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.


  1. ribosome flow model
  2. contraction
  3. random dynamical system
  4. synchronization
  5. random attractor

MSC codes

  1. 37H99
  2. 34D06
  3. 37A60

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Information & Authors


Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 65 - 93
ISSN (online): 1536-0040


Submitted: 22 January 2020
Accepted: 26 August 2020
Published online: 7 January 2021


  1. ribosome flow model
  2. contraction
  3. random dynamical system
  4. synchronization
  5. random attractor

MSC codes

  1. 37H99
  2. 34D06
  3. 37A60



Funding Information

US-Israel Binational Science Foundation
Israel Science Foundation https://doi.org/10.13039/501100003977

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