# Self-Organization When Pedestrians Move in Opposite Directions. Multi-Lane Circular Track Model

## Abstract

**:**

## 1. Introduction

## 2. The Model

- $2N$ individuals move around a circular track with L lanes. Each lane is labeled with a number. The inner lane is lane 1, the lane next to lane 1 is lane 2, and so on. Note that the outer lane is lane L.
- N individuals move in the counterclockwise direction and the other N in the clockwise direction.
- All individuals move with the same constant angular speed $\omega $ and thus, it takes each individual a time of $2\pi /\omega $ to complete a loop.
- Each individual moving counterclockwise is labeled with an integer i, where $1\le i\le N$. The position of the individual i moving counterclockwise at time t is described by an angle ${\theta}_{i}^{(+)}\left(t\right)$ and by ${\ell}_{i}^{(+)}\left(t\right)$, the number of the lane the individual is at time t. For simplicity, we require ${\theta}_{i}^{(+)}\left(t\right)$ to be a continuous function and thus, since the angular speed $\omega $ is constant, we have ${\theta}_{i}^{(+)}\left(t\right)={\theta}_{i}^{(+)}\left(0\right)+\omega t$. For convenience, we assume the initial angles to satisfy $0\le {\theta}_{i}^{(+)}\left(0\right)<2\pi $. Note that the function ${\ell}_{i}^{(+)}\left(t\right)$ will be discontinuous at the times t when the counterclockwise moving individual i changes lanes. Strictly speaking, ${\ell}_{i}^{(+)}\left(t\right)$ is not defined at the times it is discontinuous. This will not cause any problems.
- Each individual moving clockwise is also labeled with an integer j, with $1\le j\le N$. The position of the individual j moving clockwise at time t is described by the angle ${\theta}_{j}^{(-)}\left(t\right)={\theta}_{j}^{(-)}\left(0\right)-\omega t$, where $0\le {\theta}_{j}^{(+)}\left(0\right)<2\pi $, and by ${\ell}_{i}^{(-)}\left(t\right)$, the number of the lane the individual is at time t.
- We assume that initially, i.e., at time $t=0$, all the $2N$ angles defined above are different. This means not only that individuals start at different positions, but also that, at $t=0$, any two individuals do not have positions that correspond to the same angle. If an individual at $t=0$ looks to its sides, it will not see any other individual in the other lanes with the same initial angle.
- Given the last statement, two individuals moving in the same direction will never have the same angle.
- Two individuals collide at time t if they reach the same location at that time. Given the last statement, if two individuals collide, they move in opposite directions. In mathematical terms, the individual i moving counterclockwise and the individual j moving clockwise collide at time t if ${\theta}_{i}^{(+)}\left(t\right)={\theta}_{j}^{(-)}\left(t\right)+2\pi k$ for some integer k and ${\mathrm{lim}}_{s\to {t}^{-}}{\ell}_{i}^{(+)}\left(s\right)={\mathrm{lim}}_{s\to {t}^{-}}{\ell}_{j}^{(-)}\left(s\right)$. In the last equation, ${\mathrm{lim}}_{s\to {t}^{-}}$ denotes the right limit as s tends to t, that is, s approaches t but under the restriction $s<t$. This limit is taken because one of the individuals will change lanes at time t and thus, either ${\theta}_{i}^{(+)}$ or ${\theta}_{j}^{(-)}$ will not be defined at exactly that time. Note also that the angles of the colliding individuals do not have to be equal. It is enough that they differ by an integer multiple of $2\pi $, which includes the case of them being equal. Adding or subtracting $2\pi $ to an angle does not change the position.
- For all pairs i, j such that $1\le i,j\le N$, we define$$\begin{array}{c}\hfill {\tau}_{ij}=\left\{\begin{array}{cc}\frac{{\theta}_{j}^{(-)}\left(0\right)-{\theta}_{i}^{(+)}\left(0\right)}{2\omega}& \mathrm{if}{\theta}_{j}^{(-)}\left(0\right){\theta}_{i}^{(+)}\left(0\right)\\ \frac{{\theta}_{j}^{(-)}\left(0\right)-{\theta}_{i}^{(+)}\left(0\right)}{2\omega}+\frac{\pi}{\omega}& \mathrm{if}{\theta}_{j}^{(-)}\left(0\right){\theta}_{i}^{(+)}\left(0\right).\end{array}\right.\end{array}$$
- If two individuals, the individual i moving counterclockwise and the individual j moving clockwise, collide at time t, exactly one of them changes lanes at that time. The probability that the individual i changes lanes is $1/2$ and thus, $1/2$ is also the probability that j changes lanes. Assume the collision occurs in lane ℓ. If $1<\ell <L$, the individual changing lanes moves to lane $\ell -1$ with probability $1/2$ or to lane $\ell +1$, also with probability $1/2$. If $\ell =1$, the individual changing lanes moves to lane 2, and if $\ell =L$, it moves to lane $L-1$.
- An individual can only change lanes when it collides with an other individual.

