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pith:HPP72HML

pith:2025:HPP72HMLJ53YWGF5HXTXLIPO63

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Burgers dynamics for Poisson point process initial conditions of the Weibull class

Patrick Valageas

Burgers equation with Poisson point process initial conditions yields exact expressions for velocity distributions, shock statistics and correlation functions.

arxiv:2512.09813 v2 · 2025-12-10 · cond-mat.stat-mech · physics.flu-dyn

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Claims

C1strongest claim

We derive explicit analytical expressions for a broad set of statistical quantities. These include the one- and two-point probability distributions of the velocity, the multiplicity functions of voids and shocks, and the velocity and density correlation functions together with their associated power spectra. We also show that the full hierarchy of n-point distributions factorizes into a sequence of two-point conditional probabilities.

C2weakest assumption

The geometrical construction of solutions in terms of first-contact parabolas gives the exact inviscid solution for initial conditions defined by a Poisson point process whose intensity follows a power law with exponent alpha > -1.

C3one line summary

Burgers dynamics with Weibull-class Poisson point process initial conditions produces self-similar evolution and explicit analytical expressions for velocity distributions, void and shock multiplicities, and correlation functions with stretched-exponential tails whose exponents range from 1 to ∞.

References

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[1] For the first term in Eq.(24), the factorλ(ψ⋆)actually impliesψ ⋆ >0, since this configuration requires a Poisson point at the intersection(q⋆, ψ⋆)

[2] If the two parabolas share the same contact point,q1 =q 2 =q ⋆, then the density vanishes,ρ= 0, and the interval [x1, x2]contains no matter

[3] It is related to the void probabilityPvoid(x)by Pvoid(x) = Z ∞ x dx′nvoid(x′) (x′ −x),whencen void(> x) =− dPvoid dx =−R ′ α(x), n void(x) = d2Pvoid dx2 =R ′′ α(x)

[4] Configurations withψ ⋆ ≤0do not contribute, as in this case the parabolic arcsP 1 andP 2 in the upper half-plane are disjoints. Then, the integral over either symmetric arc yields a vanishing mean vel

[5] J. M. Burgers,The Nonlinear Diffusion Equation(Springer Netherlands, 1974) 1974

Formal links

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Receipt and verification

First computed	2026-05-20T00:03:00.294986Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3bdffd1d8b4f778b18bd3de775a1eef6dc914506a0d9c864d04c10c79a3ad60f

Aliases

arxiv: 2512.09813 · arxiv_version: 2512.09813v2 · doi: 10.48550/arxiv.2512.09813 · pith_short_12: HPP72HMLJ53Y · pith_short_16: HPP72HMLJ53YWGF5 · pith_short_8: HPP72HML

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Canonical record JSON

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    "abstract_canon_sha256": "9a2d502cf05e1544a964ac766027e4086c0b9c5b6dbd10369e6dc1393c8ced15",
    "cross_cats_sorted": [
      "physics.flu-dyn"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cond-mat.stat-mech",
    "submitted_at": "2025-12-10T16:33:02Z",
    "title_canon_sha256": "a2e99ac5d79e4d8f89a60ce6d41f5dc14448205820422676feeb86866ab6d14e"
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