ъ ˜ m – ¤ ° ow ŠP Ž Ö ¨ù p § T “ Ó Þ” X ¢ ‘ ¤U  \ ¥ V ê s? 0õ m Çy ¢8 ý Ä Z ؃ º 4  ˜ m

 9 Z 4, pp. 953∼957

· Ï

ъ ˜ m – ¤ °  ow ŠP Ž Ö ¨ù p § T “ Ó Þ” X ¢ ‘ ¤U    \ ¥ V ê s? 0õ m Çy ¢8 ý Ä Z ؃ º 4  ˜ m

©

¼‰ ƒ B

›

¸‚  @ /† < Ɠ §  ƒ  õ † < Æ@ /† < Æ  o† < Æõ , F  g Å Ò 501-759

 + 2­ £) כ

›

¸‚  @ /† < Ɠ §  ƒ  õ † < Æ@ /† < Æ Ó ü t o † < Æõ , F  g Å Ò 501-759

(2012¸   7 Z 4 10{ 9  ~ à Î6 £ §, 2012¸   7 Z 4 31{ 9  à º& ñ ‘ : r ~ à Î6 £ §, 2012¸   8 Z 4 30{ 9  > F  S X ‰& ñ )

M 

g- ê ø Í Ä º ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O  (Wang-Landau algorithm)`  ¦  6   x # Œ „  í ß –r Ð 3 x`  ¦ à º' Ÿ ½ + É M :, ƒ  >  © œ I

x 9 • ¸ (joint density of states)\  ¦ f ” ] X  ½ ¨  
 <  ʓ É r  © œI x 9 • ¸ (density of states)ü < y Û ¼ž ÐÕ ªÏ þ › (histogram)`  ¦  – Ð ½ ¨   H ~ ½ ÓZ O s  ŠҖ Ð  6   x ) a  . Ä ºo   H ‘ : r  7 Hë  H \ " f y Û ¼ž ÐÕ ªÏ þ ›õ   © œI x 9 • ¸\  ¦  

–

Ð ½ ¨   H ~ ½ ÓZ O Ü ¼– Ð „  í ß –r Ð 3 x`  ¦ à º' Ÿ  €   | 9 " fë “ B6 £ §  à º (order parameter) ¼ # † ¾ Ó  ) a    H   õ \  ¦   s

f ç — ¸+ þ A (Ising model) ƒ  ½ ¨\  ¦ : Ÿ x # Œ % 3 >  ÷ &% 3  .

Ù þ

˜d ” # Q: M  g-ê ø Í Ä º ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O ,  © œI x 9 • ¸, ƒ  >  © œI x 9 • ¸

Estimate of Two Different Densities of States by Using the Wang-Landau Algorithm

Seol Ryu

Department of Chemistry, Chosun University, Gwangju 501-759

Wooseop Kwak ^∗

Department of Physics, Chosun University, Gwangju 501-759 (Received 10 July 2012 : revised 31 July 2012 : accepted 30 August 2012)

We perform a simulation of the Ising model by using the Wang-Landau algorithm, and we compare the simulation results obtained from the density of state g(E

1

) with a histogram H(E

1

, E

2

) to those obtained from the joint density of state g(E

1

, E

2

). We found that the simulation results without an estimate of the joint density of states yield a biased distribution of the order parameter.

PACS numbers: 05.20.-y, 05.70.Ln, 64.60.Cn, 66.30.Dn

Keywords: Ising model, Wang-Landau algorithm, Density of states, Joint density of states

I. " e  ] Ø



7 H _ …º ú ˜– Ð „  í ß –r Ð 3 x (Monte Carlo simulation)“ É r : Ÿ x >  Ó

ü

t o † < Æ\ " f ×  æ כ ¹ô  Ç % i ½ + É [1–19] `  ¦ “ ¦ e ”  . : £ ¤ y , ¨ î + þ A

∗

E-mail: [email protected]

:

Ÿ

x > † < Æ ì  r  \ " f e ” > & h \ " f  © œ„  s  ‰ & ³ © œ`  ¦ ƒ  ½ ¨½ + É M : ( Ž

