 Okay, I thank the organizers for giving me an opportunity to talk here. I'm going to talk about the norm additivity in the quasi-equilibrium state of a short-range interacting system. Okay, so this is the outline of the talk. First, I start from the introduction of some fundamental concepts such as additivity and the quasi-equilibrium states. And I present the model which shows the norm additivity in the quasi-equilibrium state, although it is additive in the true-equilibrium state. Then I present the new mega-result and I discuss the physical implications. Okay, so I'd like to start from the introduction of the additivity. So roughly speaking, the additivity implies that the system can be regarded as a collection of independent macroscopic subsystems. Okay, because the equilibrium state is determined by the Hamiltonian, so it would be convenient to express this notion in terms of the energy. The familiar statement is that if the interaction energy between the two subsystems, say A and B, is much smaller than the energies of the two subsystems, we say that the system is additive. Okay, roughly speaking, this can be regarded as a definition of the additivity, but there is an ambiguity here. Okay, because the value of the Hamiltonian depends on the microscopic state, so the meaning of this inequality is not so clear. The one natural consideration is to compare the equilibrium averages of them. So when the equilibrium average of the interaction energy is much smaller than the internal energies of the two subsystems, can we say that the two subsystems are independent of each other? The answer is no in general. So indeed, there is a model in which this condition is satisfied but the system is not additive. Okay, I will present such an example later. But okay, this condition is a necessary condition of the additivity but not a sufficient condition. Okay, other natural consideration is to require that this inequality should hold for any microscopic states, but this turns out to be a too strong condition. So indeed, there is a model in which this condition is violated but the two subsystems are almost independent. For example, let us consider the particle systems with a sufficiently strong short-range repulsions. And we consider the configuration of the particles such that the one particle in the subsystem A is very close to another particle in the subsystem B. Then the interaction between A and B can be very large. So yeah, this obvious count example might be quite trivial, but this kind of argument indicates that we need a more appropriate definition of the additivity in order to give a precise statement on the additive or non-additive systems. Okay, now I give a definition of the additivity. But okay, I'm not claiming that this is the best one among other possible definitions of the additivity. So the terminology of the additivity tends to be used in several different meanings in the literatures. So I would like to fix the meaning of the additivity within this talk, okay? So this is the purpose of mentioning about the definition of the additivity here. Okay, my idea is to consider the quasi-static thermodynamic operation to decouple the two subsystems. So in the initial Hamiltonian, there is an interaction between A and B, but in the final Hamiltonian, there is no interaction. And I introduce a parameter lambda which connects the two situations and lambda is buried from one to zero very slowly. So if there is no thermal environment, this corresponds to a quasi-static adiabatic process in which the entropy is conserved. Okay, now the amount of work performed by the system during this thermodynamic operation is denoted by W. W is given by E minus E prime, where E is the energy in the initial state and E prime is the energy in the final state. Okay, so my proposal is that if this W is very small, we should say that the system is additive. So very small means some extensive in the volume, so it can be neglected in the thermodynamic limit. Okay, this is my definition of the additivity and indeed at least I can say that that this condition is a very good characterization of the additivity. So in order to explain it, I would like to show that this condition actually derives several important results. Okay, first of all, because the quasi-static adiabatic process, the entropy is conserved. So this condition is directly related to some property of the entropy. Okay, because of the conservation of the entropy, we have this equality. Here, this left hand side is the entropy of the combined system with the energy E. So this is the entropy in the initial state. Okay, and the right hand side is the entropy of the two independent subsystems where there's no interaction between them. And the total energy is given by E prime and E prime is distributed so that the entropy takes the maximum. So this corresponds to the entropy in the final state. Okay, now because the E prime is given by E minus W and W is very small if the system is additive. So we can neglect W in the thermodynamic limit. So we can put E prime equals E here. So I obtain this equality. Here the E prime is replaced by E and this equality means that the entropy of the combined system with the energy E is almost identical to the entropy of the two separated systems with the same energy. So this is the expression of the additivity in terms of the entropy. Okay, and from this expression of the entropy we can show that the entropy is independent of the shape of the system. Okay, so in order to understand this aspect let us consider this kind of system. Here this left half part is regarded as a subsystem A and this right half part is regarded as a