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

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 613 ◽  
Author(s):  
Quentin Jacquet ◽  
Eun-jin Kim ◽  
Rainer Hollerbach
Keyword(s):  
Probability Density ◽  
Shear Flows ◽  
Numerical Solutions ◽  
Planck Equation ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Parameter Study ◽  
Density Functions ◽  
Stochastic Forcing ◽  
Mean Values

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.

Nonholonomic Motion Planning Using Diffusion of Workspace Density Functions

2003 ◽  
Author(s):  
Yu Zhou ◽  
Gregory S. Chirikjian
Keyword(s):  
Planck Equation ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Density Functions ◽  
Value Of Time ◽  
Nonholonomic Mobile Robots ◽  
Planning Algorithm ◽  
The Fourier Transform ◽  
Nonholonomic Motion Planning ◽  
Closed Form Approximation

This paper introduces a trajectory planning algorithm for nonholonomic mobile robots which operate in an environment with obstacles. An important feature of our approach is that the planning domain is the workspace of the mobile robot rather than its configuration space. The basic idea is to imagine the robot being subjected to Brownian motion forcing, and to generate evolving probability density functions (PDF) that describe all attainable positions and orientations of the robot at a given value of time. By planning a path that optimizes the value of this PDF at each instant in time, we generate a feasible trajectory. The PDF of robot pose can be constructed by solving the corresponding Fokker-Planck equation using the Fourier transform for SE(N). A closed-form approximation of the resulting time-dependent PDF is then used to plan a trajectory based on the observation that the evolution of this “workspace density” is a diffusion process. Examples are provided to illustrate the algorithm.

Fatigue Lives of Metals up to Crack Initiation

1977 ◽  
Vol 99 (3) ◽  
pp. 212-214
Author(s):  
Chiaki Ihara ◽  
Akira Tsurui ◽  
Akito Igarashi
Keyword(s):  
Crack Initiation ◽  
Stochastic Model ◽  
Probability Density ◽  
Material Properties ◽  
Probability Density Functions ◽  
Tension Test ◽  
True Stress ◽  
Density Functions ◽  
Stress Strain ◽  
Mean Values

Recently a stochastic model for fatigue of metals has been proposed by two of the present authors. In this paper some parameters adopted in that model are related to the material properties obtained from the static true stress-strain tension test. On this basis, the probability density functions of lives up to crack initiation and their mean values are estimated.

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