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or exclusions. These influences decide a person's speed and direction of escaping. In
fact, the space with attractions is similar to the “field” in classical physics. In the
field, a particle will present different kinds of moving states as the result of the influ-
ence of the field. We define the influence generalized applied force
F
. In physics, the
random motion can be seen as a kind of thermal motion. Although the bulk movement
of the people has some what a trend, the specific moving state isn't regular and
somewhat disordered. Thus it is reasonable to introduce the concept of the statistical
physics. We introduce the influence factor —— the attractive potential energy, which
can also be called Attraction Indicators:
F
G
=
i
i
F
∑
to present the average value of the indicator.
If everyone is at the same location (of course it will never happen in the real case),
then the force
∑
. We introduce
G
F
Where
G
=
F
=
N
N
. Here
G
can be seen as the ground
state energy which can also be called the minimum energy.
F
is the same, which leads to
G
=
1
2.2 The Boltzmann Distribution
If the evacuation is treated as the thermo logy behavior of a particle in a field, then we
can introduce the Boltzmann Distribution of the statistical physics [14]. That is, in a
specific energy level, the amount of the particles
−−
=⋅
, where
w
is degen-
eracy. Though a specific person in the space has
N
kinds of states, different people
can't be at the same location. That means the evacuation of the people should be in
agreement with the no-degenerate conditions, which leads
awe
αβε
l
l
l
=
. So an individual
state which a person occupies can be seen as an energy level of this system.
l
w
1
ε
is the
1
kT
specific energy of a level and
β =
(
k
is the Boltzmann constant).
The total amount of particles satisfies the following equation:
∑∑
we
α
−−
βε
Na
=
=
⋅
l
(1)
l
l
l
l
From the view of the classical statistics, the probability of a particle at a certain en-
ergy level is:
a
we
⋅
−−
αβε
we
⋅
−
βε
l
l
P
==
l
l
=
l
(2)
∑
∑
l
−−
αβε
−
βε
N
w
⋅
e
w
⋅
e
l
l
l
l
l
l
Then the probability of a person at a certain state in the space should be similar to
equation (2).
2.3 The Probability Distribution of People
Based on the Attraction Indicators in the real circumstances and the Boltzmann Dis-
tribution, we make the following analogy:
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