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Effects of Connections and Initial Conditions
on Spreading Phenomena
24
31-116903
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
i
2013 6 1
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
ii
31-116903
1927 McKendrick
[1]. S( ),I( ),R( ) 3 ,
S, I,R .
I S ,
,
.
. . ,
1.1 ( ) ( ) .
1 [2]
, ( )
. ,SIR
SIR
.
SIR ,
λ . N
, 1 I N − 1 S
,λ ,
. ,
. 2002 Moreno [3] , (
) ( )
λ ∼ 0 ,
.
,
.
, SIR
, .
.
, ,
( ) R
. , ,
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
iii
.
, SIR ,
, 0 1
[4, 5]. ,
, 1
, .
, .
.
-1
-0.5
0
0.5
1
1.5
0 0.5 1 1.5 2 2.5 3
U(Y
)
Y
Y*
❍❍
λ=0.51.2
2 U(Y )
.N → ∞, t � O(N)
, SIR I
Y
Y = −∂Y U(Y ) +√2Dξ (1)
. , D = (λ+ 1)/8 , U(Y )
U(Y ) = −1
4(λ− 1)Y 2 +
1
8(λ+ 1) log(Y ) (2)
. 3.3 , U(Y ) λ ≤ 1 , Y (t)
0 . 1 . λ > 1( ) U(Y ) Y∗, ,Y → ∞( )
. U(Y )
. [6].
, ,
SIR ,
.
,2012 10 ( )
. .
1. Y. Kermack. Bulletin of mathematical biology, Vol. 53, No. 1, pp. 33–55, 1991.
2. V. Krebs. Mapping the spread of contagions via contact tracing, 2003.
3. Y.Moreno, et al. Eur. Phys. J. B, Vol. 26, No. 4, pp. 521–529, 2002.
4. L.K. Gallos, et al. Physica A, Vol. 330, No. 1, pp. 117–123, 2003.
5. A. Lancic, et al. Physica A, Vol. 390, No. 1, pp. 65–76, 2011.
6. J. Iwai and S. Sasa. To be submitted soon.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
v
1 1
1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 SIR . . . . . . . . . . . . . . . . . . . . . 4
1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 SIR 9
2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 11
. . . . . . . . . . . . . . . . . . . . 11
. . . . . . . . . . . . . . . . . . . . 12
2.3 . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
λc . . . . . . . . . . . 13
λ→ ∞ 〈tf 〉, σ(tf )2 . . . . . . . . . . . . 15
2.4 m=1 , . . . . . . . . . . . . . 20
2.5 m ∝ N , . . . . . . . . . . . 20
2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 . . . . . . . . . . . . . . . . . . 24
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
vi
2.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 27
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5.1 SIR . . . . . . . . . . . . . 36
Newman . . . . . . . . . . . . 37
. . . . . . . . . . . . . . . . . . 44
SIR . . . . . . . . . . . . . . . . . . . . . . 45
3.5.2 . . . 47
3.5.3 . . . . . . 47
3.5.4 . . . . . . . . . . . . . . . . . . . 51
3.5.5 . . . . . . . . . . . . . . . . 52
3.5.6 Q(Y, t) (3.10) . . . . . . . 54
3.5.7 SIS . . . . . . . . . . 54
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 59
4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 . . . . . . . . . . . . . . . . . . . . . . 62
4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 67
5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
vii
A 69
A.1 . . . . . . . . . . . . . . . . . . . . . . 69
A.1.1 . . . . . . . . . . . . . . . . . . . . . 69
A.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A.2 SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.2.2 §4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
73
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
1
1
,
, . ,
, . , ,
, , .
1.1
2 , 2
, ,
,
, , ,
. ,
, ,
. ,
, , ,
.
. [1] .
. ,
, .
, , 2
.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
2 1
1.2
,
.
. ,
[2]. ,
. ,
.
, . ,
,
,
. , .
, . ,
,
, ,
.
,
[3]. .
, .
, . 2
. . G(V,E)
V = v1, v2, · · · , vN E = e1, e2, · · · em. ei 2 ,ei = {j, k} j k
. . N × N A
Aij =
{1 for ∃k, ek = {ij}0 for ∀k, ek = {ij} (1.1)
. 2 i, j Aij = 1,
Aij = 0 . , i, j Aij
. Aij = 0 . ,
, .
. ,
,
[4] . .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
1.2 3
( ) Liu
Controllability of complex networks[4]
. .
,
. G(V,E) . i xi
, x .
:
dx(t)
dt= Ax+Bu(t) (1.2)
A G ,N ×M B N ≤ M ,
M u . (controllablity) ,N ×NM
C = (B,AB,A2B, · · ·AN−1B) rank(C) = N
, x(0) τ x(τ)
u . G
.
m 2N − 1
, N . , ,
(structural controllability) . A ,0 A
, rank(C) = N N ×M B .
B M ”maximum matching” ,
ND . ND N nD
. M u(t) ,
M , .
,
, .
, ,
,
. ,
,B , .
,ND = 1 , , .
,
, , ,
ND .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
4 1
, .
, ,
, .
§1.3 , .
1.3
.
( 1.1),
. , ,
, ,
. SIR .
1.1
[5]
1.3.1 SIR
SIR .
.
, 3 S,I,R
. S ,I S
, R I
.
Susceptible,Infected,Recoverd .
. S I
, S-I , λ S I ,I-I .
. S-I . ,I
μ R . .
. , λ , t
t + Δt Δt S-I I-I dPI
dPI = λΔt+O(Δt2) . , I R dPR
dPR = μΔt+O(Δt2) . Δt→ 0 O(Δt2) .
,N , 1
I , N − 1 S .
I ,S R . .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
1.4 5
R nR , .
N [6] , .
,
, , N N → ∞ ,
. , εN ,N → ∞ε → 0 , N m
. 1 ,
. Boguna [7] m = 1
N → ∞ ρ0 ρ0 = 1/N
, ρ0 N ρ0 = ε .
§3 .
1.4
.
SIR , SIR 1927
McKendrick [8]. S,I,R
S, I, R , .⎧⎨⎩S = −λSII = λSI − μI
R = μI
(1.3)
S + I +R = N . SIR
.
SIR
Mollison[9] Grassberger[10] . 1998 ,
[11]
[12] , .
, k P (k) P (k) ∼ k−γ
.
,
. SIR
, λ/μ ,
nR = 0 , nR 0
. 2002 , Moreno [13]
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
6 1
. ,
λ/μ ∼ 0 .
λ ,
. , SIS
[14]
, 0 . ,
,
[15].
1.4.1
SIR .
SIS SIR I R ,
SIS S , . SIR
, SIS
.
SIR
,
[16]. §3 .
λ λΔt .
,
.
, S
I SEIR [17] ,
[18] .
1.5
.
§2§2 , ,
SIR . nR tf ,
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
7
, .
§3§3 ,SIR
. ,2013 6 1
. ,
, SIR
.
§4§4 ,SIR ,
. , . S
I , I
, SIR
.
.
§5§5 , .
, .
[1] Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari,
S. Tadaki, and S. Yukawa. Traffic jams without bottlenecks–experimental evi-
dence for the physical mechanism of the formation of a jam. New Journal of
Physics, Vol. 10, No. 3, p. 033001, 2008.
[2] H. Haken. Information and self-organization: A macroscopic approach to complex
systems, Vol. 40. Springer, 2006.
[3] . .
, 3 2012.
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8 1
[4] Y.Y. Liu, J.J. Slotine, and A.L. Barabasi. Controllability of complex networks.
Nature, Vol. 473, No. 7346, pp. 167–173, 2011.
[5] V. Krebs. Mapping the spread of contagions via contact tracing, 2003.
[6] J.C. Miller. Epidemics on networks with large initial conditions or changing
structure. arXiv preprint arXiv:1208.3438, 2012.
[7] M. Boguna, C. Castellano, and R. Pastor-Satorras. Langevin approach for the
dynamics of the contact process on annealed scale-free networks. Physical Review
E, Vol. 79, No. 3, p. 036110, 2009.
[8] WO Kermack and AG McKendrick. Contributions to the mathematical theory
of epidemics-i. Bulletin of mathematical biology, Vol. 53, No. 1, pp. 33–55, 1991.
[9] D. Mollison. Spatial contact models for ecological and epidemic spread. Journal
of the Royal Statistical Society. Series B (Methodological), pp. 283–326, 1977.
[10] P. Grassberger. On the critical behavior of the general epidemic process and
dynamical percolation. Mathematical Biosciences, Vol. 63, No. 2, pp. 157–172,
1983.
[11] D. Watts and S. Strogatz. The small world problem. Collective Dynamics of
Small-World Networks, Vol. 393, pp. 440–442, 1998.
[12] A.L. Barabasi and R. Albert. Emergence of scaling in random networks. science,
Vol. 286, No. 5439, pp. 509–512, 1999.
[13] Y. Moreno, R. Pastor-Satorras, and A. Vespignani. Epidemic outbreaks in com-
plex heterogeneous networks. The European Physical Journal B-Condensed Mat-
ter and Complex Systems, Vol. 26, No. 4, pp. 521–529, 2002.
[14] C. Castellano and R. Pastor-Satorras. Routes to thermodynamic limit on scale-
free networks. Physical review letters, Vol. 100, No. 14, p. 148701, 2008.
[15] A.L. Barabasi and E. Bonabeau. Scale-free. Scientific American, 2003.
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infectious diseases in populations of moving individuals. Journal of Physics A:
Mathematical and General, Vol. 25, No. 9, p. 2447, 1999.
[17] J. Stehle, N. Voirin, A. Barrat, C. Cattuto, V. Colizza, L. Isella, C. Regis, J.F.
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[18] M.J. Keeling and K.T.D. Eames. Networks and epidemic models. Journal of the
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物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
9
2
SIR
2.1
,
[1]. Barabasi[2] , WWW
. , [3] ,
, [4], [5], SNS
[6] . ,
. ,E
, ,
,
.
