 of this term is dynamical activity because you see that, and this is a number of jumps per unit time induced by the transition from y to x and summation of all pairs of transition. So it means that total number of jumps per unit time. So we call this as a dynamical activity. So the integration of this dynamical activity from time 0 to t. So it gives a total activity, total number of jumps doing time 0 to time when delta equal to 0. And what about delta equal to 1? So delta can be either 1 or 0. So what is about delta equal to 1? So if we put here delta equal to 1 number, by using this definition of eta xy, then we can, I mean, this term, this term is equal to this one. So let's arrange it in this way. So even though we started from the same master equation, but the Fisher information becomes different, depends on delta value. So when delta equal to 0, it is this. And when delta equal to 1, the Fisher information becomes this one. So let me go further with the delta equal to 1 case. So here, let's define this square function as bxy. Then this is same as byx because there is a square here. So bxy and byx. So by dividing this whole summation into two parts, so the first part is x is larger than y, and this is x is smaller than y. So we can divide it into two parts. And then here, because bxy is equal to byx, we change this. So we change it this way. And then because x and y indices are dummy variable, so we exchange this second summation indices from xy to yx. So this is the result. And the first summation is same. The second summation, the indices are exchanged. So now we can add these two summations. So the result is like this. And now let's use the definition of bxy. So plugging this equation to here, and then this denominator is canceled out. So the remaining part is this one. OK. So now we have a simpler expression for when delta equal to 1. And we usually call this quantity as a pseudo-entropy production. Not pseudo-entropy production is not equal to real entropy production. But in the over-dental limit of this Markov-Jumper process, then this pseudo-entropy production approaches real entropy production. But anyway, in this Markov-Jumper process, it is different from real entropy production. And now I will use some famous inequality, which is called log mean inequality. So the left side is called log mean. And the right-hand side is called arithmetic mean. And we can show that this log mean is always smaller or equal to arithmetic mean. So I will use this inequality. So by changing a little bit this, we can show this inequality. Because this is this one. And you see that. So here, let's say that this one is A. This one is B. And this one is A plus B. So this form is same as this one. Then this one is smaller than this quantity by using this inequality. A minus B times logarithmic A over B. And then we can rearrange this summation. Then we can have this relation. And this relation is the expression of entropy production rate for Markov-Jumper process. So the entropy production rate and integration over from times 0 to tau, then it becomes a total entropy production. So OK, let me summarize. So we evaluate the future information in two cases. Delta equal to 0 cases. It becomes the total activity. And when delta equal to 1 case, it becomes pseudo entropy production. And we showed that pseudo entropy production is smaller than total entropy production. So when delta equal to 0, so by using this Cranero inequality, so if we use this quantity, then we can show this inequality. Of course, this inequality is different from the TOR because this is not total entropy production. So people call this inequality as a kinetic uncertainty relation. And when delta equal to 1, the future information is given in this way. So we can show that this relation, but this is not the TOR because this is just a pseudo entropy production, not a total entropy production. However, we show that this total entropy production is larger than pseudo entropy production. So we can also show that this is larger than this one. So finally, we can show this term is larger than this one, and this is a TOR. So in such a way, we can show the TOR by using the Cranero inequality by starting from the time scaling perturbation. OK, so time is over. So I think it's my time to stop my third lecture. Thank you. Any questions? So that is your case or one case. So we have different results. And I wonder, is there any physical interpretation for delta? Is there any physical meaning of delta? I mean, this delta is some parameter. So yes, it's just a random TOR, but I guess delta 0 gives the dynamical uncertainty, and delta 1 gives the non-dynamic urgency. But these two has even a different dimension, right? First one is about the kinetic rate velocity. This is just a number. Yes, just a number. But below one gives something about the energy. So I wonder why these two has very different results. Because, yeah. OK, so in the Markov-Jumper process, we can count the total number of jumps in the Markov-Jumper process. So this total activity can be one possible measurable quantity. And also, we can measure the entropy production in this. Entropy production is another possible measurable quantity. So I mean, so depending on delta, we can make some different inequality by using different measurable quantities. And usually, this TOR becomes tighter when the process is very irreversible. And this TOR becomes tighter when the process is reversible. So I mean, it has a different bound. But anyway, we can bound this relative fluctuation by using two inequalities. So we can use one of them, which gives a stronger bound. So I mean, that's the meaning of these two inequalities, as far as I understand. Yes, I understand. Thank you. If delta is so many between 0 and 1, what happens? Yeah, I mean, I don't know whether it results in a valuable result. But actually, delta is introduced here. So delta is introduced here. So this master equation and this master equation are actually there exactly same. But when we define the part of the transition matrix, then actually, they have the different form. So I mean, the delta comes from by defining these two transition matrix. So that's why it is 1 or 0. So in this formalism, we cannot choose the other value of delta. Thank you. Do you have another question? Would the dynamics change in a sense like? Dynamics are same, I mean. So what is different? One is R and another is R epsilon. They have two different transition matrices. For the same process, there are two transition matrices. That gives the same dynamics. OK, so that degeneracy is possible. Yeah, yeah, yeah, right. That's not interesting. Yeah, yeah, yeah. Very interesting point, yeah, right. Other questions? So you? Are there any finite size effects that come into these? I mean, if you consider one of these systems in a confined volume, I don't think so. This is exact for any kind of system, any kind of process. So even if it is confined in some potential or some box, I think we can come to the same conclusion. Yeah, no, I'm just wondering if there's some sort of scaling relation. Because in conformal field theories, if you have an energy momentum tensor, then in finite size, there's a specific central charge that tells you about how it scales. So I'm just wondering in these non-proto... Yeah, yeah, yeah, right. So I mean, if this scaling perturbation is about position scaling, then probably such problem will arise. But in this case, this is a time-scaling perturbation. So we are now interested in a steady state. So actually, I mean, it does not make any difference, even though there is some box potential or some kind of other constraints, positional constraints. Other questions? Your brain must be full. Okay. So why don't you start here? Let's thank Professor Lee. So now we have a banquet. The banquet place is the first floor of this building. So I want you to utilize this opportunity to know each other. Okay. Let's meet at the first floor. Thanks for visiting for both the transition dates. Same for... I'm sorry, this is my microphone. I did not...