2 #2 i p i (ɛ)., q D q lim n(ɛ) log i ɛ log(ɛ) lim ɛ log n(ɛ) log(ɛ) (238), ɛ ( 69 ), ɛ, n(ɛ) n(ɛ) ɛ Dq (239), D q (, )., q, q 2. 69: ɛ. 7.2 q (237),
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1 #2 : ( 8-3) URL : j inoue/index.html q q.,. ɛ p i, D q D q q lim ɛ log n(ɛ) i pq i log ɛ (237)..,, ɛ., 9
2 2 #2 i p i (ɛ)., q D q lim n(ɛ) log i ɛ log(ɛ) lim ɛ log n(ɛ) log(ɛ) (238), ɛ ( 69 ), ɛ, n(ɛ) n(ɛ) ɛ Dq (239), D q (, )., q, q 2. 69: ɛ. 7.2 q (237), q > i, (237)., q,., q,, q. q, q, ( q ). q (237),.,, ( ).
3 2 #2,, q,,., , [, ].,, 2, p, p., [,.5] 7:. n 2. 2,, p, p, [, 5, ] p, p. 2 ( )., 2 [,.25], [.25,.5], [.5,.75], [.75, ] 4 p 2, p( p), ( p)p, ( p) 2 ( 7 ). n [, ] 2 n, p k ( p) n k n C k., 7. [,.].,., 2, n 2 ( ) ( ) 2, ( ) 2,, ( ) 2 2., n, k p k ( p) n k, [, ] 2,.
4 カオス フラクタル 2 #2 担当: 大学院情報科学研究科 井上 純一 e-5.6 6e-5.4 4e-5.2 2e p..9 p 図 7: 二項分岐過程の密度分布. 上図では p.25 と選んである. 上図の組み込み図は同図のスケールを変えてプロッ トしたもの. 下図はそれぞれ p.,.4 の場合の二項分岐過程の密度分布. n 8 と選んでいる. この葉の 2 進数 (xn xn 2 x )2, (xi の個数が k 個) を 進数表示したものを 2n でスケー ルしなおせばよい. すなわち, この葉の [, ] 区間内の位置 X は n X i i xi 2 2n (24) で与えられる. まとめると (xn xn 2 x )2 ( )2 {z } n X n xi 2i 2 i (24) Y pk ( p)n k (242) k 個の, n k 個の となる. この (X, Y ) をプロットしたものが図 7 である3. 3 数直線上には同じ Y を持つ X が n Ck 個あることに注意. また, n の選び方には注意が必要である. 担当者のパソコ ンでは n 8 程度までが我慢できるくらいの計算時間であり, また, これを超えると出来上がる図形の容量も大きく なる. ここは 2 ページ目
5 2 # , D q : Z q (ɛ) n(ɛ) [p i (ɛ)] q (243) i D q q lim log Z q (ɛ) ɛ log ɛ (244) 4., (, ). Z q (ɛ) p n(ɛ) δ(p i (ɛ) p)[p i (ɛ)] q p i n(ɛ) δ(p i (ɛ) p)p q (245) i, n(ɛ) i δ(p i(ɛ) p), n(ɛ) p, ρ(p), (245). Z q p ρ(p)p q (246), ρ(p) p., p p k ( p) n k, ρ(p) n C k Z q n nc k [p k ( p) n k ] q k n nc k p qk ( p) q(n k) k n nc k (p q ) k {( p) q } n k [p q + ( p) q ] n (247) k., ɛ 2 n, D q log[p q + ( p) q ] n q log 2 n log[p q + ( p) q ] q log(/2) (248). p D q 72., q 2 +2, D q,, [D q+, D q ]. (248)., p < /2 q +, (248) 2 ( p) q D q+ q log( p) log( p) q log(/2) log(/2) (249) 4. q T β T, i i E i p i e E i, Z β i [p i] β i e βe i,. 3
6 2 # p. p.25 p D q.5 D q.5 p.25 D min D max q q 72: D q. p p.,.25,.4. p.25 D q q ±., : D max log p/ log(/2), D min log( p)/ log(/2)., q, (248) 2 p q D q q log p q log(/2) log p log(/2) (25). 72( ). q,, q +,.,, ,., q, , ɛ i ɛ p i (ɛ) ɛ α i (25) : ɛ., α i,. [α, α + dα] α ρ(α) ɛ f(α) dα (252) 5,,,. 4
7 2 #2., ρ(α), ɛ f(α)., α, f(α). α, Z q ρ(α) ɛ f(α) [ɛ α ] q dα ρ(α) ɛ f(α)+qα dα (253)., (253) (243) ρ(α) ɛ f(α) n(ɛ) δ(α α i ) (254) i. Z q n(ɛ) n(ɛ) n(ɛ) δ(α α i ) ɛ qα dα ɛ qαi [p i (ɛ)] q (255) i i i. (243).,, Z q α, ɛ ɛ f(α)+qα ( ). α α(q), { f(α) + qα} (256) α αα(q) q f(α) α (257) αα(q)., f(α) + qα f(α) + qα f(α(q)) + qα(q) + 2! 2 f(α) α 2 (α α(q)) 2 (258) αα(q) (, 2 f(α)/ α 2 αα(q) < ), Z q ρ(α(q))ɛ f(α(q))+qα(q) ɛ 2 f(α) 2 α 2 αα(q)(α α(q)) 2 dα (259). ɛ 2 f(α) 2 α 2 αα(q)(α α(q)) 2 dα exp 2 log f(α) ɛ 2 α 2 (α α(q)) 2 dα αα(q) π (26) 2(log ɛ) 2 f(α) α 2 αα(q) 5