,
. ,
.
, ,
[7].
, .
, ,
. [8] .
,
, . , ,
,
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
10 2 SIR
.
.
, .
SIR , m m = 1 . ,
SARS , 1
. , 1
, ,
, . ,
.
, ,
. , m = 1
.
. ( 2.1) , 10
,2 .
, 10
,
[9]. ,
. ,
,
. 10 ,
. , m = 1
m > 1 .
. , ,
. SIR
[10, 11] , ,
m = 1 . ,
. ,
. ,
. ,
.
, ,
, .
, nR
tf .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
2.2 11
. nR 〈nR〉 λ ,
λc 0,λc ,
. . ,λ→ ∞ 〈tf 〉, σ(tf )2
. m = 1 nR, tf
, .
N ,
. , N , nR tf
, ,
.
2.2
,
.
2.2.1
,
m = 1 . ,
,m = εN . ε� 1 .
2.2.2
N .
. 2
. ,
.
. ,
,
. ,
, .
, . Hi, i = 1, 2, · · ·n i
, Yj , j = 1, 2, · · ·n j .
. c . ρ1(l), ρ2(l)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
12 2 SIR
2.1
2.2 HY
[1, 2, 3, · · ·N ] → [1, 2, 3, · · ·N ] .σ(a, b) a b
. l, l = 1, 2, · · ·N Hσ(ρ1(l),c), Yσ(ρ2(l),c) .
l 2 Hσ(ρ1(l),c)Yσ(ρ2(l),c) ,
H,Y c ,
. HY .
λ . 2.2 c = 4
.
, ,
.
,
. λ/N
. .
2.3
. k ,
.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
2.3 13
2.3.1
, .
m NR tf .〈·〉,σ(A)2 A 〈A2〉 − 〈A〉2 , Σ(A)2 = Nσ(A)2 = σ(NA)2/N .
nRNR
N 〈nR〉, σ(nR),Σ(nR), 〈tf 〉, σ(tf ) , λ .
2.3 k = 3,m = 1 N = 1024, 2048, 4096, 8192 N
. 〈nR〉, σ(nR)2 N . 〈tf 〉 λ < λc
, λ→ ∞ N . σ(tf )2
λc , . 2.4 N = 8192,m = 1
k = 3, 4, 5 k . 2.3 λc ≈ 1/(k − 2)
. 2.5 k = 3 m = N/256, N = 1024, 2048, 4096, 8192
N . 2.4 Σ(nR)2 N → ∞
. m = 1 2.3.1
(〈N2R〉 − 〈NR〉2)/N2 m = N/256 2.4 (〈N2
R〉 − 〈NR〉2)/NN → ∞ . 1
( ) ,m = N/256
. m = 1 .§3.
2.3.2
λc 〈tf 〉, σ(tf )2 .
λc
k m = 1 λc
. . k
, .
. S-I I-I λ ,I R
1 . t S,I,R S, I,R
.I ∼ O(N) dI/dt
. I ∼ O(N) I I
, I ∼ O(N) . I ∼ O(N) dI/dt = 0 λ
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
14 2 SIR
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
n R
λ
m=1,N=1024N=2048N=4096N=8192
(a) 〈nR〉
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 1 2 3 4 5
σ(n R
)2
λ
N=1024N=2048N=4096N=8192
(b) σ(nR)2
0
2
4
6
8
10
12
14
16
18
0 1 2 3 4 5
tf
λ
N=1024N=2048N=4096N=8192
(c) 〈tf 〉
0
50
100
150
200
250
300
350
400
450
500
0 1 2 3 4 5
σ(tf)
2
λ
N=1024N=2048N=4096N=8192
(d) σ(tf )2
2.3 k = 3,m = 1, N = 1024, 2048, 4096, 8192
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5
n R
λ
k=3k=4k=5
(a) 〈nR〉
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 1 2 3 4 5
σ(n R
)2
λ
k=3k=4k=5
(b) Σ(nR)2
0
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5
<tf>
λ
k=3k=4k=5
(c) 〈tf 〉
0
50
100
150
200
250
300
350
400
450
500
0 1 2 3 4 5
σ(tf)
2
λ
k=3k=4k=5
(d) σ(tf )2
2.4 k = 3, 4, 5, N = 8192
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5
n R
λ
N=1024N=2048N=4096N=8192
(a) 〈nR〉
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5
Σ(n R
)2
λ
N=1024N=2048N=4096N=8192
(b) Σ(nR)2
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5
<tf>
λ
N=1024N=2048N=4096N=8192
(c) 〈tf 〉
0
10
20
30
40
50
60
70
80
90
100
0 1 2 3 4 5
σ(tf)
2
λ
N=1024N=2048N=4096N=8192
(d) σ(tf )2
2.5 m = N/256, N = 1024, 2048, 4096, 8192
.
dI
dt= [ S I ]− [ I R ] (2.1)
. . S I
,
[S − I ]× λ (2.2)
. I R ,
I (2.3)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
2.3 15
. S-I . R
t � O(1) , S I , I
m = 1 1 .
S . I kI .
I − 1 , 1 2
, S
kI − 2(I − 1)
. (2.2),(2.3) 2.1
dI
dt= λ [kI − 2(I − 1)]− I
. dI/dt = 0
λc [kI − 2(I − 1)] = I
⇒ λc =1
k − 2(1− 1/I)
.N → ∞ 1/I → 0
λc =1
k − 2(2.4)
. (2.4) 2.4 .
λ→ ∞ 〈tf 〉, σ(tf )2N 〈tf 〉N 〈tf 〉, σ(tf )2N = σ(tf )
2 N
. λ , λ � λc
〈tf 〉 . , m ≥ 1
I .
. m = N , R tf
PN (tf ) . 1 I R t p(t)
. . , μ δt
μδt . δt 1 − μδt
. t , t δt tδt
. (1− μδt)tδt
. t
p(t)
p(t) = limδt→0
(1− μδt)t/δt
= μe−μt
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
16 2 SIR
.
PN (tf ) . i R ti ,PN (tf )
PN (tf ) =
∫0≤ti≤tf ,Max[ti]=tf
dt1dt2 · · · dtNp(t1)p(t2) · · · p(tN )
= N !
∫ tN
0
dtN−1
∫ tN−1
0
dtN−2 · · ·∫ t2
0
dt1p(t1)p(t2) · · · p(tN )
.i = j ti = tj R ,
0 < t1 < t2 < · · · < tN = tf
,PN (tf ) N ! .
2.5 ,pi(t) .
p1(t) = p(t)
pi(t) =
∫ t
0
pi−1(t′)p(t′)dt′
PN (t) pi(t)
PN (t) = N !pN (t)
. pi(t) .
p1(t) = μe−μt
p2(t) =
∫ t
0
μe−μt′dt′μe−μt
= 1− e−μt
p3(t) =
∫ t
0
(1− e−μt′
)dt′μe−μt
=1
2
(1− e−μt
)2μe−μt
...
pN (t) =1
(N − 1)!
(1− e−μt
)N−1μe−μt
PN (t) = N(1− e−μt
)N−1μe−μt (2.5)
.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
2.3 17
〈tf 〉N , σN (tf )2 .
〈tf 〉N =
∫ ∞
0
tPN (t)dt
=N
μ
∫ ∞
0
x(1− e−x
)N−1e−xdx
X = e−x .
〈tf 〉N =N
μ
∫ x
0
x(1− e−x
)N−1e−xdx
=N
μ
∫ 0
1
(1−X)N−1
logXdX
∫ 0
1(1−X)
N−1logXdX
∫ 0
1
(1−X)N−1
logXdX =[(1−X)
N−1X (logX − 1)
]01
−∫ 0
1
(N − 1)(−1) (1−X)N−2
X (logX − 1)
=
∫ 0
1
(N − 1) (1−X)N−2
X (logX − 1)
= (N − 1)
∫ 0
1
[− (1−X)
N−1logXdX
]dX
+(N − 1)
∫ 0
1
[− (1−X)
N−2X + (1−X)
N−2logX
]dX
⇔ N
∫ 0
1
(1−X)N−1
logXdX = (N − 1)[− (1−X)
N−2X + (1−X)
N−2logX
]dX
= (N − 1)
[1
N (N − 1)+
∫ 0
1
(1−X)N−2
logXdX
]
=1
N+ (N − 1)
∫ 0
1
(1−X)N−2
logXdX
⇔ 〈tf 〉N =1
μN+ 〈tf 〉N−1
〈tf 〉N =1
μ
N∑j=1
1
j(2.6)
.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
18 2 SIR
. .
I N 1 R N − 1 I
∫ ∞
0
tNμe−Nμtdt =1
μN
.I N −1 〈tf 〉N−1
, 2.6 .
〈t2f 〉N .
〈t2〉N = μN
∫ ∞
0
(1− e−μt
)N−1e−μtdt
=N
μ2
∫ ∞
0
x2(1− e−x
)N−1e−xdx
∫ ∞
0
x2(1− e−x
)N−1e−xdx =
∫ 1
0
(logX)2(1−X)
N−1dX
X = e−x .∫(logX)
2dX = X
[(logX)
2 − 2logX + 2]
= −(1 − X)[(logX)
2 − 2logX + 2]+[
(logX)2 − 2logX + 2
].