subsystem B. Then because of this expression the entropy of this system is almost identical to the entropy of this separated system where there is no interaction between here and here. Okay, on the other hand let us consider the same system in a different shape like this. And this left half part is again identical to the subsystem A and this right half part is a subsystem B prime. And B prime is a translation of B. So yeah, this system is obtained by some translation of the subsystem B like this. So okay, because of this property it is also concluded that the entropy of this system is also identical to the entropy of this separated system. So therefore we can conclude that the entropy of the system in this shape is almost identical to the entropy of the same system with a different shape. Okay, so by repeating this kind of argument it is concluded that the entropy is completely independent of the shape of the system. Okay, the shape independence of the entropy implies that we can drop the subscripts of them because the functional form of them are identical. So we put them as S of ipsion here, S is the entropy density and ipsion is the energy density. Okay, and by using this expression and this property it is shown that the entropy is a concave function. Okay, so now I define X, X is defined by VA over V and VA is the volume of the subsystem A and V is the volume of the total system. Then VB over V is given by one minus X. So then from these properties we can show that the entropy density satisfies this inequality for ipsion A and ipsion B with this condition. So this inequality is nothing but the concavity of the entropy. Okay, so from the concavity of the entropy we can show that, for example, the micro-concal ensemble is equivalent to the concal ensemble, that is called the ensemble equivalence. And okay, the concavity of the entropy immediately implies that the specific heat is always non-negative in the micro-concal ensemble. Okay, so all of these properties are desired properties for the additive systems and all of them are derived from this single condition. So in this sense we can say that this is a very good characterization of the additivity and my proposal is to employ this as a definition of the additivity. Okay, I have presented some properties of the additive systems. So conversely, the non-additive systems can exhibit these properties. So the entropy may depend on the shape of the system and the entropy may be non-concave and the ensemble equivalence may be violated and the specific heat may be negative in the micro-concal ensemble and so on. Okay, as you know, the system with unscreened long-range interactions exhibit these properties. So the long-range interacting systems are a typical non-additive systems. Okay, then what happens in the short-range interacting systems? In this talk, short-range interactions means that the interaction potential decays faster than one over R to the D where R is a distance and D is a spatial dimension. So yeah, it is rigorously proven that any short-range interacting particular spin systems with the sufficiently strong short-range departions should be additive. Okay, this theorem implies that the non-additivity cannot be realized in an equilibrium state over short-range interacting macroscopic system. So this is regarded as a no-go theorem in the equilibrium statistical mechanics. Okay, then is there a possibility to realize the long-range interacting systems? One strategy is to consider the small systems. So even if the interaction potential decays very fast, it looks a long-range potential if the system size is smaller than the range of the interaction. Okay, this is a very fruitful direction and there are several attempts, but in this talk, I'd like to consider the possibility of the macroscopic long-range interacting systems. Okay, of course, the Kuro interactions and the dipole-dipole interactions are very useful, but in this talk, I'd like to focus on the use of the non-equilibrium states. Okay, because in the non-equilibrium states, the no-go theorem of the equilibrium statistical mechanics can be avoided. Okay, indeed, in some non-equilibrium systems with a broken detailed balance condition for the original Hamiltonian, the long-range correlations appear in the non-equilibrium study states. So it is particularly interesting because the long-range correlations emerge from purely local dynamics. Okay, but in this talk, I'd like to consider another approach that is the quasi-equilibrium state or the metastable-equilibrium states. Okay, in which the detailed balance condition for the original Hamiltonian is satisfied. Okay, so if we have the detailed balance condition, then if we wait for a very, very long time, the system will relax to the equilibrium state. However, sometimes we observe that the system is trapped by the some metastable state before reaching the true thermal equilibrium. And I'd like to consider such a metastable state. Okay, the concept of the quasi-equilibrium is very easy to understand. So let us consider this kind of potential and initially the particle is trapped by this local minimum. Then at low temperature, it will take a very long time to relax to the true stable state. So within a certain time scale, the system will equilibrate within this local minimum like this. Okay, and this is a very simple example of the quasi-equilibrium state and if we introduce the artificial potential like this, here it agrees very well near the local minimum, but it grows infinitely large apart from the local minimum, then this quasi-equilibrium state is described as an equilibrium state under this effective potential. So in this