. I will call the reduced density matrix of my system which I obtained by tracing the full density matrix of the entire system reservoir. And I can, this is the map that when applied on the initial condition of the system gives me the system at time T, I will be noted by this. Would you like. So, are you assuming that the system and the reservoir are independent at time zero. So in the initial condition, yes, because you see I choose a factorized state here rose tensor product with the bath. I don't the interaction could in principle be turned on already, you can think of it as a sudden switch, you can either think of it as the interaction is already on or you turn it on instantaneously at time zero the only key assumption is this factorized state here. Okay. So initially there are no correlations so you can imagine that I bring my system in contact with my reservoir at time zero so there are no pre existing correlations. Now, given this. Given this, I can. I don't see my. What I can do. Yes, I see the title thermodynamic identities. Thank you. So thermodynamic identities at the average level so I will first focus at the average level and then I will move to the fluctuating level. So now these are definitions but then I hope that I will be able to convince you of why they make sense so the work will be the total change in the system in the total system so it means that I know that that when if I have a time dependent Hamiltonian energy is conserved so now I have a time dependent Hamiltonian and so that time dependence causes the total energy to change. And that total energy change is going to be called the work. Okay, and that's why I can express it in terms of the time derivative of the Hamiltonian. Now the heat is going to be defined as the amount of energy in the reservoir and its difference between the initial time and the final time with a minus sign because I like to count it positive when it enters the system. Okay, so change of energy in the environment with a minus sign is the heat. The system entropy is the von Neumann entropy of the system. And the energy of my system is the both the energy of the system and of the interactions. That's important is the fact that the interaction is there. But of course if I initially turn it on and off I can make that term disappear, even though it's there during the evolution. I write a balance that links these different definition, the balance of energy of the system as work plus heat. And I can prove within the assumption that I showed before factorized initial states and starting at equilibrium in the bath. And the change in the entropy production defined as the change in the system entropy minus beta times beta is the inverse temperature times the heat is always greater or equal than zero, which is the structure that we would expect from thermodynamics. But of course, I haven't said anything about the rate of entry production. This really is fundamentally based on the fight that I started uncorrelated. And then I know that up to time he's going to be positive, but I'm not saying that the rate of entropy production is always going to be positive. So this was first derived by Jardinsky in 99 in the classical setup. And we derived it for quantum system as I presented it here with Catherine and back and Chris Vandenbroek in 2010. But furthermore, we could show that we could rewrite this second load a form of the entropy production as follows. Let me just remind you the definition of the relative entropy between two distribution, which is written down here and is always positive. It's a kind of measure of how different rows from Sigma. And the definition of the mutual information, which is, you take the sum of the system and the environment and you remove the total entropy. Okay, and that quantity is also always positive because it can be because it can be written also as a relative entropy. So it's a measure of how correlated the system density matrix is from the environment density matrix. And I could we could show that the entropy production can be written in in this form here this relative entropy, which compares how different the total density matrix is from the exact the system exact density matrix the one obtained by tracing over the whole one and the equilibrium one, the initial one of the bath. And I like this expression because I think it represents well conceptually the type of information that is lost when we use a thermodynamic description in term of the system, because what we keep the entire information about the system. And we disregard the correlation with the environment, and the state of the environment itself which here is taken as assumed at the equilibrium and it's precisely that loss of information that is quantified by this entropy production. And therefore, this entropy production can also be split into two information tariff quantity. One is the mutual information between the system and the environment. So this is really the part about the correlation. How many correlation there are between the system and the environment and now it's the exact density matrix. And the second one which tells me how away from equilibrium is my thermal reservoir. And we have a question in the chat. Actually was sent to me from Anupam Sarkar asks, what happens to the first law if some energies hidden in the correlation established between system reservoir during evolution. If the energy is set right. Yeah, it is stored. Energy or entropy. If some energy is okay. I think stored in the correlation established between system and reservoir during evolution. What energy will be stored. If you want in that, in the sense of the of the of these interaction. And during the evolution, it's not a problem. And if you want to get this contribution out at the end of the evolution you can turn off the interaction otherwise it will still be there. At the end, but this doesn't create any problem in the in the formalism everything that I said holds and is quantified by these quantities. Now this mutual information contribution to the entropy production. There's these Iraqi leap inequality that tells us that it is bounded by the minimum between the system entropy and environment entropy. And since the system can be quite small in many practical application that we care about doesn't need to be often it will be. It means that this is a serious constraint on these on this part of the entropy production and I will show you now in the next slide. How, especially when the, when we deal with more than one