∫ 1
0
(logX)2(1−X)
N−1dX =
[∫(logX)
2dX (1−X)
N−1
]10
+(N − 1)
∫ 1
0
∫(logX)
2dX (1−X)
N−2dX
= (N − 1)
[∫ 1
0
(logX)2{− (1−X)
N−1+ (1−X)
N−2}dX
+2
∫ 1
0
{logX (1−X)
N−1 − logX (1−X)N−2
}dX
+ 2
∫ 1
0
{− (1−X)
N−1+ (1−X)
N−2}dX
]
⇔ N
∫ 1
0
(logX)2(1−X)
N−1dX = (N − 1)
∫ 1
0
(logX)2(1−X)
N−2dX
+2 (N − 1)
NN
∫ 1
0
logX (1−X)N−1
dX
−2 (N − 1)
∫ 1
0
logX (1−X)N−2
dX
+2
N
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
2.3 19
〈t2〉N = 〈t2〉N−1 − 2 (N − 1)
μN〈t〉N +
2
μ〈t〉N−1 +
2
μ2N
= 〈t2〉N−1 +2
μN〈t〉N
⇔ 〈t2〉N =2
μ
N∑i=1
1
i〈t〉i
=2
μ2
N∑i=1
1
i
i∑k=1
1
k
=1
μ2
⎡⎢⎣⎛⎝ N∑
j=1
1
j
⎞⎠
2
+
N∑j=1
1
j2
⎤⎥⎦
. σN (tf )2
σN (tf )2 = 〈t2〉N − 〈t〉2N
=1
μ2
⎡⎣( N∑
k=1
1
k
)2
+N∑
k=1
1
k2
⎤⎦−
(1
μ
N∑i=1
1
i
)2
σN (tf )2 =
1
μ2
N∑j=1
1
j2(2.7)
.
(2.6),(2.7) . μ = 1, N = 1024, 2048, 4096, 8192
(2.6),(2.7) 〈tf 〉N , σN (tf )2 . HN,n =
∑Nk=1 1/k
n (harmonic
number) . MathWorld[12] n = 1
HN,1 ∼ lnN + γ (2.8)
. γ � 0.577 . (2.8)
H1024,1 � 7.51, H2048,1 � 8.20, H4096,1 � 8.90, H8192,1 � 9.59
H1024,2 � 1.64, H2048,2 � 1.64, H4096,2 � 1.64, H8192,2 � 1.64
, 2.3,2.4,2.5 (c),(b) .
(2.6),(2.7) .
mathematica ,∫∞0dteatPN (t)
a a = 0 .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
20 2 SIR
2.4 m=1 ,
m = 1 . 2.6, 2.7,
2.9 , ,HY .
, 〈k〉. *1 .
〈nR〉 2.4,2.6,2.7,2.8 , 〈nR〉 λc 0
, 〈nR〉 = 1 .
λc λc � 0.25,HY
λc � 0.35 , ( )
.
σ(nR)2 2.4,2.6,2.7,2.8 , σ(nR)
2 λc 0
λ � 0.6 ,
.
〈tf 〉 2.4,2.6,2.7,2.8 , 〈tf 〉 λc
, N .
σ(tf )2 2.4,2.6,2.7,2.8 , λ < λc 0 ,λc
0 . N
, .
2.5 m ∝ N ,
m = εN . 2.10,
2.11, 2.12, 2.13 , ,HY
. ε = N/128 .
〈nR〉 2.5,2.10,2.11,2.12 , 〈nR〉 λc 0
, 〈nR〉 = 1
. λc
λc � 0.25,HY λc � 0.35 ,
*1 , p ,N → ∞〈k〉N/2 = NC2p [13].
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
2.5 m ∝ N , 21
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
n R
λ
m=1,N=8192N=16384N=32768N=65536
(a) 〈nR〉
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
σ(n R
)2
λ
N=8192N=16384N=32768N=65536
(b) σ(nR)2
0
2
4
6
8
10
12
14
16
18
20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
tf
λ
N=8192N=16384N=32768N=65536
(c) 〈tf 〉
0
200
400
600
800
1000
1200
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
σ(tf)
2
λ
N=8192N=16384N=32768N=65536
(d) σ(tf )2
2.6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
n R
λ
m=1,N=8192N=16384N=32768N=65536
(a) 〈nR〉
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
σ(n R
)2
λ
N=8192N=16384N=32768N=65536
(b) σ(nR)2
0
2
4
6
8
10
12
14
16
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
tf
λ
N=8192N=16384N=32768N=65536
(c) 〈tf 〉
0
100
200
300
400
500
600
700
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2σ(
tf)2
λ
N=8192N=16384N=32768N=65536
(d) σ(tf )2
2.7
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
n R
λ
m=1,N=8192N=16384N=32768N=65536
(a) 〈nR〉
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
σ(n R
)2
λ
N=8192N=16384N=32768N=65536
(b) σ(nR)2
0
2
4
6
8
10
12
14
16
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
tf
λ
N=8192N=16384N=32768N=65536
(c) 〈tf 〉
0
100
200
300
400
500
600
700
800
900
1000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
σ(tf)
2
λ
N=8192N=16384N=32768N=65536
(d) σ(tf )2
2.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
n R
λ
m=1,N=8192N=16384N=32768N=65536
(a) 〈nR〉
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
σ(n R
)2
λ
N=8192N=16384N=32768N=65536
(b) σ(nR)2
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
tf
λ
N=8192N=16384N=32768N=65536
(c) 〈tf 〉
0
200
400
600
800
1000
1200
1400
1600
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
σ(tf)
2
λ
N=8192N=16384N=32768N=65536
(d) σ(tf )2
2.9 HY
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
22 2 SIR
.
Σ(nR)2 2.5,2.10,2.11,2.12 , Σ(nR)
2 λc 0
,
0 . N .
〈tf 〉 2.5,2.10,2.11,2.12 , 〈tf 〉 λ < λc N
λc , N
. N
.
σ(tf )2 2.5,2.10,2.11,2.12 , λ < λc 0
,λc
. N .
2.6
λc k λc
λc = 1/(k − 2) . 2.6,2.10 . HY
,k2, k ,
. ,
.
.HY ,
.
.
m = 1 σ(NR)2 Σ(nR)
2 σ(nR)2 , 2.4,2.6,2.8
, m = 1 σ(NR)2 ≈ N2 .
2.5,2.10,2.12 , m = εN σ(NR)2 ≈ N . NR
, ,σ(NR)2 ≈ N ,m = 1 NR
. §3 .
λ→ ∞ 〈tf 〉 2.4,2.6,2.8, 2.5,2.10,2.12 , λ→ ∞〈tf 〉 N . 2.3.2
.
m = εN Σ(nR)2, σ(tf )
2 2.5,2.10,2.12 , Σ(nR)2
N
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
2.6 23
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
n R
λ
m=1,N=8192N=32768N=65536
(a) 〈nR〉
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Σ(n R
)2
λ
N=8192N=32768N=65536
(b) Σ(nR)2
5
10
15
20
25
30
35
40
45
50
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
tf
λ
N=8192N=32768N=65536
(c) 〈tf 〉
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
σ(tf)
2
λ
N=8192N=32768N=65536
(d) σ(tf )2
2.10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
n R
λ
m=1,N=8192N=32768N=65536
(a) 〈nR〉
0
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Σ(n R
)2
λ
N=8192N=32768N=65536
(b) Σ(nR)2
5
10
15
20
25
30
35
40
45
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
tf
λ
N=8192N=32768N=65536
(c) 〈tf 〉
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2σ(
tf)2
λ
N=8192N=32768N=65536
(d) σ(tf )2
2.11
0
0.2
0.4
0.6
0.8
1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
n R
λ
m=1,N=8192N=16384N=32768N=65536
(a) 〈nR〉
0
2
4
6
8
10
12
14
16
18
20
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Σ(n R
)2
λ
N=8192N=16384N=32768N=65536
(b) Σ(nR)2
0
10
20
30
40
50
60
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
tf
λ
N=8192N=16384N=32768N=65536
(c) 〈tf 〉
0
20
40
60
80
100
120
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
σ(tf)
2
λ
N=8192N=16384N=32768N=65536
(d) σ(tf )2
2.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
n R
λ
m=1,N=8192N=32768N=65536
(a) 〈nR〉
0
2
4
6
8
10
12
14
16
18
20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Σ(n R
)2
λ
N=8192N=32768N=65536
(b) Σ(nR)2
5
10
15
20
25
30
35
40
45
50
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
tf
λ
N=8192N=32768N=65536
(c) 〈tf 〉
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
σ(tf)
2
λ
N=8192N=32768N=65536
(d) σ(tf )2
2.13 HY
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
24 2 SIR
. , 2.5,2.10,2.12 σ(tf )2
.
. §2.7 .
2.7
, , ,HY
. .
2.7.1
2.142.15 , , ,HY
,m = εN Σ(nR)2, σ(tf )
2 λ λc
.〈k〉 = 6, N = 16384, 32768, 65536. N < 70000
. N ,λ ≈ λc*2 .
2.8
, SIR
. m m = 1 m = εN nR, tf
. , , ,
,HY 4 ,
, ,
. HY 3 , λc
. λc,λ → ∞〈tf 〉, σ(tf )2 . m = εN ,σ(nR)
2, σ(tf )2
, .
, N
. ,
, .
*2 ,
, .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
25
0
2
4
6
8
10
12
14
16
18
20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Σ(n R
)2
λ
N=8192N=16384N=32768N=65536
(a)
0
2
4
6
8
10
12
14
16
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Σ(n R
)2
λ
N=8192N=16384N=32768N=65536
(b)
0
2
4
6
8
10
12
14
16
18
20
0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34
Σ(n R
)2
λ
N=8192N=16384N=32768N=65536
(c)
0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Σ(n R
)2
λ
N=8192N=16384N=32768N=65536
(d) HY
2.14 Σ(nR)2
0
10
20
30
40
50
60
70
80
90
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
σ(tf)
2
λ
N=8192N=16384N=32768N=65536
(a)
0
10
20
30
40
50
60
70
80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
σ(tf)
2
λ
N=8192N=16384N=32768N=65536
(b)
0
20
40
60
80
100
120
0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34
σ(tf)
2
λ
N=8192N=16384N=32768N=65536
(c)
0
10
20
30
40
50
60
70
80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
σ(tf)
2
λ
N=8192N=16384N=32768N=65536
(d) HY
2.15 σ(tf )2
[1] S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes. Critical phenomena in
complex networks. Reviews of Modern Physics, Vol. 80, No. 4, p. 1275, 2008.
