 Right, so you can see the screen and everything right. Yes. Yes. So first off, allow me to apologize because my voice just abandoned me cruelly today, and then I would like to thank all the organizers for giving me the opportunity to talk to you today here in this lovely workshop out. So I'm going to talk about the non-equilibrium energy harvesting systems. Let me just say to begin with, that all of this is going to be contained in this, in this paper that it's now available on archive. And our research question begins with the following which is, we look at all of these molecular systems that have this twofold structure in which we have some, say, in this case the Krebs cycle we have some backbone structure biochemical structure, and it's going to be driven out of equilibrium by these enzymes that are operating on top, and therefore, it's main goal is to actually extract some ATP and store it into some reservoir. Now we call these such systems energy harvesting systems. And we ask the question, are there limitations at the thermodynamic limitations that we can actually derive formally for these types of molecular energy harvesting systems, we believe so. And I will finish very quickly, how do we actually go on and tackle this problem. To begin with, we need to have some, a bit of a setup or some definitions, and we actually are going to consider the following system, where we have a non-equilibrium environment, and then we have what we call the baseline and the efficient rates that are going to be all encoded in this matrix are right. Then we also have a reservoir with which you can exchange it and we have very importantly, a work reservoir where we actually will store this energy that the system's goal is to basically harvest right. The non-equilibrium environment is actually going to drive the system which is going to be an important part. And then at this point we will say that there is one unique non-equilibrium steady state, which we call pi, for this system so far. You think of this system as the backbone so far here, we have, if you want, this biochemical backbone that I just showed before from the prep cycle. And now we add another type of transitions, which we name control, but this is just nomenclature. It's not exactly, you can think of them as control from control theory, but it's not really necessary to think of this as way, it's just nomenclature that we add. And these are going to be fine, right? So for example, the transition matrix corresponds to this new set of transitions that we put on top of the previous baseline is going to be primed. Also there will be some work exchanged with the reservoir and as well as dissipation, but importantly control, we will set it in such a way that it cannot be invoked such that it changes the driver of this non-equilibrium environment comes as part of the baseline and its driving comes as part of the baseline but not as part of this control. Let me give you a very simple example that I hope that will somehow move us beyond this formal definition into something that it's a bit more tangible and we actually solve this problem in our paper. We solve also two other problems and this particular one consists of a ring of states, there is like n states. If you forget about this thing over here for a second, if you consider the fully symmetric ring in the study field of ring, right? So we expect this to go into some equilibrium steady state which is basically unidistributed. Now let's consider how the environment is going to apply here and we are going to do it in a very, very simple form. Actually, we're going to take it to be dissipative less, although you can also add that to another problem. Suppose that our environment in this cycle basically works as follows. If there's a transition from one to two, then theta KT or theta joules of energy is going to be stored into the work reservoir and then vice versa if there's a transition from two to one. For this particular example, we take it to be dissipative less, we can also move beyond that. So how, if this is the baseline of my system, how can I imagine those control mechanisms, those control transitions? Well, we can think of them as I said, as I introduced in the beginning, just as an enzyme, right, that can be driven out of like depleting part of the energy available from the work reservoir is connected to the system. This also can include some dissipation. And basically the goal here is that if I manage to have such an enzyme, right, maybe we can think of an evolutionary process that needs to give rise to the enzyme. Then I can drive the system in a clockwise manner. They're in a way that I can store more and more energy. And then here the problem is, is there some balance because this is also self maintaining process that corresponds to this control and code mechanisms that also entail some cost, some cost, sorry. And then the question is, can we actually find some universal balance that will give us sort of like an upper bound on the maximum power that it can be stored in this in this way. Before we go to the main goal here, let me just maybe rephrase this whole situation into a bit of a more formal but also heuristic sort of approach. Remember that I said pi was our steady state for the baseline. Therefore, this is without any control mechanisms. But it doesn't have to be equidistributed all the time just like for exemplary purposes here it is. And now when I add a control mechanism, all I'm all I'm doing is I'm shifting away from this baseline distribution right to another distribution, which basically its aim is to actually map or get closer to the distribution of the states that yield the highest amount of energy into the reservoir. Right, so this is sort of what is the operation that is being carried on underneath this process of adding control mechanism. This is the goal of the system to harvest the most, the maximum amount of energy possible, the maximum amount of power. But importantly, whenever you have such a such a departure from one, from one distribution to another, then you can actually have some cost that they can be measured through entropic terms and you'll see how in a second. This actually leads me to the main, the main point of this. This talk, which is the result here with whatever we think here is on the left hand side we have the maximum increase on power that can ever be achieved for any given system with separated as I explained with the baseline and control mechanisms. But on the right hand side we have the computation for these upper bound this is an upper bound. And importantly just very briefly, these right hand side corresponds to a complex optimization problem that contains two terms is like a trade off right. On the one hand we have the cost of maintaining the speed is distributed going to be far away from pi pi remember is the baseline steady state distribution. You can, this is actually an entropic term here, and then the other term will just be the benefits, which it comes in terms of a workflow okay, just to be clear all of these V and q, these are all fluxes so these are not just gross values but actually fluxes on this system at steady state. Great. And now we see one of one important result here is that these upper bound, if you actually go in detail and observe these two terms, these upper bound is computed, regardless of the control mechanism that one should impose. What does this mean, this means that remember in our previous example right in this sort of like bio bio chemically inspired example we have an enzyme over here and we set out these and can be driving the system in a way that I get a net current over in the clockwise direction so force, and therefore I can extract more energy that if I stay in the steady state which in this case gives yields no net energy. I can't go to