# Population Balance Modeling and Opinion Dynamics—A Mutually Beneficial Liaison?

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Population Balance Model

#### 2.1. Model Formulation

#### 2.2. Initial Distribution

#### 2.3. Model Analysis

#### 2.4. Numerical Methods

`ode23t`with an equal relative and absolute tolerance of 1 × 10

^{−6}and the analytically computed Jacobian matrix. In Appendix B.7, it is shown that for at least one case ($d=1$) the steady state distribution is reached in infinite time. This makes it impossible to simulate until the system reaches the steady state. Therefore, the simulations were run for at least 1000 time units and until the norm of the derivative with respect to time was less than 1 × 10

^{−7}. From this almost steady state, the steady state was estimated. Peaks were identified as clusters with the amount of agents at a pivot never less than 1 × 10

^{−6}. The number of agents in these clusters and their mean belief was computed. From this, the variance in the estimated steady steady was computed. This variance is almost identical to the variance computed from the distribution at the end of solving the ODEs. Accordingly, the state at the end of solving the ODEs should be sufficiently close to the steady state.

## 3. Numerical Results

#### 3.1. Uniform Initial Distribution

#### 3.2. Influence of Initial Distribution

## 4. Transfer to Engineering

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DW model | Deffuant-Weisbuch model |

PB | population balance |

PBE | population balance equation |

PBM | population balance model |

## Appendix A. Derivation of Population Balance Equation

**Figure A1.**Region of x and ${x}_{1}$ that results in a valid complement ${x}_{2,c1}$; for illustration purposes, $\mu $ was set to 0.7.

## Appendix B. Moment Analysis

#### Appendix B.1. Definition of Moments

#### Appendix B.2. Transformation to a Square Integration Domain for the Source Term

#### Appendix B.3. Derivation of Ordinary Differential Equation for the Moments

#### Appendix B.4. Constant Number of Agents

#### Appendix B.5. Constant Total Belief

#### Appendix B.6. Ordinary Differential Equation for the Variance

#### Appendix B.7. Exponential Decay of Variance for d = 1

#### Appendix B.8. Variance for an Arbitrary d

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**Figure 1.**Illustration of opinion exchange according to the Deffuant–Weisbuch model; darker shades of blue correspond to higher values on the opinion scale x.

**Figure 2.**Time evolution of number density n of opinions for bounded confidence parameter $d=0.5$; results for convergence parameter $\mu =0.5$ (

**a**) and $\mu =0.1$ (

**b**).

**Figure 3.**Time evolution of number density n of opinions for convergence parameter $\mu =0.5$; results shown for bounded confidence parameter $d=0.2$ (

**a**) and $d=0.1$ (

**b**).

**Figure 4.**Variation of initial distribution; used initial distributions with mean value ${\overline{x}}_{0}=0.5$ and several values of variance ${\sigma}_{0}^{2}$ (

**a**); estimated steady state variance over initial variance for three different values of the bounded confidence parameter d (

**b**).

**Figure 5.**Fraction of agents and mean opinion of clusters over variance for three different values of bounded confidence parameter d; points used for the cluster with mean opinion 0.5, circles for the cluster with the maximal fraction of agents, and x for the cluster with the second largest fraction of agents; the dashed line marks initially uniformly distributed belief. Results shown for fraction of agents within clusters (

**a**) and mean opinion of clusters (

**b**).

**Figure 6.**Illustration of concentration exchange between two droplets according to the newly formulated model; darker shades of blue correspond to higher values of the concentration measure x.

**Figure 7.**Number density distribution at ${\sigma}^{2}\left(t\right)=\frac{{\sigma}_{0}^{2}}{10}$ for three different values of the convergence parameter $\mu $; results shown for total volume fraction of each phase ${\varphi}_{1}={\varphi}_{2}$ (

**a**) and ${\varphi}_{1}=3\xb7{\varphi}_{2}$ (

**b**).

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**MDPI and ACS Style**

Kuhn, M.; Kirse, C.; Briesen, H. Population Balance Modeling and Opinion Dynamics—A Mutually Beneficial Liaison? *Processes* **2018**, *6*, 164.
https://doi.org/10.3390/pr6090164

**AMA Style**

Kuhn M, Kirse C, Briesen H. Population Balance Modeling and Opinion Dynamics—A Mutually Beneficial Liaison? *Processes*. 2018; 6(9):164.
https://doi.org/10.3390/pr6090164

**Chicago/Turabian Style**

Kuhn, Michael, Christoph Kirse, and Heiko Briesen. 2018. "Population Balance Modeling and Opinion Dynamics—A Mutually Beneficial Liaison?" *Processes* 6, no. 9: 164.
https://doi.org/10.3390/pr6090164