 Okay, it's required, so then I will maximize this, again, we start with small, with this small activity and I have seen the papers here after break. So basically, now I'm interested in the most, what is, in your opinion, most important result or implication and phenomenon that is related to entropy and I hope that after we finish this talk, we can then have a look at the post-its and see what's been written there. So what is, in your opinion, the most important result, it doesn't have to be like result that contains entropy as a formula or something, but something that is in more general related to this entropy phenomenon. So basically, it can be like something in German, I mean, that is given by entropy or like influence by entropy or like an application, I don't know, it doesn't have to be physics, it can be biology, want to like bias you too much and when you have it, just come here somewhere, so like only those bits of space, they create it, but it doesn't matter. What patient is patient there? Mm-hmm. Our old time, the University of the Philippines. Mm-hmm. Same as our old time. Mm-hmm. Use incompressible theory. Also, our entropy of time, the universality is everywhere. Mm-hmm. That's the only implication to biology classes. Like, for example, the Markovian motors. Mm-hmm. Somebody else? I think what he contributes is for young Gibbs paradox. Mm-hmm. Good. Somebody else is writing? No. Yeah, so I think that most of the people voted for our old time, I was actually had in mind another application that you also had, the spin glasses, which is because it was the last Nobel Prize after a long time for statistical physics and all complex systems. And also because I'm now working on the various applications in this, in social systems. The arrow of time, I think it's quite nice. We might, and might have like little voting, let's, I will put a little voting on the, on my Twitter account before I have to share all my talks. So you might have seen it and you will see what the Twitter things is the most important and I will choose these ones. But the arrow of time is also one of the very important, very important applications. And now, in the last talk, I would like to go a little bit more, be a little bit more theoretical. So as the opposite approach from lecture two, where we basically didn't think much about properties of the entropy and took a system and tried to calculate it. Now I would take probably the approach that you, that you were useful in the lecture on an information theory that should basically define the set of axioms and a defined entropy. And I would like to discuss how these axioms can be built, how, what are they in relation and what we can learn from like having axiomatic approach to entropy. And also I will discuss different motivations for introducing axiomatic schemes and their possible connection. And I'm glad that you've already done this before. So I started with the repetition of the Shannon-Kinch axioms and they are basically created by the information theory. I don't know if I wrote them in exactly the same form as in your information theory lectures. So I'll just go through them very quickly. So in this formulation, I say that I define the entropy such that it's continuous of the probability distribution only, which means that it depends only on the distribution and not on any other parameters. It's maximal for the uniform distribution. It's so-called expandable, which means that probability does not change entropy. So basically, I have to talk to the microphone. Okay, this side. Okay, this is a bit tricky because then I have to put the button there. No, no, I will be doing that. I don't have too many slides so I can go back and forth with them. So yeah, so basically, if I have an event and I add one event with zero probability, the entropy doesn't change. Sounds reasonable. And then there is this most interesting axiom that is called additivities. So basically, I say that the entropy of the shown distribution is the sum of these entropies, where if these two events are dependent, then the second is conditional entropy, which is written in this form. These four axioms, as you've seen, and you also can find the proof in many places. These four axioms, in terms of entropy, have to maybe multiply by the probability of the difference, which is not very long here. Or here, maybe only the probability of the difference. So this sets the unit in which we measure. Okay, it's some echo. I have a question. You have muted it. So please mute this one. So now it should be okay? No, it's recording there. Very well. So it's recording there. I see. So I go there. So do you hear me now better? Okay. Sorry for that. Good. And now, basically, what many people try to do is that since the first three axioms sound very natural, and there is, like, very hard to think about how they changed it. People start thinking about how to generalize the four axiom. And in this case, you can have many possibilities. So one possibility is to generalize productivity. So it's the case of satisfy we've seen before. So basically, you make a deformation or a variant of additivity rule. So here I have the Q sum, which is the X plus Y plus one minus Q times the product of these two. It's called Q addition. And then I need to introduce this conditional entropy. And here the Q i, rho i Q is so-called escort distribution, which naturally appears in these cases. So basically, because there is this Q, so there must