Teoría de Campo Medio en el modelo de Bonabeau

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1/(N − 1) , j $ Pij (t) = 1 − Pij (t) =! #
N %i = 1, ..., N ' "$
hi (t + 1) = (1 − ρ)(1 − µ)hi (t) +
N
ρ(1 − µ) N −1
Pij (t) hi (t) + 1 +
j=1;j=i
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hi (t) − F
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9 N = 2 9 P12 (t) = 1 − P21 (t)! 10
0.5
N=2
0.4
5
σ
0.3
0
N=2
0.2
-5
0.1
0
0.2
0.4
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0.6
0.8
-10
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1
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# # %
(F ; µ; η)" (2; 0,1; 1) * (1; 0,1; 1) %
* (1; 0,1; 0,5) (' (1; 0,3; 1)
+ &
/ $
h1 (t+1) = (1−µ)h1 (t)+ρ(1−µ) P12 (t)(1+F )−F
-5
h(t + 1) = (1 − µ)h(t) +
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# ! !
"
! . (h∗1 , h∗2 ) ! $
h∗1 = h∗2 9 . . $
h(t) = (h1 (t)+h2 (t))/2 $
0
J = (1 − µ)
1−A
A
A
1−A
,
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ρ σ F ! η µ 3
ρc # ? 9 %F = 1' η = 1 µ = 0,1
N=2
5
5
h*
h*
0
0
-5
-10
N=2
(F = 2)
-5
-15
0.2
0.4
ρ
0.6
0.8
1
0.2
0" 1$# h∗1 h∗2 $ . F = 1 η = 1 µ = 0,1/ $ " h∗1 = h∗2 = h∗ = 0 ρ = ρc $# * * h∗1 = −h∗2 = 0
%
' ρc ≈ 0,11 , "
9 .
! h∗ = h∗2 −h∗1 3
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h∗ # $
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h =
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µ
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∗
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: # h∗ = 0! $ h∗1 = h∗2 =< h >∗ = 0! ρ > ρc $ h∗ = 0!
, ! h∗ = 0 −h∗ = 0! $
h∗1 = −h∗2 = +a/2 −h∗1 = h∗2 = −a/2
@
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$
h∗1 = h∗2 = h∗ = 0 ρ = ρc "# h∗1 = −h∗2 = 0
0.4
ρ
0.6
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1
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$ h(t) =
N
1
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N
i=1 i
N N
P = N(N−1)
!
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i=1
j=1,j=i ij
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h(t + 1) = (1 − µ)h(t) −
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2
. : #
N # 0 #
6 (h∗1 , h∗2 , . . . , h∗N ) h∗1 = h∗2 = . . . = h∗N = h∗ .
# 3 2
N=2
η = 0.5
N=2
µ = 0.3
5
h*
h*
1
0
0
-1
-5
-2
0.2
0.4
ρ
0.6
0.8
1
0.2
0.4
ρ
0.6
0.8
1
3" 1$# h∗1 h∗2 F = 1* µ = 0,3 η = 1,0
4" 1$# h∗1 h∗2 F = 1* µ = 0,1 η = 0,5
. $
⎛
⎜
J = (1 − µ) ⎝
1−A
A
N−1
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A
N−1
A
N−1
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...
A
N−1
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4
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N−1
&! "
$
ρ < ρc =
h2 (t+1) = h2 (t)+ρ(1−P12 (t)(1+F ))−µ tanh(h2 ).
7
0 ≤ ρc ≤ 1! # $
η(1 + F )
.
4 + η(1 + F )
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#
$ hi (t) $ hi (t) −
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,
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$ ! Q . 8f N = 10 ,
F = 0,7! Q = 0,7! µ = 0,0001 η = 0,001 - INEXISTENCIA DE
0.4
µ
TRANSICION
TRANSICION POSIBLE
0.2
0
0
1
2
3
η . (1+F)
4
5
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N=3
N=4
5
h*
h*
5
0
-5
0
-5
0.2
0.4
ρ
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0.8
1
N=6
N=8
5
h*
5
h*
ρ
0
-5
0
-5
0.2
0.4
ρ
0.6
0.8
1
0.2
0.4
ρ
0.6
300000
N = 10
5
200000
N = 10
relajación aditiva
h*
h*
100000
0
0
-100000
-200000
-5
-300000
0.2
0.4
ρ
0.6
0.8
1
0.1
0.2
ρ
0.3
0.4
0.5
6" 1$ -& 0 $# * $ " ./
N = 3* ./ N = 4* ./ N = 6* ./ N = 8 ./ N = 10 .$/ * N = 10* 1 &#