arXiv subjects

Makiko Sasada

Publications and source records attributed to Makiko Sasada.

At least 37 records · Page 2Linked to original sources

arXiv2022

Generalized hydrodynamic limit for the box-ball system

We deduce a generalized hydrodynamic limit for the box-ball system, which explains how the densities of solitons of different sizes evolve asymptotically under Euler space-time scaling. To describe the limiting soliton flow, we introduce a continuous state-space analogue of the soliton decomposition of Ferrari, Nguyen, Rolla and Wang (cf. the original work of Takahashi and Satsuma), namely we relate the densities of solitons of given sizes in space to corresponding densities on a scale of 'effective distances', where the dynamics are linear. For smooth initial conditions, we further show that the resulting evolution of the soliton densities in space can alternatively be characterised by a partial differential equation, which naturally links the time-derivatives of the soliton densities and the 'effective speeds' of solitons locally.

math-ph↗

arXiv2022

On the stationary solutions of random polymer models and their zero-temperature limits

We derive stationary measures for certain zero-temperature random polymer models, which we believe are new in the case of the zero-temperature limit of the beta random polymer (that has been called the river delta model). To do this, we apply techniques developed for understanding the stationary measures of the corresponding positive-temperature random polymer models (and some deterministic integrable systems). More specifically, the article starts with a survey of results for the four basic beta-gamma models (i.e. the inverse-gamma, gamma, inverse-beta and beta random polymers), highlighting how the maps underlying the systems in question can each be reduced to one of two basic bijections, and that through an `independence preservation' property, these bijections characterise the associated stationary measures. We then derive similar descriptions for the corresponding zero-temperature maps, whereby each is written in terms of one of two bijections. One issue with this picture is that, unlike in the positive-temperature case, the change of variables required is degenerate in general, and so whilst the argument yields stationary solutions, it does not provide a complete characterisation of them. On the other hand, this degeneracy does allow us to explain the appearance of atoms in the stationary measures of certain zero-temperature models. We also derive from our viewpoint various links between random polymer models, some of which recover known results, some of which are novel, and some of which lead to further questions.

math.PR↗

arXiv2022

A Decomposition Theorem of Varadhan Type for Co-local Forms on Large Scale Interacting Systems

In our previous article with Yukio Kametani, we investigated the geometric structure underlying a large scale interacting system on infinite graphs, via constructing a suitable cohomology theory called uniformly local cohomology, which reflects the geometric property of the microscopic model, using a class of functions called the uniformly local functions. In this article, we introduce the co-local functions on the geometric structure associated to a large scale interacting system. We may similarly define the notion of uniformly local functions for co-local functions. However, contrary to the functions appearing in our previous article, the co-local functions reflect the stochastic property of the model, namely the probability measure on the configuration space. We then prove a decomposition theorem of Varadhan type for closed co-local forms. The space of co-local functions and forms contain the space of $L^2$-functions and forms. In the last section, we state a conjecture concerning the decomposition theorem for the $L^2$-case.

math.PR↗

arXiv2021

Detailed balance and invariant measures for discrete KdV- and Toda-type systems

In order to study the invariant measures of discrete KdV- and Toda-type systems, this article focusses on models, discretely indexed in space and time, whose dynamics are deterministic and defined locally via lattice equations. A detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/alternately distributed configurations, characterizes those that are temporally invariant in distribution. A condition for establishing ergodicity of the dynamics is also given. These results are applied to various examples of discrete integrable systems, namely the ultra-discrete and discrete KdV equations, for which it is shown that the relevant invariant measures are of exponential/geometric and generalized inverse Gaussian form, respectively, as well as the ultra-discrete and discrete Toda lattice equations, for which the relevant invariant measures are found to be of exponential/geometric and gamma form. Ergodicity is demonstrated in the case of the KdV-type models. Links between the invariant measures of the different systems are presented, as are connections with stochastic integrable models and iterated random functions. Furthermore, a number of conjectures concerning the characterization of standard distributions are posed.

math.PR↗

arXiv2020

Convergence of a degenerate microscopic dynamics to the porous medium equation

We derive the porous medium equation from an interacting particle system which belongs to the family of exclusion processes, with nearest neighbor exchanges. The particles follow a degenerate dynamics, in the sense that the jump rates can vanish for certain configurations, and there exist blocked configurations that cannot evolve. In [Gonçalves-Landim-Toninelli '09] it was proved that the macroscopic density profile in the hydrodynamic limit is governed by the porous medium equation (PME), for initial densities uniformly bounded away from $0$ and $1$. In this paper we consider the more general case where the density can take those extreme values. In this context, the PME solutions display a richer behavior, like moving interfaces, finite speed of propagation and breaking of regularity. As a consequence, the standard techniques that are commonly used to prove this hydrodynamic limits cannot be straightforwardly applied to our case. We present here a way to generalize the \emph{relative entropy method}, by involving approximations of solutions to the hydrodynamic equation, instead of exact solutions.

