# Risk-Quantification Method for Car-Following Behavior Considering Driving-Style Propensity
## 3 Risk Analysis and Modeling Based on Security Potential Field
### Security Potential Field Modeling
#### 3.1.3 Vehicle-Actuated Potential Field  
The vehicle interaction potential field originates from other vehicles on the road, which can be analogized to molecules, and the potential field method is used to study car-following behavior. Scholars have developed various mathematical models to describe the interactions between molecules within microphysical systems, among which the Lennard-Jones potential model is extensively used due to its simple structural form. This model conceptualizes the interactions between molecules as a combination of attractive (gravitational) and repulsive forces [18]. Given that both attractive and repulsive forces are inversely related to the powers of the distance between molecules, its mathematical expression is  
\[\varphi(r)=\frac{A}{r^{\alpha}}-\frac{B}{r^{\beta}} \tag{3}\]  
where \(A\) and \(B\) are the intermediate variables of the potential function, and \(\frac{A}{r^{\alpha}}\) and \(\frac{B}{r^{\beta}}\) are the repulsive and gravitational terms, respectively, where the gravitational force on the molecule is close to zero when \(r\) is too large. Molecules close to each other present a state in which they are neither infinitely close nor infinitely far away; that is, the combined intermolecular force is zero, which is known as the equilibrium distance between molecules \(r_{0}\). At this point, the molecular potential energy was minimized and specified as \(\varepsilon\); then  
\[\left[\frac{d\varphi(r_{0})}{dr}\right]_{r=r_{0}}=0 \tag{4}\]  
\[\varphi(r_{0})=-\varepsilon \tag{5}\]  
Joining the two equations gives  
\[{r_{0}}^{\alpha-\beta}=\frac{\alpha A}{\beta B} \tag{6}\]  
The Lennard-Jones potential function expression is  
\[\varphi(r)=\frac{\varepsilon}{\alpha-\beta}\left[\beta\left(\frac{r_{0}}{r} \right)^{\alpha}-\alpha\left(\frac{r_{0}}{r}\right)^{\beta}\right] \tag{7}\]  
Drawing an analogy between vehicles traveling on the road and molecules within a potential field, there exists an equilibrium state between leading and following vehicles, characterized by a necessary safe following distance. The mathematical model is  
\[X_{r}=(\tau-\omega\Delta v)v_{n} \tag{8}\]  
where \(\tau\) and \(\omega\) are the coefficients to be determined; \(\Delta v\) is the speed difference between the leading and following vehicles; and \(v_{n}\) is the speed of the following vehicle. When the distance between the leading and following vehicles is smaller than the required safety distance, the following vehicle tends to move away from the leading vehicle; conversely, the following vehicle tends to move closer to the leading vehicle. The traveling vehicle will not be too close or too far from the leading vehicle to ensure the dynamic balance of the following vehicle. Based on this, the vehicle interaction potential function is constructed as  
\[\varphi_{n}(r)=\frac{\varepsilon}{\alpha-\beta}\left[\beta\left(\frac{X_{r}} {L}\right)^{\alpha}-\alpha\left(\frac{X_{r}}{L}\right)^{\beta}\right] \tag{9}\]  
where \(L\) is the actual distance between the leading and following vehicles. Let the mass of the vehicle be \(m\). Then, the longitudinal acceleration of the following vehicle under the action of the interacting potential field is  
\[a_{n}=\frac{\alpha\beta\varepsilon}{m(\alpha-\beta)}\left(\frac{X_{r}{}^{\beta}} {L^{\beta+1}}-\frac{X_{r}{}^{\alpha}}{L^{\alpha+1}}\right) \tag{10}\]  
Within the potential field, the intensity of the potential at any point is negatively correlated with the distance to the target vehicle. Typically, the intensity of the potential field decreases as the distance increases. The coordinates of the vehicle's center of mass are set as \((x_{0},y_{0})\), and the distance \(s\) between any point \((x,y)\) and the target vehicle can be represented as  
\[|s|=\sqrt{(x-x_{0})^{2}-(y-y_{0})^{2}} \tag{11}\]  
Vehicles exhibiting various motion states exert distinct impacts on the distribution of their potential field; notably, velocity and acceleration play crucial roles in shaping the potential field's distribution.