Spectral rigidity of vehicular streams (random matrix theory approach)

Abstract

Using the methods originally developed for Random Matrix Theory we derive an exact mathematical formula for number variance ∆N(L) (introduced in [4]) describing a rigidity of particle ensembles with power-law repulsion. The resulting relation is consequently compared with the relevant statistics of the single-vehicle data measured on the Dutch freeway A9. The detected value of an inverse temperature β, which can be identified as a coefficient of a mental strain of car drivers, is then discussed in detail with the respect to the traffic density ̺ and flow J. PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion -89.40.-a transportation -05.45.-a Nonlinear dynamics and chaos 1 Terminus a quo The main goal of this paper is to show that the statistical fluctuations of single-vehicle data (in vehicular flows) can be very well predicted by the methods known from Random Matrix Theory where used for describing the statis-tics of energy levels in quantum chaotic systems. Above that, we intend to demonstrate that the changes of statistical variances in vehicular samples depends not only on macroscopical traffic quantities and three traffic phases (as published in [1] or [2]), but also on psychological characteristics of driver's decision-making process. We will evince

Inner structure of vehicular ensembles and random matrix theory

“…Thus, the output represents a map of estimated values α and β depending on traffic density and flux. Since the parameter β admits a thermodynamic interpretation (as is discussed in [7,8]) the changes of the socio-physical coefficient β = β(̺, J) reflect a level of mental pressure which the drivers are under during a given traffic constellation. As is evident in figure 6 for fast lanes and 11 for main lanes, a driver's mental strain depends predominantly on actual density and only marginally on actual flow.…”

“…Until now, natural endeavor to interconnect Random Matrix Theory (RMT) with vehicular systems has not led to a success. However, a partial progress has been achieved in [5,6,7,8,9] where it is proved that microscopical arrangement of vehicles can be predicted by means of a certain one-dimensional gas inspired by the Dyson's gases that are well-known in RMT. Moreover, another attempts to describe (analytically) a microstructure of vehicular ensembles with help of statistical instruments (e.g.…”

Statistical rigidity of vehicular streams—theory versus reality

“…In this section we present a numerical analysis of the statistical rigidity extracted (similarly to [10]) from empirical traffic data described in section 2.2. Following the same segmentation procedure as that applied in the section 2.3, we calculate the quantity D( ) t in all the samples being sorted by a local density.…”

“…In the theory of • (Segmental data processing) Variance of real-road clearances is rapidly changing with traffic density [2][3][4]. It corresponds to the fact that statistical resistivity (measuring a level of resistance to statistical perturbations) depends strongly (as analyzed in [10,23,26]) on parameters of actual traffic state (intensity, density, mean velocity). Therefore, necessary part of intended estimation procedure should be a segmentation (i.e.…”

“…10). Another step requires expressing the Taylor polynomial of the function  f ( ) p p of the second order.…”