# Elucidating Super-Compressible States in Inner Structure of Vehicular Flow
## 7 Theoretical Derivations and Practical Consequences  
In particle systems in which only the nearest neighbors interact (the so-called _short-ranged interaction_), it holds (as proved in the paper [23]) that the probability density for the distance between the succeeding particles has the form (12). If a purely repulsive potential (13)-(17) is chosen for such a system, then the following two findings result from the relation (18). If the analyzed clearances are scaled to a unit mean value, then their variance will never exceed the unit value. Furthermore, the statistical compressibility of such a system then acquires only values from the semi-closed interval (0, 1], whereas the unit value \(\chi=1\) corresponds exclusively to the non-interaction variant of a system, i.e., the  
Figure 8: The figure depicts the evolution of the statistical rigidity in main/fast lanes inside cities. Urban data were measured in the city of Vyskov in the Czech Republic.  
Figure 7: The figure depicts the evolution of the statistical rigidity in different freeway lanes. This figure is replicated from the article [14]. Reproduced with permission from [PERSON] et.al. Physica A, published by Elsevier 2022.  
Poisson system. To summarize, it is proven analytically that all systems with short-ranged repulsive potential are sub-compressible or uni-compressible.  
However, Figures 7 and 8 show--beyond any doubt--that some constellations of traffic alignment are super-compressible. Three possible explanations follow from the aforementioned considerations:  
1. Under certain conditions, the vehicular flow represents a medium-ranged system, which means that the driver takes into account the behavior of several of his predecessors. This would violate the assumption of the short-ranged nature of inter-vehicular interactions.
2. In standard traffic interactions, force repulsion is only a partial interaction factor. If the nature of the traffic flow allows it, drivers also use attractive control components. Particularly, these components are evident at lower traffic densities for more sporty drivers, who quite naturally line up in the fast freeway lane.
3. Under certain conditions, the whole concept of the particle model fails because the basic assumption about the conservation of the number of particles in the system is violated. If the traffic density is lower, sporty drivers change lanes to a greater extent, which causes a significant deviation from the basic premise of the theoretical concept used.  
But we can eliminate the first of the above justifications. Numerical analyses of particle systems show that an increase in the interaction range causes a decrease in the statistical rigidity values, which leads to lower compressibility values. Thus, by strengthening the interaction, the sub-compressible state cannot be flipped into the region of super-compressible states.  
However, it is not possible to decide between the other two options based on theoretical considerations alone. This is well reasoned in [14], which admits both variants of the explanation. If it is not possible to decide theoretically, a suitable traffic experiment should be used to solve this open problem.