Title 絶対温度 <0となり得る点渦系の平衡分布の特性 ( オイラー方程式の数理 : 渦運動 150 年 ) Author(s) 八柳, 祐一 Citation 数理解析研究所講究録 (2009), 1642: Issue Date URL

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1 Title 絶対温度 <0となり得る点渦系の平衡分布の特性 ( オイラー方程式の数理 : 渦運動 150 年 ) Author(s) 八柳祐一 Citation 数理解析研究所講究録 (2009) 1642: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

$EDUCATION SHIZUOKA UNIVERSITY 1 Onsager 1949 2 [1] Onsager Boltzmann $\exp(-\beta E)$ $\beta$ $\searrow$ [2] Larmor $=0$ 2 Euler $[$34 $]$ [3 5 6] $\backslash$$

2 Equilibrium distribution of two-dimensional point vortices at positive and negative absolute temperature ( <0 ) YUICHI YATSUYANAGI FACULTY OF EDUCATION SHIZUOKA UNIVERSITY 1 Onsager [1] Onsager Boltzmann $\exp(-\beta E)$ $\beta$ $\searrow$ [2] Larmor $=0$ 2 Euler $[$34 $]$ [3 5 6] $\backslash$ [7]

3 $\Gamma_{i}$ 88 lib shizuoka ac jp/ 2 [8] 2 $\frac{\partial\omega(r)}{\partial t}=-(u(r)\cdot\nabla)\omega(r)$ (1) Dirac $\psi(r)$ $u(r)$ $\omega(r)$ $z$ $x-y$ $\psi(r)$ $= \sum_{i}^{n}\gamma_{i}g(r-r_{i})=-\frac{1}{2\pi}\sum_{i}^{n}\gamma_{i}\ln r-r_{i} $ (2) $u(r)$ $=$ $\nabla\cross(\psi(r)\hat{z})=\frac{1}{2\pi}\sum_{i}^{n}\gamma_{i}\hat{z}\cross\nabla\ln r-r_{i} =\frac{1}{2\pi}\sum_{i}^{n}\gamma_{i}\frac{\hat{z}\cross(r-r_{i})}{ r-r_{1} ^{2}}$ (3) $\omega(r)$ $= \nabla\cross\nabla\cross(\psi(r)\hat{z})=\sum_{:}^{n}\gamma_{i}\delta(r-r_{i})\hat{z}$ (4) $G(r)$ 2 Poisson Green [9] $r_{i}$ $i$ 2 $z$ $N$ (4) (1) $\sum_{i}^{n}\gamma_{i}(\frac{\partial}{\partial t}r_{i}-u(r))\cdot\nabla\delta(r-r_{i})=0$ (5) $k$ $r=r_{k}$ $\epsilon$ $\int_{r_{k}}[\sum_{\dot{\iota}}^{n}\gamma_{i}(\frac{\partial}{\partial t}r_{i}-u(r))\cdot\nabla\delta(r-r_{i})]=\int_{f}k[\gamma_{k}(\frac{\partial}{\partial t}r_{k}-u(r_{k}))\cdot\nabla\delta(r_{k}-r_{k})]=0$ $k$ (6) $\frac{\partial}{\partial t}r_{k}=u(r_{k})=\frac{1}{2\pi}\sum_{\neq k}\gamma_{i}\frac{\hat{z}x(r_{k}-r_{1})}{ r_{k}-r_{i} ^{2}}$ (7) Biot-Savart (7) $k$ $\sim$ $=r_{k}$