reservoir. It means that the dominant contribution will be the one due to the relative entropy between the bath and its equilibrium state. So let me furthermore do a further split that because I will use it in the next slide on the numerics. This distance between the bath at time t and the bath at equilibrium can be further split if we are dealing with a non interacting reservoir. So an only interacting reservoir can be decomposed in different modes. Because itself, the, the relative entropy that measures how far away each mode off from equilibrium and how much correlated these modes are with each other. So, how much entropy is stored in between the modes, rather than in how far they are from equilibrium. Okay, so keep in mind that this these are these two contribution. These are the distance of the entire reservoir from equilibrium. So now let's look at to reservoir and this is a very simple non interacting model. You can think of this as a quantum dots bath of fermions and a coupling between the two and we now turn on the dynamics. So what happens to the entropy production. The two temperatures are different of the two reservoir so there's going to be a heat flow between the two reservoir and entropy production you see increases linearly over time as expected because there's a continuous flux of heat. So we go very, very far because the system is finite. This is going to change but there's a long time a long regime over which we have this linear growth of the entropy production and you see that the the part due to this correlation between the system and the environment the mutual information saturates quickly, while the part that contributes the most to the entropy production is the distance of the reservoir from equilibrium. If we furthermore look at the details of that contribution, as I said before it has to contribution, the part that is the correlation between the modes and the part of the how far away each mode is from equilibrium you see interesting me that it's the correlation part that is actually dominant. So this is a non trivial observation. Okay, so it really means that the dissipation takes the form of correlation between the modes in the environment. Let's now look at a single reservoir. And this is all these work has been done by Christopher Kaczynski I should emphasize that. So if we look now at a single reservoir, there's an interesting phenomena that we observe which is has been coined by Christoph post termination. Because you see that this from the system perspective, this is the entropy production the system quite quickly relaxes to equilibrium we can say that more or less around here the system is at the equilibrium. But there's still no trivial dynamics going on in the reservoir. And this can be directly seen by this swap between the dominant contribution at short time the dominant contribution is the one due to correlations. Okay, here the Iraqi leave inequality is less stringent because we don't have this extensive behavior because there's no heat continuously flowing it's a system that is not that the equilibrium and relaxes to equilibrium so that's why otherwise the other contribution. This one would take over, but because you're, it's not the case both are important and you see that in the early phase of the dynamic of the dynamics, the entropy production is dominated by this correlation between the system and the environment. And then they decay over after the system has already relaxed and what becomes dominant is actually the distance from equilibrium in the bath. So this is also a non treatable effect which shows that things can go on in the bath that are not seen from the system perspective on it. And further, we can do as before, look at the two contribution to the relative entropy here and once again we see that it is dominated by the correlation between the modes, rather than from the distance between each mode from Now let me do a final more recent observation, which is. Okay, from the seat we know that from the system perspective the different ensembles in the sense of statistical mechanics, canonical micro canonical do not should not matter too much. And this is well studied and we know under what condition we have a kind of equivalence, but now we asked the question what about these entropy production expressed in terms of its different contribution. When we look at different ensembles and here we have two different models one is called the spin bomb it's a two level system coupled to a bath that is described by random matrix. So the problem is in Sigma X. So this can be seen as a way to mimic in a very simple way, a many body system, and the other one isn't is the previous model that we have before. So we're starting from unique system. And let's look at what happened to the different contribution to the entropy production when we consider a thermal reservoir as we did before. We compare it with a micro canonical reservoir, whose energy is chosen, of course, in such a way that it fits the temperature of the canonical one for two different energy with with of the energy shell. And we even take the extreme case of a single Eigen state, such that the energy matches with the canonical one at the given temperature team. And you see that at the level of the entropy production there's a perfect agreement with all of them. When we look at the, the two contribution correlation between system and bath and distance of the bus from equilibrium, we see significant differences. And the same is true. I will, since we are a bit late I will go I will simply say that the same is true also in the interacting model. So the lesson here is once again that the entropy production you can think of it as a feature of the system at the end we will express it only in terms of the system dynamics, but these information theoretic contribution to the entropy production. They really depend on what's happening in the bath and there is really no trivial information hidden in those quantities. The current system can display very significant differences between what is happening in the bath, although from the system perspective, everything is equivalent. That's the lesson here. Okay, so now I move to fluctuating level. This is a bit more technical but I will try not to dig too much into the technicalities. The technicalities themselves are actually well known and it's not new. The point here is to define fluctuations to do fluctuating thermodynamics. And it is known that the way to do it in quantum system is to use the two