[2] A.L. Barabasi and R. Albert. Emergence of scaling in random networks. science,
Vol. 286, No. 5439, pp. 509–512, 1999.
[3] F. Liljeros, C.R. Edling, L.A.N. Amaral, H.E. Stanley, and Y. Aberg. The web
of human sexual contacts. arXiv preprint cond-mat/0106507, 2001.
[4] R. Albert and A.L. Barabasi. Statistical mechanics of complex networks. Reviews
of modern physics, Vol. 74, No. 1, p. 47, 2002.
[5] M.E.J. Newman, S. Forrest, and J. Balthrop. Email networks and the spread of
computer viruses. Physical Review E, Vol. 66, No. 3, p. 035101, 2002.
[6] K. Yuta, N. Ono, and Y. Fujiwara. A gap in the community-size distribution of
a large-scale social networking site. arXiv preprint physics/0701168, 2007.
[7] , , , , , , , ,
, . . , Vol. 46,
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
26 2 SIR
No. 6, pp. 235–245, 2005.
[8] . ( )–(
). , Vol. 98, No. 8, pp. 1287–1291, 2010.
[9] (2012 ) ,
[10] Y. Chen, G. Paul, S. Havlin, F. Liljeros, and H.E. Stanley. Finding a better
immunization strategy. Physical Review Letters, Vol. 101, No. 5, p. 58701, 2008.
[11] R. Cohen, S. Havlin, and D. Ben-Avraham. Efficient immunization strategies for
computer networks and populations. Physical review letters, Vol. 91, No. 24, p.
247901, 2003.
[12] Jonathan Sondow and Eric W Weisstein. ”harmonic number” from mathworld–
a wolfram web resource. http://mathworld.wolfram.com/HarmonicNumber.
html, 2012.
[13] P. Erdos and A. Renyi. On the evolution of random graphs. Magyar Tud. Akad.
Mat. Kutato Int. Kozl, Vol. 5, pp. 17–61, 1960.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
27
3
[1]
. ,
§3.5 . , ,
1950 70 .
, [2].
3.1
,
. , ,
. ,
,
. , ,1
, , ,
. ,
.
SIR .
,SIR λ
. , ,
[3]. , ,
λ λc .
, .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
28 3
, [4, 5], [6, 7]
.
SIR , ,
.SIR nR ,
λ > λc 0 ,λ = λc
. , ,
1 . , SIR
. ,
SIR , 0
[8], [9]. ,SIR
[10, 11]. ,Ising
, . ,
,
. , ,
0 1 .
, ,
. , , 0
.
,SIR ,
. ,
. SIR ,
. ,
. ,
Langevin . Langevin
, 0 .
3.2
N k G . x ∈, k B(x) . σ
,σ = S, I, and R , ,Susceptible, Infectible, ,
Recovered . ,[σ(x)]x∈G , , σ
. SIR ,σ W (σ → σ′)
. , W ,I
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.3 29
λ I μ ,
W (σ → σ′) =∑x∈G
w(x),
w(x) = λ[δ(σx, S)δ(σ′x, I)
∑y∈B(x)
δ(σy, I)] + μδ(σx, I)δ(σ′x, R) (3.1)
. μ = 1 .
, , G I
, . . R
N nR ,
. , I
, S .
,k = 3, N = 8192 , λ , ,
. nR , p(nR;λ)
. 3.2 log p ,λ
. , ,nR
. 3.2 λ = 1.5
log p , .
,[10] . , 3.2
,nR < 1/16 λ . ,
, λc , 1 nR = 0 ,
q(λ) , nR ,1 − q(λ)
nR = 0 . , .
3.3
, s ≡∑x δ(σx, S)/N i ≡∑
x δ(σx, I)/N
. ,(s, i) → (s, i−1/N)
Ni , (s, i) → (s − 1/N, i + 1/N) λkNsψ
, ψ . ,ψ , σx = S x
,σy = I y x . , I
, loop
O(log(N) , ( )
.
Ni(k − 2) [12], ,ψ Nk
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
30 3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
nR
0
1
2
3
4
5
λ
-12
-10
-8
-6
-4
-2
0
0 0.2 0.4 0.6 0.8 1
nR
0
1
2
3
4
5
λ
-12
-10
-8
-6
-4
-2
0
-12-10-8-6-4-2 0
0 0.2 0.4 0.6 0.8 1nR
log(p)
λ=1.5
(a)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
p(n R
<1/1
6)
λ
N=8192N=16384N=32768N=65536
(b) λ− p(nR < 1/16)
3.1 SIR . , nR,
λ,log p . λ log p
. , λ, log p.λ = 1.5.
.N = 8192,m = 1. , λ, p(nR < 1/16). N → ∞ ,
nR < 1/16 .
. ,k = 3 .
, t (s, i) P (s, i, t) .
,P (s, i, t)
∂P (s, i, t)
∂t= N
(i+
1
N
)P
(s, i+
1
N, t
)−NiP (s, i, t)
+Nλ
(s+
1
N
)(i− 1
N
)P
(s+
1
N, i− 1
N, t
)−NλsiP (s, i, t)(3.2)
. ,N , 1/N
,
∂P (s, i, t)
∂t= −∂i
{[(λs− 1) iP (s, i, t)]− ∂i
[(λs+ 1) i
2NP (s, i, t)
]+ ∂s
[λsi
2NP (s, i, t)
]}
−∂s{− [λsiP (s, i, t)]− ∂s
[λsi
2NP (s, i, t)
]+ ∂i
[λsi
2NP (s, i, t)
]}(3.3)
+O
(1
N2
)(3.4)
.O(1/N2) ,
[13]. ,
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.3 31
. ,
3.5.2 .
(3.4) ,
. 3.5.3
. ,⎧⎨⎩ s(t) = −λsi−
√λsiN · ξ1
i(t) = λsi− i+√
λsiN · ξ1 +
√iN · ξ2
(3.5)
. ξi , 〈ξi (t)〉 = 0, 〈ξi (t) ξj (t′)〉 =
δijδ (t− t′) . (3.5) ξ1, ξ2 · .
(3.5) . (3.5)
⎧⎨⎩ s(t+Δt) = s(t)− λs(t)i(t)Δt−
√λs(t)i(t)
N ΔtN1(0, 1)
i(t+Δt) = i(t) + (λs(t)i(t)− i(t))Δt+√
λs(t)i(t)N ΔtN1(0, 1) +
√i(t)N ΔtN2(0, 1)
(3.6)
. Ni(0, 1) 0, 1 . 3.2 (s0, i0) =
(1 − 1/N, 1/N) . SIR
. ,(3.5) .
, Y =√iN ,(3.5) ,
{s = 1
N
[−λsY 2 −
√λsY 2 · ξ1
]Y = 1
2
{(λs− 1)Y − 1
4 (λs+ 1) 1Y
}+ 1
2
√λs · ξ1 + 1
2
√1 · ξ2
(3.7)
.(3.5) ξ1, ξ2 i ,(3.7)
. 3.5.4 . ,
nR ,
. , nR > 0 q(λ) .nR = 0 ,
Y � O(N1/2) . , (3.7)
.
, (3.7) s Y N
. ,N , Y s
. , O(N) ,s = 1 .
, (3.7) Y
Y = −∂Y U(Y ) +√2Dξ (3.8)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
32 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 1 2 3 4 5
n R
λ
N=512N=1024N=2048N=4096N=8192
(a) < nR >
0
0.05
0.1
0.15
0.2
0.25
0 1 2 3 4 5
sig2 (n
R)
λ
N=512N=1024N=2048N=4096N=8192
(b) σ(nR)2
0
2
4
6
8
10
12
14
0 1 2 3 4 5
<tf>
λ
N=512N=1024N=2048N=4096N=8192
(c) < tf >
0
50
100
150
200
250
300
350
400
0 1 2 3 4 5
sig2 (tf
)
λ
N=512N=1024N=2048N=4096N=8192
(d) σ(tf)2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
nR
0
1
2
3
4
5
λ
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
0 0.2 0.4 0.6 0.8 1
nR
0
1
2
3
4
5
λ
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-10-9-8-7-6-5-4-3-2-1 0
0 0.2 0.4 0.6 0.8 1
log(p)
λ=1.5
(e) log p .N = 1024,
N = 8192.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
p(n R
<1/1
6)
λ
N=1024N=2048N=4096N=8192
(f) p(nR < 1/16)
3.2 (3.5) .m = 1, N =
1024, 2048, 4096, 8192. SIR (
2.3,3.2,3.2) .
. , D = (λ+ 1)/8 , U(Y )
U(Y ) = −1
4(λ− 1)Y 2 +
1
8(λ+ 1) log(Y ) (3.9)
.ξ 1 ,√λ/2 · ξ1 + 1/2 · ξ2 =√
λ+ 1/2 · ξ . Y (0) = 1 .(3.8) N
. , Y → ∞ ,(3.7) Y � O(N1/2)
. ,(3.8) Y → ∞q(λ) , .
3.5.5 (3.4) s = 1, I = Ni
.
t Y Q(Y, t) ,Q(Y, t)
∂tQ(Y, t) = −∂Y [{∂Y U(Y )}Q(Y, t)] +D∂Y2Q(Y, t) (3.10)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.3 33
. Q(Y, 0) = δ(Y, 1) . (3.10) (3.4) s = 1, Y =√Ni .3.5.6 .
, U (Y ) . λ limY→+0
U (Y ) = −∞.λ < 1 U (Y ) lim
Y→∞U (Y ) → ∞ .1 < λ
limY→∞
U (Y ) = −∞ ,Y = Y∗ .
Y∗ =1
2
√λ+ 1
λ− 1(3.11)
. ,λ = 0.5, 1.2 3.3 .
