Connection Between Microscopic and Macroscopic Models

Abstract

This chapter is devoted to the detailed study of the relation between a microscopic cellular automation and a macroscopic partial differential equation model for the movement of pedestrians. We describe the mathematical tools allowing to derive the macroscopic from the microscopic model. Such a connection between discrete, particle based and continuous, density based models can help to improve the understanding of basic properties of human crowds. We exemplify this by applying our results to typical cases. The first one is the formation of lanes in bi-directional flow. The second is the analysis of the fundamental diagram. Our analysis provides (at least qualitatively) a connection between these phenomena and model parameters. We conclude by pointing out a number of possible directions of future research.

Appendix

The linear stability approach described in Sect. 3.5 yields the following linear system for the perturbations

$$\displaystyle\begin{array}{rcl} \partial _{t}\xi & =& P((1 - {b}^{eq})\partial _{ yy}\xi - {r}^{eq}\partial _{ yy}\eta ) + Pk_{S}\partial _{x}((1 {-\rho }^{eq})\xi - {r}^{eq}(\xi -\eta )) \\ & & +Pk_{D}{r}^{eq}(1 {-\rho }^{eq})(\partial _{ xx} + \partial _{yy})\varPsi _{r}, {}\end{array}$$

(3.27)

$$\displaystyle\begin{array}{rcl} \partial _{t}\eta & =& P((1 - {r}^{eq})\partial _{ yy}\eta - {b}^{eq}\partial _{ yy}\xi ) - Pk_{S}\partial _{x}((1 {-\rho }^{eq})\eta - {b}^{eq}(-\xi +\eta )) \\ & & +Pk_{D}{b}^{eq}(1 {-\rho }^{eq})(\partial _{ xx} + \partial _{yy})\varPsi _{b}, {}\end{array}$$

(3.28)

$$\displaystyle\begin{array}{rcl} 0& =& \kappa \partial _{xx}\varPsi _{r} -\delta \varPsi _{r} + (1 {-\rho }^{eq} - {r}^{eq})\xi + {r}^{eq}\eta,{}\end{array}$$

(3.29)

$$\displaystyle\begin{array}{rcl} 0& =& \kappa \partial _{xx}\varPsi _{b} -\delta \varPsi _{b} + (1 {-\rho }^{eq} - {b}^{eq})\eta + {b}^{eq}\xi.{}\end{array}$$

(3.30)

We denote length of the domain in y-direction by l, the length in x-direction is denoted by L, see Fig. 3.3. The perturbations are assumed as

$$\displaystyle\begin{array}{rcl} \xi = U(x)\cos \left (\frac{k\pi } {l} y\right )\exp (\lambda t),& &{}\end{array}$$

(3.31)

$$\displaystyle\begin{array}{rcl} \eta = V (x)\cos \left (\frac{k\pi } {l} y\right )\exp (\lambda t),& &{}\end{array}$$

(3.32)

$$\displaystyle\begin{array}{rcl} \varPsi _{r} = Y _{r}(x)\cos \left (\frac{k\pi } {l} y\right )\exp (\lambda t),& &{}\end{array}$$

(3.33)

$$\displaystyle\begin{array}{rcl} \varPsi _{b} = Y _{b}(x)\cos \left (\frac{k\pi } {l} y\right )\exp (\lambda t).& &{}\end{array}$$

(3.34)

where U(x), V (x), Y _r (x) and Y _b (x) denote perturbations in the x-direction, and k denotes the mode of the perturbation in y-direction. From now on, we assume ${r}^{eq} = {b}^{eq}$. Inserting this ansatz into Eqs. (3.27) and (3.28) we obtain

$$\displaystyle\begin{array}{rcl} \lambda /PU& =& -(1 - {r}^{eq})\gamma U + {r}^{eq}\gamma V + k_{ S}(1 - 3{r}^{eq})U^{\prime} + k_{ S}{r}^{eq}V ^{\prime} \\ & & -k_{D}{r}^{eq}(1 - 2{r}^{eq})(\gamma +\varGamma )Y _{ r} {}\end{array}$$

(3.35)

$$\displaystyle\begin{array}{rcl} \lambda /PV & =& -(1 - {r}^{eq})\gamma V + {r}^{eq}\gamma U - k_{ S}(1 - 3{r}^{eq})V ^{\prime} - k_{ S}{r}^{eq}U^{\prime} \\ & & -k_{D}{r}^{eq}(1 - 2{r}^{eq})(\gamma +\varGamma )Y _{ b}, {}\end{array}$$