## 3. Numerical Simulations

## 4. Possible Collision Times

## 5. Evolution Equations

- If ${u}_{I\left(n\right)}\ne {u}_{N+J\left(n\right)}$, there is no collision at time ${t}_{n}$. Thus,$$\begin{array}{c}\hfill P\left({\ell}^{\left(n\right)}=\mathbf{v}\left|{\ell}^{(n-1)}=\mathbf{u}\right.\right)=\left\{\begin{array}{cc}1& \mathrm{if}\mathbf{v}=\mathbf{u}\\ 0& \mathrm{otherwise}.\end{array}\right.\end{array}$$
- If ${u}_{I\left(n\right)}={u}_{N+J\left(n\right)}=1$, at time ${t}_{n}$, the individual $I\left(n\right)$ moving counterclockwise and the individual $J\left(n\right)$ moving clockwise collide in lane 1. Thus,$$\begin{array}{c}\hfill P\left({\ell}^{\left(n\right)}=\mathbf{v}\left|{\ell}^{(n-1)}=\mathbf{u}\right.\right)=\left\{\begin{array}{cc}1/2& \mathrm{if}{v}_{I\left(n\right)}=2\mathrm{and}{v}_{k}={u}_{k}\mathrm{for}\mathrm{all}k\ne I\left(n\right)\\ 1/2& \mathrm{if}{v}_{N+J\left(n\right)}=2\mathrm{and}{v}_{k}={u}_{k}\mathrm{for}\mathrm{all}k\ne N+J\left(n\right)\\ 0& \mathrm{otherwise}.\end{array}\right.\end{array}$$
- If ${u}_{I\left(n\right)}={u}_{N+J\left(n\right)}=L$, at time ${t}_{n}$, the individual $I\left(n\right)$ moving counterclockwise and the individual $J\left(n\right)$ moving clockwise collide in lane L. Thus,$$\begin{array}{c}\hfill P\left({\ell}^{\left(n\right)}=\mathbf{v}\left|{\ell}^{(n-1)}=\mathbf{u}\right.\right)=\left\{\begin{array}{cc}1/2& \mathrm{if}{v}_{I\left(n\right)}=L-1\mathrm{and}{v}_{k}={u}_{k}\mathrm{for}\mathrm{all}k\ne I\left(n\right)\\ 1/2& \mathrm{if}{v}_{N+J\left(n\right)}=L-1\mathrm{and}{v}_{k}={u}_{k}\mathrm{for}\mathrm{all}k\ne N+J\left(n\right)\\ 0& \mathrm{otherwise}.\end{array}\right.\end{array}$$
- If ${u}_{I\left(n\right)}={u}_{N+J\left(n\right)}=\ell $, with $1<\ell <L$, at time ${t}_{n}$, the individual $I\left(n\right)$ moving counterclockwise and the individual $J\left(n\right)$ moving clockwise collide in a lane other than lanes 1 and L. Thus,$$\begin{array}{c}\hfill P\left({\ell}^{\left(n\right)}=\mathbf{v}\left|{\ell}^{(n-1)}=\mathbf{u}\right.\right)=\left\{\begin{array}{cc}1/4& \mathrm{if}{v}_{I\left(n\right)}=\ell -1\mathrm{and}{v}_{k}={u}_{k}\mathrm{for}\mathrm{all}k\ne I\left(n\right)\\ 1/4& \mathrm{if}{v}_{I\left(n\right)}=\ell +1\mathrm{and}{v}_{k}={u}_{k}\mathrm{for}\mathrm{all}k\ne I\left(n\right)\\ 1/4& \mathrm{if}{v}_{N+J\left(n\right)}=\ell -1\mathrm{and}{v}_{k}={u}_{k}\mathrm{for}\mathrm{all}k\ne N+J\left(n\right)\\ 1/4& \mathrm{if}{v}_{N+J\left(n\right)}=\ell +1\mathrm{and}{v}_{k}={u}_{k}\mathrm{for}\mathrm{all}k\ne N+J\left(n\right)\\ 0& \mathrm{otherwise}.\end{array}\right.\end{array}$$

## 6. Self-Organized Configurations

## 7. Probabilities to Reach Self-Organization

## 8. Self-Organization Occurs with Probability 1

**Observation**

**1.**

**Observation**

**2.**

**Observation**

**3.**

- If ${u}_{I\left(n\right)}\ne {u}_{N+J\left(n\right)}$ then $\mathbf{v}=\mathbf{u}$.
- If ${u}_{I\left(n\right)}={u}_{N+J\left(n\right)}=1$ then ${v}_{N+J\left(n\right)}=2$ and ${v}_{i}={u}_{i}$ for all $i\ne N+J\left(n\right)$.
- If ${u}_{I\left(n\right)}={u}_{N+J\left(n\right)}\ne 1$ then ${v}_{I\left(n\right)}={u}_{I\left(n\right)}-1$ and ${v}_{i}={u}_{i}$ for all $i\ne I\left(n\right)$.

**Observation**

**4.**

**Observation**

**5.**

**Observation**

**6.**

**Observation**

**7.**

**Observation**

**8.**

**Theorem**

**1.**

## 9. Discussion

## Funding

## Conflicts of Interest

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**Figure 1.**Positions of the pedestrians at different times. The circles are the pedestrians moving clockwise. The exes are the pedestrians moving counterclockwise.

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**MDPI and ACS Style**

Goldsztein, G.H.
Self-Organization When Pedestrians Move in Opposite Directions. Multi-Lane Circular Track Model. *Appl. Sci.* **2020**, *10*, 563.
https://doi.org/10.3390/app10020563

**AMA Style**

Goldsztein GH.
Self-Organization When Pedestrians Move in Opposite Directions. Multi-Lane Circular Track Model. *Applied Sciences*. 2020; 10(2):563.
https://doi.org/10.3390/app10020563

**Chicago/Turabian Style**

Goldsztein, Guillermo H.
2020. "Self-Organization When Pedestrians Move in Opposite Directions. Multi-Lane Circular Track Model" *Applied Sciences* 10, no. 2: 563.
https://doi.org/10.3390/app10020563