É Ó' _  > í ß –r ç ß – (CPU time)`  ¦ é ß –» ¡ ¤ l  0 Aô  Ç ´ ú §“ É r „   í

ß –r Ð 3 x l Z O [ þ t s  > hµ 1 Ï÷ &# Q M ® o “ ¦, y Û ¼ž ÐÕ ªÏ þ › F ×  æ ~ ½ Ó Z O

 (histogram reweighting method) [1,10,11], { 9 ì ø Í o  ) a

€

© œ © œ^  ¦ ~ ½ ÓZ O  (generalized ensemble method) [14,18],  ' pà Ô

–

Ðx  Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O  (entropy sampling method)

-953-

[2,13,19] 1 p x s  Û ¼— 2 ;— ¸+ þ A (spin model)_   © œ„  s  x 9  é ß –Ñ þ ˜

| 9

 ] X j Ë µ (protein folding) ƒ  ½ ¨\  & h 6   x ÷ &# Q M ® o  . „  í ß –r  Ð

3 x`  ¦ : Ÿ x # Œ  ' pà Ԗ Ðx ü <  Ä »\  -t \  ¦ f ” ] X  > í ß –½ + É Ã º e ”

  H  ' pà Ԗ Ðx  Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O  ×  æ \  M  g- ê ø Í Ä º

³

ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O  (Wang-Landau sampling method) [2]“ É r Z

 t  î  r S X ‰  © œ$ í Ü ¼– Ð “  K   © œ y Œ • F  g`  ¦ ~ à Γ ¦ e ”   H „  í ß –r  Ð

3 x l Z O  ×  æ \  
s  . t ë ß –, \  -t  / B N ç ß – © œ_  Á º Œ • 0

A   6 £ § (random walk)`  ¦ : Ÿ x  9  © œI x 9 • ¸\  ¦ Æ ÒØ  ¦ Ù ¼

–

Ð % 3 ' õ A
>  ´ ú §“ É r > í ß –r ç ß –s  ™ èכ ¹  ) a  .   " f  © œI x 9 

•

¸ (density of states) <  ʓ É r ƒ  >   © œI x 9 • ¸ (joint density of states) Æ ÒØ  ¦Z O õ  ° ú  “ É r  € ª œô  Ç ~ ½ ÓZ O `  ¦  6   x # Œ > í ß – r

ç ß –`  ¦ ×  ¦ s  9  H ” ¸§ 4 `  ¦ # Œ M ® o  .

s

  7 Hë  H \ " f s  " é ¶ { 9 ~ ½ Ó   \ " f M  g- ê ø Í Ä º ³ ð‘ : r Æ Ò Ø

 ¦ ~ ½ ÓZ O `  ¦  6   x # Œ y © œ $ í  s f ç — ¸+ þ A_   © œI x 9 • ¸ü <

ƒ

 >   © œI x 9 • ¸\  ¦ > í ß – “ ¦, s [ þ t  © œI x 9 • ¸\  ¦  6   x # Œ

% 3

“ É r | 9 " fë “ B6 £ §  à º (order parameter) “ : r • ¸\     # Q b 

G>    É r \  ¦ ƒ  ½ ¨ % i  .

II. Ž ì ŏ ŒU ê s0 n É õ m Í T  ] Ø

1. { ¢] k ù

s

 " é ¶ { 9 ~ ½ Ó   \ " f y © œ $ í  s f ç — ¸+ þ A_  8 ú x \  -t  (Hamiltonian)“ É r

E = −J X

<i,j>

S i S j + h X

i

S i (1)

–

Ð & ñ _   ) a  . # Œl " f  s f ç — ¸+ þ A_  Û ¼— 2 ;s  +z» ¡ ¤ Ü ¼– Ð & ñ

§ >

=÷ &# Q e ” Ü ¼€   S _i = 1, ì ø Í@ / ~ ½ ӆ ¾ ÓÜ ¼– Ð & ñ § > =÷ &# Q e ” Ü ¼€   S i = −1, h  H ü @ ҁ © œ (external field), y © œ $ í  s f ç — ¸ + þ

A_    ½ + Ë © œÃ º (coupling constant) “   J  H 0 ˜ Ð   H  © œÃ º s

 . # Œl " f, Û ¼— 2 ;_  ½ + ˓ É r þ j“  ] X  s Ö  ©[ þ t (nearest neigh- bors) \  @ /K " fë ß – à º' Ÿ ÷ &“ ¦,  s f ç — ¸+ þ A_  „  í ß –r Ð 3 x`  ¦ 0

AK " f s  " é ¶ { 9 ~ ½ Ó   \   H Å Òl & h   â > › ¸|  (periodic boundary condition)`  ¦ & h 6   x ô  Ç .

