The cubic chessboard

We present a survey of recent results, scattered in a series of papers that appeared during past five years, whose common denominator is the use of cubic relations in various algebraic structures. Cubic (or ternary) relations can represent different symmetries with respect to the permutation group S_3, or its cyclic subgroup Z_3. Also ordinary or ternary algebras can be divided in different classes with respect to their symmetry properties. We pay special attention to the non-associative ternary algebra of 3-forms (or ``cubic matrices''), and Z_3-graded matrix algebras. We also discuss the Z_3-graded generalization of Grassmann algebras and their realization in generalized exterior differential forms. A new type of gauge theory based on this differential calculus is presented. Finally, a ternary generalization of Clifford algebras is introduced, and an analog of Dirac's equation is discussed, which can be diagonalized only after taking the cube of the Z_3-graded generalization of Dirac's operator. A possibility of using these ideas for the description of quark fields is suggested and discussed in the last Section.Comment: 23 pages, dedicated to A. Trautman on the occasion of his 64th birthda

Hypersymmetry: a Z_3-graded generalization of supersymmetry

We propose a generalization of non-commutative geometry and gauge theories based on ternary Z_3-graded structures. In the new algebraic structures we define, we leave all products of two entities free, imposing relations on ternary products only. These relations reflect the action of the Z_3-group, which may be either trivial, i.e. abc=bca=cab, generalizing the usual commutativity, or non-trivial, i.e. abc=jbca, with j=e^{(2\pi i)/3}. The usual Z_2-graded structures such as Grassmann, Lie and Clifford algebras are generalized to the Z_3-graded case. Certain suggestions concerning the eventual use of these new structures in physics of elementary particles are exposed

General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems

An asymptotic method for finding instabilities of arbitrary $d$-dimensional large-amplitude patterns in a wide class of reaction-diffusion systems is presented. The complete stability analysis of 2- and 3-dimensional localized patterns is carried out. It is shown that in the considered class of systems the criteria for different types of instabilities are universal. The specific nonlinearities enter the criteria only via three numerical constants of order one. The performed analysis explains the self-organization scenarios observed in the recent experiments and numerical simulations of some concrete reaction-diffusion systems.Comment: 21 pages (RevTeX), 8 figures (Postscript). To appear in Phys. Rev. E (April 1st, 1996

Traffic Network Optimum Principle - Minimum Probability of Congestion Occurrence

We introduce an optimum principle for a vehicular traffic network with road bottlenecks. This network breakdown minimization (BM) principle states that the network optimum is reached, when link flow rates are assigned in the network in such a way that the probability for spontaneous occurrence of traffic breakdown at one of the network bottlenecks during a given observation time reaches the minimum possible value. Based on numerical simulations with a stochastic three-phase traffic flow model, we show that in comparison to the well-known Wardrop's principles the application of the BM principle permits considerably greater network inflow rates at which no traffic breakdown occurs and, therefore, free flow remains in the whole network.Comment: 22 pages, 6 figure

Stability Analysis of Optimal Velocity Model for Traffic and Granular Flow under Open Boundary Condition

We analyzed the stability of the uniform flow solution in the optimal velocity model for traffic and granular flow under the open boundary condition. It was demonstrated that, even within the linearly unstable region, there is a parameter region where the uniform solution is stable against a localized perturbation. We also found an oscillatory solution in the linearly unstable region and its period is not commensurate with the periodicity of the car index space. The oscillatory solution has some features in common with the synchronized flow observed in real traffic.Comment: 4 pages, 6 figures. Typos removed. To appear in J. Phys. Soc. Jp

