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) β [2] Larmor 0 2 Euler [3,4] [3,5,6] [7] 1
http://r.lb.shzuoka.ac.jp/ 2 [8] 2 ω(r) t = (u(r) )ω(r) (1) Drac ψ(r) u(r) ω(r) z x y ψ(r) = Γ G(r r )= 1 2π u(r) = (ψ(r)ẑ) = 1 2π ω(r) = (ψ(r)ẑ) = Γ ln r r, (2) Γ ẑ ln r r = 1 ẑ (r r ) Γ (3) 2π r r 2 Γ δ(r r )ẑ (4) G(r) 2 Posson Green [9] r Γ ẑ z N (4) (1) ( ) Γ t r u(r) δ(r r )=0 (5) k r = r k ɛ [ N ( ) ] [ ( ) ] Γ t r u(r) δ(r r ) = Γ k t r k u(r k ) δ(r k r k ) =0 r k k t r k = u(r k )= 1 2π r k k (6) Γ ẑ (r k r ) r k r 2 (7) Bot-Savart (7) k r = r k 2
H I H = 1 Γ Γ j ln r r j 4π j (8) I = Γ r 2 (9) k r k =(x k,y k ) (7) (8) Γ k dx k dt = H y k, Γ k dy k dt = H x k (10) N/2 Γ 0 (> 0 ) ( ) N/2 Γ 0 ( ) R N 6724 H I H H = 1 4π j Γ Γ j ln r r j + 1 4π j Γ Γ j ln r r j 1 4π j Γ Γ j ln R r j, (11) r = R 2 r / r 2 (11) [8] (11) 3 k (11) (10) Bot-Savart t r k = 1 2π k Γ ẑ (r k r ) r k r 2 1 2π Γ ẑ (r k r ) r k r 2 (12) (12) (12) N 6000 12000 PC (2008 12 ) CPU (Core 2 Quad Q9550 ) 2 CPU (Pentum4 660 3.6GHz) 300 Vortex-In-Cell Tree code MDGRAPE-2 3
MDGRAPE-3 Bot-Savart 12000 400 20 [4, 10] 1: MDGRAPE-3 PC PCI-X Bot- Savart 3 T S E 1 T = S E = k ln W (E) B E (13) Boltzmann S = k B ln W (E) W (E) T W (E)dE (14) W (E) E W (E) 0 E 0 E >E 0 W (E) E >E 0 (13) (13) Onsager 1949 [1] (10) Onsager (x, y) A ( dω = ) N dxdy = A N (15) 4
W(E) W(E) β > 0 β < 0 E E0 E (a) 2: E W (E) 0 W (E) E 0 E >E 0 4 [4, 11] 4.1 ( ) (12) 3 3 ( ) 2 dpole 5
3: dpole 1 1 4 d ( 5 2 ) 1/2 R Emax Emax Emax dstance d d 4: 1 1 2 d d = ( 5 2 ) 1/2 R 5 2 2 6
5: 2 6 7 (plus-plus PP ) (plus-mnus PM ) 6: 2 1 ( 6) PP PM 6 1 ( 7) 7 PM PP R PM R 2 5 8 8 5 2 PP PM 7
7: 2 8: 2 2 5 2 5 6 7 8
PP PM 9 9: 0 5 0 9 β ν [12] H I (9) (11) 1 H +1 I +1 exp( βh +1 +νi +1 ) > exp( βh +νi ) +1 0 1 α exp ( β(h +1 H )+ν(i +1 I )) >α (16) +1 +1 10 ν β β β ν 9
10: 10 3 10 5 β 5 4.2 [13, 14] E(k) = 1 4πk 1 2πk + 1 2πk Γ 2 + 1 Γ Γ j J 0 (k r r j ) 4πk j ( ) l rj Γ Γ j ɛ l J l (kr)j l (k r )cos(l(ϕ ϕ j )) j l=0 R ( ) l rj Γ Γ j ɛ l Jl 2 (kr)cos(l(ϕ ϕ j )) (17) j l=0 R { 1 l =0, ɛ l = (18) 2 l 1, x = r cos(ϕ ), y = r sn(ϕ ). (19) 10
(17) 1 2 Novkov (17) 11 11: ( ) 12 E =8.51 10 3 1 12: 2 12 1 11
11 1 ( 12 10 1 ) 2 12 2 2 13 13: 2 2 1 2 ( ) 11 E =24.1 5 ( ) E = 15.9 ( ) (17) 1 Γ Γ j > 0 Γ Γ j < 0 1 1 1 11 12
5 exp( βe) β Montgomery snh-posson [15] MDGRAPE-3 [1] L. Onsager: Nuovo Cmento Suppl. 6 (1949) 279. [2] G. L. EynkandK. R. Sreenvasan: Rev. Mod. Phys. 78 (2006) 87. [3] : 56 (2001) 253. [4] : 27 (2008) 23. [5] Y. Kwamoto, N. Hashzume, Y. Soga, J. Aok, and Y. Kawa: Phys. Rev. Lett. 99 (2007) 115002. [6] Y. Yatsuyanag, Y. Kwamoto, T. Ebsuzak, T. Hator, and T. Kato: Phys. Plasmas 10 (2003) 3188. [7] Y. Yatsuyanag,, T. Ebsuzak, T. Hator, and T. Kato: Phys. Plasmas 10 (2003) 3181. [8] P. K. Newton: The N-Vortex Problem (Sprnger-Verlag, Berln, 2001) chapter 1-3. [9] : (,, 1978). [10] http://www.peta.co.jp. [11] Y. Yatsuyanag, Y. Kwamoto, H. Tomta, M. M. Sano, T. Yoshda, and T. Ebsuzak: Phys. Rev. Lett. 94 (2005) 054502. 13
[12] : (,, 1997) chapter 3. [13] T. Yoshda and M. M. Sano: J. Phys. Soc. Jpn. 74 (2005) 587. [14] E. A. Novkov: Sov. Phys. JETP, 41 (1975) 937. [15] G. Joyce and D. Montgomery: J. Plasma Phys. 10 (1973) 107. 14