Lattice random walk: an old problem with a future ahead

Giona, Massimiliano

doi:10.1088/1402-4896/aad016

Cited by 9 publications

(30 citation statements)

References 52 publications

(105 reference statements)

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“…In the case D 1 /D 2 = 1, one has D eff /D 1 = 1 from the parabolic model, independently of λ (line (c) in figure 7), while in general D eff /D 1 in parabolic Langevin-Wiener models is lower-and upper-bounded by the values attained at λ = 0 and λ = 1/2 (see the Appendix). This result indicates that the hyperbolic hydrodynamic model not only provides a more quantitatively consistent alternative to parabolic models for describing LRW at short timescales, as addressed in [20], but it is the only continuous model deriving from a continuous stochastic description of particle transport constistent with long-term dispersion data in multiphase periodic lattices.…”

mentioning

confidence: 77%

“…In [20] an hyperbolic hydrodynamic model for the classical asymmetric Lattice Random Walk (LRW) has been derived. Three parameters characterize a LRW: (i) δ the distance between nearest neighboring sites, (ii) τ the hopping time, and (iii) r = r 1 −r 2 the difference between the probabilities of moving to the right (r 1 ) and the left (r 2 ), where r 1 , r 2 > 0, r 1 + r 2 = 1.…”

Section: Lrw and Hyperbolic Hydrodynamic Modelsmentioning

confidence: 99%

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Multiphase partitions of lattice random walks

Giona¹,

Cocco²

2019

EPL

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Considering the dynamics of non-interacting particles randomly moving on a lattice, the occurrence of a discontinuous transition in the values of the lattice parameters (lattice spacing and hopping times) determines the uprisal of two lattice phases. In this Letter we show that the hyperbolic hydrodynamic model obtained by enforcing the boundedness of lattice velocities derived in [20] correctly describes the dynamics of the system and permits to derive easily the boundary condition at the interface, which, contrarily to the common belief, involves the lattice velocities in the two phases and not the phase diffusivities. The dispersion properties of independent particles moving on an infinite lattice composed by the periodic repetition of a multiphase unit cell are investigated. It is shown that the hyperbolic transport theory correctly predicts the effective diffusion coefficient over all the range of parameter values, while the corresponding continuous parabolic models deriving from Langevin equations for particle motion fail. The failure of parabolic transport models is shown via a simple numerical experiment.

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confidence: 77%

Section: Lrw and Hyperbolic Hydrodynamic Modelsmentioning

confidence: 99%

Multiphase partitions of lattice random walks

Giona¹,

Cocco²

2019

EPL

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“…The exiting flux at x = 0 is just J 0 (t) = b P − (0, t) and, consequently, the first passage time density function f θ 1 (θ 1 ) is readily obtained from the solution of eqs. (7) as…”

Section: Poisson-kac Processesmentioning

confidence: 99%

“…Therefore, in a continuous time setting the coordinate LW position process X(t) is no longer a Markov process, because the condition of bounded velocity and a fortiori the local regularity of the trajectories enforces to add the local direction of motion to the state description of the system. Exactly in the same way a lattice random walk is not Markovian if the lattice spacing δ and the hopping time τ are assumed to be finite and the trajectory of a particle is interpolated between two transitions in a continuous way [7].…”

mentioning

confidence: 99%

Age representation of Lévy walks: partial density waves, relaxation and first passage time statistics

Giona¹,

D’Ovidio²,

Cocco³

et al. 2019

J. Phys. A: Math. Theor.

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Lévy walks (LWs) define a fundamental class of finite velocity stochastic processes that can be introduced as a special case of Continuous Time Random Walks. Alternatively, there is a hyperbolic representation of them in terms of partial probability density waves. Using the latter framework we explore the impact of aging on LWs, which can be viewed as a specific initial preparation of the particle ensemble with respect to an age distribution. We show that the hyperbolic age formulation is suitable for a simple integral representation in terms of linear Volterra equations for any initial preparation. On this basis relaxation properties, i.e., the convergence towards equilibrium of a generic thermodynamic function dependent on the spatial particle distribution, and first passage time statistics in bounded domains are studied by connecting the latter problem with solute release kinetics. We find that even normal diffusive LWs, where the long-term mean square displacement increases linearly with time, may display anomalous relaxation properties such as stretched exponential decay. We then discuss the impact of aging on the first passage time statistics of LWs by developing the corresponding Volterra integral representation. As a further natural generalization the concept of LWs with wearing is introduced to account for mobility losses.

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“…(4.6) is, therefore, an artifact of the approach used to obtain the hydrodynamic limit (4.6), while it is completely absent in the original lattice particle model. It is, therefore, natural to ask whether it would be possible to derive continuous hydrodynamic equations without performing the limit for δ, τ → 0, consistently with the bounded value of the lattice velocity b 0 [44]. This effort is also reasonable in the light of elementary physical considerations.…”

Section: "A Good Old Boy": Origin Of Gpk From Lrwmentioning

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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Lattice random walk: an old problem with a future ahead

Cited by 9 publications

References 52 publications

Multiphase partitions of lattice random walks

Multiphase partitions of lattice random walks

Age representation of Lévy walks: partial density waves, relaxation and first passage time statistics

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

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