Loading conditions for self-organization in the BML model with stochastic direction choice

A dynamical system is considered such that, in this system, particles move on a toroidal lattice of the dimension $N_1\times N_2$ according to a version of the rule of particle movement in Biham–Middleton–Levine traffic model. We introduce a stochastic case with direction choice for particles. Particles of the first type move along rows, and the particles of the second type move along columns. The goal is to find conditions of self-organization system for any lattice dimension. We have proved that the BML model as a dynamical system is a special case of Buslaev nets. This equivalence allows us to use of Buslaev net analysis techniques to investigate the BML model. In Buslaev nets conception, the self-organization property of the system corresponds to the existence of velocity single point spectrum equal to 1. In the paper, we consider the model version when one notable aspect is that a particle may change its type. Exactly, we assume a constant probability $q$ that a particle changes type at each step. In the case where $q=0$, the system corresponds to the classical version of the BML model. We define a state of the system where all particles continue to move indefinitely, in both the present and the future, as a state of free movement. A sufficient condition for the system to result in a state of free movement from any initial state (condition for self-organization) has been found. This condition is that the number of particles be not greater than half the greatest common divisor of the numbers $N_1,N_2,\dots ,N_n$. It has been proved that, if $n=2$, and whether $0<q<1$ or $q=0$ and there are both at least one particle of the first type and at least particle of the second type, then a necessary condition for a state of free movement to exist is the greatest common divisor of $N_1$ and $N_2$ be not less than 3. The theorems are formulated in terms of the algebraic structures and terms of the dynamical system. The spectrum of particle velocities has been found for the net $2\times 2$. This approach allows us to hope that the spectrum of dynamical system can be studied for an arbitrary dimension of the net.