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Entropy 2018, 20(8), 613; https://doi.org/10.3390/e20080613

Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows

1
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
2
ENSTA ParisTech Université Paris-Saclay, 828 Boulevard des Maréchaux, 91120 Palaiseau, France
3
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
*
Author to whom correspondence should be addressed.
Received: 30 July 2018 / Revised: 9 August 2018 / Accepted: 16 August 2018 / Published: 17 August 2018
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Abstract

We report the time-evolution of Probability Density Functions (PDFs) in a toy model of self-organised shear flows, where the formation of shear flows is induced by a finite memory time of a stochastic forcing, manifested by the emergence of a bimodal PDF with the two peaks representing non-zero mean values of a shear flow. Using theoretical analyses of limiting cases, as well as numerical solutions of the full Fokker–Planck equation, we present a thorough parameter study of PDFs for different values of the correlation time and amplitude of stochastic forcing. From time-dependent PDFs, we calculate the information length ( L ), which is the total number of statistically different states that a system passes through in time and utilise it to understand the information geometry associated with the formation of bimodal or unimodal PDFs. We identify the difference between the relaxation and build-up of the shear gradient in view of information change and discuss the total information length ( L = L ( t ) ) which maps out the underlying attractor structures, highlighting a unique property of L which depends on the trajectory/history of a PDF’s evolution. View Full-Text
Keywords: self-organisation; shear flows; coherent structures; turbulence; stochastic processes; Langevin equation; Fokker-Planck equation; information length self-organisation; shear flows; coherent structures; turbulence; stochastic processes; Langevin equation; Fokker-Planck equation; information length
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Jacquet, Q.; Kim, E.-J.; Hollerbach, R. Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows. Entropy 2018, 20, 613.

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