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Edited by: Wenxiang Xu, Hohai University, China

Reviewed by: Zhigang Zhu, Hohai University, China; Hao Hu, Anhui University, China

This article was submitted to Soft Matter Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Counting how many people or particles pass through a specific space within a specific time is an interesting question in applied physics and social science. Here a logistic model is developed to estimate the total number of moving crowds or flowing particles. This model sheds light on a collective contribution of crowd or particle growth rate and transient probability within a specific space. This model may offer a basic concept to understand transport dynamics of moving crowds and flowing particles.

How many people or particles have passed there? This question is simple but significant in many physical, biological, and social situations [

In this article, the logistic model is developed to understand moving crowds or flowing particles for the total number estimation. This model sheds light on a collective contribution of crowd or particle mobility and growth rate to the total number. This model is applicable for both of static and mobile crowds and particles, probably offering a new framework for understanding transport dynamics of static or mobile crowds and particles.

First, consider a physical situation for flowing particles (conceptually, identical for moving crowds), where a fixed number of flowing particles occupy a limited number of positions in a space, as illustrated in

Illustration of a situation: when particles are flowing in a region of interest (ROI, gray) and their physical factors are given (

Next, to quantify the hydrodynamic aspects of flowing particles [

The transient probability is useful to characterize the nature of static or mobile particles. For instance, let's think about the following two situations. In the first case, most particles may stay to pass through for a while (e.g., for 30 min) during the entire time (e.g., for 2 h), suggesting the transient probability to be

To describe static or mobile particles with the logistic model, the logistic growth dynamics is applied prior to a peak as [^{−1} and

The logistic model:

The logistic model is appropriate to characterize the nature of static or mobile particles. The total number of particles is illustrated in

To demonstrate the validity of the logistic model, a simulation of falling balls through a triangle grid of pegs was tested with help of the Physics Education Technology (PhET) interactive simulations (

Simulation of falling balls through a triangle grid of pegs for the Plinko Probability by the PhET interactive simulations (

The logistic growth or decay dynamics is applicable to describe the number of moving crowds or flowing particles in a region of interest, based on which the total number of particles passing through the region can be estimated with

Counting the total number of particles, both

In conclusion, this study shows a theoretical frame of the logistic growth or decay dynamics that would be appropriate to estimate the total number of moving crowds or flowing particles. As demonstrated here, the model is available for both static and mobile crowds and particles. The numerical demonstration of the logistic model clearly shows how the instantaneous particle number changes with time according to the particle mobility and the growth dynamics. Practically, in physical, social, or ecological situations, the logistic model is applicable by identifying the transient probability and the growth rate to count or estimate the total number.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

BW organized the research, conducted the research, analyzed the data, and wrote the manuscript.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This manuscript has been released as a pre-print at [

The Supplementary Material for this article can be found online at:

This movie shows a simulation with falling balls.