# Self-Organization When Pedestrians Move in Opposite Directions. Multi-Lane Circular Track Model
## 5 Evolution Equations  
Given any positive integer \(n\), the time \(t_{n}\) is a possible collision time between two individuals. We denote by \(I(n)\), the integer corresponding to the individual moving counterclockwise and by \(J(n)\), the integer corresponding to the individual moving clockwise involved in the possible collision at time \(t_{n}\). In other words, the functions \(I=I(n)\) and \(J=J(n)\) are defined by the following statement: \(t_{n}\) is a possible collision time between the individual \(i=I(n)\) moving counterclockwise and the individual \(j=J(n)\) moving clockwise. Following the discussion of Section 4, it can be easily shown that \(I(n+N^{2})=I(n)\) and \(J(n+N^{2})=J(n)\) for all n.  
A simple but important observation is that each individual remains in the same lane in between consecutive possible collisions times. i.e., \(\ell_{i}^{(+)}\) and \(\ell_{j}^{(-)}\) remain constant in time intervals of the form \(t_{n}<t<t_{n+1}\) for all \(n\), \(i\) and \(j\). Thus, for any \(n\geq 0\) and \(1\leq i,j\leq N\), we define  
\[\ell_{i}^{(+,n)}=\ell_{i}^{(+)}(t)\text{ and }\ell_{j}^{(-,n)}=\ell_{j}^{(-)}(t) \text{ where }t_{n}<t<t_{n+1}. \tag{5}\]  
For each integer \(n\), we regard \(\ell_{i}^{(+,n)}\) as the \(i\)th component of a \(2N\)-vector that we call \(\ell^{(n)}\) and we regard \(\ell_{j}^{(-,n)}\) as the \((N+j)\)th component of the same vector,  
\[\ell^{(n)}=\left(\ell_{1}^{(+,n)},\ldots,\ell_{N}^{(+,n)},\ell_{1}^{(-,n)}, \ldots,\ell_{N}^{(-,n)}\right). \tag{6}\]We also denote the \(k\)th component of \(\ell^{(n)}\) by \(\ell_{k}^{(n)}\). Thus, we have \(\ell_{k}^{(n)}=\ell_{k}^{(+,n)}\) if \(1\leq k\leq N\), and \(\ell_{k}^{(n)}=\ell_{k-N}^{(-n)}\) if \(N<k\leq 2N\). We define \(\Omega\) to be the set of \(2N\)-vectors, whose components are positive integers no greater than \(L\). \[\Omega=\left\{\mathbf{v}=\left(v_{1},v_{2},\ldots,v_{2N}\right)\text{ such that }v_{i}\text{ is an integer and }1\leq v_{i}\leq L\text{ for all }i\right\}. \tag{7}\]  
Note that \(\ell^{(n)}\) is in \(\Omega\) for all \(n\). Let \(\mathbf{u}\) and \(\mathbf{v}\) be two elements in \(\Omega\). Next, we describe the probability that \(\ell^{(n)}=\mathbf{v}\) given that \(\ell^{(n-1)}=\mathbf{u}\). The standard notation for this probability, known as conditional probability, is \(P\left(\ell^{(n)}=\mathbf{v}\left|\ell^{(n-1)}=\mathbf{u}\right.\right)\). Before we proceed, note that \(\ell^{(n-1)}=\mathbf{u}\) implies that \(\ell_{I(n)}^{(+,n-1)}=\ell_{I(n)}^{(n-1)}=u_{I(n)}\) and \(\ell_{I(n)}^{(-,n-1)}=\ell_{N+I(n)}^{(n-1)}=u_{N+I(n)}\). 1. If \(u_{I(n)}\
eq u_{N+I(n)}\), there is no collision at time \(t_{n}\). Thus, \[P\left(\ell^{(n)}=\mathbf{v}\left|\ell^{(n-1)}=\mathbf{u}\right.\right)= \left\{ \begin{array}{ll}1&\text{ if }\mathbf{v}=\mathbf{u}\\ 0&\text{otherwise.}\end{array} \right.\] (8)
2. If \(u_{I(n)}=u_{N+I(n)}=1\), at time \(t_{n}\), the individual \(I(n)\) moving counterclockwise and the individual \(J(n)\) moving clockwise collide in lane 1. Thus, \[P\left(\ell^{(n)}=\mathbf{v}\left|\ell^{(n-1)}=\mathbf{u}\right.\right)= \left\{ \begin{array}{ll}1/2&\text{ if }v_{I(n)}=2\text{ and }v_{k}=u_{k}\text{ for all }k\
eq I(n)\\ 1/2&\text{ if }v_{N+I(n)}=2\text{ and }v_{k}=u_{k}\text{ for all }k\
eq N+J(n)\\ 0&\text{ otherwise.}\end{array} \right.\] (9) The first condition in the above equation corresponds to the individual \(I(n)\) moving counterclockwise, changing from lane 1 to lane 2. The second condition corresponds to the individual \(J(n)\) moving clockwise, changing from lane 1 to lane 2. 3. If \(u_{I(n)}=u_{N+I(n)}=L\), at time \(t_{n}\), the individual \(I(n)\) moving counterclockwise and the individual \(J(n)\) moving clockwise collide in lane \(L\). Thus, \[P\left(\ell^{(n)}=\mathbf{v}\left|\ell^{(n-1)}=\mathbf{u}\right.\right)= \left\{ \begin{array}{ll}1/2&\text{ if }v_{I(n)}=L-1\text{ and }v_{k}=u_{k}\text{ for all }k\
eq I(n)\\ 1/2&\text{ if }v_{N+I(n)}=L-1\text{ and }v_{k}=u_{k}\text{ for all }k\
eq N+J(n)\\ 0&\text{ otherwise.}\end{array}