# Self-Organization When Pedestrians Move in Opposite Directions. Multi-Lane Circular Track Model
## 2 The Model  
We consider the following scenario: \(2N\) individuals or pedestrians walk around a multi-lane circular track. Half of the individuals move in a clockwise direction and the other half, in a counterclockwise direction. Each time two individuals moving in opposite directions and in the same lane meet, we say they collide. When two individuals collide, exactly one of them, randomly chosen, with each having the same probability of \(1/2\) of being chosen, moves to a neighboring lane. If the collision occurs in the most inner or most outer lane, there is only one neighboring lane. If the collision takes place in another lane, the neighboring lane to which the individual changing lanes moves is also chosen randomly. Next, we enumerate a list of statements. The \(i\)th statement will be referred as statement \(i\) from Section 2. These statements are either rules that help precisely define the dynamics of our system, or observations that are consequences of those rules and will be needed in the analysis in subsequent sections. 1. \(2N\) 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 \(L\). 2. \(N\) individuals move in the counterclockwise direction and the other \(N\) in the clockwise direction. 3. 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. 4. Each individual moving counterclockwise is labeled with an integer \(i\), where \(1\leq i\leq N\). The position of the individual \(i\) moving counterclockwise at time \(t\) is described by an angle \(\theta_{i}^{(+)}(t)\) and by \(\ell_{i}^{(+)}(t)\), the number of the lane the individual is at time \(t\). For simplicity, we require \(\theta_{i}^{(+)}(t)\) to be a continuous function and thus, since the angular speed \(\omega\) is constant, we have \(\theta_{i}^{(+)}(t)=\theta_{i}^{(+)}(0)+\omega t\). For convenience, we assume the initial angles to satisfy \(0\leq\theta_{i}^{(+)}(0)<2\pi\). Note that the function \(\ell_{i}^{(+)}(t)\) will be discontinuous at the times \(t\) when the counterclockwise moving individual \(i\) changes lanes. Strictly speaking, \(\ell_{i}^{(+)}(t)\) is not defined at the times it is discontinuous. This will not cause any problems. 5. Each individual moving clockwise is also labeled with an integer \(j\), with \(1\leq j\leq N\). The position of the individual \(j\) moving clockwise at time \(t\) is described by the angle \(\theta_{j}^{(-)}(t)=\theta_{j}^{(-)}(0)-\omega t\), where \(0\leq\theta_{j}^{(+)}(0)<2\pi\), and by \(\ell_{i}^{(-)}(t)\), the number of the lane the individual is at time \(t\). 6. 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. 7. Given the last statement, two individuals moving in the same direction will never have the same angle. 8. 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 \(\theta_{i}^{(+)}(t)=\theta_{j}^{(-)}(t)+2\pi k\) for some integer \(k\) and \(\lim_{s\to t^{-}}\ell_{i}^{(+)}(s)=\lim_{s\to t^{-}}\ell_{j}^{(-)}(s)\). In the last equation, \(\lim_{s\to t^{-}}\) denotes the right limit as \(s\) tends to \(t\), that is, \(s\) approaches \(t\) but under the restriction \(s<t\).