(3.36)

where we used $\gamma = \frac{{k{}^{2}\pi }^{2}} {{l}^{2}}$, ′ denotes the derivative with respect to x, $\varGamma = \frac{{\pi }^{2}} {{L}^{2}}$ and we assume perturbations Y _i of a sinusoidal or cosinusoidal type $Y _{i}^{\prime\prime} = -\varGamma Y _{i}$. The equations for U and V finally read, using (3.29) and (3.30):

$$\displaystyle\begin{array}{rcl} & [\lambda /P& +(1 - {r}^{eq})\gamma ]U - {r}^{eq}\gamma V + k_{ D}{r}^{eq}(1 - 2{r}^{eq})\frac{\gamma +\varGamma } {\kappa \varGamma +\delta }\left [(1 - 3{r}^{eq})U + {r}^{eq}V \right ] \\ & = & k_{S}(1 - 3{r}^{eq})U^{\prime} +\mu {r}^{eq}V ^{\prime}, {}\end{array}$$

(3.37)

$$\displaystyle\begin{array}{rcl} & [\lambda /P& +(1 - {r}^{eq})\gamma ]V - {r}^{eq}\gamma U + k_{ D}{r}^{eq}(1 - 2{r}^{eq})\frac{\gamma +\varGamma } {\kappa \varGamma +\delta }\left [(1 - 3{r}^{eq})V + {r}^{eq}U\right ] \\ & = & -k_{S}(1 - 3{r}^{eq})V ^{\prime} -\mu {r}^{eq}U^{\prime}. {}\end{array}$$

(3.38)

We denote $\varTheta = \frac{\gamma +\varGamma } {\kappa \varGamma +\delta }$. The summation of (3.37) and (3.38) is given by

$$\displaystyle\begin{array}{rcl} [\lambda /P& +(1 - 2{r}^{eq})\gamma + k_{D}{r}^{eq}{(1 - 2{r}^{eq})}^{2}\varTheta ](U + V )& = k_{S}(1 - 4{r}^{eq})(U^{\prime} - V ^{\prime}).{}\end{array}$$

(3.39)

The derivatives of (3.37) and (3.38) are given by

$$\displaystyle\begin{array}{rcl} [\lambda /P& & +(1 - {r}^{eq})\gamma ]U^{\prime} - {r}^{eq}\gamma V ^{\prime} + k_{ D}{r}^{eq}(1 - 2{r}^{eq})\varTheta \left [(1 - 3{r}^{eq})U^{\prime} + {r}^{eq}V ^{\prime}\right ] \\ & =\,\,& k_{S}(1 - 3{r}^{eq})U^{\prime\prime} +\mu {r}^{eq}V ^{\prime\prime}, {}\end{array}$$

(3.40)

$$\displaystyle\begin{array}{rcl} [\lambda /P& & +(1 - {r}^{eq})\gamma ]V ^{\prime} - {r}^{eq}\gamma U^{\prime} + k_{ D}{r}^{eq}(1 - 2{r}^{eq})\varTheta \left [(1 - 3{r}^{eq})V ^{\prime} + {r}^{eq}U^{\prime}\right ] \\ & =& -k_{S}(1 - 3{r}^{eq})V ^{\prime\prime} -\mu {r}^{eq}U^{\prime\prime}. {}\end{array}$$

(3.41)

Subtracting these equation yields

$$\displaystyle\begin{array}{rcl} [\lambda /P& & +\gamma + k_{D}{r}^{eq}(1 - 2{r}^{eq})(1 - 4{r}^{eq})\varTheta ](U^{\prime} - V ^{\prime}) \\ & =& k_{S}(1 - 2{r}^{eq})(U^{\prime\prime} + V ^{\prime\prime}). {}\end{array}$$

(3.42)

Combining (3.39) and (3.42) leads to

$$\displaystyle\begin{array}{rcl} [& & \lambda /P + (1 - 2{r}^{eq})\gamma + k_{ D}{r}^{eq}{(1 - 2{r}^{eq})}^{2}\varTheta ](U + V ) = \\ & & k_{S}^{2} \frac{(1 - 4{r}^{eq})(1 - 2{r}^{eq})} {\lambda /P +\gamma +k_{D}{r}^{eq}(1 - 2{r}^{eq})(1 - 4{r}^{eq})\varTheta }(U^{\prime\prime} + V ^{\prime\prime}).{}\end{array}$$

(3.43)

In the following, we assume perturbations U and V in x-direction of a sinusoidal type, due to the homogeneous boundary conditions. This leads to

$$\displaystyle{ U^{\prime\prime} = -\frac{{m{}^{2}\pi }^{2}} {{L}^{2}} U,\qquad V ^{\prime\prime} = -\frac{{m{}^{2}\pi }^{2}} {{L}^{2}} V, }$$

where L denotes the length of the domain in x direction. In the following, we take m = 1, as we are only interested in lanes forming along the x-direction. We finally arrive at

$$\displaystyle\begin{array}{rcl} 0 = [& & {\lambda }^{2}/{P}^{2} + 2\lambda /P[\gamma (1 - {r}^{eq}) + k_{ D}{r}^{eq}(1 - 2{r}^{eq})(1 - 3{r}^{eq})\varTheta ] \\ & & {+\gamma }^{2}(1 - 2{r}^{eq}) + 2\gamma k_{ D}{r}^{eq}{(1 - 2{r}^{eq})}^{3}\varTheta \\ & & +k_{D}^{2}{({r}^{eq})}^{2}{(1 - 2{r}^{eq})}^{3}{(1 - 4{r}^{eq})\varTheta }^{2} \\ & & +k_{S}^{2}\varGamma (1 - 4{r}^{eq})(1 - 2{r}^{eq})](U + V ). {}\end{array}$$