be appearing the PQ, but we have to normalize because PQ doesn't sum up to one. We have to normalize again. And if you go, you can go through the proof and then show that this uniquely determines again up to a multiplicative constant. This can be found. This was first done by Sumir Abed. And you don't see the bottom line. Move the mouse away. It should go away in just a second or so. I usually did. Yes, great. Great, great. But it was done. It was done in 2000. So basically you say, so additivity, you say that the entropy of the joint event is not some of these entropies, but it is Q sum of these entropies. Sorry, being not explicit. So basically this is the replacement of the fourth epsilon. It is. Or yeah, you will make it. Sorry, I would ask about that symbol. Yes. Is that the right sum? No, it's just brand new symbol. It's a new symbol. People are kind of lack of these symbols. So they use this direct sum and direct product, tensor product with the skew to denote this deformation. Yeah, it doesn't have to have anything with that. It's just because we don't have enough symbols that on one hand like somehow are close to the regular plus and times and then on the other hand have these parameters. So that's the reason. Probably it would be better to write it as a function of two, basically two value function or something like that. Good. So the other option was to determine different averaging in the original Schoen-Kinchen axioms. Basically you say that the conditional entropy. So the conditional entropy is defined as the regular arithmetic beam. So it's the probability distribution times the conditional entropy given in particular event in the other set. Here one can define alternative averaging which is called cormor of Nagmoy bridge. This is kind of useful in many situations. You can think about it. If I have a simple example, I have a square of size a, square of size b. I make a square with the same, so the s1 and s2 is the s. So basically then the total area is some of the two areas. I would ask what is the c. So this is the c, this square root of a squared plus b squared. This is the generalized average. In the case of b average, you might know harmonic average. You might know geometric average. So all of these averages can be written in this form. And we then say that we have basically the entropy of the joint event is the regular sum, but the conditional entropy has this weird form of cormor of Nagmoy average. And then you can show, again, I will not be doing it here. The only choice is this exponential function e to the 1 minus q x minus 1 over 1 minus q. And basically then you end up with rainy entropy. So this is something we've seen a little bit before that this is the rainy entropy. So you can define these axioms such that they lead to rainy entropy. And this has been done in this paper in 2004. And you can play with this generalized relativity rule as you wish in a paper by Belia. This is one of the possibilities. So they were doing this thing that here, the plus here is basically most general, let's say, operator that you can think of. And then this g is, again, this cormor of Nagmoy average. And then basically black everything in that do the proof. And then we can find that basically the most general class of functions that satisfies these axioms is the function of rainy entropy or science because they are a function of each other. So basically, it means that any increasing function of solid entropy is the most general class that satisfies these axioms. This has been shown a few times, like independently by a few scientists. So here I mentioned another word by Pierre Julio Tempesta from Spain, who has this approach that he says the entropy of the joint system must be for independent systems. This function is phi of this two. So now it's this better notation that it's not the plus, but it's really the general function. And then you say this phi must have some group properties. So like symmetry, associativity, no compulsability. And what he did is that he used the group theory. And then he was able to show more as the same thing so that the G can be written in some general form. And then this entropy is written in terms of this PI times generalized look of PI, where this generalized look is basically this PI to alpha. And of course, he was considering special class of entropy where you have both trace class, so it's some of the GPI. Of course, if you take a function of this, then it's again, you just change the way of this phi, but it will still remain the same type of function. Ah, thank you. Served in a top unusual. No, maybe one can mix it with a coffee. Now I want to talk about let's say, so we have seen that that there are there are many possibilities how to generalize these channel contraptions. What we were interested in a few years ago was whether there is some classification of this, let's say, rules that are various or whether we can maybe try to see what entropy is good for which system, because these axiomatic approaches are maybe not because they really allow you to derive many entropy, but it's not that clear whether they are useful for any practical system, right? So we were interested in whether we can maybe try to classify this system of entropy a bit. And to this end, we were interested in how the multiplicity and entropy scales with that system and connected with this with the channel contraptions. And for this, we use scaling. I don't want to go too many