math.PR↗

arXiv2020

Hydrodynamic limit for a facilitated exclusion process

We study the hydrodynamic limit for a periodic $1$-dimensional exclusion process with a dynamical constraint, which prevents a particle at site $x$ from jumping to site $x\pm1$ unless site $x\mp1$ is occupied. This process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component, assuming that the initial macroscopic density is larger than $\frac{1}{2}$, or one of its absorbing states if this is not the case. It belongs to the class of conserved lattice gases (CLG) which have been introduced in the physics literature as systems with active-absorbing phase transition in the presence of a conserved field. We show that, for initial profiles smooth enough and uniformly larger than the critical density $\frac{1}{2}$, the macroscopic density profile for our dynamics evolves under the diffusive time scaling according to a fast diffusion equation (FDE). The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

math.PR↗

arXiv2020

Discrete integrable systems and Pitman's transformation

We survey recent work that relates Pitman's transformation to a variety of classical integrable systems, including the box-ball system, the ultra-discrete and discrete KdV equations, and the ultra-discrete and discrete Toda lattice equations. It is explained how this connection enables the dynamics of the integrable systems to be initiated from infinite configurations, which is important in the study of invariant measures. In the special case of spatially independent and identically distributed configurations, progress on the latter topic is also reported.

math.PR↗

arXiv2019

Dynamics of the ultra-discrete Toda lattice via Pitman's transformation

By encoding configurations of the ultra-discrete Toda lattice by piecewise linear paths whose gradient alternates between $-1$ and $1$, we show that the dynamics of the system can be described in terms of a shifted version of Pitman's transformation (that is, reflection in the past maximum of the path encoding). This characterisation of the dynamics applies to finite configurations in both the non-periodic and periodic cases, and also admits an extension to infinite configurations. The latter point is important in the study of invariant measures for the ultra-discrete Toda lattice, which is pursued in a parallel work. We also describe a generalisation of the result to a continuous version of the box-ball system, whose states are described by continuous functions whose gradient may take values other than $\pm1$.

math-ph↗

arXiv2019

Duality between box-ball systems of finite box and/or carrier capacity

We construct the dynamics of the box-ball system with box capacity $J$ and carrier capacity $K$, which we abbreviate to BBS($J$,$K$), in the case of infinite initial configurations, and show that this system is dual to the analogous BBS($K$,$J$) model. Towards this end, we build on previous work for the original box-ball system, that is BBS($1$,$\infty$), to show that when the box capacity $J$ and carrier capacity $K$ satisfy $J<K$ the dynamics can be represented by a Pitman-type transformation. These ideas are applied in the case of random initial configurations to show that the distributional properties of spatial stationarity and invariance under the BBS dynamics are dual. Moreover, for independent and identically distributed configurations, we derive a characterisation of invariant measures in terms of a detailed balance equation, which captures the duality of the system locally; this is used to find all invariant measures in this class. Finally, we deduce the speed of a tagged particle, and show that this also satisfies a natural duality relation.

math.PR↗

arXiv2019

Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures

The box-ball system (BBS) is a simple model of soliton interaction introduced by Takahashi and Satsuma in the 1990s. Recent work of the authors, together with Tsuyoshi Kato and Satoshi Tsujimoto, derived various families of invariant measures for the BBS based on two-sided stationary Markov chains. In this article, we survey the invariant measures that were presented in the latter work, and also introduce a family of new ones for periodic configurations that are expressed in terms of Gibbs measures. Moreover, we show that the former examples can be obtained as infinite volume limits of the latter. Another aspect of the previous work was to describe scaling limits for the box-ball system; here, we review these results, and also present scaling limits other than those that were covered there. One, the zigzag process has previously been observed in the context of queuing; another, a periodic version of the zigzag process, is apparently novel. Furthermore, we demonstrate that certain Palm measures associated with the stationary and periodic versions of the zigzag process yield natural invariant measures for the dynamics of corresponding versions of the ultra-discrete Toda lattice.

math.PR↗

arXiv2018

5/6-Superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field

We consider a one-dimensional infinite chain of coupled charged har- monic oscillators in a magnetic field with a small stochastic perturbation of order $ε$. We prove that for a space-time scale of order $ε$^{-1} the density of energy distribution (Wigner distribution) evolves according to a linear phonon Boltzmann equation. We also prove that an appropriately scaled limit of solutions of the lin- ear phonon Boltzmann equation is a solution of the fractional diffusion equation with exponent 5/6.

math.PR↗

arXiv2018

Dynamics of the box-ball system with random initial conditions via Pitman's transformation

The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al.\ recently showed the invariance in distribution of Bernoulli product measures with density strictly less than $\frac{1}{2}$, and gave a soliton decomposition for invariant measures more generally. We study the BBS dynamics using the transformation of a nearest neighbour path encoding of the particle configuration given by `reflection in the past maximum', which was famously shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible for all times. We give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. Finally, for Bernoulli product measures with parameter $p\uparrow\frac12$ (which may be considered the `high density' regime), the path encoding we consider has a natural scaling limit, which motivates the introduction of a new continuous version of the BBS that we believe will be of independent interest as a dynamical system.