point measurement approach. This means that we consider the system Hamiltonian and Hamiltonian of every bath, if they are and bath. And we imagine that at time zero. We measure projectively the energy of the system and all the bath. Okay, then we propagate the system with a full unitary dynamics and at time T we measure again that energy. And now these so called counting fields lambda, they will be the conjugate variable to the energy change of each of these Hamiltonians here. So land that will keep track of the changes of energy in the system and the S sorry in the system and the one in bath one, etc. The key objects, the central object of the theory is this generating function, you see, it's a modified unitary, it's a modified operator that if lambda would be equal to zero would be the unitary evolution. But it has now these e to the lambda H, which are the these counting fields lambda. And you can, if lambda is equal to zero of course this trace here would be equal to one. Because the land that it when the land are non zero, it's as if the system is that the quantity is counting the energy exchanges. And you can get the moments of these energy transfer by derivative with respect to lambda and you can also say that if you send lambda to I lambda, and you take the Fourier transform you actually reconstruct the joint probability transformation of all the energy exchanges in the system and in the different reservoirs so this has the full statistical information about the energy changes between every object system and bath in the system in the total system. We can also define the time reversed one this is going to be important we know that in quantum mechanics the dagger evolution has to do with the time reversal operation. And we will consider for the forward evolution. I assume that we start as, as before in a factorized state, but to get what is called the detail fluctuation theorem I will also assume that I initialize my system in a gift state at the different at the temperature beta s that can be different from the one of the of the bath. And then I propagate with this modified with this dressed evolution. These initial density matrix to get the generating function at 90. And for the time reversed one I start also with a factorized state but the gift state of the system now is taken to be at the final value of the forward evolution. Okay, so I re initialize the system at the equilibrium, and then I implement the backward dynamics. And one can show that there is this fundamental symmetry between the statistics in the forward process and the statistics in the backward process. This involves the change in free energy. Again, this is a result that Chris Rosinski derived for classical system in 99 already. But in quantum mechanics, it was under this exact form derived very recently in this paper but it was already implicit in various, for instance, my review on 2009 and paper by Gaspar. And thenitasaki and overall around the same time. Now, this is what we call detail but the fluctuation there because it really involves the four dynamics and the backward dynamics. integral fluctuation theorem which is slightly more general in term of the initial condition but of course less detailed than the previous one in term of the statistical information because it doesn't compare forward and a backward evolution. It will only make a statement about the statistics of the forward evolution. In this case we can take as I used before when I presented the second law the identity at the average level. I can take any density matrix for the system factorized with the bath at equilibrium and I can look at the two points statistics in the sense of the measurement as described here but now instead of measuring the energy of the system I measure this is of course how to do that is not trivial it's more of a conceptual result. You can projectively measure log rho in the basis that diagonalizes log rho and you can show that the generating function at a given value minus one of the counting field which is equivalent to taking the average of the exponential of the entropy production which is now a fluctuating quantity because we measure that quantity is equal to one and this is the so-called integral fluctuation theorem and it implies that in average the entropy production is equal to zero which is the result that is equivalent to the result I was showing you before. Okay so this is to show you that these exact identities must be there really if you consider these classes of initial condition it might sound a bit abstract it actually is a bit abstract but it has practical implication when we will try to build the theory only at the system level. Furthermore there's another important symmetry which is energy conservation in this case I restrict myself to time independent Hamiltonian when the total energy of the system and bath is conserved and this implies that not too surprisingly since lambda is really counting all the energies it means that this dressed lambda and I realize I forgot to define the dressed lambda I'm sorry for that so the dressed lambda is the object just before taking the trace here okay so it's the u lambda applied on row zero and u dagger lambda on the other side so if that that object will have that symmetry so that's one additional symmetry that needs to be conserved respected. The the strict way to implement that is a kind of trivial one is to assume that the coupling commutes with the system and the bath Hamiltonian but this is of course quite strong and it's not necessary as we will see at the level of the effective system description. So now let me say the following these are exact identities and I'm totally I understand that they might not be I mean it's a bit technical but the the important message is these identities we know that they must be satisfied and so we as we will now try to derive an effective description in term of the system only what we want to make sure is that the effective theory does not destroy those symmetries that we identified and basically this has the following consequences if we now look at the system density matrix still evolve with this lambda dressed operator we will get these effective operators acting in the system space that will need to satisfy this symmetry so this symmetry is the consequence of this previous symmetry at the level of the system evolution until now this is still exact now comes the key point is that when we want to get a close dynamics for the system density matrix we typically need