for door cells. An explicit construction of $S$ can be found in [@aki0; @aki].

The floor fields are used to translate a long-ranged spatial interaction into a local interaction, but with memory, similar to the phenomenon of chemotaxis in biology. The only other model which so far reproduces all observed collective effects of pedestrian flow, the social-force model [@social], uses exponentially decaying repulsive forces between pedestrians. In contrast, in our approach the interaction is local and attractive. However, pedestrians do not interact with the density, but with the velocity density created by the other particles.

Update rules {#update}
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The update rules of the full model, including the interaction with the two floor fields, have the following structure:

1.  The dynamic floor field $D$ is modified according to its diffusion and decay rules [@ourpaper], controlled by the parameters $\alpha$ and $\delta$. In each time step of the simulation each single boson of the whole dynamic field $D$ decays with the probability $\delta$ and diffuses with the probability $\alpha$ to one of its neighbouring cells.

2.  For each pedestrian, the transition probabilities $p_{ij}$ for a move to an unoccupied neighbour cell $(i,j)$ (fig. \[trans\]) are determined by the local dynamics and 