 Thank you for the introduction. And first, I try to thank the organizers for giving me a nice opportunity to give my talk. And my title is, Relaxation on Financier's Fractuation in Summer Equipment. So first, I say the title seems a crazy idea. What is relaxation? We start from the initial state. This is the initial state of the hot T. And then we leave this T for a long time. And then this T gets called up to the room temperature. And this is equilibrium state. And this process is called relaxation. And we take the T infinity limit. So no further evolution happened macroscopically in equilibrium state. So normally, relaxation is relaxation to equilibrium, like this. And this time, in this talk, we consider relaxation in summer equilibrium. And the purpose of this talk is like this. First, I demonstrate the relaxation in summer equilibrium numerically in the HMF model. And then I propose a simple scenario which is applicable for a vast, long-range system. And then I show the supporting evidence of the scenario via the fluctuation response duration. And I note two points. I'm concerned about the classical mechanics. And no quantum effect is included. And second one is no summer noise is included in this new mechanical simulation. So the dynamics is pure Hamiltonian dynamics. OK. And the system is a Hamiltonian mean field model, which is sometimes presented in this conference. This part is more or less xy-spin. And this one is a kinetic term. And this sum is described by the order parameter, which is the geometric mean of each xy-spin. And so this is the mean field vector. And then once we get the Hamiltonian, we get the temporal evolution of the system by the canonical equation of motion like this. And it drives the initial system to the summer equilibrium system. And after that, even after that, the canonical equation of motion drives the system like this in the same way. And the order parameter is the equilibrium value plus fluctuation in the summer equilibrium. Our initial state is here, the summer equilibrium. In the summer equilibrium, the magnetization value is like this. And there is a phase transition point here. And the temperature is the half. And at this point, low-temperature value is a low-temperature region of the clustered state. And high-temperature region has a non-clustered state. And now I'm talking about the temperature, but this is just the parameterization of the equilibrium state. And as I said, the dynamics is pure Hamiltonian, and no summers noise is included. And the fact we observe is the second moment, because the first moment is this equilibrium value, the average of equilibrium value, and it is stationary by definition. So we concentrate on the fluctuation, this part. And the fluctuation is defined as usual, the average of square minus square of average. And in this study, this bracket, t, is defined as the time average of variable. So we start from the complete time average from time zero and to t average. And then it is defined as time average of observable. So I show you, so this depends on the finite size n and the t average. So I show the t average dependence of this variance. And the result is like this. The sample point is here and here, two point. So this point is closer to the critical point. And this one is a little bit far. And horizontal axis is t average. And vertical axis is variance, otherwise fluctuation. And the first part is here. The fluctuation increase, but this is not an interesting point. Because by definition, we define the fluctuation by time average. So systems start from here, and each start from fluctuation like this, like this. So starting in the very initial time, the variance is very small. So increasing is very natural. So this is not an interesting part. The first interesting is this part. This part have the anomalous plateau before going to another level. So this level is the summer equilibrium fluctuation level. But the system is trapped at another level, lower than the summer equilibrium level. So this is the anomalous plateau. To check this numerical simulation, because I prepared the summer equilibrium state numerically. So each might be the error of the numerical simulation. So to exclude this kind of possibility, I start this result is start from t called 0. But I start from t called t to 5. So t to 5 is long enough to reach the summer equilibrium like this, like this. So I start from here, and then recompute this variance. And then the result is like this. So I start from t to 5, and take average during the t average. And the result is almost same. So I back to the original one, and the new one, original one, new one, not so different. It's almost same. So this evolution is not the cause of the numerical errors. So just the effect of dynamics. I demonstrated relaxation from one level to another level in summer equilibrium. So I explained why this kind of relaxation is possible in summer equilibrium. So antibody system is governed by canonical equation of motion. And if we take n to infinite limit, this system is described by Brasov equation. And this Brasov equation have infinite number of Casimir invariant like this in long-range system. So this is the point. So if we are in the infinite limit, if we are in the Brasov dynamics, the dynamics is constrained in a Casimir level set. So this is the Casimir level set like this, and like this. And the system should be confined on a Casimir level set, which is the initial condition is on that level set. But if we consider finite n, so this Casimir is no more exact invariant, but the approximate invariant. So I call it should Casimir invariant. So this level set is not exact, but should. So system is tend to be confined on a level set, but can escape from a level set. So the system confined our level set, but can escape and confined levels and escape and confined escape. So this is the process of the relaxation, which I showed you previously. So the relaxation, the previous relaxation is a successive escaping process from a should level set in finite size system. But this is just a scenario. So we need to support it. So how to check this scenario? So to check the scenario, I use a fluctuation response relation. So what is fluctuation response relation? So now we consider the fluctuation, and this is something like a free energy, and system fluctuates around the bottom of the free energy like this. And on the other hand, if we consider the response to the external force, then the free energy it modifies from this shape to like this shape. And the system, the previous bottom is here, but the current bottom is here. So here is some displacement. So this displacement is a response to this external force. And both of the fluctuation and the response are related to the bottom curvature of the free energy. So this fluctuation and the response have the relation. And in the Brasov theory, we know the Casimir's restrict the response or decrease the response because of the imbalance constraint. So this is the known part, and this is known part. So by using this part and this part, I check the Casimir's restrict the fluctuation also. But the checking point is we have two checking points. Now this is set in the statistical mechanics. So is it true in dynamics