details, but it's like when you do a critical phenomena, then what you do is that you rescale the system or like distance of the system. So in this case, we rescale the size of the system. So from n-particles to lambda n-particles, then if you do and then you can show by a mathematical theorem that if you do it for large n, then the ratio of the multiplicity goes like lambda to some constant. So this is a mathematical theorem that is not so difficult to prove. And then you have this exponent. So basically, the leading term is always n to c0. And then you can do the second scaling, which is that you rescale the n to lambda n, but also n to lambda. So we have this power rescaling. And then you can do the same with, which is the correction to the first term. And then you have this first correction, which is n to c0, log n to c1, et cetera. So you see the pattern here. Yes, yes, yes. Can you repeat what was n? And it's the size of the system. Okay, okay, but that can change a little from consistent to consistent. No, yeah. So we are now interested in like kind of thermodynamic limits. So exactly. So this will be, these exponents will be describing the properties of the system. So for each system, we can get this characteristic scaling exponents that will determine how the system behaves if it's very large. Like renormalization. Exactly, exactly. That's the idea. So, and here we're typically in renormalization or a critical phenomena. You do, you only are interested in this first exponent. We are, we want to go like a little bit further because it might be also interesting. And you will see that it is really interesting. Yes, sorry. So this was taken for another CDentropies, classentropies that was introduced by my colleagues, Rudy Hanell and Stefan Turner. And it's basically the first step in doing this classification. So I will show you that these are included in this, this general, in this general scheme. So it is this class of entropies that are like restricted for this first two scaling each one. So we would call it C0, C1, but in their paper they use the CD and then some other papers citing them, we spare them as the CDentropies. So it's like already the name that, that makes its own life. Relationships within this relation to Caramata function. Yeah, yeah. So basically these ideas appear here and there in like condensate theory, critical phenomena, which is something that people are interested in in general of how your system behaves when you rescale it, right, when you change the size. This is an interesting to mention because I don't know what is the year of the result in Stefan and Rudy. Did you write his classification paper? The first one would be say 2010-2012. And in this their paper they write his relationship, which we already know in Caramata's theory and I don't know, when Caramata theory started, this form of 30s. So it was the result of 30s, it was rediscovered about the years after that. I mean, in a way, yes, but of course I think that here we apply it to the end, to the multiplicity and then find entropy and friends do the, like they provided the examples of the four different also exponents, which I'll put through later. But you have real examples. So it's not just, you know, just a mental exercise that the three different exponents describe different systems that have something with the real physics systems. I'm sure w is number of matrices. Yeah, it's a multiplicity as we always in this talk. Yeah, and here I, before I was using this n1 and 2 here, I just, you know, the number of part, the total number of part equals and consider that the structure is the same. So as I said, this is not new. So basically you can define this set of n3 scaling, which is that basically you take the n-fold exponential times lambda times n-fold logarithm. And you see that the zero of rescaling is the lambda x. The first is the x to lambda. The second is the e to log to the lambda, et cetera, et cetera. And then they have some nice group properties and everything. And then you can repeat the procedure. And then you see the structure that what you basically get is that you have the, that the exponents are like n to c0, log n to c1, double log to c2, et cetera. And if you want to, you can go as far as you want, if you wish. Of course, normally you take only the first few exponents because these are the ones that are important. But what can happen is that, so yeah, so we can like derive this fancy formulas. What can happen is that the c0 is infinity because this is, this is not forbidden. That means that the, that w n grows faster than any polynomial. So it means that it grows exponentially. So basically what we can do is that we replace the w by log w and then do the same thing. And if it's not enough, we replace by double log, et cetera. So basically then the, the most general expansion of this type is that you take the health exponential of this n c to c0, log n to c1, log log n to c2, et cetera. And this, this we found that this is the, if you, in any kind of rescaling theory, this is the most general rescaling that you can. But I'm sorry, I'm lost. Like how do you do the rescaling? I don't get the idea, sorry. So, so basically we find this general, so we, we generalize these two, these two simple rescaling. So like this multiplicative x to lambda and x lambda times x and x to lambda. So these are the, the second line. So basically this is what we were considering at the beginning. So the rescaling