2017/11/08 by Marc Pradas, Alain Pumir, Michael Wilkinson · 1 citation
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Diffusion and Search Dynamics #Distribution (mathematics) #Independent and identically distributed random variables #Intensity (physics) #Phase (matter) #Phase transition #Point (geometry) #Random lasers and scattering media #Transition point #physics.optics #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/1751-8121/aab161
14 pages, 3 figures
arxiv created 2017/11/08 · openalex created_date 2017/11/17 · openalex publication_date 2018/02/22 · arxiv updated 2018/04/04 · openalex updated_date 2026/08/05
Abstract This paper analyses a model for the intensity of distribution for rays propagating without absorption in a random medium. The random medium is modelled as a dynamical map. After N iterations, the intensity is modelled as a sum S of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">N</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:mstyle> </mml:math> contributions from different trajectories, each of which is a product of N independent identically distributed random variables x k , representing successive focussing or de-focussing events. The number of ray trajectories reaching a given point is assumed to proliferate exponentially: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">N</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:mstyle> </mml:math> , for some <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mstyle> </mml:math> . We investigate the probability distribution of S . We find a phase transition as parameters of the model are varied. There is a phase where the fluctuations of S are suppressed as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>N</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mstyle> </mml:math> , and a phase where the S has large fluctuations, for which we provide a large deviation analysis.