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Random attraction in TASEP with time-varying hopping rates

Lars Grüne, Kilian Pioch, Thomas Kriecherbauer, Michael Margaliot

TL;DR

The paper addresses how individual stochastic paths of TASEP on a finite lattice behave under time varying hopping rates. It develops a nonautonomous random dynamical systems (NRDS) framework with a finite state space to model TASEP with time dependent rates and nonhomogeneous Poisson clocks. It proves sufficient conditions for global pullback attractors to exist and for these attractors to be singleton, implying almost sure synchronization of paths and robustness to perturbations; it also provides tight counterexamples showing the conditions are necessary for forward attraction. In addition, it specializes the theory to periodic and bounded rate regimes, shows how autonomous TASEP results are recovered as special cases, and presents numerical simulations that illustrate synchronization and non synchronization under different rate structures. These results deepen our understanding of long time behavior of driven transport processes under nonautonomous randomness and offer a rigorous mechanism for path level entrainment in time varying environments.

Abstract

The totally asymmetric simple exclusion principle (TASEP) is a fundamental model in nonequilibrium statistical mechanics. It describes the stochastic unidirectional movement of particles along a 1D chain of ordered sites. We consider the continuous-time version of TASEP with a finite number of sites and with time-varying hopping rates between the sites. We show how to formulate this model as a nonautonomous random dynamical system (NRDS) with a finite state-space. We provide conditions guaranteeing that random pullback and forward attractors of such an NRDS exist and consist of singletons. In the context of the nonautonomous TASEP these conditions imply almost sure synchronization of the individual random paths. This implies in particular that perturbations that change the state of the particles along the chain are "filtered out" in the long run. We demonstrate that the required conditions are tight by providing examples where these conditions do not hold and consequently the forward attractor does not exist or the pullback attractor is not a singleton. The results in this paper generalize our earlier results for autonomous TASEP in https://doi.org/10.1137/20M131446X and contain these as a special case.

Random attraction in TASEP with time-varying hopping rates

TL;DR

The paper addresses how individual stochastic paths of TASEP on a finite lattice behave under time varying hopping rates. It develops a nonautonomous random dynamical systems (NRDS) framework with a finite state space to model TASEP with time dependent rates and nonhomogeneous Poisson clocks. It proves sufficient conditions for global pullback attractors to exist and for these attractors to be singleton, implying almost sure synchronization of paths and robustness to perturbations; it also provides tight counterexamples showing the conditions are necessary for forward attraction. In addition, it specializes the theory to periodic and bounded rate regimes, shows how autonomous TASEP results are recovered as special cases, and presents numerical simulations that illustrate synchronization and non synchronization under different rate structures. These results deepen our understanding of long time behavior of driven transport processes under nonautonomous randomness and offer a rigorous mechanism for path level entrainment in time varying environments.

Abstract

The totally asymmetric simple exclusion principle (TASEP) is a fundamental model in nonequilibrium statistical mechanics. It describes the stochastic unidirectional movement of particles along a 1D chain of ordered sites. We consider the continuous-time version of TASEP with a finite number of sites and with time-varying hopping rates between the sites. We show how to formulate this model as a nonautonomous random dynamical system (NRDS) with a finite state-space. We provide conditions guaranteeing that random pullback and forward attractors of such an NRDS exist and consist of singletons. In the context of the nonautonomous TASEP these conditions imply almost sure synchronization of the individual random paths. This implies in particular that perturbations that change the state of the particles along the chain are "filtered out" in the long run. We demonstrate that the required conditions are tight by providing examples where these conditions do not hold and consequently the forward attractor does not exist or the pullback attractor is not a singleton. The results in this paper generalize our earlier results for autonomous TASEP in https://doi.org/10.1137/20M131446X and contain these as a special case.

Paper Structure

This paper contains 10 sections, 7 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Nonautonomous random dynamical systems
  3. TASEP with time-varying rates as a nonautonomous random dynamical system
  4. Homogeneous and non-homogeneous Poisson processes
  5. Random attractors for finite state nonautonomous random dynamical systems
  6. Nonautonomous random attractors
  7. Random attractors consisting of single trajectories
  8. Random attraction in the nonautonomous TASEP model
  9. Numerical simulations of synchronization
  10. Conclusion

Key Result

Theorem 6

Consider an NRDS with a finite state space. Then $A(t,\omega)$ from eq:Adef is nonempty for all $\omega\in\Omega$, $t\in\mathbb{R}$, and defines a global random pullback attractor. Moreover, for all $\omega\in\Omega$ and $t\in\mathbb{R}$ there exists $T_0(t,\omega)< t$ such that $\phi(t,t_0,X, \omeg then $A(t,\omega)$ also defines a global random forward attractor and for almost every $\omega\in\O

Figures (2)

  • Figure 1: Six trajectories of TASEP with $n=20$ sites, varying rate $\lambda_{11}(t)$ for three different random jump time sequences (left to right) and two different initial conditions (depicted on top of each other for each $t$). The red light indicates that $\lambda_{11}(t)=10^{-6}$, while the green light indicates that $\lambda_{11}(t)=1$. Synchronization occurs for each run between $t=30\text{sec}$ and $t=40\text{sec}$.
  • Figure 2: Two trajectories of TASEP with $n=2$ sites and with rates varying as in Example \ref{['weirdTASEP']} with modifications as described in Section \ref{['sec:NumSim']}. No synchronization occurs in forward time.

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 6
  • proof
  • Remark 7
  • Lemma 8
  • proof
  • ...and 14 more