was that from x to lambda x or from n to lambda n or the second was from n to n to lambda. And this is actually just the first two cases of other types of rescaling that you can, you know, with n. Actually, if you plug in the minus one to this, it's giving you a different scan. So the r minus one x is x plus lambda. And you can also go the negative ones if you want. Okay. So that's why we use this, this formalism. And again, yeah, so you can derive general formulas, but basically the most important is this last last line where we have this, the W n can be approximate for large n, like as the L times exponential, like L fourth exponential n to c0, log n to c1, etc. And you can do the same with the entropy actually. So you basically write the scaling entropy as the, in terms of W. And then you say, let's consider that entropy depends on W. And what I have is that I have the same expansion. And then I've been discussing that, that the n of W n, if we want to have it extensive should go linearly with n. And from this, you can determine the relation between the coefficients c and d where the c are the coefficients for the for the multiplicity and d are the coefficients for the entropy. And again, without going to the details, you can derive the relation between them. So once you know the scaling coefficients of your sample space or your multiplicity, then you know your coefficients of your, of the extensive entropy. And to give you some examples, we can think about several processes. So maybe start with random walks or random walks is like really n times the senior quality, whether you go to the left or to the right. Then the number of possible states is 2 to n. So then you see that basically the, the, you can calculate that the d1 is 1. And what we get is that in this case, is really the most one entropy that is the most useful, because it's extensive. So when I was talking at the beginning that while log of W is the entropy, and I said because log is easy to that makes the product a sum, this one easy argument, but the other is that this is then the extensive entropy for exponential systems. So exponentially growing systems, this is the most simple case of system and a unique key to us in the nature. This log W is extensive. You can think about another example where you have so ancient random walks. So then we have a walk where you throw a coin, do one step, some direction, then you throw a coin again, and you have to do two steps in the same direction. Then you throw a coin and you have to do three steps in the same direction, and then four and five and et cetera. And if you then do the calculation, then you see that the state base grows approximately like 2 to square root of n. So it's like 2 to n half. And then by the calculation, the scaling exponents two, and then the extensive entropy is the log W squared. So then for the magnetic coins, what I call it magnetic coins, this is the structure forming systems from the premise stocks. You can find that the sample space goes super exponential. So it goes like n to n or like e to n log n. For this, the scaling exponents are one and minus one. And here the entropy would be log W over log log W. Here I have to mention that we didn't use this entropy. And the reason for that is that it doesn't fulfill the Shannon-Kinchen axiom two because the Shannon-Kinchen axiom two tells you that the entropy is maximized for the uniform distribution or in other ways it must be symmetric in all the distributions of probability of the elements. But this is not the case for structure forming systems because first it's not maximized by uniform distribution. And second, it's not symmetric because the probability of finding a particle in a free state and finding a particle in like molecule are not interchangeable. These states are basically fundamentally different. So this is an example of a system where the second Shannon-Kinchen axiom doesn't hold and it's natural because it's not just relieving or relieving because of the balls or something, it's really changing the physical state. So that's the reason why in this case we still use the log W because the log W is extensive if we do not use this approach or this assumption that the entropy must be maximized by the uniform distribution, otherwise you would need to use this entropy. Then what you can think of is a random network. So basically I have n-nodes and then I say each link between the nodes is my state. So basically I can have a weight of each link or something like that. And then of course the number of links goes like to be a middle factor n over 2. And so then we get that the number of states goes like n to 2 to n squared. So then basically this d is one half and then it means that extensive entropy is log W to one half. Then we can think about another model where we have the so-called random walk cascade, which means that to the normal left and right I add the third possibility to display. So it's like I don't know, like a decay of atoms or something like that. Basically I spread the worker to two independent workers. They then do independent steps. They can also be in the same position for simplicity. And then basically you can find that the number of possible configurations is 2 to 2 to n. So then basically the first two exponents are zero. And the third one is one because it grows like this double exponentially. And then you have to use this double lock. And so then what you can do is that you can