The Aw-Rascle-Zhang (ARZ) model can be interpreted as a generalization of the Lighthill-Whitham-Richards (LWR) model, possessing a family of fundamental diagram curves, each of which represents a class of drivers with a different empty road velocity. A weakness of this approach is that different drivers possess vastly different densities at which traffic flow stagnates. This drawback can be overcome by modifying the pressure relation in the ARZ model, leading to the generalized Aw-Rascle-Zhang (GARZ) model. We present an approach to determine the parameter functions of the GARZ model from fundamental diagram measurement data. The predictive accuracy of the resulting data-fitted GARZ model is compared to other traffic models by means of a three-detector test setup, employing two types of data: vehicle trajectory data, and sensor data. This work also considers the extension of the ARZ and the GARZ models to models with a relaxation term, and conducts an investigation of the optimal relaxation time.2000 Mathematics Subject Classification. 35L65; 35Q91; 91B74.
The Aw–Rascle–Zhang (ARZ) model can be interpreted as a generalization of the first-order Lighthill–Whitham–Richards (LWR) model, with a family of fundamental diagram (FD) curves rather than one. This study investigated the extent to which this generalization increased the predictive accuracy of the models. To that end, two types of data-fitted LWR models and their second-order ARZ counterparts were systematically compared with a version of the test for the three-detector problem. The parameter functions of the models were constructed with historic FD data. The models were then compared with the use of time-dependent data of two types: vehicle trajectory data and single-loop sensor data. These partial differential equation models were studied in a macroscopic sense (i.e., continuous field quantities were constructed from the discrete data, and discretization effects were kept negligibly small).
In this paper, we study the discrete-time quantum random walks on a line subject to decoherence. The convergence of the rescaled position probability distribution p(x, t) depends mainly on the spectrum of the superoperator L kk . We show that if 1 is an eigenvalue of the superoperator with multiplicity one and there is no other eigenvalue whose modulus equals to 1, thenP ( ν √ t , t) converges to a convex combination of normal distributions. In terms of position space, the rescaled probability mass function p t (x, t) ≡ p( √ tx, t), x ∈ Z/ √ t, converges in distribution to a continuous convex combination of normal distributions. We give an necessary and sufficient condition for a U (2) decoherent quantum walk that satisfies the eigenvalue conditions. We also give a complete description of the behavior of quantum walks whose eigenvalues do not satisfy these assumptions. Specific examples such as the Hadamard walk, walks 1 under real and complex rotations are illustrated. For the O(2) quantum random walks, an explicit formula is provided for the scaling limit of p(x, t) and their moments. We also obtain exact critical exponents for their moments at the critical point and show universality classes with respect to these critical exponents.
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