way, the quasi-equilibrium state is described by the equilibrium statistical mechanics of the effective Hamiltonian. Okay, so there's a big advantage to consider the thermodynamics of such a state because the mechanism of the metastability is just a deep local minimum of the potential. So this state is almost independent of the detail of the dynamics as long as the dynamics satisfy the detailed balance condition. So it means that this state is almost independent of the type of the thermal environment. Oh, no, here, I call it a short-range interacting system. Ah, yeah, yeah, yeah, this red one, yeah, red one is a long-range potential, yes. Okay, this, yeah, but this blue one, blue one I call this as a short-range interaction, but this red one can be a long-range interaction. Yeah, this is the point, actually, yes. Okay, so yes, short time, yeah. So this is a transient description, yes. Okay, so in this way, the quasi-equilibrium state is described by the equilibrium statistical mechanics of the sum effective Hamiltonian. Okay, and as I said, this is very stable against several perturbations, so yeah, there is a big advantage to consider the thermodynamics of this state. But of course, there is a finite lifetime of this state, so eventually the system will relax to the equilibrium state. Okay, so now I present the model which shows the non-additivity in the quasi-equilibrium state. Okay, I consider the classical particles in the two or three spatial dimension. Here, in this talk, for simplicity, I consider the two-dimensional case. And the interaction potentials are given by this kind of renaissance-like potential, but the result is independent of the precise form of the interaction potential. The important point is that the sufficiently strong short-range repulsion should be present here, and this attractive part should be short-range, so this should decay very fast. Okay, now if we denote the radius of the particle by R i and R j like this, then the minimum of the interaction potential is at R i plus R j. Okay, this corresponds to this situation. And the special feature of this model is that each particle has an internal degree of freedom denoted by sigma, and sigma takes a value of plus one or minus one, which is called the spin degrees of freedom. Okay, and the important point is that depending on the value of the spin, the radius of the particle changes. So the radius of the particle is a function of sigma, and it is assumed that the radius in the downspin state is smaller than the radius in the upspin state. So then this R i and R j are the functions of sigma i and sigma j, respectively. So this interaction potential depends on the spin degrees of freedom, so this is the special feature of this model. Okay, so this is the Hamiltonian of the model. So the first term is the kinetic energy, and the second term is the interaction energy, and the third term is the external field for the spin degrees of freedom. Okay, so in order to show the dependence on sigma i and sigma j explicitly, I write the subscript like this. Okay, this is the Hamiltonian. And the dynamics is given by the hybrid of the Hamilton dynamics and the Monte Carlo dynamics. So the position and the moment are available in time according to the Hamilton dynamics of this Hamiltonian. And the spin degrees of freedom available in time according to the stochastic single spin flip. So this is the Monte Carlo dynamics. Okay, so the theoretical, okay, the motivation of considering this kind of model is stems from the spin crossover material. So in the spin crossover material, each molecule has two stable internal states. One is the high spin state and the other one is the low spin state. And it is experimentally verified that according to the high spin or low spin states, the molecular size changes. And the difference of the molecular size is about 10%. So this is a visible effect. So therefore, this kind of model can be regarded as a theoretical model of the spin crossover materials. Okay, now I explained the initial states. So yeah, in order to consider the crazy equilibrium states, the choice of the initial state is very important. Yeah, I consider this kind of initial states. So all the particles constitute the triangular lattice like this. And all the particles are initially in the, for example, down spin state. And the distance between the nearest neighbor particles is chosen as a minimum of the interaction potential. So then, if the temperature is much smaller than this potential depth, this lattice structure will be stable up to the time given by the Arrhenius law like this. So then, within this time scale, the system will reach the quasi equilibrium state with this lattice structure held kept. So after some time evolution, okay, the particles move and some spins are flipped, but this lattice structure is maintained. So this is a quasi equilibrium state. Okay, after a longer time scale, some particles are evaporated from the lattice like this. So by getting over the activation energy from here to here. But the remaining particles are still in the quasi equilibrium state. But, okay, finally, all the activation processes take place and all the particles are scattered out and the initial lattice structure is completely broken. So this is the final state. Okay, and it is numerically confirmed that this kind of scenario actually takes place. So yeah, this is, okay, I consider the canonical setup in which the energy is exchanged between the system and the thermal environment. Okay, and this is the time evolution of the energy density. So as you see, there is a clear two-step relaxation. And okay, the first initial quick