 Ah. Do you have enough left? Are you ready to start again? Okay. For the third lecture, shall we start? Okay. Thank you. Okay, so let me continue my lecture too, even though this is a third lecture. Okay, so in lecture two, so let me first briefly review what we learned in the second lecture. So in the second lecture, we learned how to define entropy production, so which is a logarithmic ratio between time forward and time reverse path probability. And then we learned the fluctuation theorem. Actually, the fluctuation theorem means that this average is equal to one. So because R is equal to total entropy production, it means that total entropy production satisfies this fluctuation theorem. And from this fluctuation theorem, we can derive the thermodynamic second law. And of course, there is a freedom to choose the dual path probability. So by choosing some useful dual path probability, then we can show various fluctuation theorems. So the first one was Dresden's key equality, and second one is Krupp's relation. And then now I will continue to the third one, which is called the Hattano-Sassoff fluctuation theorem. Okay, so what is the Hattano-Sassoff fluctuation theorem? So it is about the separation of the total entropy into two parts. Okay? So this is a total entropy production, which is given in this way. Forward and time reverse path probability. But one natural separation of the total entropy into two parts is that, now we know that, I mean, this is a system entropy change, and this is a reservoir entropy change. So this is a one, I mean, natural separation of the total entropy production into two parts. So then next question will be like this. Now we know that this total entropy production satisfies the fluctuation theorem. Then what about these two separated entropy production? So this system entropy production satisfies the fluctuation theorem, or this heat satisfies the fluctuation theorem. So this is the next question. So let's look at, I mean, step by step. So by definition, this average can be written in this way. So this is an exponential factor, and this is a time forward path to probability. And now here we use this equation. So here we plug this equation to this. Then we can show that this becomes this one. And you know, I mean, you see that this is not the path to probability. So this is not the normalization condition. It means that this summation is not equal to one. So it means that the system entropy production does not satisfy the fluctuation theorem. Any question here? Okay. And then the second one. So by definition, so it can be written in this way. So here this is exponential part, and this is a path to probability. And now same thing. By using this equation, so here we plug this equation into this one, and then we can show that this becomes this one. So this is not the probability. This is not the path to probability. So the summation of this one, this is not the normalization condition. So actually it is not equal to zero. It is not equal to one. So it means that this average also that the heat does not satisfy the fluctuation theorem either. So this separation actually does not, I mean, the separated these two entropy production do not satisfy the fluctuation theorem separately. Then the next question will be then, is there any some clever way such that two separated entropy production, entropy productions satisfy the fluctuation theorem simultaneously. So there is a way. So Hatana also found it. Okay, so this is a total entropy production. For example, I mean, this is a total entropy production for Markov's jumper process. But actually the derivation process is essentially same for long-term dynamics, but here I will only focus on the Markov's jumper process. So this is a total entropy production. And here we add this term. This term is, SS means that this is a probability in the steady state. So this is a probability in the steady state. So we add this term. And here look this. And this is actually the upside down. This is the upside down fleet. So actually this term is minus this term. So it is like something minus something. So actually they are canceled out. So these terms are actually vanishing time. But anyway, we add this term and we subtract this term. Okay, so here we define this first part as a non-idiotic entropy production. And the second part, we define it as an idiotic entropy production. And this non-idiotic entropy production actually is related to the excess heat. And this idiotic entropy production is related to housekeeping heat. And sometimes this non-idiotic entropy production is called HATANA-SASA entropy production. Okay, so this is the definitions. And now let's look at whether these two entropy productions satisfy the fluctuations here and separately. So let's first look at this non-idiotic entropy production. So we can write this term in this way. Actually, this part are same. And this term becomes this one here. This one, R-degas, R-degas is defined as this quantity. You see that here X and X prime order is reversed, and X prime X. So the order is reversed. So you see that if we plug this equation into this one, and the transition rates are canceled out, and only this ratio, steady-state probability ratio remains. So that's why this term is exactly same as this term. Okay, now this beta transition rate has an important property. So this is it. So this one is actually the transition rate of the original dynamics. So the summation over X, actually, this is escape rate of the original dynamics. And you can also show that the escape rate of this beta dynamics is actually the same as that of the original dynamics. You can show easily, I mean, by summing over X here. But anyway, after