to assume this semi group property of the evolution of the system and that's really where the most uncontrolled approximation appear and as a result of that approximation we get what is called the super operator of the evolution of the reduced density matrix of the system which fully characterizes the dynamics of the system we closed the dynamics for the system we don't need to care anymore about the bar if this is valid if these assumptions are valid we can only resolve the system dynamics and we get one can show that these symmetries that I mentioned before have constrained at the level of this effective dynamic so if you don't want your fundamental symmetries to be broken you need to make sure that this super operator has this symmetry here which is the fluctuation theorem symmetry which is reminiscent of a detail balance condition that's what we call generalized quantum detail balance condition and also the energy conservation condition which is that one and that's now a kind of guide to check that the effective theories are correct and one can show that they imply the stochastic thermodynamics basically at the system level we can really show that now we can write the balance of energy for the system in term of heat entropy production rate is positive now not only the integrated one but also the rate is now positive and I can for those who are familiar with that I can make contact with the traditional definition in the literature that were obtained a long time ago in the case of a single reservoir for the heat in term of in the context of quantum master equations and we can also show that the Gibbs state is the fixed point of the super operator meaning that the dynamics will actually relax to equilibrium so we really have all the requirements that I mentioned at the beginning of phenomenological thermodynamics and now a few comments about the well-known quantum master equation do they satisfy or not these conditions the redfield equation we know it's not a lean platform so it has already fundamental issues in terms of preserving the hermiticity and the positivity of the of the matrix it has not the fluctuation theorem symmetry and it doesn't conserve energy so strictly speak of course it doesn't mean you know you can always violate this in a weak way but the structure of the equation has none of those symmetries recently some symmetries version of this master of this redfield master equation which are now of lean platform have been considered they do have the fluctuation theorem symmetry but they don't conserve energy at the fluctuating level only in average and then the one this is kind of known in a sense we are only recovering a known result but with this more systematic machinery is that the redfield equation when we apply the rotating wave approximation or when the levels are degenerate that this these are this is a level of theory which is fully consistent with all the symmetry and it preserves all the fundamental symmetries for from first law and second law the fluctuating level also at the effective level and I think what what did we gain well I think we gained an easy way in term of these if you derive a master equation rather than having to explicitly recalculate everything to check that all the fluctuation theorem etc are satisfied you can immediately check if those two fundamental symmetries are preserved and if they are you know that stochastic thermodynamics is ensured at the level of this effective theory okay so I hope that with this I could kind of show you that at least for the system reservoir classical setup we have these fundamental identities in the full space that can be derived and as we trace out the path we can construct a reduced description only in term of the systems dynamics and we can have we can build thermodynamics in term of this reduced description and we need to make sure that the those fundamental symmetries are preserved at the effective level and if they are we basically succeeded in construct in deriving an effective stochastic an effective theory which is called stochastic thermodynamics for open systems where we don't need to explicitly track what is happening in the bath so I will now move to the second part if there are questions maybe it's a it's a good time to ask and also how much time do I have given the mess of starting later and all that okay so we have to start with the next I mean in the program we have to start with the next tutorial in a half six so there is half an hour but maybe if Hugo is okay with this maybe we can eat up a little bit of his time if there are many questions and so there was one so essentially I think maybe you can go on I think you you can go on for other 20 minutes if that's okay yes okay and then we see how it goes and there was a question in the chat from before and so is this formulation consist for work okay I read it so is this formulation for work considered possible exchange of energy through work between two subsystems environment and systems so so let me try to repeat so if I understand the question is whether they can be work between the exchange between the the system and the environment right yeah yes so yes in the sense that the the work contains these two contribution in this formulation it contains the because the this is the work is expressed into the time dependent change of the full Hamiltonian and the full Hamiltonian in the most general case has time dependence both in the system and in the coupling meaning that even if there's no time dependence in the system but you turn on and turn off the coupling for instance then they will be work due to this turning on and off of the coupling so there is work really at the interface between the system and the reservoir okay I don't see other questions in the chat so now well there is a question what is Linblad Linblad is the most general form of master equation that reserves some basic properties of the density matrix so this is a known result in the literature on open quantum system of the the the symmetries I mean entering into the detail would be too long but the symmetries that the most general form of a master equation that reserves the positivity of the density matrix I mean TCT the trace equal to 1 and the complete positivity which is a bit more involved but yeah so basically if it doesn't it's you need to be careful and then you need to understand that there might be problems of course the amount of the problem is can be related to the validity of your approximations but and it is