Équation de Burgers avec conditions initiales à accroissements indépendants et homogènes

The ground state of an elastic interface in a disordered medium undergoes collective jumps upon variation of external parameters. These mesoscopic jumps are called shocks, or static avalanches. Submitting the interface to a parabolic potential centered at w, we study the avalanches which occur as w is varied. We are interested in the correlations between the avalanche sizes S_{1} and S_{2} occurring at positions w_{1} and w_{2}. Using the functional renormalization group (FRG), we show that correlations exist for realistic interface models below their upper critical dimension. Notably, the connected moment 〈S_{1}S_{2}〉^{c} is up to a prefactor exactly the renormalized disorder correlator, itself a function of |w_{2}-w_{1}|. The latter is the universal function at the center of the FRG; hence, correlations between shocks are universal as well. All moments and the full joint probability distribution are computed to first nontrivial order in an ε expansion below the upper critical dimension. To quantify the local nature of the coupling between avalanches, we calculate the correlations of their local jumps. We finally test our predictions against simulations of a particle in random-bond and random-force disorder, with surprisingly good agreement.

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“…(ii) We give in Appendix C an alternative derivation of Eq. (71) that uses the Carraro-Duchon formalism [32,42].…”

Section: Generating Function For All Momentsmentioning

confidence: 99%

Universal correlations between shocks in the ground state of elastic interfaces in disordered media

2016

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“…We will see how one can recover formally the results of [4]. The initial velocity u 0 is here supposed to be a Lévy process (which means that it has independent and stationary increments) of finite variance having no negative jumps.…”

Section: The Case Of Lévy Processesmentioning

confidence: 93%

“…We will closely follow [4] (see also [10]). Let E be the space of càdlàg real functions equipped with the smallest σ-algebra C(E) such that for each x ∈ R, u → u(x) is measurable.…”

Section: Statistical Solutions Of Burgers Equationmentioning

confidence: 99%

“…Carraro and Duchon [4] have checked that if φ 0 is the exponent of a Lévy process of finite variance with negative jumps, (11) has a smooth solution for all time t ≥ 0, which is still the exponent of a homogeneous Lévy process with negative jumps. Hence such Lévy processes are conserved by the Burgers equation.…”

Section: The Case Of Lévy Processesmentioning

confidence: 99%

“…Carraro and Duchon [3,4] showed that Lévy processes are conserved by Burgers equation. They also obtained the explicit evolution equation for the characteristic function of the Lévy process solutions of Burgers.…”

Section: Introductionmentioning

confidence: 99%

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2004

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Abstract. For solutions of (inviscid, forceless, one dimensional) Burgers equation with random initial condition, it is heuristically shown that a stationary Feller-Markov property (with respect to the space variable) at some time is conserved at later times, and an evolution equation is derived for the infinitesimal generator. Previously known explicit solutions such as FrachebourgMartin's (white noise initial velocity) and Carraro-Duchon's Lévy process intrinsic-statistical solutions (including Brownian initial velocity) are recovered as special cases.

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Stochastic Solutions to Hamilton-Jacobi Equations

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Équation de Burgers avec conditions initiales à accroissements indépendants et homogènes

Abstract: Équation de Burgers avec conditions initiales à accroissements indépendants et homogènes

Cited by 24 publications

References 3 publications

Universal correlations between shocks in the ground state of elastic interfaces in disordered media

Universal correlations between shocks in the ground state of elastic interfaces in disordered media

Markovian Solutions of Inviscid Burgers Equation

Stochastic Solutions to Hamilton-Jacobi Equations

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