the lecture, you can show it. So by using this property, we can say that the same probabilities are same for each transition matrix. So we can now write this non-ideactive entropy production in this logarithmic ratio of these two path probabilities, time for the path of probability and time reverse path probability. And not the time reverse path probability, but this is kind of some dual path probability. And this dual path probability satisfies a normalization condition. Because actually the escape rate are same. So the same probabilities are also same. So the same probabilities are canceled out. So only actually if we calculate this one, then it becomes this one. So in such a way, we can write the non-idearity entropy production. So because this dual path probability satisfies the normalization condition, then the non-ideactive entropy production satisfies the fluctuation theorem. So in such a way, we can show that this non-ideactive entropy production satisfies fluctuation theorem. And this is the same for IWT part. So this is IWT entropy production. Actually, you see that this part is actually canceled out. So this term is actually zero. So if we plug this definition into this one, then you can check that this ratio becomes this one. And this two transition matrix ratio becomes this one. So actually this term and this term are same. And because it has also same properties, same escape rate, so we can also write this as a logarithmic ratio of forward path probability and some kind of a dual path probability. And also this dual path probability satisfies the normalization condition. So this IWT entropy production also satisfies the fluctuation theorem. So the thing is that the total entropy production can be divided into two parts, IWT and non-ideactive entropy production, and each one satisfies the fluctuation theorem. So it tells us that total entropy production has some kind of a hierarchy. Okay, so this is about the hot and the sauce of fluctuation theorem. So here, any questions here? Ah, measure, I mean you mean excessive housekeeping. So actually it is not easy to experiment and measure this quantity, but the housekeeping entropy production is actually when in the steady state, in the steady state actually the IWT entropy production is equal to total entropy production because in excess entropy is actually some entropy necessary for making some system transition. So in the steady state actually this term goes out and only this IWT entropy production term remains. So in the steady state we can easily measure the housekeeping here. But in a general case, I mean it is not easy to measure experimentally. Actually it is, I mean this term is related to non-ideability. This term is related to some process, I mean process. In the steady state process actually it is related to some steady state process entropy production. So I mean that's why I mean they call it IWT entropy production and because this is not so they call it non-ideability entropy production. But that's what I know, but yeah, yeah. Okay, so any other question? Okay, so now I want to show you the expression of entropy production rate. So I mean this is different from the entropy production. Entropy production is a total entropy production during finite time, but this is the entropy production rate. So let's look at the over-adent long-term system first. So this is a long-term equation, given long-term equation. And here this stochastic differential equation can be converted to this partial differential equation and this partial differential equation is called focal Planck equation. And this here P is a probability distribution function of X at time t and this J is a probability current which is defined in this way. Force F and some temperature and another partial derivative. So probably there may be some, if you are unfamiliar with this focal Planck equation, in this lecture due to the time limit, I cannot explain how this converted into this partial differential equation. But if you are interested in, you can read the book of Ritzkin. Okay, so here let's, if you don't know what about this then first let's accept that this can be converted into this partial differential equation now. Then the system entropy is given by this Shannon entropy form, minus kB log this probability density function. So its time derivative can be written in this way. So it can be calculated in this way and because this total time derivative, because this P is a function of X and T, so by using the chain rule, this is a partial derivative of X, partial derivative of T. And as I mentioned when we make a partial, I mean the chain rule for the over-damped electron dynamics, we have to use this Stradenovitz product. And here now by using this equation, we can convert this partial derivative of X of P. We can replace this term by using this formula. Okay, from this one. So, okay let me arrange the terms. So this is the first term and this is a second term and this last term gives this third term. And here now we replace this f function by using this equation of motion in terms of x dot and Krissi. Then we can replace this f function in this way and the same thing, same thing. So and this term, the meaning of this term is actually the heat rate. So now let's move this term, Q dot over T, move this term to the left-hand side. Then the left-hand side becomes this one and right-hand side becomes this one. And the meaning of this term is actually the total entropy production rate. Okay, so it means that the total entropy production rate is expressed in this way. So the average value of this total entropy production rate can be written in this way. And the meaning of this average value is this one, average