relaxation, the system reaches a quasi equilibrium state, and but here the lattice structure persists. And after an exponentially long time scale in this potential depth, the second relaxation occurs and after that the lattice structure is completely broken and the particles are scattered out. Okay, so this is the graph of the time evolution. Okay, and it is also found that in the quasi equilibrium state, the momentum distribution obeys the Maxwell-Boltzmann distribution at the temperature identical to the thermal bus. Okay, and this implies that the momentum degrees of freedom have already summarized in the quasi equilibrium state, although the position and the spin degrees of freedom have not summarized yet. So in this way, in the quasi equilibrium state, the partial summarization occurs. Okay, okay, and the dynamics is, yeah, the position and the momentum are evolving time according to the Hamiltonian dynamics. Think flip is, yeah, I employ the metropolis loop, metropolis method. Ah, yes, yes, so yeah, the, yeah, yes. Time evolution of the particles and time evolution of the spins and so on. Yes? Ah, yes? Oh, but okay, even if there is no room, okay. So it means that there are very large interaction energy. So it's such a, sorry? Yeah, yeah, such a flipping is very hardly to occur, yes. So yeah, now I'd like to consider the effective Hamiltonian, describing the quasi equilibrium state. Okay, so yeah, effective Hamiltonian is easily guessed because, okay, in the quasi equilibrium state, the lattice structure is maintained. So the distance between the nearest neighbor particles is always near this local minimum. So we can approximate the interaction potential by this kind of quadratic one. And, okay, we can simply neglect the contributions except for the nearest neighbor pairs because of this short range nature of the original interactions. Therefore, the effective Hamiltonian is given like this. So the interaction potential is replaced by the quadratic potential. And the summation is restricted only over the nearest neighbor pairs. So this is the effective Hamiltonian describing the quasi equilibrium states. Okay, and it is noted that this Hamiltonian is not the Hamiltonian of a short range interacting systems. Okay, because this quadratic interactions becomes stronger and stronger as the distance between the two particles increases. So this is, in this sense, this is a long range potential. So the important point is that although the original Hamiltonian is given by the short range interactions, the effective Hamiltonian describing the quasi equilibrium state is not necessarily a short range interacting system. So this is the important point. Okay, and it is numerically confirmed that the effective Hamiltonian introduced in the previous slide actually describes the quasi equilibrium state very accurately. Okay, these red points are the average in the quasi equilibrium states calculated from the exact dynamics. And the green points correspond to the average in the equilibrium state of the effective Hamiltonian. And okay, this left figure shows the temperature as a function of the energy and the right figure shows the magnetic field as a function of the magnetization. Okay, as you see, the red points and the green points agree very well. So it implies that this effective Hamiltonian actually describes the quasi equilibrium state very well. And the important point is that, okay, as you see here, there is a region of the negative specific heat. And here, there's a region of the negative susceptibility. Both of them are clear indications of the non-additivity of the system. So yeah, in this model, although the original Hamiltonian contains only the short range interactions, the non-additivity emerges in the quasi equilibrium states. Okay, and we can also obtain the direct evidence of the non-additivity by measuring the walk performed by the system during the thermodynamic process to decouple the two subsystems. Okay, I compare this system and I decouple these two subsystems very slowly. And I numerically measure the amount of walk performed by the system. Okay, this is the result and this is the walk per particle and this is the one over system sites. And as you see, it does not banish even in the thermodynamic limit. So it means that the walk performed by the system is extensively large. So this is the direct evidence of the non-additivity of the quasi equilibrium states. Okay, and here it is remarked that in this model, the equilibrium average of the interaction energy is always much smaller than the equilibrium average of the, yeah, okay, the internal energies of the two subsystems. Okay, because in this effective Hamiltonian, the interactions are present only for the nearest neighbor pairs. Therefore, the interaction between A and B is proportion to the surface area between A and B. Although, okay, and why the internal energies are proportion to the volume of the system? So in this sense, the, yeah, the equilibrium average of the interaction energy is always much smaller than the internal energies. So in this model, the, yeah, interaction energy is in this sense very weak, but the system exhibits the non-additivity. So this is an interesting point of this model. Okay, and I'd like to consider the effective spin-spin interactions. So, yeah, the original Hamiltonian is described by the degrees of freedom of the position and the momentum and the spins. Okay, but by integrating out over the position and the momentum degrees of freedom, we can get the effective Hamiltonian for the spin degrees of freedom. Okay, but, okay, yeah, of course, it is very difficult to obtain the effective