8 2 #2 6, D q (q )D q lim ɛ log Z q log ɛ log ρ(α(q)) 2(log ɛ) 2 f(α) α lim f(α(q)) + qα(q) + lim αα(q) ɛ log ɛ ɛ log ɛ f(α(q)) + qα(q) (26). lim ɛ (log ɛ), lim ɛ (log ɛ) 3/2.,. τ(q) (q )D q (262) τ(q) f(α(q)) + qα(q) (263) f(α(q)) qα(q) τ(q) (264) π dτ dq τ α + ( τ q + α { q f α ) ( ) α q } α q α (265)., f(α(q)) qα(q) τ(q) (266) dτ dq α (267) 7., 2 f(α)/ α 2 αα(q) <, f(α)., α (267) τ q (266) f(q(α)) qα(q) τ(q) α() τ() τ() D q f() (268)., f q D q., f(α), (257) q + α α min, q α α max., (267), τ(q) αq τ(q + ) α min (+ ) (269) τ(q ) α max ( ) (27) 6 ɛ bα2 dα e log ɛ bα2 dα e b(log ɛ)α2 dα 2 π b log ɛ. 7, (266)(267) f τ., df/dα y P (y) αy f α df dα f αq f τ, τ f. 6
9 2 #2, D q τ(q)/(q ) D q {α max ( )}/( ) α max, D q + {α min (+ )}/(+ ) α min, α min, α max f. f(α min ) f(α(q + )) (+ ) α(q + ) τ(q + ) (+ ) α min α min (+ ) (27) f(α max ) f(α(q )) ( ) α(q ) τ(q ), (257)(266) ( ) α max α max ( ) (272) f(q(α)) qα(q) τ(q) f(α) α αα(q) α(q) τ(q) (273),, : q f(α) α αα(q) (274) : f α (266) α, ( q)α τ(q) ( q)d q α D q (275), f f(α()) α D q (276) ,.,. (248), D q, τ(q) p q + ( p) q q log(/2) (277)., (267) τ(q) (q )D q pq + ( p) q α dτ dq log(/2) log(/2) [ p q log p + ( p) q ] log( p) p q + ( p) q (278) (279). 7
10 2 #2 (279) α ξ ξ,, (266) f p q p q + ( p) q (28) ( p) q p q + ( p) q (28) (ξ log p + ( ξ) log( p)) (282) log(/2) q log(/2) (ξ log p + ( ξ) log( p)) log(/2) log(pq + ( p) q ) (ξ log ξ + ( ξ) log( p)) (283) log(/2)., α.8 p. p.25 p p.25 f α D q D q.6 f f α α.9 p. p.25 p p. p.25 p f α q q 73: f α. p p.,.25,.4 ( ). p.25, f D q, D q. f q ( ),, α q ( ). α f (ξ log p + ( ξ) log( p)) log(/2) (284) (ξ log ξ + ( ξ) log( ξ)) log(/2) (285) 8
11 2 #2., ξ, p 73( )., D q, D q,, α. (28)(28) ξ ξ ( ) q p (286) p., q, ξ /2,, f (285) f D q, α (284) α log p( p) 2 log(/2) (287)., q, (286) ξ p,,, (285)(284) f α p log p + ( p) log( p) log(/2). p.25 73( ). D q (288)., q. 9
12 2 #2, A4.,.,,., #. : ( ) 5 ( ), 8-3. : 7/26 ( ): 74: ( ). 2
13 2 #2.,,. [], (992). [2] Fractals for the classroom, by H. Peiger, H. Jurgen and D. Saupe, part I,II, Springer-Varlag (993). [3] Modeling reality: How computer mirror life, by Iwo Bialynicki-Birula and Iwona Bialynicka- Birula (24). [4] : J.M.T. Thompson, H.B. Stewart,, (988). [5],,,, (2). [6] (986). [7] (987). [8],, (99). [9] (22). [] (99). [] (993). [2] Introduction to percolation theory : revised 2nd edition, by D. Stauffer and A. Aharonoy, (Tayler & Francis, 994). [3] : (996). [4] (995). [5] (997). [6] (987). [7], (986). 2
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