relate it to this other channel pinching axiom. So if you say I want that the axioms one, two, three are valid. Then it gives you some constraints to the scaling exploring. So basically here it would, so for negative c, we violate the channel pinching axiom three. And here for this bigger than one, we evaluate the channel pinching axiom two. And then we see that there are a few other cases where you get this different. And now I refer to it CD entropy is because originally the C0 and C1 were denoted as CD. So that's why I call it here CD entropy. And it's remained in the ages. So that's why I still have it. And you see, I think this is the first paper you were asking, but yeah, it's 2011. And then you can generalize. So this was done for first two exponents, you can do it for more than one exponent. So then basically you get that in this D0, D1, D2, have this random walkout skater. So you have this space where you have all the processes we've been discussing between entropy somewhere in between. And you can have a look. And then again, you have this space. Good. So maybe are there some questions to this? So then what you basically have is not like classification of particular entropy, but process of entropy that scale the same. So for each entropy function, you can always calculate, depending on like really the particular function, you can calculate the scaling exponents. And if they belong to the same class, then their difference is only in the like finite size effects. So we have, then you can have many entropies that fall into the same class, but the asymptotic properties are the same. So it's okay. So yeah, so the classes are all the values of these parameters. So yeah, so there are many, theoretically many classes. Practically you see that the real systems fall into certain classes that are probably more important or somehow more useful than the others because certain types of correlations in the multiplicity space are more common in the systems. But theoretically yes, then we have to hold set of these numbers and it can be anything that is allowed by the term pension axioms. All these are the four axioms. So they satisfy the three axioms and then the fourth is replaced by the speaker version that you say it scales with these exponents. It's similar to the situation in geometry. When it's a triangle, when they remove the fourth axiom, Euclidean axiom, and they reach this classification with three geometry. Kind of, yeah, it's kind of similar. Exactly. Deep class is an information theory. Yeah, yeah, yeah, something like that. Exactly. Yeah. So then the particular functions just influence how the system behaves for finite scales. In fact, you basically have these universality classes, which will like in the critical phenomenon that you have these universal classes that only are determined by the exponents. Okay, now I want to talk a bit about another approach that is from the statistical inference that, so since change was proposing all these approaches, information theoretic approaches and statistical approaches to entropy, then sometimes later people are thinking, okay, so let's consider the maximum entropy principle as basically the inference principle. But if you have any inference principle and statistics, there should be some consistent requirements. So for example, and this was done by Shor and Johnson in 80s, that they said that if I have the maximum entropy principle, so basically I take my constraints that are made by data or my model and my entropy, I maximize the entropy such that the constraints are put. So again, this method of factoring multipliers, then of course what we require that the result should be unique, then permutation invariance, it is something that is like the symmetry of entropy that the permutation of states should not matter. So basically if you label the state, so if you label them one, two, three, four, five, it doesn't matter if I label them five, one, three, two, one. So this is natural. So then there is subset independent, so it should not matter whether one treats the joint subsets of a system states in separate conditional distribution or in terms of full distribution. So this means that you have a full system and you have some constraint on a subsystem. So then it should not matter whether you take the conditional distribution on the subsystem or a full distribution. It should give you the same result. And then there is the fourth that is most controversial. So it should not matter whether one accounts for independent constraint related to disjoint subsystems separately in terms of marginal distributions or in terms of full system constraints and joint distribution. So consider that you have two disjoint systems, you have the constants of both of them. So they say it should not matter whether I basically do the maximum entropy principle for each of the subsystem separately or if I do it for the joint distribution and the maximality. So in absence of any prior information, uniform distribution should be the solution. Again, this is what we were considering in the change. Is it possible to have independent constraints or the systems or the subsystems? Of course, of course. So independent constraints means constraints about disjoint systems. So if you have a system and another system and basically they don't share any states. So whenever they are really physically separated, so let's say I have two boxes constrained on one box, constrained on the other