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Gaussian Convergence for Stochastic Acceleration of Particles in the Dense Spectrum Limit

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Abstract

The velocity of a passive particle in a one-dimensional wave field is shown to converge in law to a Wiener process, in the limit of a dense wave spectrum with independent complex amplitudes, where the random phases distribution is invariant modulo π/2 and the power spectrum expectation is uniform. The proof provides a full probabilistic foundation to the quasilinear approximation in this limit. The result extends to an arbitrary number of particles, founding the use of the ensemble picture for their behaviour in a single realization of the stochastic wave field.

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Notes

  1. Note that this model differs from stochastic acceleration problems in a random potential, for which the field E(x,t) reduces to a static random E(x).

  2. The contrast between this conclusion and Bénisti–Escande’s [4] might be attributed to the asymptotic nature of our result, as s→∞. We do not provide estimates for the “convergence rate” of the empirical distribution to its Fokker-Planck limit.

  3. As pointed out by a referee, this argument reduces to Bessel’s inequality, when one views \(u_{n}^{M}\) as a sum of 2M+1 basis functions, in the Hilbert space (whose elements are stochastic processes u) with scalar product \((u,v) = \mathbb{E}\int_{0}^{2\pi} u^{*}(s) v(s) \, \mathrm {d}s\). Our assumptions on the r.v.’s α m,n ensure orthonormality of our basis.

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Acknowledgements

This work benefited from many discussions with D. Escande and members of équipe turbulence plasma, with E. Pardoux, and with participants to the 107th statistical mechanics conference at Rutgers. Stimulating comments by D. Bénisti and D. Escande, and an explanation by A. Lejay are gratefully acknowledged, as are the careful reading and constructive comments by the anonymous referees.

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  1. Equipe turbulence plasma, case 321, PIIM, UMR 7345 CNRS, Aix-Marseille université, campus Saint-Jérôme, 13397, Marseille cedex 13, France

    Yves Elskens

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Correspondence to Yves Elskens.

Appendix

Appendix

In Ref. [17] we introduced the auxiliary process (X t ,Y t ), describing the relative position and velocity of two particles evolving in the same wave field. This process solves

(33)
(34)

in the state space \(E = \mathbb{T}\times\mathbb{R}\setminus\{(0,0), (\pi,0)\}\), where \(\mathbb{T}= \mathbb{R}/(2\pi\mathbb{Z})\) and B t is the standard Brownian motion in C(ℝ+,ℝ). We proved there in Proposition 5.1 that, for any (x,y)∈E, this process a.s. does not reach the points {(0,0),(π,0)} in finite time. The proof in Ref. [17] does not identify points modulo 2π for their x component; one can streamline it as follows.

Proposition 1

For any (x,y)∈E, \(\inf\{ t>0 : \sin^{2}(X_{t}) + Y_{t}^{2} = 0 \} = + \infty\) a.s., and \(\inf\{ \theta>\nobreak 0 : \limsup_{t \to\theta^{-}} (\sin^{2}(X_{t}) + Y_{t}^{2}) = + \infty\} = + \infty\) a.s.

Proof

Let \(R_{t} = \sin^{2}(X_{t}) + Y_{t}^{2}\) and define Z t =logR t . Denote by τ either of these stopping times, corresponding respectively to Z t →−∞ and Z t →+∞. Then Itô calculus on [0,τ[ yields

(35)
(36)
(37)

Noting that 2|ab|≤a 2+b 2 and that |cosx|≤1 yields the estimates \(Y_{t}^{2} \sin^{2}(X_{t}) \leq R_{t}^{2} / 4\) and 2Y t sin(X t )cos(X t )+sin2(X t )≥−R t , so that on the time interval [0,τ[

$$ Z_t \geq Z_0 - \frac{3t}{2} + \int _0^t \varphi_s \, \mathrm{d}B_s $$
(38)

where |φ s |≤1. This ensures that Z t is bounded from below on any finite time interval since B t is bounded. Hence inf{t>0:R t =0}=+∞ a.s.

Similar upper estimates imply

$$ Z_t \leq Z_0 + 2 t + \int_0^t \varphi_s \, \mathrm{d}B_s $$
(39)

ensuring that Z t is bounded from above on any finite time interval. Hence \(\inf\{ \theta>0 : \limsup_{t \to\theta^{-}} R_{t} = + \infty\} = + \infty\) a.s. □

The second claim of the present statement does not supersede Lemma 5.4 of Ref. [17], which proves that Y t does a.s. not diverge as t→∞. The present statement only proves that (X t ,Y t ) remains in E for all t>0 a.s.

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Elskens, Y. Gaussian Convergence for Stochastic Acceleration of Particles in the Dense Spectrum Limit. J Stat Phys 148, 591–605 (2012). https://doi.org/10.1007/s10955-012-0546-2

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