math.PR↗

arXiv2018

Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field

We introduce a $d$-dimensional system of charged harmonic oscillators in a magnetic field perturbed by a stochastic dynamics which conserves energy but not momentum. We study the thermal conductivity via the Green-Kubo formula, focusing on the asymptotic behavior of the Green-Kubo integral up to time $t$ (i.e., the integral of the correlation function of the total energy current). We employ the microcanonical measure to calculate the Green-Kubo formula in general dimension $d$ for uniformly charged oscillators. We also develop a method to calculate the Green-Kubo formula with the canonical measure for uniformly and alternately charged oscillators in dimension $1$. We prove that the thermal conductivity diverges in dimension $1$ and $2$ while it remains finite in dimension $3$. The Green--Kubo integral calculated with the microcanonical ensemble diverges as $t^{1/4}$ for uniformly charged oscillators in dimension $1$, while it is known to diverge as $t^{1/2}$ without magnetic field. This is the first rigorous example of the new exponent $1/4$ in the asymptotic behavior for the Green-Kubo integral. We also demonstrate that our result provides the first rigorous example of a diverging thermal conductivity with vanishing sound speed. In addition, employing the canonical measure in the Green-Kubo formula, we prove that the Green-Kubo integral for uniformly and alternately charged oscillators respectively diverges as $t^{1/4}$ and $t^{1/2}$. This means that the exponent depends not only on a non-zero magnetic field but also on the charge structure of oscillators.

math.PR↗

arXiv2017

Heat transport via low-dimensional systems with broken time-reversal symmetry

We consider heat transport through systems with broken time-reversal symmetry. We apply strong magnetic fields to weakly charged particle systems, where the dynamics are dominated by the Lorentz force and spring forces. The standard momentum conservation is not satisfied and the sound wave disappears depending on the charge structure. We introduce an exactly solvable model and demonstrate the appearance of a new universality class. The absence of sound waves affects the relation between equilibrium correlation and the diverging thermal conductivity in the open system.

cond-mat.stat-mech↗

arXiv2017

The gradient condition and the contribution of the dynamical part of Green-Kubo formula to the diffusion coefficient

In the diffusive hydrodynamic limit for a symmetric interacting particle system (such as the exclusion process, the zero range process, the stochastic Ginzburg-Landau model, the energy exchange model), a possibly non-linear diffusion equation is derived as the hydrodynamic equation. The bulk diffusion coefficient of the limiting equation is given by Green-Kubo formula and it can be characterized by a variational formula. In the case the system satisfies the gradient condition, the variational problem is explicitly solved and the diffusion coefficient is given from the Green-Kubo formula through a static average only. In other words, the contribution of the dynamical part of Green-Kubo formula is 0. In this paper, we consider the converse, namely if the contribution of the dynamical part of Green-Kubo formula is 0, does it imply the system satisfies the gradient condition or not. We show that if the equilibrium measure μ is product and {L^2} space of its single site marginal is separable, then the converse also holds. As an application of the result, we consider a class of stochastic models for energy transport studied by Gaspard and Gilbert in [1, 2], where the exact problem is discussed for this specific model.

math.PR↗

arXiv2016

Thermal conductivity for stochastic energy exchange models

We consider a class of stochastic models for energy transport and study relations between the thermal conductivity and some static observables, such as the static conductivity, which is defined as the contribution of static correlations in Green-Kubo formula. The class of models is a generalization of two specific models derived by Gaspard and Gilbert as mesoscopic dynamics of energies for two-dimensional and three-dimensional locally confined hard-discs. They claim some equalities hold between the thermal conductivity and several static observables and also conjecture that these equations are universal in the sense that they hold for mesoscopic dynamics of energies for confined particles interacting through hard-core collisions. In this paper, we give sufficient and necessary conditions for these equalities to hold in the class we introduce. In particular, we prove that the equality between the thermal conductivity and other static observables holds if and only if the model obeys the gradient condition. Since the gradient condition does not hold for models derived by Gaspard and Gilbert, our result implies a part of their claim is incorrect.

math-ph↗

arXiv2015

Algebraic Structure of Vector Fields in Financial Diffusion Models and its Applications

High order discretization schemes of SDEs by using free Lie algebra valued random variables are introduced by Kusuoka, Lyons-Victoir, Ninomiya-Victoir and Ninomiya-Ninomiya. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical methods. The authors found a special Lie algebraic structure on the vector fields in the major financial diffusion models. Using this structure, we can solve the flows associated with vector fields analytically and efficiently. Numerical examples show that our method saves the computation time drastically.

math.PR↗

arXiv2015

Diffusive scaling in energy ginzburg-Landau dynamics

Ginzburg-Landau energy models arise as autonomous sto-chastic dynamics for the energies in coupled systems after a weak coupling limit (cf. [3, 6]). We prove here that, under certain conditions, the energy fluctuations of these stochastic dynamics are driven by the heat equation, under a diffusive space time scaling.

cond-mat.stat-mech↗