An Evacuation Simulation Based on Boltzmann Distribution - Advanced Research on Computer Education, Simulation and Modeling - page 329

1

→→→ β , G ε , P P (3)

i l i l

G

So in the “attractive field” which is equivalent in the evacuation, the probability dis-
tribution of the

th

i

person is:

G
G

−

i

we ⋅

p =

l

(4)

i G

G

i

−

∑

we ⋅

l

l

w

= 1

Where :

G
G

−

i

e

p = ∑

(5)

i G

G

i

−

e

l

The equation (5) is the probability distribution in the “attractive field”, we can ap-
proximately consider the distribution is also available in the specific evacuation space.

2.4 The Specific Description of the Evacuation Model

We treat the specific area of the evacuation as an attractive field, in which the poten-
tial energy decides the speed and direction of people's moving. In the real case, there
are two means factors which lead the people's moving, one is the attractive force of
the exits and the other is the influence the other people because of the group psychol-
ogy. Based on the Boltzmann Distribution mentioned above, we come up with a prob-
ability vector

th

i

to describe the next moving state of the people:

PPP = (, )

12

l
l

i

l

−

− K

ci
c

e

l

c

e

P

i

= ∑

(6)

i

= ∑

P 1 l

l

2 K l

i
c

−

ci
c

−

e

e

i i

P represents the influence
from the other people. l c represents the bar centric coordinates of the person at one
moment. l ci represents that bar centric coordinates all the other people.
As the coordinates of the exits will not change as time goes by, so the influence of
the exits can be seen as an absolute influence and the influence of the people is a
relativistic influence. So we introduce the parameter K to differ

P represents the influence of the exits on the

th

person,

i

i

i

1

2

P . In fact, K
represents the intensity of the attractiveness. We call it the indicator constant, so the
intensity will increase if K increases, and vice versa.
From the above, the moving direction of the th

P from

i i

2 1

person is the direction of

i

PP , and the value of the speed is the maximum a person can reach.
As congestion will happen near the exits, so there will be very serious overlap
situations if we keep the people's speed to be their maximum all the time. So we in-
troduce a distance indicator

max{ , }

12

to represent the extent of the people's congestion.

ε

In the range of ε , we give the person who has the minimum distance to the exit the