Probability theory of active suspensions

Abstract

A new approach to studying active suspensions is presented. They exhibit a specific behavior pattern, sometimes referred to as active turbulence. Starting from first principles, we establish a description for an active suspension, consisting of a Newtonian fluid and active Janus particles. The fluid phase is described by Navier–Stokes equations and the particles by Newton–Euler equations. A level set approach is used to separate the two phases, well-known from the representation of sharp interfaces in various numerical schemes. By introducing the multi-point probability density function (PDF)-approach known from hydrodynamic turbulence, we obtain a hierarchical ordered infinite set of linear statistical equations. However, the equations for the K-point PDF depend on the K + 1 and K + 2-point PDF, exposing the closure problem of active turbulence. As all statistical moments can be formed from the PDF, the latter set of equations already includes every statistical model for an active suspensions. To illustrate this, we derive the Eulerian spatial averaging theory from the hierarchy of multi-point PDF-equations.

Incorporating Multidimensional Data Analysis Methods for Probability Theory and Mathematical Statistics Teaching Reform and Practical Exploration

“…Estimating the number of neurons in the output layer. Set the output layer neurons as 1, take the number of output layer neurons m=1, the evaluation value is [0, 10], set the evaluation set as: excellent, good, qualified, unqualified four grades, and their evaluation values are: [9,10], (8,9], (7,8], (7,1]. The number of network input nodes is 25, i.e., the quantitative scores of 25 secondary indicators of classroom teaching quality.…”

“…Probability theory and mathematical statistics is a course that studies random phenomena and their laws, and it is a general mathematics course for undergraduates majoring in economics, management and engineering, which mainly introduces the basic concepts of stochastic mathematics and its theories as well as how to apply the relevant theories to solve practical problems [1][2][3]. Probability statistics is widely used in economics, management, information science and technology, aerospace and biomedicine, etc., and it is constantly integrating and penetrating with other disciplines, and it is making its presence felt in emerging disciplines and cutting-edge fields such as big data, artificial intelligence, machine learning, etc.…”