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Kolesnik, Alexander D.

doi:10.1023/a:1011167815206

Cited by 13 publications

(6 citation statements)

References 8 publications

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“…This property follows also from the observation that the Green function for the Cattaneo hyperbolic transport model in R n , n ≥ 2 does not present positivity and attains negative values [17] (which is deprecable in a probabilistic context). The definition of GPK processes is closely connected with the class of higher-dimensional stochastic models studied by Kolesnik [18,19,20] A GPK process in R n is defined by a finite number N of stochastic states, by a family of N constant velocity vectors {b h } N h=1 , b h ∈ R n , by a vector of transition rates Λ = (λ 1 , . .…”

Section: Generalized Poisson-kac Processesmentioning

confidence: 99%

Generalized Poisson-Kac processes and hydrodynamic modeling of systems of interacting particles I - Theory

Giona

2018

Preprint

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This article analyzes the formulation of space-time continuous hyperbolic hydrodynamic models for systems of interacting particles moving on a lattice, by connecting their local stochastic lattice dynamics to the formulation of an associated (space-time continuous) Generalized Poisson-Kac process possessing the same local transition rules. The hyperbolic hydrodynamic limit follows naturally from the statistical description of the latter in terms of the system of its partial probability density functions. Several cases are treated, with particular attention to: (i) models of interacting particles satisfying an exclusion principle, and (ii) models defined by a given interparticle interaction potential. In both cases, the hydrodynamic models may display singularities, dynamic phase-transitions and bifurcations (as regards the flux/concentration-gradient constitutive equations), whenever the Kac limit of the model (infinite propagation velocity limit) is considered.

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Section: Generalized Poisson-kac Processesmentioning

confidence: 99%

Generalized Poisson-Kac processes and hydrodynamic modeling of systems of interacting particles I - Theory

Giona

2018

Preprint

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“…As a two-dimensional model of a Poisson-Kac stochastic process we consider that proposed by Kolesnik and Turbin [27] and Kolesnik [28], and therefore referred to as the Kolesnik-Kac model. It is a particular case of the class of Generalized Poisson-Kac processes introduced in [29].…”

Section: Kolesnik-kac Stochastic Processmentioning

confidence: 99%

Relativistic analysis of stochastic kinematics

Giona

2017

Phys. Rev. E

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The relativistic analysis of stochastic kinematics is developed in order to determine the transformation of the effective diffusivity tensor in inertial frames. Poisson-Kac stochastic processes are initially considered. For one-dimensional spatial models, the effective diffusion coefficient measured in a frame Σ moving with velocity w with respect to the rest frame of the stochastic process is inversely proportional to the third power of the Lorentz factor γ(w)=(1-w^{2}/c^{2})^{-1/2}. Subsequently, higher-dimensional processes are analyzed and it is shown that the diffusivity tensor in a moving frame becomes nonisotropic: The diffusivities parallel and orthogonal to the velocity of the moving frame scale differently with respect to γ(w). The analysis of discrete space-time diffusion processes permits one to obtain a general transformation theory of the tensor diffusivity, confirmed by several different simulation experiments. Several implications of the theory are also addressed and discussed.

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“…where χ 1 (t, λ) and χ 2 (t, λ) are two independent Poisson processes characterized by the same transition rate λ, the resulting overall probability density function does not satisfy a Cattaneo equation [41]. This result is even more evident from the mathematical elaborations by Kolesnik [23,24] for relatively simple processes in the plane.…”

Section: Introductionmentioning

confidence: 98%

“…In point of fact, positivity problems may arise also in the one-dimensional case, whenever bounded domains (intervals) are considered, depending on the way boundary conditions are set [22]. The statistical properties of stochastic processes possessing finite propagation velocity have been mathematically approached by Kolesnik in a brilliant way in a series of articles [23][24][25][26]. Focusing on the evolution equations for the overall probability density function, Kolesnik showed that these equations are governed by extremely complex (hyperparabolic) operators.…”

Section: Introductionmentioning

confidence: 99%

Generalized Poisson--Kac Processes and the Regularity of Laws of Nature

Giona¹

2018

Acta Phys. Pol. B

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The meaning and the features of Generalized Poisson-Kac processes are analyzed in the light of their regularity properties in order to show how the finite propagation velocity, characterizing these models, permits to eliminate the occurrence of singularities in transport models. Apart from a brief overview on their spectral properties, on the regularization of boundary-value problems, and on their origin from simple Lattice Random Walk models, the article focuses on their application in the study of stochastic partial differential equations, and how their use permits to eliminate the divergence of low-order moments that characterizes the corresponding field equations in the presence of spatially δ-correlated stochastic perturbations, and to ensure positivity whenever needed. A simple reaction-diffusion system subjected to a stochastically intermitted flux and the Edwards-Wilkinson model are used to show these properties.

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Untitled

Cited by 13 publications

References 8 publications

Generalized Poisson-Kac processes and hydrodynamic modeling of systems of interacting particles I - Theory

Generalized Poisson-Kac processes and hydrodynamic modeling of systems of interacting particles I - Theory

Relativistic analysis of stochastic kinematics

Generalized Poisson--Kac Processes and the Regularity of Laws of Nature

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