,λ < 1 . ,U(Y ) Y ,
(3.8) Y → ∞ . ,
1 Y = 0 . ,q(λ) = 0 .
,λ > 1 . (3.8) Y Y∗, Y → ∞ . 3.3 , Y Y∗ Y → ∞
(3.8) , Y = Y∗ Y → ∞. , λ , Y Y∗
. Y = 1 Y = Y∗ W ,
t = 0 Y = 1 ,t = τ Y = Y∗ p(Y (τ) = Y∗|Y (0) = 1)
p(Y (τ) = Y∗|Y (0) = 1) = Wτ τ τ .
3.4 (3.8) τ .λ ≈ λc
, p(Y (τ) = Y∗|Y (0) = 1) = Wτ τ .
τ , SIR Y = 1 Y = 0 1
, W Y∗ ,q = W/(1 +W ) . ,
q(λ) 0 1 .
. λc ,
λc = 1 (3.12)
. 3.3 (3.8) ,Y = 1
Y∗ λ .
,Y∗ < 1 λ > 5/3 1 .
, q(λ) . , , λ = λc + ε
ε . Y∗ � O(ε−1/2), U(Y∗) � O(log ε)
ε .(Y∗, U(Y∗)) (1, U(1)) ε → 0
U(Y∗)− U(1))/(Y∗ − 1) � √ε→ 0 , Y = 1 Y = Y∗ ,
D = (λ+1)/8 . , τtr
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
34 3
-1
-0.5
0
0.5
1
1.5
0 0.5 1 1.5 2 2.5 3
U(Y
)
Y
Y*
❍❍
λ=0.51.2
3.3 U(Y ) .λ = 0.5
red line , Y ,
1 Y = 0 .
λ = 1.2 green line , Y∗ =√11/2
, Y Y∗ Y →∞ .
0
0.002
0.004
0.006
0.008
0.01
0.012
0 5 10 15 20 25 30 35 40
p(Y
(τ)=
Y*|Y
(0)=
1)
τ
λ=1.004λ=1.008λ=1.016
λ=1.02
3.4 (3.8)
, Y = 1 Y = Y∗τ pp(Y (τ) =
Y∗|Y (0) = 1) .
pp(Y (τ) = Y∗|Y (0) = 1) ∝ Wτ
τ .
0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
p(Y
->∞
|Y(t=
0)=Y
*)
λ
p(Y-> ∞|Y=Y*)
(a)
0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
p(Y
-> Y
*|Y(t=
0)=1
)
λ
p(Y=Y*|Y=1)
(b)
3.5 (3.8) p(Y∗ → 0).λ >
λc λ p(Y∗ → 0) = 0.4 .
(3.8) . ,Y = 1 Y∗p.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.4 35
Y 2∗ /(2D) = 1/ε+O(ε0) . W ,W = ε+O(ε2)
. ,
q(λ) = ε+O(ε2). (3.13)
. 3.4 (3.7) nR > 0.003 p(nR > 0.003)(�q(λ)) ε = λ − 1 . ,
p(nR > 0.003) = ε + O(ε2) . 3.4 (3.7)
nR > 0.003 p(nR > 0.003)(� q(λ)) ε = λ− 1 .
(3.13) p(nR > 0.003) � ε1 .
3.4
, ,
. ,
, U(Y ) ,
. ,
.
, ,
. , m 1 ,
. Y∗ , N
. ,m = cN ,
,Y (0) , .
, . , 3.4 SIR
,c = 1/2561 nR < 1/16 p λ
. λ q(λ) = 1 , .
,
. , ,
. , ,
. , .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
36 3
0.001
0.01
0.1
1
0.001 0.01 0.1 1
n R
λ-1
N=16384N=32768N=65536fit (λ-1)1
(a)
0.001
0.01
0.1
1
0.001 0.01 0.1
n R
λ-1
N=131072N=262144
N=2097152N=16777216
fit (λ -1)1
(b)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
p(n R
<1/1
6)
λ
N=8192N=16384N=32768
(c)
3.6 , SIR , m = 1
, p(nR > 0.003) ε = λ−1 . p(nR > 0.003) < ε
, . , (3.7)
, p(nR > 0.003) ∼ q(λ) ε = λ − 1 . N → ∞q(λ) � ε . , SIR
, m N/128 nR < 1/16 p λ . λ
1 q(λ) = 1 .
3.5
3.5.1 SIR
Newman SIR
[6] , SIR
,Newman ,
. ,SIR
nR
.
, 1 .
, 1 .
. Newman .
, λc
. ,
. λ < λc 0,λ > λc 1
.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 37
1 SIR ,SIR
. ,SIR
, , τ
,
, .
Newman
Newman[6] Durrett[14] , SIR
. .
i, j ,i I ,j S . i s
γij ,i τi
. i j i 1− Tij
1− Tij = limδt→0
(1− γijδt)τi/δt = e−γijτi (3.14)
. i j Tij
Tij = 1− e−γijτi (3.15)
.
i, j γij τi i, j , P (γ), P (τ)
. T (γ, τ) = 1 − e−γτ . SIR λ,
1 ,{P (γ) = δ(λ− γ)P (τ) = 1− e−τ) (3.16)
, 〈T 〉 =∫dγ∫dτT (γ, τ) .
〈T 〉 =∫ ∞
0
dγ
∫ ∞
0
dτP (γ)P (τ)(1− e−γτ )
= 1−∫ ∞
0
dτP (τ)e−(λ+1)τ
= 1− 1
λ+ 1
=λ
λ+ 1(3.17)
. T = T (γ, τ) .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
38 3
p(k)
.
N → ∞ ,
.
, ,
.
G0(x) .
G0(x) =∑k
p(k)xk (3.18)
< k >,< k2 > .
∂xG0(1) = < k >
∂2xG0(1) = < k2 > − < k >
G0(x) .
. 0 , n
n . 0
k 1 p1(k) , 1
k p(k) ,
p1(k) ∝ kp(k)
. .
p1(k) =kp(k)∑k k
′p(k′)=
kp(k)
< k >(3.19)
. p1(k) ∑k kp(k)x
k
< k >=x∑
k kp(k)xk−1
G0(1)= x
G0(x)
G0(1)(3.20)
. 1 0 2
k p′1(k) , 1 k + 1
(k + 1)p(k + 1) p′1(k)
p′1(k) =(k + 1)p(k + 1)∑k(k + 1)p(k + 1)
=(k + 1)p(k + 1)
G′0(1)
(3.21)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 39
.p′1(k) G1(x)
G1(x) =
∑k(k + 1)p(k + 1)xk
G0(1)=G′
0(x)
G′0(1)
(3.22)
.
2 k p′2(k) G2(x) .
p′2(k) =∑n
p(n)∑
k1+k2+···+kn=k
p′1(k1)p′1(k2) · · · p′1(kn) (3.23)
G2(x) =∑k
p′2(k)xk =
∑k
∑n
p(n)∑
k1+k2+···+kn=k
p′1(k1)p′1(k2) · · · p′1(kn)xk(3.24)
, (3.24)∑
k
∑k1+k2+···+kn=k
∑k1
∑k2
· · ·∑kn
G2(x) =∑n
p(n)∑k1
∑k2
· · ·∑kn
p′1(k1)p′1(k2) · · · p′1(kn)xk1+k2+···+kn
=∑n
p(n)∑k1
p′1(k1)xk1
∑k2
p′1(k2)xk2 · · ·
∑kn
p′1(kn)xkn
=∑n
p(n)n∏
i=1
∑ki
p′1(ki)xki
=∑n
p(n)
(∑k
p′1(k)xk
)n
=∑n
p(n)G1(x)n
= G0 (G1(x))
. G0(x) x 1 , 1
0 G1(x) x
.
T (γ, τ) , T (γ, τ)
. T (γ, τ)
, 1− T (γ, τ) .
G0(x;T ) k m
kCmTm(1− T )k−m (3.25)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
40 3
.G0(x;T )
G0(x;T ) =∞∑k
k∑m=0
p(k)kCmTm(1− T )k−mxm
=
∞∑k=0
p(k)
k∑m=0
kCm(xT )m(1− T )k−m
G0(x;T ) =∞∑
m=0
p(k) (1− T + xT )k
(3.26)
.G1(x;T )
G1(x;T ) =
∑∞k=0(k + 1)P (k + 1)
∑km=0 T
m(1− T )k−mxm∑∞k=0(k + 1)P (k + 1)
=
∑∞k=0(k + 1)P (k + 1) (1 + (x− 1)T )
k
G′0(1)
.< a >=∫∞0ap(T )dT G0(x) =< G0(x;T ) >, G1(x) =< G1(x;T ) >{G0(x) = <
∑∞k=0 p(k) (1 + (x− 1)T )
k>
G1(x) = <∑
k p(k)k(1+(x−1)T )k−1
G′0(1)
>(3.27)
.
0
s P0(s) , H0(x)
H0(x) =∑s
P0(s)xs (3.28)
. 1 s
P1(s), H1(x)
H1(x) =∑s
P1(s)xs (3.29)
. O(N) .
, 1 s
, 2 . 1 0
(3.22) G1(x) . 2
s P1(s) . H1(x)
H1(x) = xG1 (H1(x)) (3.30)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 41
. 1 1 x . H0(x)
H1(x), G0(x)
H0(x) = xG0 (H1(x)) (3.31)
. 0 x .
< s > . (3.31)
< s > = H ′0(1)
=[G0 (H1(x)) + G′
0 ([H1(x)]x=1)H′1(x)
]x=1
=[G0 (1) + G′
0 (1)H′1(x)
]x=1
< s > = 1 + G′0 (1)H
′1(1) (3.32)
. (3.30)
[H ′1(x)]x=1 =
[G1 ([H1(x)]x=1) + G′
1 (H1(x))H′1(x)
]x=1
=[1 + G′
1(x)H′1(x)
]x=1
H ′1(1) =
1
1− G′1(1)
(3.33)
(3.32),(3.33)
< s > = 1 +G′
0(1)
1− G′1(1)
(3.34)
.G′0(1), G
′1(1) .<> ∂x .