over this probability density function. So let's calculate this average first. And then from this definition, it can be written in this way. And because this integration is nothing but a normalization condition, so it is simply one. So the time derivative of this one is just simply zero. So it means that this second term vanishes. And now I will use some general equation relation. So I mean for a general function g, then g multiplied by x tau. And here the product is Strato-Novich. So this average can be, we can show that this average is equal to this integration. So g times this j, the probability current j and integration of our all position. Then it is, they are saying, we can show that. In the next slide, there is a derivation in the next slide. So anyway, at this point, let's use this equation relation. Then we can calculate this average by using this equality. So this is it. So this is a p, a j square over p and integration over a whole range of x. So this is an expression of entropy production rate in the over-dental algebra system. And for under-dental algebra system, the expressions are almost the same. Except instead of this j, we have to replace j into irreversible current. So the same form but a little bit different function. But you don't care about that. Anyway, so almost similar functional form. And then I will use this expression to derive thermodynamic uncertainty relations. So please memorize this form. Okay, and this is a derivation of this relation. But due to the time limit, so if you are interested in, then you can read my lecture notes. So here I will skip this. Okay, then what is the expression for entropy production rate for the Markov-Jumper process? So I mean, in this case, it is easy to find entropy production rate expression. Because in the Markov-Jumper process, entropy is produced only when jump occurs. When it stays at the same state, then actually no entropy is produced. Only when jump occurs, entropy is produced. So let's say that jump from y state to x state, let's say that this jump occurs. So the during time t, so when this jump occurs once, then the probability of serving this jump from y state to x state is given by this way. And the time reverse, time reverse path probability is given in this way. So the logarithmic ratio gives the entropy production induced by this jump during time delta t. But this delta t are same, so actually we can write in this way. And the number of jumps from y to x during delta t is given by this way. So the entropy production, total entropy production during delta t is given in that way. So the entropy production rate divided by delta t and time delta t goes to zero limit, then it becomes the entropy production rate. So it can be written in this way. So this is an entropy production rate expression in the Markov-Jumper process. Okay, so this is a summary of the whole lecture too. So now I just talked about the Hatana-Sasa entropy production. So the entropy can be divided into two parts, and each one satisfies the fluctuation theorem separately. And then this is an entropy production rate expression over the entire dynamics. And this is an expression of entropy production rate for Markov-Jumper process. Okay, so I'll use these two expressions later. So okay, so now let's turn to my lecture, the original lecture three. Okay, so I'll skip this summary. Okay, so let's summarize what we have learned up to this point by using this schematic restochastic system. So there is a stochastic system. And let's say that t0, this is the initial time, and this is the final time tau, and this is the initial state, and this is the final state, x1 prime. And gamma1 is one single stochastic trajectory. And there are many, there are many initial states, there are many final states, so this initial state constitutes this initial distribution, and final states constitute final distribution, and there are many stochastic trajectories. So here we are interested in measuring some observable theta. Of course, this theta is a function of trajectory gamma. So for example, in lecture one, we learned how to define heat and how to define work, or theta can be displacement. So they are actually the observables we are interested in. And one special observable is entropy production. So entropy production is also a function of gamma trajectory. So we learned this entropy production in lecture two. So in the first lecture, I mentioned that the thermodynamics is physics, which is about the relation, which is about to study relations between these observables. So the important question for this stochastic system is that, is there any general relation for entropy production and these measurable observables? So in lecture two, as a one important relation, we learned this fluctuation theorem. Here, from the lecture three, I will use this notation sigma instead of a delta as total for simplicity. So this sigma now means the total entropy production from the lecture three. So in lecture two, I talked about the fluctuation theorem. This is an important relation for entropy production. So by using the Johnson's inequality, we can derive this thermodynamic second law. And so it can be called as a generalized second law. This is what we learned in lecture two. So by choosing the dual dynamics, we can derive many other fluctuation theorems. Johnson's inequality, Kirch's fluctuation theorem, Hatano's as a fluctuation theorem, and information fluctuation theorem. Even though I did not talk about this information fluctuation theorem, then I mean Tadaito will talk about this information fluctuation theorem in his lecture. And then the second relation is about a thermodynamic, this is called a thermodynamic uncertainty