Asymptotic Derivation of Langevin-like Equation with Non-Gaussian Noise and Its Analytical Solution

Kanazawa, Kiyoshi; Sano, Tomohiko G.; Sagawa, Takahiro; Hayakawa, Hisao

doi:10.1007/s10955-015-1286-x

Cited by 37 publications

(43 citation statements)

References 88 publications

(216 reference statements)

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“…MCT has been applied to granular systems driven by Gaussian noises [45][46][47]. It qualitatively predicts a liquid-glass transition, though the noise in granular systems is non-Gaussian in general [48][49][50][51]. MCT also has been applied to sheared dense granular systems [52][53][54].…”

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confidence: 99%

Divergence of Viscosity in Jammed Granular Materials: A Theoretical Approach

Suzuki

Hayakawa

2015

Phys. Rev. Lett.

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A theory for jammed granular materials is developed with the aid of a nonequilibrium steady-state distribution function. The approximate nonequilibrium steady-state distribution function is explicitly given in the weak dissipation regime by means of the relaxation time. The theory quantitatively agrees with the results of the molecular dynamics simulation on the critical behavior of the viscosity below the jamming point without introducing any fitting parameter. DOI: 10.1103/PhysRevLett.115.098001 PACS numbers: 45.70.-n, 05.20.Jj, 64.70.ps, 83.50.Ax Introduction.-A description of granular rheology has been a long-term challenge for both science and technology. The problem extends to a vast range, from solidlike creep motion, gaslike, to liquidlike phenomena [1]. Similar to solid-liquid transitions, granular materials acquire rigidity when the density exceeds a critical value [2][3][4][5]. This phenomenon, referred to as the jamming transition, is universely observed in disordered materials such as colloidal suspensions [6], emulsions, and foams [7], as well as granular materials. The jamming transition and its relation to the glass transition have attracted much interest in the last two decades, and various aspects have been revealed [8][9][10][11][12]. In particular, characteristics in the vicinity of the jamming point, including the critical scaling behavior, have been extensively investigated by experiments and numerical simulations [2][3][4][13][14][15][16][17][18][19][20][21][22][23][24][25]. It has been shown that the shear stress, the pressure, and the granular temperature can be expressed by scaling functions with exponents near φ ∼ φ J , where φ J is the jamming transition density. The shear viscosity exhibits a form η ∼ ðφ J − φÞ −λ with λ ≈ 2 for non-Browninan suspensions, foams, and emulsions [26][27][28][29], and a recent careful analysis demonstrated that λ is in the range between 1.67 and 2.55 [30]. It seems that the exponent λ for granular flows takes a larger value than that for suspensions [18,19,25,31], although a value in the range mentioned above has been reported as well [17]. However, these studies are based on numerical simulations or phenomenologies without any foundation of a microscopic theory.Even when we focus only on the flow properties below the jamming point φ J , which can be tracked back to Bagnold's work [32], we have not yet obtained a complete set in describing the rheological properties of dense granular flows. One of the remarkable achievements is the extension of the Boltzmann-Enskog kinetic theory to inelastic hard disks and spheres [33][34][35][36][37][38]. However, it has been recognized that the kinetic theory breaks down at densities with volume fraction φ > φ f ¼ 0.49 [39][40][41][42],

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confidence: 99%

Divergence of Viscosity in Jammed Granular Materials: A Theoretical Approach

Suzuki

Hayakawa

2015

Phys. Rev. Lett.

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“…Note that this limit would not be possible without external friction, because it is this friction that is responsible for the relaxation of the rotor after a kick. There exist a few studies that describe the angular velocity distribution of the rotor in this limit, both theoretically and numerically [4,12,14], but they are not in agreement with experimental results [10].…”

Section: Experimental Setup and Limiting Behavioursmentioning

confidence: 91%

“…Several studies strived after understanding and modeling the granular motor in these limits, which has lead to an adequate description in the Brownian Limit [11,7,12,9,13]. However, the few theoretical studies that exist in the Single Kick Limit compare well with particle simulation but do not have a good agreement with experimental results [9,10,14]. Moreover, analysing the behaviour of the rotor in between these two limits appears to bea very hard problem to address in general.…”

Section: Introductionmentioning

confidence: 99%

Granular motor in the non-Brownian limit

Gálvez¹,

Meer²

2016

J. Stat. Mech.

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Abstract. In this work we experimentally study a granular rotor which is similar to the famous Smoluchowski-Feynman device and which consists of a rotor with four vanes immersed in a granular gas. Each side of the vanes can be composed of two different materials, creating a rotational asymmetry and turning the rotor into a ratchet. When the granular temperature is high, the rotor is in movement all the time, and its angular velocity distribution is well described by the Brownian Limit discussed in previous works. When the granular temperature is lowered considerably we enter the so-called Single Kick Limit, where collisions occur rarely and the unavoidable external friction causes the rotor to be at rest for most of the time. We find that the existing models are not capable of adequately describing the experimentally observed distribution in this limit. We trace back this discrepancy to the non-constancy of the deceleration due to external friction and show that incorporating this effect into the existing models leads to full agreement with our experiments. Subsequently, we extend this model to describe the angular velocity distribution of the rotor for any temperature of the gas, and obtain a very good agreement between the model and experimental data.

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“…These are situations where the noise probability density function does not decay in the tails like a Gaussian. Also, when the system under consideration is coupled to multiple stochastic environments the driving noise term may exhibit non-Gaussian behavior, see for example references Kanazawa et al (2015a) and Kanazawa et al (2015b). A number of approaches for modeling time-series in a Bayesian nonparametric context have been proposed in the literature.…”

Section: Introductionmentioning

confidence: 99%

Joint reconstruction and prediction\break of random dynamical systems under\break borrowing of strength

Hatjispyros

Merkatas

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We propose a Bayesian nonparametric model based on Markov Chain Monte Carlo (MCMC) methods for the joint reconstruction and prediction of discrete time stochastic dynamical systems, based on m-multiple time-series data, perturbed by additive dynamical noise. We introduce the Pairwise Dependent Geometric Stick-Breaking Reconstruction (PD-GSBR) model, which relies on the construction of a m-variate nonparametric prior over the space of densities supported over R m . We are focusing in the case where at least one of the time-series has a sufficiently large sample size representation for an independent and accurate Geometric Stick-Breaking estimation, as defined in . Our contention, is that whenever the dynamical error processes perturbing the underlying dynamical systems share common characteristics, underrepresented data sets can benefit in terms of model estimation accuracy. The PD-GSBR estimation and prediction procedure is demonstrated specifically in the case of maps with polynomial nonlinearities of an arbitrary degree. Simulations based on synthetic time-series are presented.

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Asymptotic Derivation of Langevin-like Equation with Non-Gaussian Noise and Its Analytical Solution

Cited by 37 publications

References 88 publications

Divergence of Viscosity in Jammed Granular Materials: A Theoretical Approach

Divergence of Viscosity in Jammed Granular Materials: A Theoretical Approach

Granular motor in the non-Brownian limit

Joint reconstruction and prediction\break of random dynamical systems under\break borrowing of strength

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