G′0(1) =
[G′
0(x)]x=1
=∞∑k=0
p(k)[< ∂x (1 + (x− 1)T )
k>]x=1
=
∞∑k=0
p(k)k[< (1 + (x− 1)T )
k−1T >
]x=1
=∞∑k=0
kp(k)k−1∑m=0
k−1Cm [(x− 1)m]x=1 < Tm+1 >
= 〈T 〉∞∑k=0
kp(k)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
42 3
G′0(1) = 〈T 〉G′
0(1) (3.35)
.
G′1(1) =
[G′
1(x)]x=1
=
[∂x <
∑k p(k)k (1 + (x− 1)T )
k−1
G′0(1)
>
]x=1
=
[<
∑k p(k)k∂x (1 + (x− 1)T )
k−1
G′0(1)
>
]x=1
=
[<
∑k p(k)k(k − 1)T (1 + (x− 1)T )
k−2
G′0(1)
>
]x=1
=
[<
∑k p(k)k(k − 1)T
∑k−2m=0(x− 1)mTm
G′0(1)
>
]x=1
=
∑k p(k)k(k − 1)
∑k−2m=0 [(x− 1)m]x=1 < Tm+1 >
G′0(1)
=
∑k p(k)k(k − 1)
∑k−2m=0 [(x− 1)m]x=1 < Tm+1 >
G′0(1)
=〈T 〉∑k p(k)k(k − 1)
G′0(1)
G′1(1) =
〈T 〉G′′0(1)
G′0(1)
(3.36)
. (3.35),(3.36) (3.34) . G′′0(1) =< k2 > − < k >,G′
0(1) =< k >
< s > = 1 +〈T 〉G′
0(1)
1− 〈T 〉G′′0 (1)
G′0(1)
= 1 +〈T 〉 < k >
1− 〈T 〉(<k2>−<k>)<k>
.< s > 〈T 〉 = Tc
1− Tc(< k2 > − < k >)
< k >= 0
⇔ Tc =< k >
< k2 > − < k >(3.37)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 43
. (3.37),(3.17) < s > λ < k >
< k >
< k2 > − < k >=
λcλc + 1
⇔ λc =< k >
< k2 > −2 < k >(3.38)
.
< k >= k,< k2 >= k2 ,
λc =1
k − 2(3.39)
. (3.12) 2.3.2 . (
) §2.
,p(k′) = δ(k − k′) (3.27) G0(x), G1(x) ,
{G0(x) = 〈(1 + (x− 1)T )
k〉G1(x) = 〈k(1+(x−1)T )k−1
G′0(1)
〉 (3.40)
(3.35)
G′0(1) = 〈T 〉k (3.41)
. (3.30),(3.31)
H1(x) = x〈[1 + (H1(x)− 1)T ]
k−1〉〈T 〉 (3.42)
H0(x) = x〈[1 + (H1(x)− 1)T ]k〉 (3.43)
(3.42) k . k = 3 ,
H1(x)k=3 =1− 2(1− T )x−√1− 4(1− T )x
2Tx(3.44)
,
H0(x)k=3 =
[1−√1− 4(1− T )x
]38x2
(3.45)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
44 3
. x = 1 H0(1) =∑
s P0(s) = 1 ,
H1(1)k=3 =1− 2T +
√4T − 3
2T(3.46)
H0(1)k=3 =1− 2T +
√4T − 3
8(3.47)
,T ≥ 3/4 λ ≥ 2 .
λ > 0 , . ,
[15] .
,
, 〈T 〉 > Tc , 1 . ,
1
. , ,SIR 1
R 1 , SIR
. k ≥ 3
,
[15] , .
3.5.1 ,
. k .
〈T 〉 . q
. q , 1 − 〈T 〉 ,
k − 1 ,
〈T 〉qk−1 ,
q = 1− 〈T 〉+ 〈T 〉qk−1
⇔ (1− q)− 〈T 〉(1− qk−1) = 0
⇔ (1− q)(1− 〈T 〉k−2∑j=0
qj) = 0 (3.48)
. q k ≥ 3 1 1 . 1 q∗.
1 − p . p , k
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 45
p = qk
⇒ p =
{1 for q = 1qk∗ for q = q∗
(3.49)
.
:k = 3 k = 3 . (3.48)
q = 1,1− 〈T 〉〈T 〉 (3.50)
. (3.49) p
p =
{1 for q = 1(
1−〈T 〉〈T 〉
)3for q = 1−〈T 〉
〈T 〉(3.51)
.q = 1, p = 1 〈T 〉 < Tc λ < λc , q = q∗, p = q3∗〈T 〉 > Tc λ > λc .
SIR
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
p
λ
percSIR
3.7 (3.53)
p.
.k = 3.
SIR nR <
1/16 p(nR < 1/16).
k = 3,m = 1, N = 65536.
(3.17)
〈T 〉 = λ
λ+ 1(3.52)
. p =(
1−〈T 〉〈T 〉
)3
p =
{1 for λ < λc(1λ
)3for λ > λc
(3.53)
.
SIR ( 3.2) ( 3.7).
λ > λc
p .
. .
. .
. , S I
. ,λ > λc
. ,S I a
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
46 3
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000
P(N
R)
NR
N=8192,lam=0.4N=4096,lam=0.4N=2048,lam=0.4N=1024,lam=0.4N=512,lam=0.4
(a) λ = 0.4
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000
P(N
R)
NR
N=8192,lam=1.0N=4096,lam=1.0N=2048,lam=1.0N=1024,lam=1.0N=512,lam=1.0
(b) λ = 1
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000
P(N
R)
NR
N=8192,lam=2.0N=4096,lam=2.0N=2048,lam=2.0N=1024,lam=2.0N=512,lam=2.0
(c) λ = 2
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000
P(N
R)
NR
N=8192,lam=3.0N=4096,lam=3.0N=2048,lam=3.0N=1024,lam=3.0
N=512,lam=3.0
(d) λ = 3
3.8 SIR NR . m = 1, N =
1024, 2048, 4096, 8192.
I , I b , a S
. 〈T 〉a, 〈T 〉b,
. λ > λc
.
. nR
, . ,
I S
.
O(logN) , N → ∞ .
3.2) N , . 3.8
SIR NR p(NR;λ,N) . λ > λc
N . .
. .
, SIR
.SIR ,τ I
, τ P (τ) τ , I
.
SIR .
,SIR 1
, . ,
,
, .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 47
3.5.2
(3.4) i = 0 , ,
. .
(3.4) P (s, i)
0 = −∂i{[(λs− 1) iP (s, i)]− ∂i
[(λs+ 1) i
2NP (s, i)
]+ ∂s
[λsi
2NP (s, i)
]}
−∂s{− [λsiP (s, i)]− ∂s
[λsi
2NP (s, i)
]+ ∂i
[λsi
2NP (s, i)
]}(3.54)
. SIR , i = 0 .
i > 0 P (s, i) = 0 .
Pε (s, i) = P (s) δ(i− ε) (3.55)
. i = 0 , i = ε Pε (s, i)
, ε→ 0 . (3.54)
i . i [0, 1] .
f(s) = −{[(λs− 1) iP (s) δ(i− ε)]− ∂i
[(λs+ 1) i
2NP (s) δ(i− ε)
]+ ∂s
[λsi
2NP (s, i)
]}i=1
i=0
−∫ 1
0
di∂s
{− [λsiP (s) δ(i− ε)]− ∂s
[λsi
2NP (s) δ(i− ε)
]+ ∂i
[λsi
2NP (s) δ(i− ε)
]}
= −∂s{− [λsεP (s)]− ∂s
[λsε
2NP (s)
]}(3.56)
.f(s) , P (s) . f(s) = 0
c
P (s) = ce−2Ns
s(3.57)
, . i = 0 δ(i)
.
3.5.3
2 .
2 x(t), y(t){x(t) = ax + bxx · ξx + bxy · ξyy(t) = ay + byx · ξx + byy · ξy (3.58)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
48 3
. ξi , .
〈ξi〉 = 0 (3.59a)
ξi(t)ξj(t′) = δijδ(t− t′) (3.59b)
ξj · . ,2 x, y
∂tP (x, y, t) = −∂x [PAx + ∂xPBx + ∂yPCx]− ∂y [PAy + ∂yPBy + ∂yPCy](3.60)
. x, y ax, ay, bxx, bxy, byx, byy Ax, Bx, Cx, Ay, By, Cy
.
P (x, y, t) x, y .
P (x, y, t) = 〈δ(x− x(t))δ(y − y(t))〉 (3.61)
< · > . P (x, y, t +Δt) Δt . ,xi(t+
Δt) = xi(t) + Δxi Δxi .
Δxi = xi(t+Δt)− xi(t)
= [ai + byj · ξj ] Δt+O(Δt3/2) (3.62)
Δt→ 0 O(Δt3/2) .
P (x, y, t + Δt) = 〈δ(x− x(t+Δt))δ(y − y(t+Δt))〉 (3.62)
.