relation. And shortly we call this TUR. And TUR is a relation between entropy production and observable. So what is TUR? So TUR looks like this. Fluctuation theorem is equality, is given by equality, but TUR is given by inequality. So it looks like this, and this is a variance of observable. So this term is called relative fluctuation. And this is entropy production average. And this KB is Boltzmann constant. So what it means? So let me give you an example. So here, let's say that observable. Here, let's say that we set theta as a displacement of this molecular motor. Of course, because of the stochasticity, X-variable is a stochastic variable. So it will have some kind of distribution, and it will have some finite variance. Okay, so here. Sometimes we want to reduce this relative fluctuation. For example, let's say that we want to make this relative fluctuation zero. But if we want to decrease this term as zero, then from this inequality, the entropy production should be infinity. So it means that if we want to reduce this relative fluctuation, then entropy production should be increased. And if we want to decrease this entropy production, then relative fluctuation should increase. So it means that this TUR is kind of a trade-off relation between the relative fluctuation and entropy production. And entropy production is, we can understand, it is a thermodynamic cost. So we have to pay more thermodynamic costs for reducing fluctuation. This is the meaning of this thermodynamic uncertainty relation. Okay, and if we rearrange this thermodynamic uncertainty relation, then we can have this relation. Then you see that the thermodynamic second law just tells us that entropy production is larger than zero. But this TUR tells us that entropy production is larger than some positive value. So it means that TUR gives us a stronger bound than the thermodynamic second law. And this is an importance of TUR in theoretical viewpoints. Okay, so I'll turn out clear. Okay, so in lecture three, I'll talk about TUR. So in the first section, I'll talk about how to derive TUR. So for over-the-enter long-term system and for Markov-Jumper process. And in the second section, I'll talk about how to apply this TUR to our some experimental system. Okay, so this is a brief history of TUR discovery. So it was first reported in 2015, so seven years ago, very recent one. And here what I mean by report is that in this paper, actually they discover. I mean, they discovered this TUR, but they couldn't derive. They couldn't prove it. In the one-year layer, this group first derived this TUR by using large-division theory. We learned this large-division theory in the morning by Vipul's lecture. And then in 2018, this Japanese group derived TUR by using generating function method. And the next year, 2019, this group derived TUR by using Kramer-Rau inequality. And this Kramer-Rau inequality, I mean, this is very famous inequality in information science. So it looks like this. And here, I, I is means Fisher information. And this is variance. Okay, so this Kramer-Rau inequality greatly simplifies the TUR derivation. So I will skip all this derivation, but I'll only focus on this Kramer-Rau inequality derivation. Okay, then what is the Kramer-Rau inequality? So here, let's say that there is some general observable theta. And theta is a function of z. And there is some probability distribution, which is also function of z. And this is parameterized by some parameter epsilon. So it means that P also depends on epsilon. And here, so this bracket, epsilon, means that average over this distribution. So if I write in this way that it means that the observable average, indeed by using this probability distribution. Okay, so now let's look at this one. So this is observable average and partial derivative of epsilon. And square. So by using this, we can write in this way. Because theta has no dependence on epsilon, so this partial derivative only applies to this probability distribution. And total square, right? And then here, let's look at this note. So now let's consider this term. And because this part is already integrated our z variable, so actually it has nothing to do with this integration. So it is kind of a constant, so we can take this term out of this integration. Then the remaining term can be written in this way. And because this is simply one, because this is a normalization condition, so this partial derivative is simply zero. So actually this term, a little bit looks complicated, but actually this is just a zero number. So here we just add this zero number here. So these are the same one. And then here we bind these two quantities with this common factor in this way. And here this one can be rewritten in this way. And so by using this equality, now this equation can be written in this way. So now I will use the Cauchy-Schwarz inequality. Cauchy-Schwarz inequality is something that if there is a two vector, and this vector product square is smaller than each vector square product. So Cauchy-Schwarz is essentially the same as this one. So in this quantity, there are two vectors. If let's say that this is a vector and this is a c vector, then this a vector square and this is c vector square. So we can separate in this way. And this product, these two products is larger than this one. This is a Cauchy-Schwarz inequality. If you are not familiar with this Cauchy-Schwarz inequality, you can, after the lecture, you can sit in your chair and follow this line. So this first term is nothing but a variance of theta. And the second term, we can rewrite this integrand in this way. Because this is a square, so we write second time and distribution. And by using this equality, now