Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
ENSTA ParisTech Université Paris-Saclay, 828 Boulevard des Maréchaux, 91120 Palaiseau, France
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
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 (
), 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 (
) which maps out the underlying attractor structures, highlighting a unique property of
which depends on the trajectory/history of a PDF’s evolution.
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MDPI and ACS Style
Jacquet, Q.; Kim, E.-J.; Hollerbach, R. Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows. Entropy 2018, 20, 613.
Jacquet Q, Kim E-J, Hollerbach R. Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows. Entropy. 2018; 20(8):613.
Jacquet, Quentin; Kim, Eun-jin; Hollerbach, Rainer. 2018. "Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows." Entropy 20, no. 8: 613.
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