dorsal/arxiv
View SchemaDiffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field
| Authors | A. Asselah, P. Dai Pra, J. L. Lebowitz, Ph. Mounaix |
|---|---|
| Categories | |
| ArXiv ID | physics/0011010 |
| URL | https://arxiv.org/abs/physics/0011010 |
| DOI | 10.1023/A:1010470231689 |
Abstract
We investigate solutions to the equation $\partial_t{\cal E} - {\cal D}\Delta {\cal E} = \lambda S^2{\cal E}$, where $S(x,t)$ is a Gaussian stochastic field with covariance $C(x-x',t,t')$, and $x\in {\mathbb R}^d$. It is shown that the coupling $\lambda_{cN}(t)$ at which the $N$-th moment $<{\cal E}^N(x,t)>$ diverges at time $t$, is always less or equal for ${\cal D}>0$ than for ${\cal D}=0$. Equality holds under some reasonable assumptions on $C$ and, in this case, $\lambda_{cN}(t)=N\lambda_c(t)$ where $\lambda_c(t)$ is the value of $\lambda$ at which $<\exp\lbrack \lambda\int_0^tS^2(0,s)ds\rbrack>$ diverges. The ${\cal D}=0$ case is solved for a class of $S$. The dependence of $\lambda_{cN}(t)$ on $d$ is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, ${\cal D}\to i{\cal D}$, the case of interest for backscattering instabilities in laser-plasma interaction.
{
"annotation_id": "24042d02-bd1b-4926-8211-896369d795d6",
"date_created": "2026-03-02T18:00:32.595000Z",
"date_modified": "2026-03-02T18:00:32.595000Z",
"file_hash": "d0d6b4aaa080d842c873308456431cd507cd56a57009b9fb85f383f23221ac24",
"private": false,
"record": {
"abstract": "We investigate solutions to the equation $\\partial_t{\\cal E} - {\\cal D}\\Delta\n{\\cal E} = \\lambda S^2{\\cal E}$, where $S(x,t)$ is a Gaussian stochastic field\nwith covariance $C(x-x\u0027,t,t\u0027)$, and $x\\in {\\mathbb R}^d$. It is shown that the\ncoupling $\\lambda_{cN}(t)$ at which the $N$-th moment $\u003c{\\cal E}^N(x,t)\u003e$\ndiverges at time $t$, is always less or equal for ${\\cal D}\u003e0$ than for ${\\cal\nD}=0$. Equality holds under some reasonable assumptions on $C$ and, in this\ncase, $\\lambda_{cN}(t)=N\\lambda_c(t)$ where $\\lambda_c(t)$ is the value of\n$\\lambda$ at which $\u003c\\exp\\lbrack \\lambda\\int_0^tS^2(0,s)ds\\rbrack\u003e$ diverges.\nThe ${\\cal D}=0$ case is solved for a class of $S$. The dependence of\n$\\lambda_{cN}(t)$ on $d$ is analyzed. Similar behavior is conjectured when\ndiffusion is replaced by diffraction, ${\\cal D}\\to i{\\cal D}$, the case of\ninterest for backscattering instabilities in laser-plasma interaction.",
"arxiv_id": "physics/0011010",
"authors": [
"A. Asselah",
"P. Dai Pra",
"J. L. Lebowitz",
"Ph. Mounaix"
],
"categories": [
"physics.plasm-ph",
"math-ph",
"math.MP"
],
"doi": "10.1023/A:1010470231689",
"title": "Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field",
"url": "https://arxiv.org/abs/physics/0011010"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "c8726e5b-4213-488a-bb4e-45fa364f2b2c",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}