## 3 Dynamic Disturbance Model of Crowd Panic Behaviors
### Dynamic Model of Internal Crowd Disturbance  
In time domains, the dynamic behavior of a sudden internal disturbance of a crowd is similar to a sudden white noise disturbance in the crowd movement. Its randomness cannot be completely expressed by deterministic variables but can be expressed by a stochastic process. However, the numerical solutions of stochastic equations and the inherent mathematical complexity of stochastic calculus are challenging problems. Based on a probability density function, a Langevin equation can be used to describe the turbulent velocity of fluid particles, which is modeled as Brownian motion. Initially, this equation was applied to homogeneous and isotropic turbulence. The Brownian motion model was widely used in internal random disturbance modeling.  
If the random process \(B(t),\ t\in[0,\infty)\) with value R satisfies the following properties:  
(1) The orbit \(t\to B(t)\) is almost necessarily continuous:  
\[B(0)=0\ a.s; \tag{2}\]  
(2) For any \(0<t_{1}<\cdots<t_{n}\) and Borel set \(A_{1},\cdots,A_{n}\subset R\)  
\[\begin{array}{c}P\{B(t_{1})\in A_{1},\cdots,B(t_{n})\in A_{n}\}\\ \qquad=\int_{A_{1}}\cdots\int_{A_{n}}p(t_{1},0,x_{1})p(t_{2}-t_{1},x_{1},x_{2} )\cdots p(t_{n}-t_{n-1},x_{n-1},x_{n})dx_{1}\cdots dx_{n}\end{array} \tag{3}\]  
where if \(p(t,x,y)=\frac{1}{\sqrt{2\pi t}}\mathrm{e}^{-\frac{(x-y)^{2}}{2}}\), then \(B(t),t\in[0,\infty)\) is called the standard Brownian motion. The standard Brownian motion satisfies a Gaussian distribution. For the sake of simplicity, this study only considers the disturbances of single pedestrian behaviors, so the probability distribution of the random disturbance intensity can be expressed by a single Gaussian function. Figure 1a shows the two-dimensional Gaussian distributions of four groups of different expectation values (mu) and standard deviations (sigma). Figure 1b shows the standard three-dimensional Gaussian distribution when the expectation value is 0 and the standard deviation is 5.  
When a crowd moves forward in a crowded state, it will produce a certain \"pressure\" among the crowd. In this study, the crowd is regarded as fluid, and a crowd flow model is established based on fluid dynamics. Therefore, the pressure among a crowd can be established according to the characteristics of fluid dynamics, as shown in Equation (4) as follows:  
\[P=\rho^{\gamma},0.1<\gamma<1.2 \tag{4}\]  
Here, \(\gamma\) is a pressure factor. In the case of sudden abnormal situations occurring in crowds, such as pedestrian falls and terrorist attacks, a disturbing force will be produced and disturb the balanced crowd flow; this can be directly assumed as the change in \"pressure\" among the crowd. To construct the dynamic model of a disturbance caused by an abnormal behavior in a pedestrian crowd, we assumed that the disturbance burst point is \((a,b)\), then the crowd pressure at this point is the largest, and the crowd pressure around it decays exponentially in the form of \(e\), with same character of a Gaussian distribution in Figure 1. This result showed a single disturbance without damping from the neighboring pedestrians. Therefore, under the influence of random disturbances in the crowd, an equivalent pressure \(P_{\xi}(x,y,t)\) at anytwo - dimensionalpoint\((x,y)\) can be defined as follows:  
\[P_{\xi}(x,y,t)=\rho^{\gamma}\cdot\xi_{p}\cdot\frac{1}{\sqrt{2\pi t}}e^{-\frac {(x-a)^{2}+(y-b)^{2}}{2}} \tag{5}\]  
Here, \(\rho\) is the pedestrian density, \(\gamma\) is the power of density, \(\xi_{p}\) is the pressure intensity coefficient of the random disturbance, \((a,b)\) is the position of the disturbance center, and \(t\) is the duration of a disturbance burst. Therefore, the equivalent pressure is a compre  
Figure 1: Standard Gaussian distribution.  
hensive and dimensionless definition considering the pedestrian density, scenario-based pressure intensity, disturbance propagation time, and two-dimensional layout of observed measurement points.