2. V ê s? 0õ m Çy ¢Ñ ÷ Z ­ Ž‚ º§ Žq œ 



s f ç — ¸+ þ A ¢ ¸  H Û ¼— 2 ;— ¸+ þ A_   © œ„  s  ƒ  ½ ¨\ " f  H  ' p à

Ԗ Ði ”  ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O “   M  g- ê ø Í Ä º ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O `  ¦ s

6   x # Œ  © œI x 9 • ¸\  ¦ ½ ¨½ + É M : ( Ž É Ó' _  > í ß –r ç ß –`  ¦ é ß –

»

¡

¤ l  0 A # Œ  6 £ § _  ~ ½ ÓZ O s  V , o   6   x ÷ &“ ¦ e ”  . Á º



Œ

•0 A   6 £ §`  ¦ l  0 AK " f  H — ¸Ž  H  © œI x 9 • ¸ (density of

states)\  ¦ Û ¼— 2 ;  © œI  \  -t  E 1 `  ¦ t   H  © œI  A(E 1 ) \ 

"

f Û ¼— 2 ;  © œI  \  -t  E 1 0 `  ¦ ° ú   H  © œI  A ⁰ (E ₁ ⁰ ) – Ð_  „  s  S X

‰Ò  ¦ (transition probability)`  ¦  6 £ § õ  ° ú  s  j þ t à º e ”  .

p (A → A ⁰ ) = min  g(A) g(A ⁰ ) , 1



, (2)

g(A) ≥ g(A ⁰ ) s €   min  _g(A)

g(A

⁰

) , 1 

= 1 s Ù ¼– Ð „  s  S X ‰Ò  ¦ p = 1 s  ÷ &# Q" f D h– Ðî  r \  -t  E 1 0 \  ¦ ° ú   H D h– Ðî  r  © œI 

 † ½ Ó © œ ‚  × þ ˜s  ÷ &“ ¦, ë ß –{ 9  g(A) < g(A ⁰ ) Ü ¼€   „  s  S X ‰Ò  ¦ p = _g(A ^g(A)

0

) \     D h– Ðî  r  © œI  ‚  × þ ˜ ) a  .

„

 í ß –r Ð 3 x`  ¦   H 1 l x î ß – \  -t  y Û ¼ž ÐÕ ªÏ þ › (histogram) H(A) ü < à º& ñ “    (modification factor) f\  ¦  6   x ô  Ç  © œI  x 9

• ¸ g(A)  H > 5 Å q » ¡ ¤' ‘  ) a  . » ¡ ¤' ‘  ~ ½ ÓZ O “ É r  6 £ § õ  ° ú   ;

H(A) + 1 → H(A) (3) g(A) × f → g(A) (4)

#

Œl " f, œ íl  f = e ¹ `  ¦  6   x ô  Ç . \  -t  y Û ¼ž ÐÕ ªÏ þ › s

 \  -t  / B N ç ß –0 A\ " f_  Á º Œ •0 A   6 £ §`  ¦ : Ÿ x # Œ ¨ î ¨ î K  t

€   y Û ¼ž ÐÕ ªÏ þ › H(A)\  ¦ „  Â Ò 0Ü ¼– Ð F [ O & ñ “ ¦, D h– Ð î

 r à º& ñ “    f new = f ^1/2 \  ¦ • ¸{ 9  # Œ, þ j7 á x à º& ñ “   

f f inal = e ⁻⁹  | ¨ c M : t  „  í ß –r Ð 3 x`  ¦ à º' Ÿ  # Œ g(A)\  ¦

½

¨ô  Ç . Õ ªo “ ¦, ] jë ß – † ½ Ó (Zeeman term)_  | 9 " fë “ B6 £ §  à º

\

 -t  E 2 = P

i S _i “ É r 0 A_  d ”  (2) `  ¦ ë ß –7 á ¤ €    – Ð y  Û

¼ž ÐÕ ªÏ þ ›\  $  © œ # Œ é  H  .