we change this term into this one. So this can be written in this way. And here we use integration by parts. So we move this partial integration in front of this one and take a minus sign. Then it becomes this one. So this is the integration by parts. And this one is so we can write, this is an average of this integrand. So we can write in this way. So here we define this average value is equal to feature information. So this is the definition of feature information. So this, I mean this equation can be written, simply written in this way. So variance times feature information. And you see that this variance times feature information is larger than this value. So we are arranging the terms. So we now finally have this Kramer-Rau inequality. Okay, this is a derivation. I mean this is a derivation for this Kramer-Rau inequality. So this Kramer-Rau inequality holds for any observable, any probability distribution. This is very general inequality. So now I will use this Kramer-Rau inequality to the stochastic thermodynamic system. So let's consider there is some original dynamics over them to run the system. Here the external force is f here. And now in this original dynamics I will add some perturbation force. So here the perturbation force is epsilon g of x. So g of x can be any arbitrary perturbation force. And here so you can regard this epsilon as a perturbation parameter. So small number, epsilon is a small number. So when epsilon goes to zero limit then this perturb dynamics returns back to the original dynamics. So when you now think of this Kramer-Rau inequality then you can regard this epsilon as some perturbation parameter from now on. Because I mean it always holds for any observable, any probability distribution, any epsilon. And so now from now on I will consider some observable which is a function of gamma instead of z. And I will consider path probability which is a function of gamma also. And because we are now considering this perturb dynamics so this path probability also depends on epsilon. Because this holds for any theta, any probability density function so I mean it also holds for this observable and path probability. So now take epsilon goes to zero limit. It means that so we evaluate the Kramer-Rau inequality at this original dynamics limit then it will becomes like this. Then here let's compare this Kramer-Rau inequality and this T-R. Then you see that there is a similarity right? So if this one, this one and this one so these three relations are satisfied then actually this Kramer-Rau inequality exactly same as a T-R. So and the first relation is actually automatically true. So we don't care about this first relation and actually this second relation, I mean second and third relation. So by choosing some proper perturbation force g of x so by choosing this proper function g x we can show these two relations are satisfied then we can derive this T-R from this Kramer-Rau inequality. We already derived this Kramer-Rau inequality. So if we only show these two relations by choosing some proper g of x then we can show this T-R. So this is the basic strategy to derive the T-R by using Kramer-Rau inequality. Okay then the remaining task is that what kind of perturbation leads to these two relations we have to find such a perturbation force. Okay so there are many, I think there will be many possible trials but one intuitive one is like this. So this is an observable average in the perturbation dynamics and this is an observable average in the original dynamics. So if these two averages have this relation then the first relation is automatically true. Right? This is trivial. So what kind of perturbation leads to this relation? So from now on our first simplicity I will only focus on steady state. Okay so in the steady state we can define the steady state rate. So let's say that this theta dot steady state means that this is an observable rate. Then the total, I mean the accumulated observable in the steady state is simply given by multiplying by total time tau times rate. So here now here we make this time scaling perturbation. It means that time is scaled by this quantity, one plus epsilon quantity, here epsilon is a small number. Then total time is also scaled by the same factor. Because the total time is increased so the total accumulated average observable is scaled by the same factor. Right? So it means that in the steady state if we make this scaling perturbation then this relation holds. Then the first relation can be satisfied. Okay so the remaining task we have to do is that if we make this scaling perturbation then what is the value of this feature information? It is really same as the total entropy production. So if it is so then we can derive the TOR. Okay so let's check it. Okay so this is the original dynamics. And the original dynamics of Focke-Planck equation as I mentioned that we can write this Focke-Planck equation. Actually they are same dynamics. So here let's make this time scaling perturbation. So instead of T we put one plus epsilon T. Here one plus epsilon T. And this one plus epsilon comes from the derivative of this time. Okay now let's consider only the steady state. Then in the steady state actually the probability density function does not no longer depends on time. So actually we can just write the steady state in this way. And in the partial derivative of this steady state probability density function is simply zero because it does not change in time. So the remaining thing is this one. And one important thing is that the steady state solution of this perturbative dynamics is the same as the original dynamics steady state distribution. Okay so this one is a probability density function