4 $\frac{\partial}{\partial t}r_{k}$ 89 $H$ $I$ $H$ $=$ $- \frac{1}{4\pi}\sum_{i}^{n}\sum_{j\neq i}^{n}\gamma_{i}\gamma_{j}\ln r_{i}-r_{j} $ (8) $I= \sum_{i}^{n}\gamma_{i} r_{i} ^{2}$ (9) $k$ $r_{k}=(x_{k} y_{k})$ (7) (8) $\Gamma_{k}\frac{dx_{k}}{dt}=\frac{\partial H}{\partial y_{k}}$ $\Gamma_{k}\frac{dy_{k}}{dt}=-\frac{\partial H}{\partial x_{k}}$ (10) $N/2$ $\Gamma_{0}(>0$ $)$ : ( ) $N/2$ $-\Gamma_{0}$ ( ) $R$ $N$ 6724 $H$ $I$ $H$ $H=- \frac{1}{4\pi}\sum_{*}^{n}\sum_{j\neq:}^{n}\gamma_{i}\gamma_{j}\ln r_{i}-r_{j} +\frac{1}{4\pi}\sum_{i}^{n}\sum_{j}^{n}\gamma_{i}\gamma_{j}\ln r_{i}-\overline{r}_{j} -\frac{1}{4\pi}\sum_{i}^{n}\sum_{j}^{n}\gamma_{i}\gamma_{j}\ln\frac{r}{ r_{j} }$ (11) $\overline{r}_{\dot{*}}=r^{2}r_{i}/ r_{i} ^{2}$ (11) [8] (11) 3 $k$ (11) (10) Biot-Savart $=$ $\frac{1}{2\pi}\sum_{*\neq k}^{n}\gamma_{i}\frac{\hat{z}\cross(r_{k}-r_{i})}{ r_{k}-r_{i} ^{2}}-\frac{1}{2\pi}\sum_{i}^{N}\Gamma_{i}\frac{\hat{z}\cross(r_{k}-\overline{r}_{i})}{ r_{k}-\overline{r}_{1} ^{2}}$ (12) (12) (12) $N\approx 6000$ PC. ( ) CPU (Core 2 Quad Q9550 ) 2 CPU (Pentium46603.$6GHz$) 300 Vortex-In-Cell Tree code MDGRAPE2.

5 90 MDGRAPE3 Biot-Savart [4 10] 1: MDGRAPE3 PC PCI-X Biot- Savart 3 $T$ $S$ $E$ $\frac{1}{t}=\frac{\partial S}{\partial E}=k_{B}\frac{\partial\ln It^{\gamma}(E)}{\partial E}$ (13) Boltzmann $S=k_{B}\ln W(E)$ $W(E)$ $T$ $\int lv(e)de$ (14) $W(E)$ $Earrow\infty$ $W(E)arrow 0$ $E_{0}$ $E>E_{0}$ $W(E)$ $E>E_{0}$ (13) (13) Onsager 1949 [1] (10) Onsager $(x y)$ A $\int d\omega=(\int dxdy)^{n}=a^{n}$ (15)

6 91 (a) (b) 2: $Earrow\infty$ $W(E)arrow 0$ $W(E)$ $E_{0}$ $E>E_{0}$ $=$ 4 [4 11] 4.1 ( ) (12) 3 3 ( ) 2 dipole

$92 3: dipole 1 1 4 $d$ $($ $5-2)^{1/2}R$ Emax$

7 92 3: dipole $d$ $($ $5-2)^{1/2}R$ Emax Emax Emax 4: $d=($ $5-2)^{1/2}R$ $d$ 5 2 2

8 93 5: (plus-plus : PP ) (plus-minus : PM ) 6: 2 $-$ 1 ( 6) PP PM 6 1 ( 7) 7 PM PP $R$ PM $R$ PP PM

9 94 7: 2 8:

$95 PP PM 9 9: $0$ 5 9 $0$ $\beta$ $\nu$ [12] $H_{i}$ (9) (11) $\sim$ $i$ 1 $H_{i+1\prime}$ $I_{*+1}$ $\exp(-\beta H_{i+1}+\nu$

10 95 PP PM 9 9: $0$ 5 9 $0$ $\beta$ $\nu$ [12] $H_{i}$ (9) (11) $\sim$ $i$ 1 $H_{i+1\prime}$ $I_{*+1}$ $\exp(-\beta H_{i+1}+\nu I_{\dot{*}+1})>\exp(-\beta H_{t}+\nu I_{i})$ $i+1$ $0$ 1 $\alpha$ $cxp(-\beta(h_{i+1}-h_{i})+\nu(i_{i+1}-i_{i}))>\alpha$ (16) $i+1$ $i+1$ 10 $\nu$ $\beta$ $\beta$ $\beta$ $\nu$

