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Register a new userExact domain wall theory for deterministic TASEP with parallel update
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Domain wall theory (DWT) has proved to be a powerful tool for the analysis of one-dimensional transport processes. A simple version of it was found very accurate for the Totally Asymmetric Simple Exclusion Process (TASEP) with random sequential update. However, a general implementation of DWT is still missing in the case of updates with less fluctuations, which are often more relevant for applications. Here we develop an exact DWT for TASEP with parallel update and deterministic (p=1) bulk motion. Remarkably, the dynamics of this system can be described by the motion of a domain wall not only on the coarse-grained level but also exactly on the microscopic scale for arbitrary system size. All properties of this TASEP, time-dependent and stationary, are shown to follow from the solution of a bivariate master equation whose variables are not only the position but also the velocity of the domain wall. In the continuum limit this exactly soluble model then allows us to perform a first principle derivation of a Fokker-Planck equation for the position of the wall. The diffusion constant appearing in this equation differs from the one obtained with the traditional `simple DWT.
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Motivated by interest in pedestrian traffic we study two lanes (one-dimensional lattices) of length $L$ that intersect at a single site. Each lane is modeled by a TASEP (Totally Asymmetric Exclusion Process). The particles enter and leave lane $sigma$ (where $sigma=1,2$) with probabilities $alpha_sigma$ and $beta_sigma$, respectively. We employ the `frozen shuffle update introduced in earlier work [C. Appert-Rolland et al, J. Stat. Mech. (2011) P07009], in which the particle positions are updated in a fixed random order. We find analytically that each lane may be in a `free flow or in a `jammed state. Hence the phase diagram in the domain $0leqalpha_1,alpha_2leq 1$ consists of four regions with boundaries depending on $beta_1$ and $beta_2$. The regions meet in a single point on the diagonal of the domain. Our analytical predictions for the phase boundaries as well as for the currents and densities in each phase are confirmed by Monte Carlo simulations.
We study the dynamical evolution toward steady state of the stochastic non-equilibrium model known as totally asymmetric simple exclusion process, in both uniform and non-uniform (staggered) one-dimensional systems with open boundaries. Domain-wall theory and numerical simulations are used and, where pertinent, their results are compared to existing mean-field predictions and exact solutions where available. For uniform chains we find that the inclusion of fluctuations inherent to the domain-wall formulation plays a crucial role in providing good agreement with simulations, which is severely lacking in the corresponding mean-field predictions. For alternating-bond chains the domain-wall predictions for the features of the phase diagram in the parameter space of injection and ejection rates turn out to be realized only in an incipient and quantitatively approximate way. Nevertheless, significant quantitative agreement can be found between several additional domain-wall theory predictions and numerics.
We study the flux of totally asymmetric simple exclusion processes (TASEPs) on a twin co-axial square tracks. In this biologically motivated model the particles in each track act as mobile bottlenecks against the movement of the particles in the other although the particle are not allowed to move out of their respective tracks. So far as the outer track is concerned, the particles on the inner track act as bottlenecks only over a set of fixed segments of the outer track, in contrast to site-associated and particle-associated quenched randomness in the earlier models of disordered TASEP. In a special limiting situation the movement of particles in the outer track mimic a TASEP with a point-like immobile (i.e., quenched) defect where phase segregation of the particles is known to take place. The length of the inner track as well as the strength and number density of the mobile bottlenecks moving on it are the control parameters that determine the nature of spatio-temporal organization of particles on the outer track. Variation of these control parameters allow variation of the width of the phase-coexistence region on the flux-density plane of the outer track. Some of these phenomena are likely to survive even in the future extensions intended for studying traffic-like collective phenomena of polymerase motors on double-stranded DNA.
Exact analyses are given for two three-dimensional lattice systems: A system of close-packed dimers placed in layers of honeycomb lattices and a layered triangular-lattice interacting domain wall model, both with nontrivial interlayer interactions. We show that both models are equivalent to a 5-vertex model on the square lattice with interlayer vertex-vertex interactions. Using the method of Bethe ansatz, a closed-form expression for the free energy is obtained and analyzed. We deduce the exact phase diagram and determine the nature of the phase transitions as a function of the strength of the interlayer interaction.
We investigate the non-equilibrium dynamics of a one-dimensional spin-1/2 XXZ model at zero-temperature in the regime $|Delta|< 1$, initially prepared in a product state with two domain walls i.e, $|downarrowdotsdownarrowuparrowdotsuparrowdownarrowdotsdownarrowrangle$. At early times, the two domain walls evolve independently and only after a calculable time a non-trivial interplay between the two emerges and results in the occurrence of a split Fermi sea. For $Delta=0$, we derive exact asymptotic results for the magnetization and the spin current by using a semi-classical Wigner function approach, and we exactly determine the spreading of entanglement entropy exploiting the recently developed tools of quantum fluctuating hydrodynamics. In the interacting case, we analytically solve the Generalized Hydrodynamics equation providing exact expressions for the conserved quantities. We display some numerical results for the entanglement entropy also in the interacting case and we propose a conjecture for its asymptotic value.
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