debated whether this is a major issue or not but you need to be cautious when it's not of Linblad to make a long story short a debate of 34 years in one minute okay so I don't see other questions in the chat and so maybe you can go ahead and maybe for the rest maybe we can collect questions in the chat and then we will ask we will address them at the end okay okay and I will try to give more I think go a bit quicker so that I don't eat too much time for the others okay to emphasize the the main points okay so now I move to the second part which is repeated interaction this is now it's not anymore that we put the system in contact with the bath and we leave it there in contact with the bath the picture that you should have in mind is that the system is put repeatedly in interaction with things that I call units and so you know there's a first interaction with one unit then the next unit comes in and and the units are always fresh so there are no initial correlations between the system and the unit and and then the evolution of the system unit happens with the unitary evolution given by the total Hamiltonian system plus unit plus coupling and then every for every interaction there's an evolution and so you iterate these different evolutions and and that's kind of a repeated interaction repeated interaction setup okay in the traditional I mean this has been also studied a lot uh some some years ago and in in this paper there's really a complete description of that but in the non-autonomous setup by this I mean that the way in which we imagine that the interaction occurs is by turning on and off the interaction between the system and the unit but turning it on and off with a time dependent Hamiltonian okay there's no notion of space at this level and it actually relates to the question that the person was asking about the work between the system and the environment because if we start with thermal units we can derive a first law and a second law and I I send you to this paper here for the details but and this is a point that was emphasized by first by Philippe Barra one should be aware that because we turn on and off the interaction we are doing work on our system and as a result of that it is not surprising and this was a puzzle during some time in the literature it is not surprising that the system does not relax to equilibrium because we are continuously driving it due to this work you know we expect in thermal dynamics the system to relax to equilibrium if it is not driven but if we keep driving it it's not going to relax to equilibrium and so these models can be thermodynamically consistent but generically they don't correspond to a system even if we initialize the unit in thermal states they're not going to mimic the effect of a thermal bath or they're not going to thermalize the system and therefore I mean it's motivated by that that actually um Felipe Baja Juan Parondo Sam Jacob we started um thinking about a more fundamental description of these repeated interaction and we use the scattering approach and so here there's no time dependence uh and the time dependence of before is explicitly we expect to to to describe it more microscopically more realistically by incorporating space and uh you see now the the Hamiltonian looks as follows we have the system the the the unit which you you should now think of HU as the Hamiltonian of the internal degrees of freedom of the unit but we add kinetic degrees of freedom to the unit so there's space that is introduced the unit has a kinetic energy um and that's why it can move so it's not that we are turning on and off to mimic the effect of the interaction it's really that the the units are moving and they collide with the system and and that creates the factor this repeated interaction and the interaction now depends of course on space and will happen only in a region where the potential is is localized and before and after we leave the interacting region and that's why we can use scattering theory to look at the dynamics of these type of setups and it's I mean we I don't go into any detail of scattering theory but these scattering operators they they preserve the the unitarity of the full dynamics they preserve also energy so the scattering matrix commutes with the total energy of the system in that sense we have a first law saying that the the change of energy in the system and unit is always compensated by a change a corresponding change in kinetic energy okay the kinetic energy of incoming and outgoing wave packets and the unitarity of the of the scattering matrix implies that the mutual information can be split into the entropy change into system and unit again the internal part and the the one due to the kinetic degrees of freedom and this is what we are going to use to the energy balance and the entropy balance to to look at different limiting case of how we prepare these kinetic degrees of freedom and also the internal degrees of freedom to mimic either a work source or a heat bath a heat source so the key point will be to distinguish between two types of wave packets so these kinetic degrees of freedom in one case we refer to narrow wave packets this is depicted picture here as wave packets which are narrow in momentum and narrow is compared to the typical energy of the system okay and system here think of it as system units for the moment I really all the internal parts of the system and of the units are this SU this joint SU system and it has some energies and frequencies associated to this energy and so if the wave packet is narrow compared to this typical energy that you have the following picture holding that you have a certain wave packet with a certain momentum going in and then it exchanges energy with the system unit internal degrees of freedom and since these energies are large compared to the width you basically get well separated peaks after the interaction to be distinguished from the broad situation where there the width is large compared to this typical energy scale of the system internal system unit system and and there you have more this picture you still have overlap and you it's harder to tell what happened really in terms of exchanges due to this overlap okay so this will be the central concept broad and narrow wave packets we can show that if we consider narrow wave packets the narrow wave packet always induce a decoherence process so the decoherence in the system unit internal space will die off and we will be left with a mark of chain for the population