box, these are independent constraints. But then the two boxes are on a coordinate. There might be, of course, the question about the system. So you have an energy of one system, energy of the other system, and then this is the question whether total energy is some of these energies or not. Mainly it is, but there might be some interaction energy between these two systems, right? I'm somewhat confused. I also thought of these two boxes and then some seems kind of trivial to me. But if they are correlation, that seems like a very strong assumption, very strong assumption. Exactly. You are having it very correct because I will show you that this is exactly the controversy about this action because what you have is mainly, if you have a box of gas, for example, then it's natural that we have energy of the one box, energy of the other box, and then the total energy is some of these energies, right? Because it's just gases. But there might be other systems you can think about, I don't know, quantum entanglement where you have particles. And of course, if you are talking about entanglement, then basically they are correlated. Or if you have some other systems like gravitational systems, then of course they are correlated. So if you take the energy of, I don't know, one part of the system that is gravitating and the other part to like one black hole, second black hole, then we know that the total energy is not the sum of these energies that have the energy. So then the energy in black holes plays the role of the size of the black hole, right? Or the area. So then we know that the area of the black hole when two are merged, it's not the sum of the areas, right? So then in these cases, it's a bit too strict as you correctly pointed out. And what I want to mention is that there's been really some controversy about this. And it's quite a funny because also it led to one of my first papers. So basically this was done in a work by two statisticians, Shor and Johnson, where they find these consistency requirements. They also derived that only channel entropy is the correct one. And this is something that they really wanted to show because at that time in information during there were some other measures that were sometimes called entropies that might start to be relevant. A guy, Yosuke Ufink, and then 15 years later, was carefully going through the proof and he showed that they basically in their proof used one assumption that is exactly what you mentioned that if the two systems are disjoint, there must be independent, which is not included in the original sets of axioms that they use in the proof. So they basically are restraining themselves by much smaller classes of entropies. And he showed that basically the axioms as they originally formulate, as they were originally formulated, is fulfilled by a large class of entropies included in any studies, etc. Basically any function of any entropy, an increasing function of any entropy satisfies axioms. And then in 2013, a group by Steve Press said that I mentioned earlier in the previous talk came and wrote a paper that no narrative entropies because there was a boom of this salis entropy, particularly yield probability distributions with biases because they are not warranted by the data. They basically say by this that they don't satisfy Schoen-Kinchen axioms and they repeat the proof of Schoen-Jonesen. And then Konstantino Salis saw this paper, got angry, of course, because they were ruining his entropy and wrote a paper that maybe then the Schoen-Jones and axioms are not adequate. And then Steve Press replied to them that they are adequate and then sorry, your entropy is not useful in any way. And then there were other scientists that tried to do the same thing with the entropy, with the rainy entropy and then me and my former supervisor, Peter Chisba, tried to hopefully resolve this issue once forever, where in this paper in physical or religious, we try to really discuss the solution and also show the systems like entangled systems or quantum systems or gravitational systems. We also discuss high energy collisions where basically you have like particles, where you say one particle is my system and the rest of the particles is the bath. But then the bath is not much larger than the system because it's only a few particles, so typically like 20. And then you have these fine size effects that basically the system of my interest and the bath, which is the rest of the particles get correlated. So then you see very often, if you look at the papers from CERN that they observe the power, the digital distribution, if Q going very close to 1, so it's something like 1.5, 1.0 by 5. I think now this is really, we've discussed this with few people, they think that it's really just the statistics that we have this small bar, because it's universal for different types of particles. So it's not only one type of particles, but also the exponent is pretty universal and it's something like 1.05. And you can calculate by easy calculations, so that basically the Q is something like 1 plus 1 over n or something like that. So if it's n is 20, so then it's like 1 plus 1 over 20 and then it really vaguely corresponds. So it probably is fine size effect, but there are people that still try to dispute this. Yeah, so it's never ending show. And what we were then trying to show in our next paper is that the Shannon-Kinchen axioms and short Johnson axioms basically define the same class