3. Ž ì Å4  V ê s? 0õ m Çy ¢



s f ç — ¸+ þ A_  ƒ  >   © œI x 9 • ¸\  ¦ s 6   x ô  Ç M  g- ê ø Í Ä º

³

ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O “ É r  6 £ § õ  ° ú   . Û ¼— 2 ;  © œ  ñ \  -t ü <

]

jë ß – † ½ Ó_  | 9 " fë “ B6 £ §  à º \  -t \  ¦ t   H ƒ  >  © œI  B(E 1 , E 2 ) \ " f r ' Ÿ  ƒ  >  © œI  B(E ^{1 0} , E 2 0 ) – Ð_  „  s  S X ‰ Ò

 ¦“ É r  6 £ § õ  ° ú   .

p (B → B ⁰ ) = min  g(B) g(B ⁰ ) , 1



, (5)

g(B) ≥ g(B ⁰ ) s €   min  _g(B)

g(B

⁰

) , 1 

= 1 s Ù ¼– Ð „  s  S X ‰Ò  ¦ p = 1 s  ÷ &# Q" f D h– Ðî  r ƒ  >  © œI  B ⁰ s  † ½ Ó © œ ‚  × þ ˜s  ÷ &

“

¦, ë ß –{ 9  g(B) < g(B ⁰ ) Ü ¼€   „  s  S X ‰Ò  ¦ p = _g(B ^g(B)

0

) \     D

h– Ðî  r ƒ  >  © œI  ‚  × þ ˜÷ &# Q ”   .

„

 í ß –r Ð 3 x`  ¦   H 1 l x î ß – ƒ  >  \  -t  y Û ¼ž ÐÕ ªÏ þ › (his- togram) H(B) ü < à º& ñ “    (modification factor) f\  ¦   6

 

x ô  Ç ƒ  >   © œI x 9 • ¸ g(B)\  ¦ ½ ¨ # Œ > 5 Å q » ¡ ¤' ‘ r †   . » ¡ ¤ '

‘  ~ ½ ÓZ O “ É r  6 £ § õ  ° ú   ;

H(B) + 1 → H(B) (6)

g(B) × f → g(B) (7)

Fig. 1. (Color online) Ln(g(A)) as a function of E ₁ and E ₂ .

#

Œl " f, œ íl  f = e ¹ `  ¦  6   x ô  Ç . ƒ  >  \  -t  y Û ¼ž Ð Õ

ªÏ þ ›s  ƒ  > \  -t  / B N ç ß –0 A\ " f_  Á º Œ •0 A   6 £ §`  ¦ : Ÿ x # Œ

¨ î

¨ î K t €   ƒ  >  \  -t  y Û ¼ž ÐÕ ªÏ þ › H(B)\  ¦ „  Â Ò 0Ü ¼

–

Ð F [ O & ñ “ ¦, D h– Ðî  r à º& ñ “    f ^new = f ^1/2 \  ¦ • ¸{ 9  

#

Œ, þ j7 á x à º& ñ “    f ^{f inal} = e ⁻⁹  | ¨ c M : t  „  í ß –r Ð 3 x

`

 ¦ à º' Ÿ ô  Ç Ê ê g(B)\  ¦ ½ ¨ô  Ç .



© œI x 9 • ¸\  ¦ ½ ¨   H ~ ½ ÓZ O “ É r II 2 _  ~ ½ ÓZ O s  > í ß –5 Å q • ¸



Ø ÔÙ ¼– Ð V , o   6   x ÷ &# Q t “ ¦ e ”  .

4. ö n ÚP S ë s

“

: r • ¸ T \  _ ” > r   H ì  r C † < Êà º (partition function) Z(T ) = P ge ^−βE   H  © œI x 9 • ¸ ¢ ¸  H ƒ  >   © œI x 9 • ¸– ÐÂ Ò '

 ½ ¨½ + É Ã º e ” “ ¦, ì  r C † < Êà º Z(T )\  ¦ · ú ˜€   # Q‹ "  Ó ü t o | ¾ Ó Q_  l

Î . ° ú כ < Q >= P Qge ^−βE /Z(T ) • ¸ ~ 1 >  ½ ¨½ + É Ã º e ”  .

