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1 ( ) review Onsager [1] PPM
2 図 1: 実験装置の図 写真中央にある円筒形の容器が超電導コイルで囲まれた真空 容器で この中に電子を閉じ込める 左側の四角い箱の中には光学系が設置されて おり 電子の像を箱左端の CCD カメラへ導く役割を担う このようにして超電導マ グネットから CCD カメラを遠ざけないと 強磁場の影響を受け正しい撮像が行え ない 2 本研究の背景 本研究のスターティングポイントは 京都大学の際本研究室で行われていた非中 性プラズマを用いた渦実験である [2 4] 図 1 に実験装置の写真を示す 普通 プラ ズマ状態といえば 元々電気的に中性だった原子 分子が電離してイオンと電子に 分離した状態なので 巨視的に見れば電気的には中性となる 非中性プラズマとは この電気的な中性条件が破れた状態のプラズマを指し 電気的に負の粒子である電 子のみから構成されたプラズマ (純電子プラズマ) などが該当する この実験装置では 純電子プラズマを円筒容器の軸方向に超電導コイルで印加し た最大 2T の磁場により径方向に 両端の電極に印加した負電位により軸方向に そ れぞれ閉じ込める (図 2) すると 導体の円筒容器と電子の間に径方向内向きの電 場が発生する この電場と軸方向磁場により 電子は E B 方向に回転運動をする (理由は後述) 一方 電子は磁場に平行方向には自由に運動することができるので 両端の負電位の間を高速に往復運動する すなわち 電子は全体として軸方向に高 速な往復運動を繰り返しながら 軸の周りにゆっくり回転していく 2T における典型的パラメタを表 1 に示す ここで注目すべき点は 磁場に垂直面 内での運動の時間スケールが 100ms のオーダなのに対して 磁場方向の往復運動の 周期が 3ms と約 2 桁短いことである これは すなわち 磁場に垂直な方向のどこ PPM
3 B E Rotation Trap and hold 2: 2T E B 1: 2T 100 ms /m 3 360mm 600mm 3ms 0.3 ev 1/e 0.15 m PPM
4 P θ P θ = p θi = i i m e r 2 i θ i + e B 0 2 r2 i (1) m e e B 0 i p θi r i θ i 1 2 m e r 2 i θ i eb 0 r 2 i 2 m ev eb =2 0 r i =2 r L r i (2) r i θi = v r L r i (1) 1 2 P θ i e B 0 2 r2 i = const. (3) 1 2 ẑ B = B 0 ẑ (4) dv m e = e(e + v B) (5) dt ( 0) v = E B B 2 (6) E B E B (4) (6) E = φ (7) v = 1 B 0 ẑ φ (8) PPM
5 3: (a) 2(b) 3(c) 4 Kelvin-Helmholtz ω z ẑ = v = ẑ B 0 2 φ = en ɛ 0 B 0 ẑ (9) (9) n (ẑ ) φ v = = 0 (10) n + v n = 0 (11) t e/(ɛ 0 B 0 ) ω z t + v ω z = 0 (12) 2 ( ) 1 2 Kelvin-Helmholts(Diocotron) 3 CCD ! B 0 5 PPM
6 3 3.1 N/2 N/2 R 2 i (i =1, 2, N) r i =(x i y i ) Ω i Dirac ω z (r,t)= Ω i δ(r r i (t)) (13) i G(r) Green ψ(r,t) = i Ω i G(r r i (t)) (14) u(r,t) = ẑ ψ(r,t) (15) ω z (r,t)ẑ = 2 ψ(r,t)ẑ = u(r,t) (16) u(r,t)) = (ψ(r,t)) (17) 2 Ω 0 Ω 0 (Ω 0 ) 2 H I H = 1 4π I = 1 4π i j i Ω i Ω j ln r i r j + 1 4π i Ω i Ω j ln r i r j j Ω i Ω j ln R r j, (18) i j Ω i r i 2 (19) i (18) 3 R 2 r j r j = R2 r j 2 (20) (18) H 2! Ω i dx i dt Ω i dy i dt = H y i, (21) = H x i (22) PPM
7 4: dr i dt = 1 2π j i Ω j (r i r j ) ẑ r i r j π j Ω j (r i r j ) ẑ r i r j 2 (23) (23) 2 (23) 2 2 MDGRAPE- 2 MDGRAPE-3 2 Core 2 Quad 4 CPU GPU(Graphic Processing Unit) MDGRAPE-3 7 PPM
8 2: CPU Memory FSB Clock Dual Core Core 2 Duo E6750 (2.66GHz) DDR GB 1333MHz Quad Core Core 2 Quad Q6600 (2.4GHz) DDR GB 1066MHz MDGRAPE-2 Pentium4 2.4GHz DDR MB 533MHz MDGRAPE-3 Pentium4 660 (3.6GHz) DDR GB 800MHz W(E) E 5: 3.2 β β = ds de = d log W (E) de W (E) E β ( 5) E W 0 E 0 E >E 0 d ln W/dE < 0 ( 6) E >E 0 Onsager [1,5] (21) (22) Onsager A dγ (24) dγ =dx 1 dy 1 dx N dy N (25) PPM