of entropy. So if we say that what is the class of entropy that satisfies the Shannon-Kinchen axioms and the generalized Shannon-Kinchen axioms either in form of what Bellymer was doing or Pierre Giulio was doing and the short Johnson axioms, then we see that this function, this is basically the function of preny entropy and a function of preny entropy, increasing function of preny entropy for first both of them. So basically from this perspective the R equivalent and Shannon-Kinchen did some what they did in information theory and short Johnson in statistical inference is equivalent. So now they can shake their hands, although maybe Shannon-Kinchen cannot do it anymore. Good, so then I want to talk about one more thing that might not be might be a bit surprising to you because in your course on stochastic thermodynamics what you were doing is that you basically took Shannon entropy and took the assumption so that there is a detailed balance and that there is linear Markov evolution and then you showed that the second law of thermodynamics. Well you can reverse this by saying let's say that I have the second law of thermodynamics, I have detailed balance and I have linear Markov evolution and my question is whether the Shannon entropy is the only entropy that fulfills these three conditions because it's not a priori clear so we show the other way around doesn't mean that this this this must also hold and what you can show is that actually by these three conditions you again up to maybe more typically constant the Shannon entropy is the unique entropy that satisfies these three if you want to put axioms so these three axioms this is a special case of more general result that connect the nonlinear mass equation and generalized entropy so then what we were considering is that we said that in this mass equation we have not p but the function of p and then this function of p can be related to the function of generalized entropy so then what we were discovering is that if you have nonlinear dynamics then it's naturally related to generalized entropy. What is q? q is heat. Heat is defined sorry there should be q dot so normally so so it's like in stochastic thermodynamics you have the internal energy sum of p i epsilon i so the heat is the sum of p i dot epsilon i and the work is sum of p i epsilon dot i. This is good just for thermodynamics yes this is the thermodynamic one just for hold on yes so yeah so so then then it's like linear thermodynamics with detailed balance but we were just we were interested in this connection and also the question whether it's the only entropy that can be used because it's yeah so now you can change okay you cannot change the third axiom which is the second law of thermodynamics but you can change the first and the second one in terms you can generalize the third axiom the third axiom which which hurt axiom the second law of thermodynamics okay does it make sense to consider a pseudo addition not well depends so if we say that this s e the entropy flow is the is the change of entropy of the infinite reservoir that's always in equilibrium then it doesn't make any sense then you would need to consider another system of that basically here we say that since the and since the heat bath is infinite then it's independent from the system of interest and then these two entropies must to be additive if you if you have some other system then may be yes but then it's very hard to make the concentrating over the fine the bath because you cannot use the this simplification that you use for infinite okay so the third axiom is the first and the second so so yeah the first yes so that's what we did so you can consider non-linear Markov dynamics what we plan to do is to consider maybe non-Markovian dynamics for the second one you can also try to generalize this but always because this you needed to get the thermonetic interpretation because this this stationary or equilibrium distribution gives you the relation between thermodynamics and dynamics and if you go with this standard way for example to detail balance to add power to alpha for example you can you can do that I don't know what you get um the thing is that here the reason why why we choose it is because so so if if you do the the following thing that you say I want to I change it to the non-linear dynamics then basically the detail balance balance looks like w m n function of this p so basically you always say that in this bracket in this round bracket is the so-called probability flow and we say that the probability flow vanishes because this is what defines equilibrium in equilibrium all flows vanish so it's not about the the fact that it's time independent because this is not enough this might be on equilibrium steady state but also about the fact that all currents vanish this is what defines equilibrium you can play with that but here the it's this natural reason that that you have this probability flow so then if you generalize the the linear evolution and you write us some of probability flows then you basically say I want to have the probability flow vanish for the equilibrium I have some question for the si yes the only constraint is the positivity and the thing that is vanish at the table yes yes exactly is the rule that or not if you have thermodynamics the equations are not enough you can find such systems but can't really uh yeah so normally equation so normally it's it's the master equation is linear but we found