δ(xi − xi(t+Δt)) = δ(xi − xi(t)−Δxi)
= δ(xi − xi(t))− δ′(xi − xi(t))Δxi +1
2δ′′(xi − xi(t))Δx
2i
= δ(xi − xi(t))− δ′(xi − xi(t)) [ai + byj · ξj ] Δt+1
2δ′′(xi − xi(t)) [ai + byj · ξj ]2 Δt2 (3.63)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 49
(3.59) δ(x− x(t+Δt))δ(y − y(t+Δt))
δ(x− x(t+Δt))δ(y − y(t+Δt))
=
[δ(x− x(t))− δ′(x− x(t)) [ax + bxj · ξj ] Δt+ 1
2δ′′(x− x(t)) [ax + bxj · ξj ]2 Δt2
]
×[δ(y − y(t))− δ′(y − y(t)) [ay + byj · ξj ] Δt+ 1
2δ′′(y − y(t)) [ay + byj · ξj ]2 Δt2
]= δ(x− x(t))δ(y − y(t))
−δ(x− x(t))δ′(y − y(t)) [ay + byj · ξj ] Δt+ δ(x− x(t))1
2δ′′(y − y(t)) [ay + byj · ξj ]2 Δt2
−δ(y − y(t))δ′(x− x(t)) [ax + bxj · ξj ] Δt+ δ(y − y(t))1
2δ′′(x− x(t)) [ax + bxj · ξj ]2 Δt2
+δ′(x− x(t))δ′(y − y(t)) [ax + bxj · ξj ] [ay + byj · ξj ] Δt2 (3.64)
. j j = x, y . (3.64) < · >P (x,y,t+Δt)−P (x,y,t)
Δt . (3.64) Δt
,
〈δ′(xi − xi(t))δ′(xj − xj(t))〉 = ∂xi∂xjP (xi, xj , t) (3.65a)
ξjΔt = 0 (3.65b)
ξjξjΔt = 1 (3.65c)
.Δt→ 0
∂P (x, y, t)
∂t= −∂Pax
∂x− ∂Pay
∂y
+1
2
∂2P(byx
2 + byy2)
∂y2+
1
2
∂2P(bxx
2 + bxy2)
∂x2
+∂2P (bxxbyx + bxybyy)
∂x∂y+O(Δt1/2) (3.66)
.O(Δt1/2 , . ∂∂y ,
∂∂x
∂P (x, y, t)
∂t= −∂x
[Pax − 1
2∂x[P(bxx
2 + bxy2)]− 1
2∂y [P (bxxbyx + bxybyy)]
]
−∂y[Pay − 1
2∂y[P(byy
2 + byx2)]− 1
2∂y [P (bxxbyx + bxybyy)]
](3.67)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
50 3
.
Ax = ax (3.68a)
Bx = −1
2
(bxx
2 + bxy2)
(3.68b)
Cx = −1
2(bxxbyx + bxybyy) (3.68c)
Ay = ay (3.68d)
By = −1
2
(byy
2 + byx2)
(3.68e)
Cy = −1
2(bxxbyx + bxybyy) (3.68f)
.
x→ s, y → i (3.4) (3.68)
−λsi = as (3.69a)
− 1
2N(λs+ 1)i = −1
2
(bss
2 + bsi2)
(3.69b)
1
2Nλsi = −1
2(bssbis + bsibii) (3.69c)
(λs− 1)i = ai (3.69d)
1
2Nλsi = −1
2
(bii
2 + bis2)
(3.69e)
1
2Nλsi = −1
2(bssbis + bsibii) (3.69f)
. as, ai, bss, bsi, bis, bii . as, ai
as = −λsi (3.70a)
ai = (λs− 1)i (3.70b)
.bss, bsi, bis, bii ,
bss = −√λsi
N(3.71a)
bsi = 0 (3.71b)
bis =
√λsi
N(3.71c)
bii =
√i
N(3.71d)
.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 51
3.5.4
x, y ,(x, y) (u(x, y), v(x, y)
. x, y {x(t) = ax + bxx · ξx + bxy · ξyy(t) = ay + byx · ξx + byy · ξy (3.72)
.
,dx(t), dy(t) .{dx(t) = axdt+ bxx · ξxdt+ bxy · ξydt+O(dt3/2)dy(t) = aydt+ byx · ξxdt+ byy · ξydt+O(dt3/2)
(3.73)
,dx2, dy2, dxdy dt1 .⎧⎨⎩dx2 = (b2xx + b2xy)dtO(dt2)dy2 = (b2yx + b2yy)dt+O(dt2)dxdy = (bxxbyx + bxybyy)dt+O(dt2)
(3.74)
, u, v .du, dv x, y
. {du = uxdx+ 1
2uxxdx2 + uydy +
12uyydy
2 + uxydxdydv = vxdx+ 1
2vxxdx2 + vydy +
12vyydy
2 + vxydxdy(3.75)
. uxy = ∂2u∂x∂y . (3.74)
⎧⎪⎪⎨⎪⎪⎩du =
[uxax + 1
2uxx(b2xx + b2xy) + uyay +
12uyy(b
2yx + b2yy) + uxy(bxxbyx + bxybyy)
]dt
+(uxbxx + uybyx) · ξxdt+ (uxbxy + uybyy) · ξydt+O(dt3/2)dv =
[vxax + 1
2vxx(b2xx + b2xy) + vyay +
12vyy(b
2yx + b2yy) + vxy(bxxbyx + bxybyy)
]dt
+(vxbxx + vybyx) · ξxdt+ (vxbxy + vybyy) · ξydt+O(dt3/2)
(3.76)
. dt u, v .⎧⎪⎪⎨⎪⎪⎩u = uxax + 1
2uxx(b2xx + b2xy) + uyay +
12uyy(b
2yx + b2yy) + uxy(bxxbyx + bxybyy)
+(uxbxx + uybyx) · ξx + (uxbxy + uybyy) · ξy +O(dt1/2)v = vxax + 1
2vxx(b2xx + b2xy) + vyay +
12vyy(b
2yx + b2yy) + vxy(bxxbyx + bxybyy)
+(vxbxx + vybyx) · ξx + (vxbxy + vybyy) · ξy +O(dt1/2)
(3.77)
uxbxy + uybyy, vxbxy + vybyy u, v ,
.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
52 3
x→ s, y → i u = s, v =√Ni .
us = 1 (3.78a)
uss = ui = uii = usi = 0 (3.78b)
vs = vss = vsi = 0 (3.78c)
vi =1
2
√N
i=N
2
1
v(3.78d)
vii = −1
4
√N
i3= −N
2
4
1
v3(3.78e)
(3.70),(3.71)
{u = 1
N
[−λuv2 −
√λuv2 · ξ1
]+O(dt1/2)
v = λu−12 v − λu+1
81v +
√λu2 · ξ1 + 1
2 · ξ2 +O(dt1/2)(3.79)
. (3.7) (3.79)(u, v) (s, Y ) .
3.5.5
(3.4) s = 1, I = Ni
∂tP (I, t) = −∂I[(λ− 1)IP (I, t)− ∂I
[(λ+ 1)I
2P (I, t)
]](3.80)
. . (3.80) ,{∂tP (I, t) = −∂IJ(I, t)J(I, t) = (λ− 1)IP (I, t)− ∂I
[(λ+1)I
2 P (I, t)]
(3.81)
.
. ∂+f(x) = (f(x + Δx) − f(x))/Δx, ∂−f(x) =
(f(x)− f(x−Δx))/Δx , (3.81){∂tP (I, t) = −∂+J(I, t)J(I, t) = (λ− 1)IP (I, t)− ∂−
[(λ+1)I
2 P (I, t)]
(3.82)
. .
P (I, t+Δt) = P (I, t)− ∂+J(I, t)Δt (3.83)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 53
1e-005
0.0001
0.001
0.01
0.1
1
0 2 4 6 8 10 12 14 16 18 20
P(I)
I
lam=0.5,t=0t= 1.2t= 2.4t= 3.6t= 4.8t= 6.0t= 7.2t= 8.4t= 9.6
t=10.8
(a) λ = 0.5
1e-005
0.0001
0.001
0.01
0.1
1
0 2 4 6 8 10 12 14 16 18 20
P(I)
I
lam=1,t=0t=0.6t=1.2t=1.8t=2.4t=3.0t=3.6t=4.2t=4.8t=5.4
(b) λ = 1
1e-005
0.0001
0.001
0.01
0.1
1
0 2 4 6 8 10 12 14 16 18 20
P(I)
I
lam=2,t=0t=0.3t=0.6t= 0.9t= 1.2t= 1.5t= 1.8t= 2.1t= 2.4t=2.7
(c) λ = 2
3.9 λ (3.80) .
. λ = 0.5 . I = 0
. λ = 1, λ = 2 .λ
I = L .
I [0,∞] , L [0, L] .
I = 0, I = L .{J(0, t) = 0J(L, t) = 0
(3.84)
. (3.82)⎧⎨⎩J(0, t) = [(λ− 1)IP (I, t)]I=0 −
[(λ+1)I2ΔI P (I, t)− (λ+1)(I−ΔI)
2ΔI P (I −ΔI, t)]I=0
J(L, t) = [(λ− 1)IP (I, t)]I=L −[(λ+1)I2ΔI P (I, t)− (λ+1)(I−ΔI)
2ΔI P (I −ΔI, t)]I=L
⇔{P (−ΔI, t) = 0
P (L, t) = (λ+1)/2(λ−1)Δt−(λ+1)/2
L−ΔIL P (L−ΔI, t)
(3.85)
. .
, I [0, 2] μ = 1, σ2 = 0.1
, 0 .ΔI = L/128, L = 20,Δt = 1/4096, N = 128
.
λ = 0.5, 1, 2 3.9 .
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54 3
3.5.6 Q(Y, t) (3.10)
(3.4) s = 1, Y =√Ni (3.10) . s .s = 1 ,s
0 . P (i, t) = P (1, i, t) .
∂P (i, t)
∂t= −∂i
{[(λ− 1) iP (i, t)]− 1
2N∂i [(λ+ 1) iP (i, t)]
}(3.86)
Y =√Ni . ∂i = N
2Y ∂Y , P (i, t)di = Q(Y, t)dY Q(Y, t) =2YN P (Y
2
N , t) .Q(Y, t) .
∂P(
Y 2
N , t)
∂t= − N
2Y∂Y
{[(λ− 1)
Y 2
NP
(Y 2
N, t
)]− 1
2N
N
2Y∂Y
[(λ+ 1)
Y 2
NP
(Y 2
N, t
)]}(3.87)
2YN . ∂tQ(Y, t) .
∂ 2YN P
(Y 2
N , t)
∂t= −∂Y
{[λ− 1
2Y2Y
NP
(Y 2
N, t
)]− λ+ 1
8Y∂Y
[Y2Y
NP
(Y 2
N, t
)]}∂Q (Y, t)
∂t= −∂Y
{[λ− 1
2Y Q (Y, t)
]− λ+ 1
8Y∂Y [Y Q (Y, t)]
}∂Q (Y, t)
∂t= −∂Y
{[λ− 1
2Y − λ+ 1
8Y
]Q (Y, t)−
[λ+ 1
8
]∂YQ (Y, t)
}(3.88)
(3.10) .