#

Œl " f, β = 1/k ^B T   H % i “ : r • ¸ (inverse temperature) s 



.

III. + s ÇÊ Ý õ m Í w Š²  o

r

Û ¼% 7 › ß ¼l  N“    t f ç — ¸+ þ A_  0 p x ô  Ç — ¸Ž  H  © œI  Ã

º  H s  : r& h Ü ¼– Ð 2 ^N s  . ‘ : r ƒ  ½ ¨\ " f  H & ñ S X ‰ • ¸\  ¦ Z  } s  l

 0 AK  N = 100“   Û ¼— 2 ;s  s  " é ¶ { 9 ~ ½ Ó   _     & h \  0

Au ô  Ç “ ¦ & ñ “ ¦ Å Òl & h   â > › ¸| `  ¦ & h 6   x # Œ „  í ß – r

Ð 3 x`  ¦ à º' Ÿ  % i  .

Õ

ªa Ë > 1“ É r  © œI x 9 • ¸\  ¦ E 1 õ  E ² _  † < Êà º– Ð Õ ª 2 ;  כ s  .

\

 -t  E 1 \  @ /6 £ x   H E 2 _  ì  r Ÿ í ¼ # † ¾ Ó÷ &# Q e ” 6 £ §`  ¦ · ú ˜ Ã

º e ”  .

Õ

ªa Ë > 2  H ƒ  >   © œI x 9 • ¸\  ¦ E 1 õ  E 2 _  † < Êà º– Ð Õ ª 2 ;  כ s

 .

Fig. 2. (Color online) Ln(g(B)) as a function of E ₁ and E ₂ .

Fig. 3. (Color online) Contour plot of Ln(g(A)) as a function of E ₁ and E ₂ .

Õ

ªa Ë > 3“ É r  © œI x 9 • ¸_  1 p x “ ¦‚  `  ¦ E 1 õ  E 2 _  † < Êà º– Ð Õ ª



2 ;  כ s  .

Õ

ªa Ë > 4“ É r ƒ  >   © œI x 9 • ¸_  1 p x “ ¦‚  `  ¦ E 1 õ  E 2 _  † < Êà º

–

Ð Õ ª 2 ;  כ s  .

Õ

ªa Ë > 5ü < ° ú  s  ^  ¦ Þ Ôë ß – ×  æ (Boltzmann weight) e ^−E/k

^B

^T \  ¦  6   x ½ + É M :, “ : r • ¸ Z  }  | 9  à º2 Ÿ ¤ > _  8 ú x \ 



-t \  l # Œ   H ± ú “ É r \  -t  ï  r 0 A[ þ t _  l # Œ  Œ • t 



 H  כ `  ¦ · ú ˜ à º e ”  . s – РÒ'  “ : r • ¸ Z  }  t €   „  % ò % i _  l

# Œ• ¸  _  ° ú    ± ú “ É r \  -t  ï  r 0 A_   © œI x 9 • ¸  © œ

@

/& h Ü ¼– Ð ×  æ כ ¹ t  · ú §t ë ß –, ± ú “ É r “ : r • ¸\ " f  H ± ú “ É r \  - t


`  ¦  © œI x 9 • ¸_  & ñ S X ‰$ í s  ˜ Ð  ×  æ כ ¹K ”     H  כ

`

 ¦ · ú ˜ à º e ”  .

Õ

ªa Ë > 6“ É r | 9 " fë “ B6 £ §  à º (order parameter) h|m|i\  ¦ “ : r

•

¸_  † < Êà º– Ð Õ ª 2 ;  כ s  . $ “ : r \ " f  H  © œI x 9 • ¸ g(A)\  ¦ s

6   x # Œ > í ß –ô  Ç | 9 " fë “ B6 £ §  à º ƒ  >   © œI x 9 • ¸ g(B)\  ¦

Fig. 4. (Color online) Contour plot of Ln(g(B)) as a function of E 1 and E 2 .

