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Register a new userInter-particle gap distribution and spectral rigidity of totally asymmetric simple exclusion process with open boundaries
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Added by
Milan Krbalek Ph.D.
Publication date
2010
fields
Physics
and research's language is
English
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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.
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We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the process exhibits pre-cutoff and in some cases cutoff. Our main contribution is to study mixing times for the asymmetric simple exclusion process with open boundaries. We show that the order of the mixing time can be linear or exponential in the size of the segment depending on the choice of the boundary parameters, proving a strikingly different (and richer) behavior for the simple exclusion process with open boundaries than for the process on the closed segment. Our arguments combine coupling, second class particle and censoring techniques with current estimates. A novel idea is the use of multi-species particle arguments, where the particles only obey a partial ordering.
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.
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.
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.
We study the driven Brownian motion of hard rods in a one-dimensional cosine potential with an amplitude large compared to the thermal energy. In a closed system, we find surprising features of the steady-state current in dependence of the particle density. The form of the current-density relation changes greatly with the particle size and can exhibit both a local maximum and minimum. The changes are caused by an interplay of a barrier reduction, blocking and exchange symmetry effect. The latter leads to a current equal to that of non-interacting particles for a particle size commensurate with the period length of the cosine potential. For an open system coupled to particle reservoirs, we predict five different phases of non-equilibrium steady states to occur. Our results show that the particle size can be of crucial importance for non-equilibrium phase transitions in driven systems. Possible experiments for demonstrating our findings are pointed out.
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