in the system unit that's the first important result but at the moment still not enough to connect to thermodynamics the the the second important result is that if we prepare these wave packet distributed in the momenta using the so-called effusion distribution so effusion is well known in classical kinetic theory it's the this the velocity distribution if you look at the particle coming out from a a box with an equilibrium gas what is the velocity at which they come out at the at the level of a very small hole that you poked into the box it's essentially the Maxwell velocity distribution but reweighted by the fact that the fast particles are oversampled and that's why you have this additional term here so it's a well known thing for things coming out of a of a thermal reservoir so if you assume that these wave packets now are distributed according to such a in velocity momentum according to such an effusion distribution with narrow wave packets this is also important you can prove that the decoherence is still valid because of the narrow wave packets and furthermore because of the effusion distribution the map satisfies the property of detail balance which ensures that the system will relax to equilibrium keep in mind that now system it's still system and unit together so these internal degrees of freedom will thermalize and we really can say that the kinetic these kinetic degrees of freedom played a role of a heat reservoir because that energy that is exchanged that that comes from the changing kinetic energy of the incoming and outgoing packets is really the heat okay and also the entropy balance you see that the heat appears as a change of entropy so this is exactly what we what we would expect what we had in stochastic thermodynamics for heat bath but it's still system and unit internal degrees of freedom but we can go further we can also show that if furthermore we assume that the internal part of the unit is thermal okay also the system will thermalize everything now that I said before for system unit applies now for the system alone so the system itself will thermalize and satisfy a second law first law and will relax to a well-defined Gibbs distribution respect to its own amulet tunnel okay we can also show that if the internal degrees of freedom are at a different temperature with respect to the kinetic ones that we the system will remain out of equilibrium and will be dissipating so this is to to show you it's a funny application you basically send these particles particles in the sense of here the particle was the internal unit plus its kinetic degrees of freedom but you could prepare both of them you imagine that they've been prepared in different temperatures and the interaction with the system will now allow a heat flowing between the internal degrees of freedom of the particle and the kinetic ones and you can look at the entropy production and you see that you only reach zero when the temperature of the internal degrees of freedom matches the one of the kinetic degrees of freedom so this is a kind of funny application okay now I want to convince you that we can this was for heat now I want to convince you that we can also do work and here the message is the following if we prepare the wave packets at high velocity this is this corresponds to a kind of semi classical limit so the the kinetic energy is high compared to the interaction potential and also the the the the typical spatial scale over which the potential varies must not be too small compared to this incoming velocity so that the wave packet cannot resolve these fine details of the potential the potential must be sufficiently smooth in a sense and furthermore we consider broad wave packets as defined before in that I and I should also emphasize that now I can define a time that is given by the typical distance of the interaction that the this is the time the spatial scale over which the interaction happens and the velocity of the incoming wave packets that gives me a time and that time will be important to compare now with this non-autonomous description that I was referring before because we can show that in that case the the map the scattering map really takes the form of what we would expect for a time dependent map so without space but only time provided that the the potential now is represented by the spatial averaging of the real spatial potential and so you see this is V is this spatial average potential and the tau is the tau obtained using this quite natural picture and you can really explicitly show that if these conditions are met the scattering map the scattering maps coincide with the non-autonomous map that I mentioned when I was describing this non-autonomous description of repeated interaction so we could really connect the two type of description and derive the one that was mostly used in the past to this one in a certain limit that's also the same limit that justifies using time dependent Hamiltonians and we can also explicitly show that the the energy balance in this case and the entropy balance match with what we would expect for a work source because now the change in energy the kinetic energy change corresponds to the to the work and and there's no entropy change in the wave packet because it is semi-classical so that there's kind of a very negligible effect on the entropy change of the interaction with the system okay I will skip the numerics the numerics confirms what I said theoretically and let me just repeat and conclude this part saying that we basically identified two different regimes that in this scattering approach to repeated interaction where we can use wave packets to describe a heat reservoir so in that sense they really play the role of a heat bath that thermalizes the system and we and the other extreme limit is the limit where we have broad wave packets semi-classical that really do work when they interact with the with the system and we think that this is interesting this should be seen as two extreme case of course there are plenty of in-between case and and these will be neither heat nor work they will have more non-trivial effects from an energetic point of view because there's going to be a non-trivial information effects in intermediate cases and and we hope that this could be an interesting way to approach problems of energetics in systems like this cavity QED where we one sends atoms through cavities and one can control how one can prepare them before they interact with the cavity and so one could maybe play