Fig. 5. (Color online) Plot of Boltzmann weight as a function of energies at T = 1.8 and at T = 2.6. Shown in the insert is the linear-log plot of Boltzmann weight.

s

6   x # Œ > í ß –ô  Ç | 9 " fë “ B6 £ §  à º˜ Ð  ± ú “ É r ° ú כ`  ¦ t “ ¦ e ”  6

£

§`  ¦ › ' a8 £ ¤ ½ + É Ã º e ”  .

IV. + s Ç Â ] Ø

r

Û ¼% 7 ›_  ß ¼l  N = 100“   y © œ $ í  s f ç — ¸+ þ A_   © œ I

x 9 • ¸ü < ƒ  >   © œI x 9 • ¸\  ¦ M  g- ê ø Í Ä º ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O `  ¦ :

Ÿ

x # Œ > í ß – “ ¦, “ : r • ¸\    É r | 9 " fë “ B6 £ §  à º_  l @ /° ú כ

`

 ¦ ½ ¨ % i  . \  -t  / B N ç ß –0 A\ " f_  Á º Œ •0 A   6 £ §`  ¦ : Ÿ x 

#

Œ ƒ  >   © œI x 9 • ¸\  ¦ ½ ¨   H M  g- ê ø Í Ä º · ú ˜“ ¦o 7 £ §“ É r ( Ž É Ó '

 > í ß –r ç ß –s   -Á º ´ ú §s  [ þ t l  M :ë  H \  ´ ú §“ É r ƒ  ½ ¨[ þ t s  ( Ž  É

Ó' _  > í ß –r ç ß –`  ¦ é ß –» ¡ ¤ l  0 A # Œ  © œI x 9 • ¸\  ¦ ½ ¨ €  

"

f Á º Œ •0 A   6 £ § ×  æ \  % 3 “ É r | 9 " fë “ B6 £ §  à º\  ¦  – Ð $  © œ 



 H „  í ß –r Ð 3 x l Z O `  ¦  6   x ô  Ç .

Fig. 6. (Color online) Plot of h|m|i as a function of T where thin lines show the results estimated by using of G(A) and thick ones estimated by using of G(B).

‘

: r ƒ  ½ ¨\ " f  H  © œI x 9 • ¸ü < | 9 " fë “ B6 £ §  à º\  ¦  – Ð y  Û

¼ž ÐÕ ªÏ þ ›\  $  © œ # Œ > í ß –ô  Ç E ₁ õ  E 2 _  ì  r Ÿ í (Õ ªa Ë >

1)  ¼ # † ¾ Ó  ) a    H  כ `  ¦ · ú ˜ Í Ç x . s  Qô  Ç ì  r Ÿ í_  ¼ # † ¾ Ó

“ É

r  © œI x 9 • ¸– Ð > í ß –ô  Ç | 9 " fë “ B6 £ §  à º_  l @ /° ú כ h|m|i = P

A mg(A)e ^−βE õ  ƒ  >   © œI x 9 • ¸– Ð > í ß –ô  Ç | 9 " fë “ B6 £ §   Ã

º_  l @ /° ú כ P

B mg(B)e ^−βE _  s _  " é ¶ “   ) a    H  כ

`

 ¦ · ú ˜€ Œ ¤  (Õ ªa Ë > 6). Õ ªo “ ¦, s  Qô  Ç > í ß –  õ _  s   H E  ± ú `  ¦ à º2 Ÿ ¤  8 & ”     H  כ `  ¦ Õ ªa Ë > 5`  ¦ : Ÿ x # Œ · ú ˜ à º

 e ” % 3  .

 

 : r& h Ü ¼– Ð,  € ª œ >    + þ A ) a  © œI x 9 • ¸\  ¦  6   x # Œ à º '

Ÿ ÷ &# Q“ : r ´ ú §“ É r  7 Hë  H[ þ t _  > í ß –   õ _  & ñ S X ‰ • ¸\  ë  H ] j

e ”

`  ¦ t • ¸ — ¸ É r    H  z  ´`  ¦ · ú ˜>  ÷ &% 3  . · ú ¡Ü ¼– Ð  © œI x 9 

•

¸ <  ʓ É r   + þ A ) a  © œI x 9 • ¸\  ¦ s 6   x ô  Ç ƒ  ½ ¨  õ [ þ t s  # Q* ‹ ô