3.5.7 SIS
SIS
P (I, t) =λ
N(S + 1) (I − 1)P (I − 1, t)− λ
NSIP (I, t) (3.89)
+ (I + 1)P (I + 1, t)− IP (I, t) (3.90)
. S = N − I, i = IN
1N .
P (i, t) = −∂i[(λ(1− i)− 1) iP (i, t)− ∂i
(λ(1− i) + 1) i
2NP (i, t)
](3.91)
i = a+ b · ξ ⇔ P (i, t) = −∂[aP (i, t)− ∂i
b2
2P (i, t)
](3.92)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
3.5 55
(3.91)
a(i) = (λ(1− i)− 1) i (3.93)
b(i) =
√(λ(1− i) + 1) i
N(3.94)
.
3.5.4 .
df(i) = f ′di+1
2f ′′di2 (3.95)
df(i)
dt= f ′a+
1
2f ′′b2 + f ′b · ξ (3.96)
f ′(i)b(i) = const. y = f(i)
y(i) = ArcTan
(√i
λ+1λ − i
)(3.97)
i(y) =λ+ 1
λsin2(y) (3.98)
.
a(i(y)) =(λ+ 1) (λ− 3 + (λ+ 1)cos(2y)) sin2(y)
2λ(3.99)
b(i(y)) =(λ+ 1) cos(y)sin(y)√
λN(3.100)
f ′(i(y)) =λ
2 (λ+ 1) cos(y)(3.101)
f ′′(i(y)) = − λ2(1− 2sin2(y)
)4 (λ+ 1)
2cos3(y)sin3(y)
(3.102)
(3.103)
. f ′(i(y))a(i(y)), f ′′(i(y))b(i(y))2, f ′(i(y))b(i(y))
f ′(i(y))a(i(y)) =sin(y)
4cos(y)(λ− 3 + (λ+ 1) cos(2y)) (3.104)
f ′′(i(y))b(i(y))2 = − λcos(2y)
4Nsin(y)cos(y)(3.105)
f ′(i(y))b(i(y)) =1
2
λ
N(3.106)
. y
y = f ′(i(y))a(i(y)) +1
2f ′′(i(y))b(i(y))2 + f ′(i(y))b(i(y)) · xi (3.107)
=sin(y)
4cos(y)(λ− 3 + (λ+ 1) cos(2y))− 1
2
λcos(2y)
4Nsin(y)cos(y)+
1
2
λ
N· xi (3.108)
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
56 3
.u(y)
u(y) =((1 + λ)Ncos(2y) + (λ− 8N)log(cos(y)) + λlog(sin(y)))tan(y)
8Ntan(y)(3.109)
. Q(y) ∝ e−u(y)
λ8N , i
P (i) P (i) = Q(y)dydi . ,
.
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物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
59
4
, SIR
, ,
. ,
.
4.1
4.1.1
,
, ,
. ,
, , ,
,
. ,
,
, .
,
, .
, , ,
.
,
. ,
,
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
60 4
, .
3 , , SIR
SIS .
, . ,
, .
1:twitter 4.1.1 twitter
. 16
. , .
, .
2: 4.1.1 Google[1]
.
[2] [3] . ,
. (2012 12 ) ,
. 2002 ,
2007 , .
2008 ,
. ,
. [4],
. .
1300 [5].
2 2008
, [6].
SNS
, 2
. . ,
. , .
4.1.2
,
. ,
SIR , .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
4.1 61
(a) twitter . [7] .
(b) google [8]. , .
(c)
[2]
(d)
[3]
.
. ,
.
.
. ,
.
,
. , ,
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
62 4
[9].
,
.
.
, , .
4.2
,
, 3
. , .
4.2.1
, SIR . ,
S,I,R , S ,I
,R . S I
I .I R
( 4.1). .
4.2.2
, .
.
, λ, μ , k I
kI kIkI
k λ(kI), μ(kI) . λ(kI), μ(kI)
4.2.1 kI .
λ(kI) =
⎧⎨⎩
λk2
(kI
cλ
)αλ
for kI ≤ cλ
λk − λk2
(1−kI
1−cλ
)αλ
for kI > cλ(4.1)
μ(kI) =
⎧⎨⎩μ− μ
2
(kI
cμ
)αμ
for kI ≤ cμ
μ2
(1−kI
1−cμ
)αμ
for kI > cμ(4.2)
. 0 ≤ cλ ≤ 1, 0 ≤ cμ ≤ 1, αλ ≥ 0, αμ ≥ 0 .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
4.2 63
4.1 .
,
. (a) λ(kI), μ(kI)
(b) ( ) ( ) .
4.2 (a) λ(kI), μ(kI) . α > 1. (b)
. .
. .
.
SIR ,S j I kI ,
j I I
λkI (4.3)
, I . ,I
, μ . (4.2) cλ = 1/2, αλ =
1, cμ = 1, αμ = 0 .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
64 4
4.2.3
2.2.2 HY .
.
, , HY
, [10]
.
. 4.2.1 ,
. I , S
I ,R , I
R .
.
I ,
. 4.2.1 . I
, ,μ(kI) ,R
.
.
4.3
,
,
.
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65
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物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
67
5
5.1
§2 , ,
SIR . nR tf ,
, .
nR , nR tf
, . ,
λc λ → ∞ tf, σ(tf)2 .
. , ,
εN nR ,
,
.
§3 ,SIR
. SIR
, , ,
. 1 ,
.
U(Y ) λ > 1 Y = 12
√λ+1λ−1 ,
Y = 1 , nR = 0
nR > 0 . , (λ = λc + ε)
q(ε) q(ε) = ε ,
. , ,
, .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
68 5
, λ .
,
. ,SIS
, SIR .
§4 ,SIR ,
. ,S ,I
I . ,
, , ,
.
5.2
.
§2 , nR, tf
,
. , ,
, , .
,§3 , ,
. ,SIS
. ,
.
§4 , ,
. ,
,
, ,
.
,§1 .
,
. §3 ,
,
. ,
, ,
,
. , .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
69
A
, .
A.1
. N 〈k〉, 〈k〉N/2 .
, 〈k〉N . N = 4 , 1
2,3 4, 1 3 , {1, 2, 3, 4, 1, 3} .
A.1.1
N k .
. ,k � N ,k � N
i k .i ei[j], j = 0, 1, · · · k − 1
. f [i] i , .
k , f = k, k, · · · , k .
g .
0, 1, · · · , N − 1 k kN . rand(j) 0 ∼ j
. kN/2 .
1. j = 0 .
2. j .
3. 0 ≤ i ≤ N − 1 f [i] f i , imax
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70 A
0
50
100
150
200
250
300
350
400
450
0 10 20 30 40 50 60 70 80 90 100
sec.
k
N=1024N=2048N=4096
(a)
0
500
1000
1500
2000
2500
0 10000 20000 30000 40000 50000 60000 70000
sec.
N
k=3k=4k=5
(b)
A.1 N, k .
.
4. r = rand(kN − 2j − 1) .
5. g[r] eimax , r = imax 4
. r j′ .
6. eimax j′ ,ej′ imax .f [imax]
−1 . g imax, j′ . (imax, j
′) .
.
7. f [imax] = 0 4 . f [imax] = 0 j+ = 1 2
.j = kN/2 .
c++ g [multiset] ,ei [set] .f [i] = k − ei.size() .
,f .
k � N ,
,2∼7 N
, . A.1 , N, k
. ,
N = 65536, k = 3 15 ,N = 32768, k = 40 8 , N = 16384, k = 80 7
, .
A.1.2
.
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A.2 SIR 71
1. j = 0 .
2. j .
3. 2 r0 = random(N − 1), r1 = random(N − 1) .
4. r0 = r1 er0 r1 3 .
5. er0 r1 ,er1 r0 . (r0, r1)
j . .
6. j+ = 1 2 .j = kN/2 .
A.2 SIR
A.2.1
I 1, S-I
λ . I I , I .
S − I , I P [I − 1]
λP [I − 1] . P [i] P [i] =∑i
0 k[Ilist[i]] . ,I
0 ∼ I − 1 , j
j = Ilist[i] .
1. I = m,S = N − I .
.P, Ilist .
2. I = 0 . rate . rate = λP [I−1]+ I
.
3. . rate
, τ (0, 1) r τ = −log(1 − r)/rate
. τ .
4. (0, rate) rrate ,rrate < I 5 . rrate ≤ I
6 .
5. Ilist[(int)(rrate) R . Ilist, P, I .2
.
6. P [ir] ≥ (rrate− I)/λ < P [ir + 1] ir .
7. Ilist[ir] .
8. S 2 . S I
.Ilist, P, I .2 .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
72 A
A.2.2 §4,
.
, . . kI
I ,λpsj [kI , k] S I
. pij [kI , k] I R .st[j]
j , rate =∑
j [λpsj [kI , k]δ(S, st[j]) + pij [kI , k]δ(I, st[j])]
. δ(S, st[j]), δ(I, st[j]) , Ilist Slist ,
P .
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
73
, , ,
.
,
, A.3 ,
, , .
A.2 2012
.
,
15 16
,
.
2 ,
,
.
,
. ,
,
.
,
,
. ,
,
.
, ,
.
.
. ( A.2:sasa ...)
.
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
74
A.3
. ,
( )
,
.
, ,
.
, ,M2
5 ,
. ,
,
. 3
, .
,
. , , C C++
. ,
, ,
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, , ,
,
. 3 , ,
.
A.4
.
. ,
, ,
,
, , ,
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,
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.
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,
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》
75
( ) , *1
. , .
.
2013 6 1
*1 2012 9 13 ( ) 14 ( )CMRU ” ”
物性研究・電子版 Vol, 2, No. 3, 023602 (2013年8月号)《修士論文》