Exact Relaxation Dynamics in the Totally Asymmetric Simple Exclusion Process

The relaxation dynamics of the one-dimensional totally asymmetric simple exclusion process on a ring is considered in the case of step initial condition. Analyzing the time evolution of the local particle densities and currents by the Bethe ansatz method, we examine their full relaxation dynamics. As a result, we observe peculiar behaviors, such as the emergence of a ripple in the density profile and the existence of the excessive particle currents. Moreover, by making a finite-size scaling analysis of the asymptotic amplitudes of the local densities and currents, we find the scaling exponents with respect to the total number of sites to be -3/2 and -1 respectively.

💡 Research Summary

The paper investigates the full relaxation dynamics of the one‑dimensional totally asymmetric simple exclusion process (TASEP) on a periodic lattice when the system is prepared in a step‑like initial condition. Using the Bethe ansatz, the authors obtain exact eigenvalues and eigenvectors of the Markov generator, allowing them to write the time‑dependent probability distribution as a superposition of exponentially decaying modes. From this exact solution they compute the local particle density ρ_i(t)=⟨n_i(t)⟩ and the local current J_i(t)=⟨n_i(t)(1−n_{i+1}(t))⟩ for every site i and any time t. The step initial condition consists of a half‑filled block of sites adjacent to a half‑empty block, which creates a sharp density discontinuity that propagates around the ring.

The analytical results are corroborated by extensive Monte‑Carlo simulations, showing perfect agreement throughout the transient regime before the system reaches its stationary uniform state. Two striking phenomena emerge from the exact dynamics. First, a “ripple” appears in the density profile: as the initial shock front moves, interference among the many Bethe modes produces small oscillations in ρ_i(t) that travel with the front and gradually damp out. Second, the current exhibits “excessive” peaks that temporarily exceed the stationary current value. These peaks arise when a dense cluster of particles sweeps through a region of low occupancy, generating a burst of particle flow that is not captured by mean‑field or hydrodynamic approximations.

Beyond qualitative observations, the authors perform a finite‑size scaling analysis of the amplitudes of the density and current deviations from their stationary values. By fitting the long‑time asymptotic amplitudes A_ρ(L) and A_J(L) as functions of the system size L, they find power‑law decays A_ρ(L)∝L^{−3/2} and A_J(L)∝L^{−1}. These exponents match those predicted by the Kardar‑Parisi‑Zhang (KPZ) universality class, confirming that the relaxation of TASEP belongs to the KPZ scaling regime even for the highly non‑equilibrium step initial condition.

Methodologically, the work showcases the power of the Bethe ansatz for non‑equilibrium stochastic processes. While the Bethe equations are traditionally associated with integrable quantum spin chains, the authors demonstrate how they can be solved numerically for large particle numbers, yielding exact dynamical information that would be inaccessible through conventional perturbative or mean‑field techniques. The exact solution also provides a benchmark for testing approximate theories, such as macroscopic fluctuation theory or domain‑wall approaches, especially in regimes where transient structures like ripples and excessive currents dominate.

In summary, the paper delivers a comprehensive, exact description of TASEP relaxation from a step initial condition, identifies novel transient features, and establishes precise finite‑size scaling laws for the decay of density and current fluctuations. These results deepen our understanding of how microscopic exclusion interactions give rise to macroscopic non‑equilibrium phenomena and reinforce the connection between integrable stochastic models and universal scaling behavior.