 Ç > í ß – © œ_  ë  H ] j& h `  ¦ ? /Ÿ í “ ¦ e ”   H t \  ¦  € ª œô  Ç — ¸+ þ A

\

 @ / # Œ  [ jy  q “ § ƒ  ½ ¨ “ ¦, M  g- ê ø Í Ä º ³ ð‘ : r Æ ÒØ  ¦

~

½ ÓZ O _  > í ß –r ç ß –`  ¦ é ß –» ¡ ¤ † < Êõ  1 l x r \    õ \  ¦ & ñ S X ‰ y  \ V 8

£

¤   H · ú ˜“ ¦o 1 p u (algorithm) `  ¦ > hµ 1 Ͻ + É >  S \ ‰ s  .

P

c p 8 ý ò k >

s

  7 Hë  H“ É r 2012¸  • ¸ & ñ  Ò(“ §¹ ¢ ¤ õ † < Æl Õ ü t  Ò)_  F " é ¶ Ü

¼– Ð ô  Dz D Gƒ  ½ ¨F é ß –_  t " é ¶`  ¦ ~ à Î  à º' Ÿ  ) a ƒ  ½ ¨e ” (No.

2012-0003927).

Y

c p w Š à U Ø ”  ô

[1] Computer Simulation Studies in Condensed Matter

Physics II, edited by D. P. Landau, K. K. Mon and

H.-B. Sch¨ uttler (Springer-Verlag, Berlin, 1990) ; A.

M. Ferrenberg and D. P. Landau, Phys. Rev. B 44, 5081 (1991); K. Chen, A. M. Ferrenberg and D. P.

Landau, Phys. Rev. B 48, 3249 (1993).

[2] F. Wang and D. P. Landau, Phys. Rev. Lett. 86 2050 (2001).

[3] A. M. Ferrenberg, D. P. Landau and R. H. Swend- sen, Phys. Rev. E 51, 5092 (1995).

[4] K. Binder, Computational Methods in Field Theory, edited by C. B. Lang and H. Gausterer, (Springer, Berlin, 1992).

[5] K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics, (Springer, Berlin, 1992).

[6] W. Kwak, D. P. Landau and B. Schmittmann, Phys.

Rev. E 69, 66134 (2004).

[7] W. Kwak and D. P. Landau, Int. J. Mod. Phys. C, 17, 15 (2006).

[8] W. Kwak, J.-S. Yang and I.-M. Kim, Phys. Rev. E 75, 41108 (2007).

[9] W. Kwak, J.-S. Yang, J.-I. Sohn and I.-M. Kim, Phys. Rev. E 75, 61110 (2007).

[10] W. Kwak, J.-S. Yang, J.-I. Sohn and I.-M. Kim, Phys. Rev. E 75, 61130 (2007).

[11] W. Kwak, J.-S. Yang, K.-I. Goh and I.-M. Kim, Eu- rophys. Lett. 84, 36004 (2008).

[12] J.-S. Yang, I.-M. Kim and W. Kwak, Europhys.

Lett. 88, 20009 (2009).

[13] W. Kwak and U. H. Hansmann, Phys. Rev. Lett.

181, 2010 (2010).

[14] J.-S. Yang and W. Kwak, Comput. Phys. Commun.

95, 138102 (2005).

[15] H. E. Stanley, D. Stauffer, J. Kertesz and H. J. Her- rmann, Phys. Rev. Lett. 59, 2326 (1987).

[16] H. J. Herrmann, Physica A 168, 516 (1990).

[17] F. Wang, N. Hatano and M. Suzuki, J. Phys. A 28, 4543 (1995).

[18] E. Marinari and G. Parisi, Europhys. Lett. 19 451 (1992); U. H. E. Hansmann, Chem. Phys. Lett. 281 140 (1997); V. Aleksenko, W. Kwak and U. H. E.

Hansmann, Physica A 350 28 (2005).

[19] B. A. Berg, Int. J. Mod. Phys. C 4 249 (1993); B. A.

Berg and T. Neuhaus, Phys. Rev. Lett. 68 9 (1992);

P. M. C. de Oliveira, T. J. P. Penna and H. J. Her-

rman, Brazilian J. Phys. 26 677 (1996); J.-S. Wang,

Physica A 281 147 (2000).