Velocity trapping in the lifted TASEP and the true self-avoiding random walk

Abstract

We discuss non-reversible Markov-chain Monte Carlo algorithms that, for particle systems, rigorously sample the positional Boltzmann distribution and that have faster than physical dynamics. These algorithms all feature a non-thermal velocity distribution. They are exemplified by the lifted TASEP (totally asymmetric simple exclusion process), a one-dimensional lattice reduction of event-chain Monte Carlo. We analyze its dynamics in terms of a velocity trapping that arises from correlations between the local density and the particle velocities. This allows us to formulate a conjecture for its out-of-equilibrium mixing time scale, and to rationalize its equilibrium superdiffusive time scale. Both scales are faster than for the (unlifted) TASEP. They are further justified by our analysis of the lifted TASEP in terms of many-particle realizations of true self-avoiding random walks. We discuss velocity trapping beyond the case of one-dimensional lattice models and in more than one physical dimensions. Possible applications beyond physics are pointed out.

Figures (2)

• Figure 1: Time evolution of the lifted Tasep ($N=10^6, L=2N)$ at $\alpha = \alpha_\text{crit} = \frac{1}{2}$, from an initial "step" (at $t=0$, $\rho(x< L/2)=1$ and $\rho(x> L/2)=0$). (a): Pointer position $x_{\text{p}}$vs time $t$, showing the dome-shaped critical region $\mathcal{C}_{t}$ expanding at the expense of the step. (b): Ensemble-averaged density $\left\langle \rho(x,t) \right\rangle$, for rescaled positions $(x - L/2)/\sqrt{t}$ (the pointer position is excluded from the histogram of densities). The critical region appears clearly, and the ensemble-averaged interfaces have width $\sim t^{1/4}$. (c): Close-up of trajectory, between boundaries $L/2 \pm \sqrt{t}$ of the critical region. A pair of events defining a transfer is highlighted.
• Figure 2: Correspondence of TSAW's with particle models related to the lifted Tasep. (a): Zero-temperature version of the TSAW introduced in Ref. Toth1995, with local times indicated by horizontal lines, initialized as in Ref. TothWerner1998. (b): Particle--hole representation of (a) TothWerner1998. From the doubly occupied site, a particle moves to the right, or a hole to the left. (c): Equivalent representation of (a) in terms of the nearest-neighbor lifted Tasep of eqs. (\ref{['equ:Fleches1']}) to (\ref{['equ:Fleches3']}). (d): Oriented TSAW with directional local times as initialized in Ref. Veto2008. (e): Equivalent representaton of (d) in terms of the nearest-neighbor lifted Tasep, with a "step" initial configuration.