of the original dynamics. So we can rewrite this term into this way. This is nothing but this first part comes from this one factor. And this second part comes from this epsilon factor by definition of this probability current. And then here I insert this term into this parenthesis. Then we can write function in this way. Okay I mean this term and this term are same one. So now let's define this function as g function g of x in this way. Then let's look at I mean now this focal plan equation in the steady state. Then the corresponding long-term dynamics of this focal plan equation is this one. So it means that I started from this time-scaling perturbation. But come to a conclusion that actually this time-scaling perturbation is equivalent to this perturbive dynamics in the steady state. So this is a point. Okay so now we have this perturb dynamics and the function of g of x looks like this. So now because now we know what is the perturbation force and we can now calculate the future information. So this is a path probability so we know how to write the path probability for this perturb dynamics. So this is it and here I used the product notation e to product for convenience. So by plugging this equation to here actually this term and this term does not have any epsilon dependence. So because of this second derivative of epsilon so actually these two terms goes away. We don't care about these two terms. So only remaining term, the remaining part is this one. So this is it. Now we expand this square in this way. So this term square and this term square and their cross product becomes this one. And because this is an e to product so actually this is just a normal product. And look at this second partial derivative of epsilon. Then if it applies to this one actually this is vanishes because there is no epsilon dependence. This also vanishes because it only has a linear dependence. So the remaining term is only this one. This has a second and a squared term, epsilon squared term. So only this term remains by using this, applying this second derivative. So the result is this one. And now we know what is g function. G function is given in this way. So if you plug in this equation into this one and then now and evaluate this average value, then it becomes this one. So probably you are familiar with this expression. Previously I mean learn this expression is expression for entropy production rate. So the entropy production rate and integration over from zero time to tau, then actually this term is nothing but total entropy production. So what we show is that this fissure information is equal to the entropy production. So it means that this relation I mean now is satisfied. So by making the time perturbation, by making the time scale perturbation, so it satisfies these two relations. So because these two, because now these three relations are satisfied, so we derive the TOR. Okay, so this is the derivation of how to derive TOR. Okay, so are you now following me? So is there any question up to this point? I mean you can check by yourself by reading my lecture note, but the important thing is that some big picture how to derive a TOR. To derive TOR we use this Cramor-Rau inequality, and by using the time scaling perturbation, then we can show these three relations are satisfied. In such a way we can derive TOR. This is some whole picture, big picture of this derivation. Okay, can you microphone? Thank you. Maybe you will call this in the following, but the question arises since the Cramor-Rau and the fish information come from essentially developing to second order flows to a certain probability distribution, one might wonder whether there exists sort of higher order, more linear expressions of this. Yes, right. As I said, and this is actually when epsilon goes to zero, then the fish information is order of epsilon squared. So I mean, so this derivation is we call this linear perturbation, I mean the inequality, but we can also find more higher order inequality. Yes, as I said, it is possible. For example, Haydn, I forget his name. Dunsko-Varadan inequality, so such an inequality is kind of such a higher order inequality. So is this TOR is only valid for molecular equilibrium in steady state or it can be valid for the time dynamic equation also? Yes, very important question. So to derive this TOR, it should be steady state, non-equilibrium steady state. Then the next question will be like this. Then what happens if the system is not in the steady state? So for example, non-steady state situation, then in such a case, this TOR form is changed a little bit. I'll show you later how to change. In the proof, it seems that the TOR is a special case of taking special perturbation function. Then I wonder why we use TOR rather than the left-hand side of that arrow? So your question is that we can also use this inequality. Why do we have to use this one? That's a good question. But in experiments, so let's do some experiments. Then measuring the average of some observables I think are easier than to get this some partial, some different. So in such a way, directly using this inequality is difficult in experiments. So that's why we want to use just this average value, not the derivative value. Because if we get this number, then we have to change the perturbation force a little bit. And then we have to evaluate this one from the experiment, but it is not easy matter, so that's why. Thank you. I'm of course a g of x function. The perturbation function is quite a fictitious function. It is not real function. I mean fictitious because it looks like a very strange way. The form is actually we have to understand why is the steady-state probability density