### 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, aswell 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.

Original language | English |
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Article number | 613 |

Journal | Entropy |

Volume | 20 |

Issue number | 8 |

DOIs | |

Publication status | Published - 17 Aug 2018 |

Externally published | Yes |

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### Keywords

- Coherent structures
- Fokker-Planck equation
- Information length
- Langevin equation
- Self-organisation
- Shear flows
- Stochastic processes
- Turbulence

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Entropy*,

*20*(8), [613]. https://doi.org/10.3390/e20080613