the system, but the point of this result is that if I for example imagine any other set of control mechanisms. These are actually going to be bounded exactly by the same computation here which is given by this complex optimization. Right. Now this complex optimization problem which basically aims is that finding this solution P, which we show with the show can be can be always found. And also, that it can be saturated by using Markovian control mechanisms, but then you, your first need to show need to solve this problem this complex optimization problem which is not a trivial thing to do. And in fact, not even for a two state system you'll find close general close from solutions. However, there are close from solutions in three distinctive regimes that we that we show in the paper and describe how to, how to arrive at them. So briefly and again from a heuristical standpoint, these three regimes with their return them linear response microscopic and far from equilibrium in each of the cases for example here it's kind of easy to understand what's going on. Right here we have the baseline again in the same sort of like a realistic approach have a baseline and deviating from the baseline is rather costly so this means that this term here is very high. And therefore you can only go ever so slightly away from the baseline and that actually give rise to some approximation that you can solve in the close form solution, the microscopic case is kind of the other extreme. I'll give you some hint of the towards the end of the talk of how do you actually make sure whether you are in the microscopic or the linear response regime, but very naively, this corresponds to the case in which you have a high abundance of glucose in the system, for example, and therefore you can spend as much energy as you want to pay this cost, and you just go into a delta function. It is very extreme of course but it's just like a mathematical approach. All right. So, finally, far from equilibrium is just an operative expansion of this microscopic regime. These three cases can be explicitly put in close form for any, for any system for the case of the ring that I showed you. We actually did so and it shows that it is very well in accordance with numerical solutions here exact means that I am performing some numerical analysis on this complex optimization problem. I don't have the linear response kids perfectly where you expect it to be which is low energies and so on, the microscopic regime will all will only take a whole once you go to very high energies. Also, the, the optimal probability distributions can be shown in close form for the three limiting regimes, and then it all can also be computed and checked for this particular case. And finally, last remark in this in the sense is that one of the interesting aspects that you can figure out is for example that if you're considered this is being and you go for large and this means that you have a very large number of states, you find interesting relations but not only the increase in power that you can, that you have upper bounded for whatever control mechanisms you can come up with goes like feet a square this is remember the energy that is input by the non-equilibrium environment but also you have this scaling as one of our end which somehow somehow if you can, you can think of this kind of constraints what type of chemical cycles you would like to go to if your goal is basically to maximize energy harvest. And this is my main point here so basically I talked about how there are limits in energy harvesting systems and this can be actually figured out for any type of control mechanisms as well as long as you have this distinction between control, so then you have this type sort of universality. There are close form solutions that are in pre regimes in the paper we described with a more accuracy that I've been able to have the time to do here. How do you can actually distinguish these three regimes and they're actually there is a dimensional parameter that you can always compute for a system, which allows you to do that. You can generate a generalized these for fluctuating environments so here I talked about environments that are not fluctuating but you can go beyond that and it's all explicit in the paper. So just allow me to thank again all the organizers and more collaborators and all these funding agencies here and thank you very much for your attention. Thank you very much already for a great talk. And there was a question during the chat by Tom, so maybe he wants to. Hey, so a question. Thanks for the talk a question about what alterations you see. So the default system I come up with your baseline system that includes the work reservoir doesn't. Yes, yes. Sorry. Yeah. Yeah. And presumably some temperature bath. Yes, in India. And you then imagine adding something to it. What do you allow yourself to add. Do you allow new transitions new states, or, or varied transition rates of the between the states. That's a great question. Thank you very much for asking it because this is the bit of like the need to give you the other paper. I encourage you to maybe go find it because it's clear there but let me give you a few answers to this. Yeah, the work reservoir he belongs to the system. So if we go here, before we actually have any control mechanisms, the water reservoir is part of the system the environment is also part of the system. Okay, and the, there's a heat reservoir and you exchange heat with that. Now control mechanisms, you can add other other transitions between states you cannot add new states. In general, not in the, in the, in the, I think that you know, you couldn't do that in general for for our system here. And importantly, and I think I mentioned these, the control mechanism is, I think it to give you an example like imagine that your, your environment is concentration of glucose. So like I say, and I'm going to control the system by increasing the amount of glucose that is involved in the environment so therefore you would have like some, if you want some vector here D prime, that would be non zero that we don't allow. Okay. Now, if you actually consider more complicated environment, you need to be more careful in how you define control mechanisms as well, because maybe the way that we define all of these, all of the system which is using my my coven. Although let me just say that you can go beyond that as well, and more terms will need to be added in Europe in Europe around. But in my coven dynamics, you will have to be careful when you allow to have a non equilibrium. I'm sorry fluctuating environment, because basically what you're going to be doing is you're going to be adding the fluctuations of the environment in this matrix here. Okay, because it's again all part of the same system. Okay, and then sorry. Well I was going to say just just getting back to your like your inspiration was this biochemical energy harvesting thing. So my experience of biochemical energy harvesting mechanism or like, maybe not energy harvesting but energy transduction. Yeah, is that is that what what happens is effectively you have a system. And then you couple it to I don't know ATP turnover. And basically what you're doing is providing another pathway to get from a to be via a catalyst that couples you to ATP turnover for example, and I'm just, I'm wondering how that relates to your description of the system. I think this is a great question. In fact, you can contain all of these phenomena in a single model. These, these description over here, let me just say, this is sort of like, let's say an observer dependent situation you really need to be explicit here what what are you doing. And of course that's going to be on the on the part of the scientists to decide which goes where. But, for example, in this case.