a few examples they are not like purely physical or like there are some especially more for the case of continuous dynamics for focal point equations then you can have then in force media in fractal dynamics this is more difficult for let's say evolutionary systems or financial systems to have this one in the Markov equation but you can find examples they are not super uh ubiquitous but you can find those examples and that's it so we did it to the end and then I would like to hear your comments opinions what do you think what you've learned anything you want to say and then you can also drink in a while so cheers and thank you for being here yes yes I'm sorry maybe I didn't get some major points but I know yeah I I don't understand what what makes you uh I mean if you have some phenomenology that makes you um yeah think about these postulates of the entropy or I mean if I have a system in a very practical way yes like what yeah what information so yeah so that's a very good question and it's also our problem that many people were driving these axioms and were getting different entropy functions I don't see an interpretation so that's why um I really like the approach in the second lecture that basically you take your system that is maybe more complicated than than than the regular gas or like um some some some some throwing off coins and then you calculate it basically and you get the answer and then you then you are interested in the system some people do it because of the mathematics for beauty so to say some people do it because you know they are interested in general phenomena um but yeah it's the last question that I also had when it's actually okay can we come back to the last slide if you can put it yeah I have some problem for me in this story because I say yeah but you have to remember that you have to limit market solutions so when you take that the s i is the function of probability then the time relative of probability appears which gives which puts you in this this form and then basically you like it seems like the detail branch seems like a small constraint but it's actually a big constraint because you it really tells you it must be zero only at this point and if all other points it must be positive and this is kind of you can then show that that this is kind of the constraint that that gives you this solution yeah so I know it seems that this is like subtle but also remember that this means that this this holds for all pairs of m and m and m so this is actually quite strong then so I know it seems that it's it's just you know almost no no constraints which is I think nice about this that that very natural constraints nothing special give you this Shannon entropy yeah you can you can have a look to the paper there is also the proof yeah so maybe I would have one suggestion or closing the book because we have a lot of papers here a lot of words written maybe we can read it quickly to see where we started the game yes if you can do that because I'm a little bit yes you know after okay so what I understood you he was putting these sticks like I tried to try to order them according to the days Michael you 100% okay so now I will just read the names and I will read responses and this is actually from the first day to the last day by young theoretical bias physics he said about entropy and measure of disorder that can only increase some deep from machine learning also is written here moment mentioned about entropy in quantum physics and variable which tell us about what is the direction of process then I said about entropy as disorder then we have t and I did put this nice formula delta s equals it up you over t then we have this can only increase for the second time my team said the measure of non-maintenance of the system alexandre is no of atoms number of atoms of information then Javier said this famous bone plasma formula we have also a lack of knowledge interesting answer like was that pece mentioned this as a measure of chaos of the system uh nicole neshich said that it is a measure of disorder of the system so we have a lot of answers about measure of disorder also delta s equals it up you over t is very often mentioned also by Elena milkitsa again nivana lack of order then we had mentioned a lot of applications a lot of information theory then in time series services then like in intrusion properties or appearance of entropy we had the black hole thermodynamics we had also mentioned the entropy cybernetics i'm just going quickly because it was a lot of answer also connection with error of time in irreversibility connection with second law of thermodynamic dynamics again entropy as error of time and also several possible applications time series analysis biological systems and so on i think that ryan actually supported all of these answers with his talk and he actually gave us a very good retrospective of entropy from all possible use so thank you very much for giving us this great lectures so with this we are closing today's lectures and also like to make one announcement for in-person students but those are very important for all all of the participants so so you will receive the questionary about the school and we will greatly appreciate if you can uh fill in the question as soon as possible ideally tomorrow during the day please find some five minutes to make it while memory is still fresh and yes with this i think that you can all the lectures thank you all for the zoom