(3.44)

Accordingly, the equation for λ is given by

$$\displaystyle\begin{array}{rcl} \lambda _{1/2}& =& -P[(1 - {r}^{eq})\gamma + k_{ D}{r}^{eq}(1 - 2{r}^{eq})(1 - 3{r}^{eq})\varTheta ] \\ & & \pm P\sqrt{{({r}^{eq } )}^{2 } {[\gamma -k_{D } {r}^{eq } (1 - 2{r}^{eq } )\varTheta ]}^{2 } - k_{S }^{2 }\varGamma (1 - 4{r}^{eq } )(1 - 2{r}^{eq } )}.{}\end{array}$$

(3.45)

The parameter λ is supposed to be real-valued for all k, particularly for k = 1. From that we conclude

$$\displaystyle\begin{array}{rcl}{ ({r}^{eq})}^{2}{[\gamma -k_{ D}{r}^{eq}(1 - 2{r}^{eq})\varTheta ]}^{2} \geq k_{ S}^{2}\varGamma (1 - 4{r}^{eq})(1 - 2{r}^{eq}).& &{}\end{array}$$

(3.46)

As r ^eq ≤ 1∕2, (3.46) is always fulfilled in case that r ^eq ≥ 1∕4. This means that instabilities arise only in case r ^eq ≥ 1∕4. To obtain instabilities increasing in time, λ > 0 has to be satisfied. This means

$$\displaystyle\begin{array}{rcl} & & {[(1 - {r}^{eq})\gamma + k_{ D}{r}^{eq}(1 - 2{r}^{eq})(1 - 3{r}^{eq})\varTheta ]}^{2} \\ & & \quad < {({r}^{eq}){}^{2}\gamma }^{2} - 2\gamma k_{ D}{({r}^{eq})}^{3}(1 - 2{r}^{eq})\varTheta \\ & & \quad + k_{D}^{2}{({r}^{eq})}^{4}{(1 - 2{r}^{eq}){}^{2}\varTheta }^{2} - k_{ S}^{2}\varGamma (1 - 4{r}^{eq})(1 - 2{r}^{eq}){}\end{array}$$

(3.47)

Assuming (1 − 2r ^eq) > 0, which means that the overall density is below maximum, we obtain

$$\displaystyle\begin{array}{rcl} & & {\gamma }^{2} + 2\gamma k_{ D}{r}^{eq}{(1 - 2{r}^{eq})}^{2}\varTheta \\ & & \quad + k_{D}^{2}{({r}^{eq})}^{2}{(1 - 2{r}^{eq})}^{2}{(1 - 4{r}^{eq})\varTheta }^{2} + k_{ S}^{2}\varGamma (1 - 4{r}^{eq}) < 0.{}\end{array}$$

(3.48)

The mode of the cosinusoidal perturbation in y-direction is given by k, hence it gives the number of lanes of particles moving in opposite direction which are amplified during time. If k = 1, we obtain one lane in each direction. Accordingly, we obtain as inequality for $\gamma = \frac{{k{}^{2}\pi }^{2}} {{l}^{2}}$

$$\displaystyle\begin{array}{rcl} & & {\gamma }^{2}\left [1 + 2k_{ D}{r}^{eq}{(1 - 2{r}^{eq})}^{2} \frac{1} {\kappa \varGamma +\delta }\right. \\ & & \quad \left.+k_{D}^{2}{{r}^{eq}}^{2}{(1 - 2{r}^{eq})}^{2}(1 - 4{r}^{eq}) \frac{1} {{(\kappa \varGamma +\delta )}^{2}}\right ] \\ & & \quad +\gamma \left [2k_{D}{r}^{eq}{(1 - 2{r}^{eq})}^{2} \frac{\varGamma } {\kappa \varGamma +\delta }\right. \\ & & \quad \left.+2k_{D}^{2}{{r}^{eq}}^{2}{(1 - 2{r}^{eq})}^{2}(1 - 4{r}^{eq}) \frac{\varGamma } {{(\kappa \varGamma +\delta )}^{2}}\right ] \\ & & \quad + k_{D}^{2}{{r}^{eq}}^{2}{(1 - 2{r}^{eq})}^{2}(1 - 4{r}^{eq}) \frac{{\varGamma }^{2}} {{(\kappa \varGamma +\delta )}^{2}} + k_{S}^{2}\varGamma (1 - 4{r}^{eq}) \\ & & < 0. {}\end{array}$$

(3.49)

The evaluation of (3.49) leads to a condition on k which determines under which conditions instabilities, which lead to lane formation, appear.