function. So it is not easy to experimentally apply this kind of force to the experiment. It is not easy matter. Is there any other perturbation function g that makes some inequality or entropy production? I cannot say this is a unique way because I have to not try all possible perturbation force. But I think it is almost a unique way to derive. Because we know what is the expression for, I mean we know what is the expression of entropy production rate. So to derive such expression, then I think the possible trial function, I think it should be very limited. So, yeah. Thank you. Okay, so is there any other question? Okay, so this is a derivation of TOR over the influence of dynamics. And let's turn to then how to derive TOR for Markov-Champ process. But strategy is simply same. So we will use this time scaling perturbation. So if we make this time scaling perturbation in the steady-state, then this is automatically holds. So the first relation is automatically true. So the remaining thing is that by using this scaling perturbation, when we calculate this feature information, which is really gives this total entropy production. So the remaining point we have to check is this second relation. Okay, so this is original dynamics. This is original master equation. So here let's make it this time scaling perturbation. So instead of T, we put at 1 plus epsilon T. 1 plus epsilon T here and there. And this 1 plus epsilon originate from this T. So in the steady-state, this simply becomes 0. And this one actually does not depends on time in the steady-state. So we can write in this way. And this 1 plus epsilon terms come into this summation. So we can write in this way. And here let's... Okay, so this is the same one. Same one. But here let's subtract this value and add this value. And you see that the denominators are same. And the numerator are also same. So it means that minus something plus something. And actually this is a 0. So here we define this function as eta xy. Then because this yx and xy is reversed. So if it is eta xy, then it is eta yx. So now add this term and this term. Then this term, addition of this term gives this first part. And addition of this term, this term gives the second part. Okay, so I mean these two master equations are equivalent to one. But it looks like different, but they are equivalent actually. So we can write, in general way, we can write this master equation in this way. So here now transition rate depends on epsilon. So in this case, transition rate is this one. And in this case transition rate is this one. So in general way we can write the transition rate in this way. So here when delta equal to 0, actually it becomes this one. And when delta equal to 1, it becomes this one. Probably you are wondering why they are same, but we have this kind of strange things. But you can see why later. So it means that this time-standing perturbation is equivalent to this perturb dynamics. Here the transition matrix is perturbed by the factor of epsilon. Okay, so with this perturbed transition matrix, now let's evaluate this Fisher information. So this is the definition of Fisher information. And then to evaluate this Fisher information, we have to write the path probability of this Markov-Jumper process. And so the change is nothing but this transition matrix depends on epsilon. And here now let's look at this too. I mean the whole staying probability. And we can write this whole staying probability in this way. So here x of t denotes the state at time t. So this delta function picks the state at time t. So we can write this whole staying probability in this way. And this transition part. So transition part can be written in this way. So actually this term comes from this logarithmic function. And there is some m dot x, y, m x, y dot here. So m x, y dot means that actually here t x, y means that time at which transition from y to x occurs. So this delta function picks the time at which some transition occurs. So it only counts when the transition occurs. So actually these two expressions are same. So anyway we can rewrite this path probability by using this delta function and this m dot function. And the thing is that the average value of this delta function is equal to the probability. And this m dot x, y, this picks the transition time. So actually this average value is equal to this value. Number of transitions per unit time. Okay so I mean anyway this is an expression rewritten of the previously known path probability. Okay so now let's calculate this Fisher information. And because actually this term does not have any epsilon dependence so we can ignore this term because there is a second derivative of epsilon here. So the remaining part is given in that way. So second derivative applies to this first part and the second derivative applies to this second part. And we know that from the definition of this r epsilon this is a linear function in epsilon. So it means that this second derivative of this actually vanishes. So we don't need to care about this. So the only the remaining term is this one, second term. So if we take this second derivative, if we calculate this second derivative, then the result looks like this. And for the same reason this second derivative of r epsilon vanishes and the remaining term is only this term. And we can explicitly calculate this part because this is I mean linear in epsilon. This is the first derivative of epsilon which gives only this part. Okay so this is the result. And this term is cancelled out. So only the remaining part is this one. Okay so we obtain the expression for Fisher information. So when delta equal to zero, then this term is neglected and only this term remains.