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whether two particles “see” each other or whether the interaction is blocked by
some obstacle. All this is not necessary here.
For some applications it is useful to introduce a
matrix of preference
which
encodes the preferred walking direction and speed of each pedestrian. It is a
3
×
3 matrix (see Fig. 3) where the matrix elements
M
ij
can directly be
related
to observable quantities
, namely the average velocity and its fluctuations [11].
M
−
1
,−
1
M
−
1
,
0
M
−
1
,
1
M
0
,−
1
M
0
,
0
M
0
,
1
M
1
,−
1
M
1
,
0
M
1
,
1
Fig. 3.
A particle, its possible transitions and the associated matrix of preference
M
=(
M
ij
).
The area available for pedestrians is divided into cells of approximately 40
×
40
cm
2
which is the typical space occupied by a pedestrian in a dense crowd [20].
Each cell can either be empty or occupied by exactly one particle (pedestrian).
Apart from this simplest variant it is also possible to use a finer discretization,
e.g. pedestrians occupying four cells instead of one.
In contrast to vehicular tra
c the time needed for acceleration and braking
is negligible in pedestrian motion. The velocity distribution of pedestrians is
sharply peaked [21]. These facts naturally lead to a model where the pedestri-
ans have a maximal velocity
v
max
= 1, i.e. only transitions to neighbour cells
are allowed. Furthermore, a larger
v
max
would be harder to implement in two
dimensions and reduce the computational e
ciency.
The stochastic dynamics of the model is defined by specifying the transition
probabilities
p
ij
for a motion to a neighbouring cell (von Neumann or Moore
neighbourhood) in direction (
i, j
). The
transition probability p
ij
in direction (
i, j
)
is determined by the contributions of the static and dynamic floor fields
S
ij
and
D
ij
and the matrix of preference
M
ij
at the target cell:
p
ij
=
Ne
k
D
D
ij
e
k
S
S
ij
M
ij
(1
− n
ij
)
ξ
ij
.
(1)
N
is a normalization factor to ensure
(
i,j
)
p
ij
= 1 where the sum is over the
possible target cells. The factor 1
− n
ij
, where
n
ij
is the occupation number
of the neighbour cell in direction (
i, j
), takes into account that transitions to
occupied cells are forbidden.
ξ
ij
is a geometry factor (obstacle number) which
is 0 for forbidden cells (e.g. walls) and 1 else. The coupling constants
k
D
and
