## Abstract

The totally asymmetric simple exclusion process (TASEP) is a basic model of statistical mechanics that has found numerous applications. We consider the case of TASEP with a finite chain where particles may enter from the left and leave to the right at prescribed rates. This model can be formulated as a Markov process with a finite number of states. Due to the irreducibility of the process, it is well-known that the probability distribution on the states is globally attracted to a unique equilibrium distribution. We extend this result to the more detailed level of individual trajectories. To do so we formulate TASEP as a random dynamical system. Our main result is that the trajectories from all possible initial conditions contract to each other yielding the existence of a random attractor that consists of a single trajectory almost surely. This implies that in the long run TASEP ``filters out"" any perturbation that changes the state of the particles along the chain.

Original language | English |
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Pages (from-to) | 65-93 |

Number of pages | 29 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 20 |

Issue number | 1 |

DOIs | |

State | Published - 2021 |

## Keywords

- Contraction
- Random attractor
- Random dynamical system
- Ribosome flow model
- Synchronization