9 W(E) β β E0 E 6: ( dγ= ) N dxdy = A N (26) Onsager Joyce sinh-poisson 2 ψ = λ sinh(βψ) (27) [6,7] λ ψ [8] [9] [10,11] [12, 13] ( 3362 ) E I (E,I) 9 PPM
10 7: E 8: I =0 PPM
11 β>0 β<0 9: ( ) E I =0 E E = E 0 E 0 E >E 0 I =0 E ( ) 9 (23) 1 ( ) 1 11 PPM
12 10: (Physica D, 51 (1991) ) 11: 10 sinh-poisson (Physica D, 51 (1991) ) Leonard review [14] 3 Physica D, 51 (1991) Matthaeus 2 Navier-Stokes Navier-Stokes ω z t + u ω z = ν 2 ω z (28) R = ν 1 = (29) Δt = (30) 3 Leonard It now appears that using an increased number of point vortices of decreased strength will not yield a converged solution.... Ironically, best results with the point vortex method often are achieved by using only a few vortices with a diffusive time integration scheme. PPM
13 12: (Physica D, 51 (1991) ) sinh-poisson [7] Leonard Matthaeus 2 Euler ω z (r,t) t + u ω z (r,t) = 0 (31) (13) t ω z(r,t) = ( ) Ω i δ(r r i (t)) t i = ( ) Ω i i t r i(t) δ(r r i (t)) = u(r,t) ω z (r,t) (32) 13 PPM
14 ω z t + u ω z = viscous term (33) ˆ ˆω z (r,t) ω z (r,t) ˆω z (r,t) SE (34) S E Λ ˆω z (r,t) S = 1 dr ˆω z (r,t) (35) Λ Λ(r) r ˆω z (r,t) = i Ω i δ(r r i (t)) = ω z (r,t)+δω z (r,t) (36) ˆψ(r,t) = i Ω i G(r r i (t)) (37) û(r,t) = ẑ ˆψ(r,t) (38) 5.3 ˆω z t + û ˆω z = 0 (39) (36) t ω z(r,t)+ [u(r,t)ω z (r,t)] = δu(r,t)δω z (r,t) SE (40) ( ) PPM
15 δω z (r,t) (39) (36) 1 t δω z(r,t)+u(r,t) δω z (r,t)= δu(r,t) ω z (r,t) (41) 2 u(r,t) ω z (r,t) δω z (r,t)= t dτδu (r (t τ)u,τ) ω z (r,t) (42) δω z (r,t= ) =0 (40) δu(r,t)δω z (r,t) SE = ( η ω z ) η = t dτ δu(r,t)δu(r (t τ)u,τ) SE (43) 6 2 Klimontovich formula Leonard [1] L. Onsager: Nuovo Cimento Suppl. 6 (1949) 279. [2] : 56 (2001) PPM
16 [3] Y. Kiwamoto, K. Ito, A. Sanpei and A. Mohri: Phys. Rev. Lett. 85 (2000) [4] Y. Kiwamoto, N. Hashizume, Y. Soga, J. Aoki and Y. Kawai: Phys. Rev. Lett. 99 (2007) [5] G. L. EyinkandK. R. Sreenivasan: Rev. Mod. Phys. 78 (2006) 87. [6] G. Joyce and D. Montgomery: J. Plasma Phys. 10 (1973) 107. [7] D. Montgomery and G. Joyce: Phys. Fluids 17 (1974) [8] S. Kida: J. Phys. Soc. Jpn. 39 (1975) [9] R. A. Smith and T. M. O Neil: Phys. Fluids B 2 (1990) [10] D. J. Johnson: Phys. Fluids 31 (1988) [11] O. Bühler: Phys. Fluids 14 (2002) [12] T. S. Lundgren and Y. B. Pointin: J. Stat. Phys. 17 (1977) 323. [13] Y. B. Pointin and T. S. Lundgren: Phys. Fluids 19 (1976) [14] A. Leonard: J. Comput. Phys. 37 (1980) 289. PPM
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Equlbrum dstrbuton of two-dmensonal pont vortces at postve and negatve absolute temperature ( < 0 ) Yuch YATSUYANAGI Faculty of Educaton, Shzuoka Unversty 1 Onsager 1949 2 [1] Onsager Boltzmann exp( βe)
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