Cellular automata approach to three-phase traffic theory

The cellular automata (CA) approach to traffic modeling is extended to allow for spatially homogeneous steady state solutions that cover a two dimensional region in the flow-density plane. Hence these models fulfill a basic postulate of a three-phase traffic theory proposed by Kerner. This is achieved by a synchronization distance, within which a vehicle always tries to adjust its speed to the one of the vehicle in front. In the CA models presented, the modelling of the free and safe speeds, the slow-to-start rules as well as some contributions to noise are based on the ideas of the Nagel-Schreckenberg type modelling. It is shown that the proposed CA models can be very transparent and still reproduce the two main types of congested patterns (the general pattern and the synchronized flow pattern) as well as their dependence on the flows near an on-ramp, in qualitative agreement with the recently developed continuum version of the three-phase traffic theory [B. S. Kerner and S. L. Klenov. 2002. J. Phys. A: Math. Gen. 35, L31]. These features are qualitatively different than in previously considered CA traffic models. The probability of the breakdown phenomenon (i.e., of the phase transition from free flow to synchronized flow) as function of the flow rate to the on-ramp and of the flow rate on the road upstream of the on-ramp is investigated. The capacity drops at the on-ramp which occur due to the formation of different congested patterns are calculated.Comment: 55 pages, 24 figure

Microscopic features of moving traffic jams

Empirical and numerical microscopic features of moving traffic jams are presented. Based on a single vehicle data analysis, it is found that within wide moving jams, i.e., between the upstream and downstream jam fronts there is a complex microscopic spatiotemporal structure. This jam structure consists of alternations of regions in which traffic flow is interrupted and flow states of low speeds associated with "moving blanks" within the jam. Empirical features of the moving blanks are found. Based on microscopic models in the context of three-phase traffic theory, physical reasons for moving blanks emergence within wide moving jams are disclosed. Structure of moving jam fronts is studied based in microscopic traffic simulations. Non-linear effects associated with moving jam propagation are numerically investigated and compared with empirical results.Comment: 19 pages, 12 figure

Interpreting the Wide Scattering of Synchronized Traffic Data by Time Gap Statistics

Based on the statistical evaluation of experimental single-vehicle data, we propose a quantitative interpretation of the erratic scattering of flow-density data in synchronized traffic flows. A correlation analysis suggests that the dynamical flow-density data are well compatible with the so-called jam line characterizing fully developed traffic jams, if one takes into account the variation of their propagation speed due to the large variation of the netto time gaps (the inhomogeneity of traffic flow). The form of the time gap distribution depends not only on the density, but also on the measurement cross section: The most probable netto time gap in congested traffic flow upstream of a bottleneck is significantly increased compared to uncongested freeway sections. Moreover, we identify different power-law scaling laws for the relative variance of netto time gaps as a function of the sampling size. While the exponent is -1 in free traffic corresponding to statistically independent time gaps, the exponent is about -2/3 in congested traffic flow because of correlations between queued vehicles.Comment: For related publications see http://www.helbing.or

Motions and world-line deviations in Einstein-Maxwell theory

We examine the motion of charged particles in gravitational and electro-magnetic background fields. We study in particular the deviation of world lines, describing the relative acceleration between particles on different space-time trajectories. Two special cases of background fields are considered in detail: (a) pp-waves, a combination of gravitational and electro-magnetic polarized plane waves travelling in the same direction; (b) the Reissner-Nordstr{\o}m solution. We perform a non-trivial check by computing the precession of the periastron for a charged particle in the Reissner-Nordstr{\o}m geometry both directly by solving the geodesic equation, and using the world-line deviation equation. The results agree to the order of approximation considered.Comment: 23 pages, no figure

Derivation, Properties, and Simulation of a Gas-Kinetic-Based, Non-Local Traffic Model

We derive macroscopic traffic equations from specific gas-kinetic equations, dropping some of the assumptions and approximations made in previous papers. The resulting partial differential equations for the vehicle density and average velocity contain a non-local interaction term which is very favorable for a fast and robust numerical integration, so that several thousand freeway kilometers can be simulated in real-time. The model parameters can be easily calibrated by means of empirical data. They are directly related to the quantities characterizing individual driver-vehicle behavior, and their optimal values have the expected order of magnitude. Therefore, they allow to investigate the influences of varying street and weather conditions or freeway control measures. Simulation results for realistic model parameters are in good agreement with the diverse non-linear dynamical phenomena observed in freeway traffic.Comment: For related work see http://www.theo2.physik.uni-stuttgart.de/helbing.html and http://www.theo2.physik.uni-stuttgart.de/treiber.htm