these kinds of game of preparing objects in work-like or heat-like states and see if everything I said can be reproducing those experiments I was this I will skip for the sake of time yeah let me simply skip that and and let me conclude I hope I I managed to show you that at least in these two setups that I consider system reservoir setup and repeated interaction setup both having common that we always at time zero have no correlation between the system and the reservoir or the units that we can really derive from microscopic quantum mechanics with assumption of initial equilibrium for certain degrees of freedom the phenomenological laws of thermodynamics or of stochastic thermodynamics and we understand also how system thermalized or not and I also want to say by this that there's of course much more to say about this topic because I only focus on these uncorrelated states but there's been a lot of work also on correlated initial states the simplest case is we start really from a full deep state between the system and the reservoir and we drive them with a time dependent Hamiltonian or a more refined approach that works only for classical system the generalization to quantum is still more open where we consider conditionally equilibrated system reservoir states where the system at time zero interacts with the reservoir but one one fixes degrees of freedom in certain states so this is a kind of conditional non-equilibrium state and one can also derive first law second laws which are actually quite different from the ones I showed so this is intriguing it shows the importance of the second law of the initial condition sorry on these different formulation of the second law and I also want to say that there's there are all these other types of approaches which are interesting I think I would have love to comment more on that because there are interesting connections between what I said and these other approaches but this is where we basically consider a many body system that is isolated and one tries to to describe the thermodynamics of certain observables typically in kinetic theory like Boltzmann equation single particle densities and I think there I can show that there are many concepts from this correlation entropy that that play a similar role as in the system reservoir setup and also ETH which is yet another approach so there's many much more out there but I focused on what I could cover within an hour thank you very much for your attention okay thank you very much so now the session is open for questions and so either you can write it on the chat I don't know whether participants are able to unmute themselves and ask directly a question so I have a say I have a question on on the first part but which maybe goes a little bit off from what you have been discussing so one particular application I've been interested in is training of say neural networks and then you can think that you have say a system which is put in contact with with an environment which is the data and I was wondering whether this law that you were because you can think that you start from a factorized state where essentially your machine that you are training and the data are independent and then when you put them in contact then you and you train you in a state which is in tango so I was wondering whether in general one can apply these ideas and in particular the way of computing thermodynamic quantities from these relative entropies also in that case it's an interesting question and I thought about this too in a sense it goes beyond what I presented because it goes into here I really I was sticking to heat and work sources but but many of the things this formalism can be extended to more complicated reservoirs which are in non-equilibrium states and this is a bit what your the picture that you're describing if you think of learning you have a data set there is information in that data set and that would if you're it's interesting to ask whether one can think of training the the energetics of training but that would one would need to enter the realm of of this information thermodynamics where we start looking at entropy and energy balance with not only heat and work reservoir but also with more general non-equilibrium reservoirs which can exchange both information and energy and try to see if one can derive useful balance and and say interesting things I think at the moment it's hard to to say whether one can say one will be able to say something interesting but it's certainly an appealing idea thank you very much so that there is the shelving party can you unmute yourself oh yes thank you camera also so we can yeah absolutely thank you so I want to know what if you studied time evolution now time evolution of the condition of distribution of system condition on observation of environment so let's say that I am observing the microscopic state of my environment so I want to study p s condition over e yeah um so let me first say that discounting field is a is a simple form of conditioning because I keep track of what is exchanged in the environment by projecting at time zero and projecting at time t so in a sense when I extend my dynamics by keeping track of this land this is already what I do but one can also look at that in a more systematic way I did this from a dynamical point of view in in my thesis I was interested in tracking energy in the in the path and and recently philips has been has looked at the thermodynamics of that um but that's only my question my question is that in this picture when your condition over your environment if the environment is at equilibrium then environment turns to ideal work source right because you are driving because you are observing your environment uh and that's acts like a protocol that's drive your system x yeah so so so my question my question is that it seems to me there is a dual rule for environment either if you if you go to the ensemble picture when you when you see the environment as a system or you you measure it you reduce to the microscopic state of the environment and so so in that picture you say heat reservoir and in the in the in when you measure it it reduced to the work parameter this this is my question yeah so it's not as symbolized it reduced to the work parameter because uh I can say the following if you if the interaction is weak uh a new measure energy this is called uh for um counting statistics it's used in in photon counting statistics people do this in experiment you have an atom for instance