Initial flux fluctuations of random walks on regular lattices Sooyeon YOON, Byoung-sun AHN, Yup KIM...
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Initial flux fluctuations of random walks on regular lattices Sooyeon YOON, Byoung-sun AHN, Yup KIM (Dept. of Phys.,Kyung Hee Univ.)
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Transcript of Initial flux fluctuations of random walks on regular lattices Sooyeon YOON, Byoung-sun AHN, Yup KIM...
Initial flux fluctuations of random walks
on regular lattices
Sooyeon YOON, Byoung-sun AHN, Yup KIM(Dept. of Phys.,Kyung Hee Univ.)
Abstract
We study a characteristic coupling between average flux <fi> and dispersion σi at each site i on 1 and 2-dimensional lattices when random fluctuated incoming walkers from the outside undergoes random diffusion. We assume that variations of the number of walkers, W, define a dynamic variable chosen from an uniform distribution in the interval[W-ΔW, W+Δ W]. Each walker does M-step random walks from a location which they are initially placed at random on 1, 2-dimensional lattices. After M steps, we measured the average flux vs dispersion at each lattice site. We study the dispersion depends on the average flux as σ~<f>α. The two scaling regimes are found.
one is the regime with α ≃ 0.5 and the other is one with α=1.
Disorder System Research Group Kyung Hee University
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1. Motivation of Study
(1) Fluctuation in Complex Network Structure. M. Atgollo de Menezes and A.-L. Barabasi, Phys. Rev. Let
t. 92, 028701(2004)
The coupling of the flux fluctuations with the Total flux on individual nodes obeys a unique scaling law.
- Internet, Microchip, WWW, River Network, Highway
(2) How about on Regular Lattice?
(3) How does dispersion depend on the time?
σ~<f>α
Disorder System Research Group Kyung Hee University
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2. Model
• We assume that variations of the number of w
alkers, W, define a dynamic variable chosen from an uniform distribution in the interval [W-ΔW, W+Δ W].
• We start with 1D, 2D lattice of size N.
(3) Each walker does T-step random walks from a location which they are initially placed at random.
(4) After T-steps, we measured the average flux <fi> vs dispersion σi at each lattice site.
22 iii ff
Disorder System Research Group Kyung Hee University
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3. Simulation Results
(1) 1D N=10000, W=10000, R=100, T=100
)()( ~ TiTi f
Disorder System Research Group Kyung Hee University
80 85 90 95 100 105 110 115 120
30
40
50
60
70
80
101 102 103 1040.5
0.6
0.7
0.8
0.9
1.0
101 102 103 104
30
40
50
60
70
W = 0 W = 20 W = 40 W = 80 W = 100 W = 200 W = 800 W = 1000 W = 4000 W = 10000
i
<fi>
< i>
W
4
i
TiTi N)()(
1
Disorder System Research Group Kyung Hee University
t~
100 101 102 103 104
100
101
102
103
104
101 102 103 1040.5
0.6
0.7
0.8
0.9
1.0
W = 0 W = 20 W = 40 W = 80 W = 100 W = 200 W = 800 W = 1000 W = 4000 W = 10000
< i>
T
W
5
(2) 2D N=100, W=10000, R=100
if~
Disorder System Research Group Kyung Hee University
80 85 90 95 100 105 110 115 120
20
30
40
50
60
70
101 102 103 1040.5
0.6
0.7
0.8
0.9
1.0
101 102 103 10410
20
30
40
50
60
W = 0 W = 20 W = 40 W = 80 W = 100 W = 200 W = 800 W = 1000 W = 4000 W = 10000
i
<fi>
< i>
W
6
Disorder System Research Group Kyung Hee University
100 101 102 103 104
100
101
102
103
104
101 102 103 1040.5
0.6
0.7
0.8
0.9
1.0
T
< i>
W = 0 W = 20 W = 40 W = 80 W = 100 W = 200 W = 800 W = 1000 W = 4000 W = 10000
W
t~
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(3) Scale-free Network (=3)
N10000, W=10000, R=100
if~
Disorder System Research Group Kyung Hee University
0 50 100 150 200 250 3000
20
40
60
80
100
120
140
160
101 102 103 1040.5
0.6
0.7
0.8
0.9
1.0
101 102 103 10410
20
30
40
50
60
W = 0 W = 20 W = 40 W = 80 W = 100 W = 200 W = 800 W = 1000 W = 4000 W = 10000
i
<fi>
< i>
W
8
Disorder System Research Group Kyung Hee University
100 101 102 103 104
100
101
102
103
104
101 102 103 1040.5
0.6
0.7
0.8
0.9
1.0
W = 0 W = 20 W = 40 W = 80 W = 100 W = 200 W = 800 W = 1000 W = 4000 W = 10000
< i>
T
W
t~
9
Disorder System Research Group Kyung Hee University
(1) W=0 : only internal fluctuations which comes from the random diffusion
(2) W0 :The total incoming flux is changed at each experiment.The external driving force dr(W) contributes to the fluctuations with a dispersion.
22 )()()( tWtWWdr
2ext2int2 )()( iii The total fluctuations for node i,
)(ext WA drii factorgeometricaAi :
W=0 , 2/1int iii fa .vanishedisexti
int)( idr
i WA For,
, the fluctuation are dominated by the changes in the external driving force.
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Disorder System Research Group Kyung Hee University
Assume, )()( tWAtf ii
)()( tWAtf ii
)()( 22 tWAtf ii
driiii Aff 22
i
drextii f
tW
)(
ii f~ )1( 2dr
22
)(
iiii f
tWf
Therefore, increasing the fluctuations W induce a change from the =1/2 to the =1.
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4. Summary and Discussion
Disorder System Research Group Kyung Hee University
We study a characteristic coupling between average flux <fi> and dispersion σi at each site i on 1 and 2-dimensional lattices when random fluctuated incoming walkers from the outside undergoes random diffusion.
1. We found the coupling of the average flux and the dispersion satisfies the scaling relation σ~<f>α .
(1) When the small initial flux comes in the system, 1/2. (2) But the initial incoming flux becomes larger, 1. These results do not depend on the network stru
ctures. 2. We also found that the effect of the incoming flux
through the relation of dispersion and time. The time dependence of the dispersion changes from to as increasing the incoming flux. 2
1
t 1t
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