or not? And the second point is this theory, the Brasov theory, is in the system of infinite N. But we are considering the finite N. So is it true even in the finite N? So we check this and this point by N body simulations. Before going to the result, I revisit the fluctuation of the HMF model. As I said, in the low energy part, the free energy is like this, like a Mexican hat. And the fluctuation is like this. And this model is a Goldstone model and not so interesting. So we focus on just fluctuation for the longitudinal axis. So we took the absolute value of order parameter. But in high energy side, the free energy is like this. And Mx and My fluctuate in the two-dimensional Mx and My plane. So in this case, this part, so I remove this absolute value symbol. And the fluctuation is defined like this and this. And by the rotational symmetry of the system, this burn it. So this is a slightly different definition of fluctuation in below and above the critical point. OK, so anyway, we can compute this fluctuation in N body system. And we check the response relation between the fluctuation and the response, both in the off-critical and on-critical. And at off-critical, we get a linear response like this. And on-critical, we get a non-linear response. So we check both at the off-critical and on-critical. Off-critical, the linear response is written like this in the limit of small external force, H. And this delta M is the response. And the response is proportional to the small external force. And the coefficient chi sets susceptibility. And the important point is this chi in brass of dynamics is known by this paper. And the relation is like this. So this is a susceptibility. And this is a fluctuation. And if everything is OK, this equality should hold. So we check that we put a product that this chi is derived by salary. And this variance fluctuation obtained by N body numeric, we compare this term and this term. And this is that we have three lines. So we have three types of susceptibility. The one is the canonical susceptibility, like this gray line. And the light blue line is the micro canonical susceptibility, like this, like this. And the black side line is the susceptibility in brass of dynamics. Because the micro canonical system have the energy constraint, so the susceptibility is suppressed. And brass of dynamics have additional custom invariant, so further suppressed. And the line from theory and points from N body simulation and points are on the brass of the line, like this. And above the critical point, these three lines collapse in one line, like this. So there is only one line. And numerical point is a little bit far from line. But if we increase the number of particles, so 10 to 2, 3, 4, 5, then the point approach to the theoretical line, like this. So we can conclude the fluctuation response relation whole, even in the finite N system in dynamics. So this is the first check. So I skip this insight. Now we can understand what happened in the previous relaxation process. So I draw the two lines, the black solid line and the light blue line here and here, black line and light blue line here. And this black line corresponds to the brass of susceptibility level. So for this line, for this picture, the point is here. And this micro canonical level corresponds to this point here. So this relaxation corresponds to the process from brass of level to the micro canonical level. OK, so the second check is a nonlinear response at the critical point. What we know is linear response question, susceptibility diverge at the critical point. So the response is no more linear response. So the nonlinearity of the response is described by the critical exponent delta. And in statistical mechanics, delta is 3. But in brass of system, previously we derived, it is 3 over half, 3 half. So we use this difference to check the scenario at the critical point. And the problem is how to relate and how to make a relation between this critical point and the fluctuation. The idea is to use the Landau's phenomenological sugar free energy. The Landau's free energy like this. So first term is M squared. And normally this is M to force due to the symmetry of the system. And this is the external force term. But I keep this delta in the generic side. And suppose the fluctuation with the G equals, so this level is G equals constant over N. Because G is the free energy per one particle. So this is the assumption, a phenomenological assumption. And at T equal to Tc, so at critical point, and no external force. So I bunch this term and this term. And then with this assumption, we get how large M is at the critical point. And a short derivation gives the scaling, finalized scaling for the fluctuation. And the statistical mechanics, so delta equals 3, predict N to minus half. But Blast Dynamics, delta equals 3 half, represent N to minus 4 over 5. So this is a strange exponent. So the checking point is this exponent is true or not. So I show the N dependence of the fluctuation at the critical point. So result is like this. So this is N in log scale. And this is the fluctuation in log scale. And this line is 0.779 and 0.779 is close to 4 over 5. And the difference of the color correspond to the difference of the averaging time. So anyway, if the averaging time is small enough, the finalized scaling have the scale exponent minus 4 over 5 instead of the normal half exponent. And this is the vertical axis of this inset is this. So the horizontal line correspond to the exponent of minus 4 over 5. And we can confirm the exponent 4 over 5 from this picture also. Yes? This time, temperature? At the current? Just a critical point. At the critical point. OK, so we could confirm this strange exponent at the critical point. So we have checked the scenario or the relation between the fluctuation and the response at the critical point and out of the critical point like this. So we have checked this point, this point, and this point, the relation between the Casimir and the response we know. And we checked the numerically the relation between fluctuation and the response like this. So the two checkpoints are cleared. So this is a summary and discussion. And the main message of this talk is the long-range system with finite n have the should Casimir constraint. And this should Casimir constraint should Casimir constraint give an anomalous finite size fluctuation even in thermal equilibrium. And the anomality is like a relaxation in thermal equilibrium or a strange scaling at the critical point. So this mechanism is very simple. So I expected that this anomalous finite size fluctuation appeared in very large class of long-range systems. And the essential point of this relaxation in thermal equilibrium is the existence of the should constraint. So if the should constraint exist, we can expect the relaxation beyond the long-range systems. So for instance, if we consider the particles linked by hard springs. And in this case, a hard spring plays a role of the should constraint because this is approximate bond length. So I expect this kind of strain anomalous finite scaling in this model. So this model is like molecules, protein, or something like that. But this is the future work. OK, so thank you for your attention.