Quatrième Rencontre de Physique Statistique de Paris

We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of shortrange infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as 1/R d+σ . By means of Wilson's momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an ε-expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection σ = σc > 2.

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“…In order to develop the renormalization group analysis we recast the Langevin equation (9) as a dynamic functional [31][32][33]6]…”

Section: A Renormalization Group Analysismentioning

confidence: 99%

Lévy-flight spreading of epidemic processes leading to percolating clusters

et al. 1999

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“…This is the third of a series of papers where we investigate the noisy Burgers equation in the context of modelling a growing interface; for a brief account of the present work we also we refer to [1]. In the previous two papers, in the following denoted paper I [2] and paper II [3]; a brief account of paper II also appeared in [4], we discussed the originally proposed one dimensional noiseless Burgers equation [5,6] from a solitonic point of view and the noisy one dimensional Burgers equation [7,8] in terms of a Martin-Siggia-Rose path integral [9][10][11][12][13][14], respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Here ∆ constitutes an essential singularity in the same manner as the temperature entering the usual Boltzmann factor. Recasting the stochastic Langevin equation (1.7) in terms of a Martin-Siggia-Rose path integral [9][10][11][12]14,13] we proposed a principle of least action in the nonperturbative weak noise limit ∆ → 0 and derived canonical saddle point or field equations (note that in paper II the noise field ϕ was rotated, ϕ → −iϕ) ∂ ∂t − λu∇ u= ν∇ 2 ϕ , (1.10)…”

Section: Introductionmentioning

confidence: 99%

Canonical phase-space approach to the noisy Burgers equation: Probability distributions

Fogedby

1999

Phys. Rev. E

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We present a canonical phase-space approach to stochastic systems described by Langevin equations driven by white noise. Mapping the associated Fokker-Planck equation to a Hamilton-Jacobi equation in the nonperturbative weak noise limit we invoke a principle of least action for the determination of the probability distributions. We apply the scheme to the noisy Burgers and Kardar-Parisi-Zhang equations and discuss the time-dependent and stationary probability distributions. In one dimension we derive the long-time skew distribution approaching the symmetric stationary Gaussian distribution. In the short-time region we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer-replica and asymmetric exclusion model results. We also comment on the distribution in higher dimensions.

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“…Nevertheless we can use the functional that generates the technique since it is a non-perturbative object. Such generating functional was introduced in [8,9], for the equation (1.1) it has the form Z(λ) ≡ exp i dt dr λu = DuDp exp iI + i dt dr λu , (1.3) where p is an auxiliary field and the effective action is I = dt dr p ∂ t u + L(u)…”

Section: Introductionmentioning

confidence: 99%

Instantons and intermittency

et al. 1996

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We propose the new method for finding the non-Gaussian tails of probability distribution function (PDF) for solutions of a stochastic differential equation, such as convection equation for a passive scalar, random driven Navier-Stokes equation etc. Existence of such tails is generally regarded as a manifestation of intermittency phenomenon. Our formalism is based on the WKB approximation in the functional integral for the conditional probability of large fluctuation. We argue that the main contribution to the functional integral is given by a coupled field-force configuration -instanton. As an example, we examine the correlation functions of the passive scalar u advected by a large-scale velocity field δ-correlated in time. We find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of u. We discuss the simplest instantons for the Navier-Stokes equation.PACS numbers 47.10.+g, 47.27.-i, 05.40.+j

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Quatrième Rencontre de Physique Statistique de Paris

Cited by 42 publications

References 0 publications

Lévy-flight spreading of epidemic processes leading to percolating clusters

Lévy-flight spreading of epidemic processes leading to percolating clusters

Canonical phase-space approach to the noisy Burgers equation: Probability distributions

Instantons and intermittency

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