Hamiltonian. Exactly, so I'd like to guess the interaction, effective spin-spin interactions under the ansatz of this effective Hamiltonian. So I assume that the effective Hamiltonian is given by this form, and I'd like to estimate Jij from the numerical data. Okay, this is possible by using the data of the correlation functions. So under a certain approximation, we can show that the effective spin-spin interactions is related to the inverse matrix of the correlation function like this. So by using this formula, we can estimate the effective spin-spin interactions. And this is the result. Okay, this is Jij, multiplied by the number of the particles as a function of the distance between i and j divided by the system size. And the, okay, different colors imply the different system size. So, and as you see, all the data are collapsed very well. And this data collapse implies that the effective spin-spin interactions obey this kind of scaling. And it is also found that the Jij is almost independent of the temperature. So it implies that the effective spin-spin couplings of energetic origin and not of the entropic origin. Okay, now, yeah, I'd like to explain the meaning of this scaling. The dependence on the distance divided by the system size implies that the interaction range is comparable with the system size. So this represents the ultra-long range interactions. On the other hand, the scaling of the one over L to the D in front of this function implies that the interaction between the two particular spins is very weak. So, yeah, interaction is very weak, but the range of the interaction is very long. And as a result, the interaction energy per particle becomes independent of the system size in this scaling. So in this sense, the system is extensive, although it is not additive. Okay, and I, it is noted that this scaling is the same one as the scaling of the cut prescription. So, yeah, this interaction is nothing but a long range interaction with the cut prescription. Okay, so, yeah, usually the cut prescription is put by hand in order to make the system extensive. However, in this model, the cut prescription is not necessary. The effective spin-spin interactions naturally obey the scaling corresponding to the cut prescription. Okay, this stems from the fact that originally the system is a short range interacting system. So for example, let us look at the effective Hamiltonian. So as I said, the long range nature of this quadratic potential leads to the non-addictivity of the system. On the other hand, the locality in the sense that the interactions are present only for the nearest neighbor pairs. The locality in this sense leads to the extensivity of the system. So in this model, the locality and the non-locality coexist. So this is an interesting feature of this model. Okay, so, yeah, now I'd like to discuss what is equilibrium and quasi-equilibrium. So far, we have considered that the equilibrium and quasi-equilibrium are completely different concepts. However, practically, we cannot distinguish the equilibrium and the quasi-equilibrium in the finite observation time. For example, let us consider the gas in the container like this. And if the container is modeled by the potential barrier, then these particles are in the container forever. So the system will reach the equilibrium state in this container. So, yeah, on the other hand, if we consider the exact Hamiltonian of all the atoms of the gas and the container. So here, now, the container is not the potential barrier but the ensemble of atoms. Then, this state is not the true equilibrium state. So if we wait for a very, very long time, the container will be eroded and broken and the particles inside it will go out. So in this sense, yeah, this equilibrium state is a quasi-equilibrium state if we consider this exact Hamiltonian. But of course, it does not mean that the concept of the equilibrium cannot be applied to this case. Rather, we should consider that the concept of the equilibrium depend on the time scale we observe. So, yeah, Richard Feynman said in his textbook that if all the first things have happened and all the slow things not, the system is said to be in some equilibrium. Okay, so, yeah, of course, in the previous example, the first things correspond to the equilibration within the container and the slow things correspond to the erosion of the container. So in this sense, okay, we, practically, we cannot distinguish, yeah, it is very hard to distinguish the equilibrium and the quasi-equilibrium. And the concept of the equilibrium depends on the time scale. So in this sense, what I have shown is that in a certain finite time scale, the equilibrium state of the short-range interacting systems can exhibit the normativity. So this is a message of this talk. Okay, finally, I'd like to comment on the dynamics of this model. So as a future problem, it is, yeah, it will be very interesting to consider the dynamics of this model because, okay, in this model, the spin-spin effective interactions are long-drenched, but the interactions are spread with a finite speed because, yeah, the interactions are present only for the nearest neighbor pairs. So, yeah, the effective spin-spin interactions are long-drenched, but the spread of the correlation is, yeah, finite speed. So, yeah, therefore, in the short-time dynamics, the system will behave as a short-range interacting system. So in this sense, it is expected that the dynamical phenomena in this system will differ from those in usual long-range interacting systems. So, yeah, it is interesting to explore the dynamics of this model. Okay, this is a summary of this talk. Yeah, thank you very much.