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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2402.09986 |
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| _version_ | 1866917848170889216 |
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| author | Mounaix, Philippe |
| author_facet | Mounaix, Philippe |
| contents | The validity of the instanton analysis approach is tested numerically in the case of the diffraction-amplification problem $\partial_zψ-\frac{i}{2m}\partial^2_{x^2} ψ=g\vert S\vert^2\, ψ$ for $\ln U\gg 1$, where $U=\vertψ(0,L)\vert^2$. Here, $S(x,z)$ is a complex Gaussian random field, $z$ and $x$ respectively are the axial and transverse coordinates, with $0\le z\le L$, and both $m\ne 0$ and $g>0$ are real parameters. We consider a class of $S$, called the `one-max class', for which we devise a specific biased sampling procedure. As an application, $p(U)$, the probability distribution of $U$, is obtained down to values less than $10^{-2270}$ in the far right tail. We find that the agreement of our numerical results with the instanton analysis predictions in Mounaix (2023 {\it J. Phys. A: Math. Theor.} {\bf 56} 305001) is remarkable. Both the predicted algebraic tail of $p(U)$ and concentration of the realizations of $S$ onto the leading instanton are clearly confirmed, which validates the instanton analysis numerically in the large $\ln U$ limit for $S$ in the one-max class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Testing the Instanton Approach to the Large Amplification Limit of a Diffraction-Amplification Problem Mounaix, Philippe Statistical Mechanics The validity of the instanton analysis approach is tested numerically in the case of the diffraction-amplification problem $\partial_zψ-\frac{i}{2m}\partial^2_{x^2} ψ=g\vert S\vert^2\, ψ$ for $\ln U\gg 1$, where $U=\vertψ(0,L)\vert^2$. Here, $S(x,z)$ is a complex Gaussian random field, $z$ and $x$ respectively are the axial and transverse coordinates, with $0\le z\le L$, and both $m\ne 0$ and $g>0$ are real parameters. We consider a class of $S$, called the `one-max class', for which we devise a specific biased sampling procedure. As an application, $p(U)$, the probability distribution of $U$, is obtained down to values less than $10^{-2270}$ in the far right tail. We find that the agreement of our numerical results with the instanton analysis predictions in Mounaix (2023 {\it J. Phys. A: Math. Theor.} {\bf 56} 305001) is remarkable. Both the predicted algebraic tail of $p(U)$ and concentration of the realizations of $S$ onto the leading instanton are clearly confirmed, which validates the instanton analysis numerically in the large $\ln U$ limit for $S$ in the one-max class. |
| title | Testing the Instanton Approach to the Large Amplification Limit of a Diffraction-Amplification Problem |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2402.09986 |