and you look at the photons that are emitted by the atom or you have a quantum dot and you look at the electrons that are tunneling in and out of the quantum dot you basically can condition your dynamics on these events that you observe uh and if your ensemble average you recover what I said but at the fluctuating level that would lead you if you measure in energy so energy measurement in the weak coupling limit would totally be consistent with what I described as fluctuating thermodynamics where you enter a different paradigm is if you start measuring observables in the bath that are not commuting with the energy because then you start having non-trivial uh effects from a thermodynamic point of view uh and it's yeah you there I think it's much less easy to make clear thermodynamic sense of what is happening in term of energy balance because the the energy balance is affected the entropy balance is affected but it's hard to give a meaningful interpretation of that but then the message is for whatever commutes with energy in the weak coupling limit perfectly fits with this so you can do it it's simply looking the the the ensemble average theory will coincide with what I presented when you care about integrated observable where you will get this fluctuation theorem but you can go even further uh look at first passage time and things like that if it's things that do not commute with the energy I think this is much more open and in by weak coupling you mean that uh the reservoir evolve really from the system I didn't understand the word I want to by weak coupling you mean that the environment is not affected with the system yes so I actually the the precise meaning is is this semi-group property that I was referring to here this and it has to do it can be easily justified when the interaction strength between the system and the bus is weak and it has to do with saying that every dt it is okay to assume that my bath is again re-initialized as if it was at equilibrium okay thank you okay so thank you so we have two other questions one from Jin Fu Chen can you hello hello I have a question on this page uh that's you you add a counting field on the sorry can you turn your camera on uh I just have a question question on this page on this night here here you add a counting field on the on the new real new real super operator which you act on the system and I see that you you just add the counting field symmetrically and I just wondering is there any principle to tell you how to add the counting field for for such a quantum system because because I think this is okay for new blood uh operate in the in blood master equations but I have considered you this ways to calculate the work statistics of the quantum Brownian motion master equation but I but it seems that sometimes you will find that the the work distribution will become so quasi-probability instead of probability I just want to ask is there whether we can whether there are some knowledge to tell you how to add the counting field yes the systematic procedure is this two-point projective measurement if you do that you really get a probability you don't get a quasi-probability but what you mentioned is a known in in the beginning of counting statistics in 2007-08 people were they didn't have really a rigorous rationale to how to put them and and they would sometimes get quasi-probabilities so I recommend to to really try to start from the two-point projective measurement you will make sure that this is a real probability and then you can see what's the difference with what you did and which led to quasi-probabilities yes I if you if you start from the unitary evolution it is always probability but if you use a master equation to describe the evolution and sometimes you'll find you will find it becomes complex or negative but you need the difficulty is how do you add the counting field in the at the level of the master equation in a controlled way this is not a trivial question and that's why you can get surprises and to be sure it's better to always start from the the full description trace out your reservoir there you know where to put your your accounting field then you trace out the reservoir and you see where they end up in your master equation okay thank you thank you okay so uh we have another question from uh Vladimir Villegas you wrote it in the chat maybe Vladimir if you are there you can just unmute yourself and ask the question directly it was hi hi yeah hi I would just want to ask because it seems that the first and the third laws were pointed out for the inclusive formalism how about the zero law would there be I mean with all these things coming in hand would there be effects on the zero law of the thermodynamics okay by the the zero slow the traditional formulation is the existence of an equilibrium such that if you assign a temperature to it and you put A and B at same temperature there will be no heat flow and and then you can use a triangular A and B and B and C and A will all not exchange heat if the temperature is appropriate well defined this is what you have in mind yeah that's it would there be an effect with all of these things would there be an effect on it I think in the weak coupling the zero slow will be well respected when you go to the strong coupling of course you're gonna have to turn on the interactions and and then it becomes a bit more tricky because the zero slow usually has neglects effects of interaction so I would say by default when you when you can derive effective effectively stochastic thermodynamics the zero slow is built in but if you want to go beyond weak coupling it might be a bit more tricky okay got it thank you okay thank you very much thank you Massimiliano so that was really great and but I think we can move to the next tutorial by Hugo, Hugo are you there? yes so thank you very much for joining so we move to um can you share the screen now uh Hugo um very good I assume you see the full screen so Matteo can you confirm you see you see the slide yes yes we see the slide yes yes thank you okay should I start then yeah please okay good thank you well first thanks to the organizers for the invitation and organizing these tutorials I'm quite glad to see that there are so many participants it's nice to have such a big audience from different places I'm not going to say good afternoon or I guess it's morning for many people so I'll be talking I'll give a tutorial on large deviation theory and how it's we can apply it to calculate distribution