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Register a new userDomain wall of the totally asymmetric exclusion process without particle number conservation
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Yunxin Zhang
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In this research, the totally asymmetric exclusion process without particle number conservation is discussed. Based on the mean field approximation and the Rankine-Hugoniot condition, the necessary and sufficient conditions of the existence of the domain wall have been obtained. Moreover, the properties of the domain wall, including the location and height, have been studied theoretically. All the theoretical results are demonstrated by the computer simulations.
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We apply the bi-moment determinant method to compute a representation of the matrix product algebra -- a quadratic algebra satisfied by the operators $mathbf{d}$ and $mathbf{e}$ -- for the five parameter ($alpha$, $beta$, $gamma$, $delta$ and $q$) Asymmetric Simple Exclusion Process. This method requires an $LDU$ decomposition of the ``bi-moment matrix. The decomposition defines a new pair of basis vectors sets, the `boundary basis. This basis is defined by the action of polynomials ${P_n}$ and ${Q_n}$ on the quantum oscillator basis (and its dual). Theses polynomials are orthogonal to themselves (ie. each satisfy a three term recurrence relation) and are orthogonal to each other (with respect to the same linear functional defining the stationary state). Hence termed `bi-orthogonal. With respect to the boundary basis the bi-moment matrix is diagonal and the representation of the operator $mathbf{d}+mathbf{e}$ is tri-diagonal. This tri-diagonal matrix defines another set of orthogonal polynomials very closely related to the the Askey-Wilson polynomials (they have the same moments).
We revisit the totally asymmetric simple exclusion process with open boundaries (TASEP), focussing on the recent discovery by de Gier and Essler that the model has a dynamical transition along a nontrivial line in the phase diagram. This line coincides neither with any change in the steady-state properties of the TASEP, nor the corresponding line predicted by domain wall theory. We provide numerical evidence that the TASEP indeed has a dynamical transition along the de Gier-Essler line, finding that the most convincing evidence was obtained from Density Matrix Renormalisation Group (DMRG) calculations. By contrast, we find that the dynamical transition is rather hard to see in direct Monte Carlo simulations of the TASEP. We furthermore discuss in general terms scenarios that admit a distinction between static and dynamic phase behaviour.
We consider the one-dimensional totally asymmetric simple exclusion model (TASEP model) with open boundary conditions and present the analytical computations leading to the exact formula for distance clearance distribution, i.e. probability density for a clear distance between subsequent particles of the model. The general relation is rapidly simplified for middle part of the one-dimensional lattice using the large $N$ approximation. Both the analytical formulas and their approximations are successfully compared with the numerical representation of the TASEP model. Furthermore, we introduce the pertinent estimation for so-called spectral rigidity of the model. The results obtained are sequentially discussed within the scope of vehicular traffic theory.
As a solvable and broadly applicable model system, the totally asymmetric exclusion process enjoys iconic status in the theory of non-equilibrium phase transitions. Here, we focus on the time dependence of the total number of particles on a 1-dimensional open lattice, and its power spectrum. Using both Monte Carlo simulations and analytic methods, we explore its behavior in different characteristic regimes. In the maximal current phase and on the coexistence line (between high/low density phases), the power spectrum displays algebraic decay, with exponents -1.62 and -2.00, respectively. Deep within the high/low density phases, we find pronounced emph{oscillations}, which damp into power laws. This behavior can be understood in terms of driven biased diffusion with conserved noise in the bulk.
We consider the exclusion process on a ring with time-dependent defective bonds at which the hoping rate periodically switches between zero and one. This system models main roads in city traffics, intersecting with perpendicular streets. We explore basic properties of the system, in particular dependence of the vehicular flow on the parameters of signalization as well as the system size and the car density. We investigate various types of the spatial distribution of the vehicular density, and show existence of a shock profile. We also measure waiting time behind traffic lights, and examine its relationship with the traffic flow.
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