# Elucidating Super-Compressible States in Inner Structure of Vehicular Flow
### Vehicular Stream Micro-Structure: Prior Art  
In this part of the text, we try to roughly summarize the current state of the art in the field of statistical analysis of traffic micro-quantities (quantities associated with individual vehicles, i.e., non-aggregated quantities). The physics of traffic ([4; 15]) in the flow of time came to the conclusion that the statistics of traffic quantities are strongly sensitive to the current values of traffic macroscopic quantities (traffic density \(\varrho\), traffic intensity \(I\) and average speed of the traffic sample \(V\)). And indeed, the mean values of the selected micro-quantities depend quite directly on the aforementioned macroscopic values, which is more or less trivial knowledge. A much more interesting finding is the fact that also the characteristics of higher orders (variance, typically) are strongly dependent on the location of the data sample in the so-called fundamental diagram, which is the two-dimensional graph of the binary relation between intensity and density (see [16]). A typical shape of such a graph is shown in Figure 1.  
It also soon became apparent that with a thoughtful way of analyzing empirical traffic data (when data extracted under different traffic conditions are not mixed), the hypothesis that vehicle speeds are normally distributed has been statistically confirmed several times (e.g., [17]). This means that the instantaneous speed \(\mathcal{V}\) of vehicles (with realization \(v\)) analyzed in a small sub-region \(\Omega\) (see Figure 1) is distributed with a probability density  
\[q(v)=\frac{1}{\sqrt{2\pi}\sigma}\mathrm{e}^{-\frac{(v-V)^{2}}{2\sigma^{2}}}, \tag{1}\]  
which implies that the average value \(E\mathcal{V}=V\) and the variance \(D\mathcal{V}=\sigma^{2}\), both of them depending on traffic density and intensity. Much more interesting, however, are questions regarding the spatiotemporal arrangement of vehicles on roadways. Complex traffic dynamics (controlled by the decision-making processes in the driver's brain) generate extremely interesting variants of the inner arrangement of a vehicular system. Surprisingly, some of them have no analogy to any of the classical physical systems. This is mainly due to a significant force repulsion, the main reason for which is the attempt to avoid a collision with other vehicles, predominantly with the preceding vehicle. Common variants of the inner structure of a one-dimensional system can only be found in two special variants of the vehicular stream. The first of them is a situation where the traffic density is very low and therefore nothing forces the drivers to interact with each other. Such a situation, then (from a mathematical point of view), represents a so-called _Poisson system_, which is characterized by a total absence of interactions between elements  
Figure 1: Fundamental diagram of vehicular traffic—schematic representation of binary relation between traffic density \(\varrho\) and intensity \(I\), extracted from typical empirical data samples. \(\Omega\) symbolizes a small sub-region with similar values of macroscopic quantities, where empirical data can be considered homogeneous. of the system. In this type of system, the interval frequency \(\mathcal{N}_{L}\) (i.e., the number of particles occurring in an interval of fixed length \(L\)) follows the Poisson distribution  
\[P[\mathcal{N}_{L}=k]=\frac{(\mu L)^{k}}{k!}\,\mathrm{e}^{-\mu L};\quad k=0,1,2, \ldots, \tag{2}\]  
where \(P[\mathcal{N}_{L}=k]\) stands for probability that in an interval of length \(L\) there is \(k\) elements/particles/vehicles. The mutual distances between neighboring particles in the Poisson system are then subject to the exponential distribution  
\[g(x)=\Theta(x)\lambda\mathrm{e}^{-\lambda x}, \tag{3}\]  
where \(\Theta(x)\) represents the Heaviside unit-step function. With regard to the above-mentioned properties, the Poisson system is understood as a system with the maximum possible freedom of movement, i.e., with the maximum degree of statistical fluctuations.