11 $\epsilon_{\ell}$ $=$ 96 $v=10^{2}$ $v=10^{arrow 2}$ 10: $10^{3}$ $10^{5}$ $\beta$ [13 14] $E(k)$ $=$ $\frac{1}{4\pi k}\sum_{*}^{n}\gamma_{i}^{2}+\frac{1}{4\pi k}\sum_{i}^{n}\sum_{j\neq:}^{n}\gamma_{i}\gamma_{j}j_{0}(k r_{i}-r_{j} )$ $- \frac{1}{2\pi k}\sum_{i}^{n}\sum_{j}^{n}\gamma_{i}\gamma_{j}\sum_{\ell=0}^{\infty}\epsilon_{\ell}(\frac{ r_{j} }{[\{})^{\ell}\cross J_{\ell}(kR)J_{\ell}(k r_{i} )\cos(\ell(\varphi_{i}-\varphi_{j}))$ $+ \frac{1}{2\pi k}\sum_{:}^{n}\sum_{j}^{n}\gamma_{i}\gamma_{j}\sum_{\ell=0}^{\infty}\epsilon_{\ell}(\frac{ r_{j} }{R})^{\ell}xJ_{\ell}^{2}(kR)\cos(\ell(\varphi_{i}-\varphi_{j}))$ (17) $\{\begin{array}{l}1 \ell=02 \ell\geq 1\end{array}$ (18) $x_{i}$ $=$ $ r_{i} \cos(\varphi_{i})$ $y_{i}= r_{i} \sin(\varphi_{i})$. (19)

12 97 (17) 1 2 Novikov (17) 11 11: ( ) 12 $E=8.51\cross 10^{3}$ $-1$ 12: $-2$ 12 $-1$

$98 11 $10^{1}$ 1 ( 12 ) $-2$ 12 $-2$ $-2$ 13 13: $-2$ $-2$ 1 $-2$ ( $)$ 11 $E=24.$

13 98 11 $10^{1}$ 1 ( 12 ) $-2$ 12 $-2$ $-2$ 13 13: $-2$ $-2$ 1 $-2$ ( $)$ 11 $E=24.1$ 5 ( ) $E=-15.9$ ( ) (17) 1 $\Gamma_{i}\Gamma_{j}>0$ $\Gamma_{i}\Gamma_{j}<0$ 1 $-1$ $-1$ $h^{a}$ 11

14 99 5 $\exp(-\beta E)$ $\beta$ Montgomery sinh-poisson [15] MDGRAPE3 [1] L. Onsager: Nuovo Cimento Suppl. 6 (1949) 279. [2] G. L. Eyink and K. R. Sreenivasan: Rev. Mod. Phys. 78 (2006) 87. [3] : 56 (2001) 253. [4] : 27 (2008) 23. [5] Y. Kiwamoto N. Hashizume Y. Soga J. Aoki and Y. Kawai: Phys. Rev. Lett. 99 (2007) [6] Y. Yatsuyanagi Y. Kiwamoto T. Ebisuzaki T. Hatori and T. Kato: Phys. Plasmas 10 (2003) [7] Y. Yatsuyanagi T. Ebisuzaki T. Hatori and T. Kato: Phys. Plasmas 10 (2003) [8] P. K. Newton: The N-Vortex Problem (Springer-Verlag Berlin 2001) chapter 1-3. [9] : ( 1978). [10] http: $//www$.peta.co.jp. [11] Y. Yatsuyanagi Y. Kiwamoto H. Tomita M. M. Sano T. Yoshida and T. Ebisuzaki: Phys. Rev. Lett. 94 (2005)

15 $[$ 12] 100 : ( 1997) chapter 3. [13] T. Yoshida and M. M. Sano: J. Phys. Soc. Jpn. 74 (2005) 587. [14] E. A. Novikov: Sov. Phys. JETP 41 (1975) 937. [15] G. Joyce and D. Montgomery: J. Plasma Phys. 10 (1973) 107.

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proc.dvi 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)