ŒW Ä Ä Z ؃ º] K ¤• ¤õ u § “ Ó Þ” X ¢ KLS { ¢] k ù8 ý ! aŒ £ ? ŒW Ä ù m Ç" e8 04 ì ŕ ¤



ŒW Ä Ä Z ؃ º] K ¤• ¤õ u §  “ Ó Þ” X ¢ KLS { ¢] k ù8 ý ! aŒ £ ? ŒW Ä ù m Ç" e8 04 
ì ŕ ¤

… è

¡< ) o 

“

¦ 9@ /† < Ɠ § Ó ü t o † < Æõ , " fÖ  ¦ 136-713

 + 2­ £) כ ^∗

›

¸‚  @ /† < Ɠ § Ó ü t o † < Æõ , F  g Å Ò 501-759 (2008¸   2 Z 4 13{ 9  ~ à Î6 £ §)

!

QF K ½ ¨% i  | 9 " fB > h  à º Ising    l ^ ‰ — ¸+ þ Aõ  Katz-Lebowitz-Sphon (KLS) — ¸+ þ A\  & h ½ + Ëô  Ç | 9 

"

fB > h  à º“  t \  ¦ S X ‰ “   l  0 AK  ½ ¨% i ì  r Ÿ í† < Êà º\  ¦  6   x % i  . ½ ¨% i ì  r Ÿ í† < Êà º\  ¦  6   x ô  Ç ! QF K ½ ¨% i  | 9 

"

fB > h  à º  H | 9 " fB > h  à º– Ð & h ½ + Ë  9,  € ª œô  Ç „  s Ö  ¦ \  @ /K " f• ¸ { 9 › ' a$ í e ”   H   õ 
 M ® o  . ¢ ¸ ô

 Ç ½ ¨% i ì  r Ÿ í† < Êà º\  ¦  6   x ô  Ç ’ < H  © œ ! QF K ½ ¨% i  B > h  à º\  @ /K " f• ¸ ° ú  “ É r „  í ß – z  ´+ « >`  ¦ à º' Ÿ  % i Ü ¼ 9 s 

%

i r   € ª œô  Ç „  s Ö  ¦ x 9  ¿ º ¨ î + þ A x 9  q ¨ î ' Ÿ > \  @ /K  | 9 " fB > h  à º– Ð" f & h ½ + ˆ < Ê`  ¦ S X ‰ “   % i  .

PACS numbers: 05.20.-y, 05.70.Ln, 64.60.Cn, 66.30.Dn Keywords:    l ^ ‰, | 9 " fB > h  à º, ’ < H  © œ ( f ” 

I. " e  ] Ø

Monte Carlo „  í ß –r Ð 3 x“ É r : Ÿ x > Ó ü t o † < Æ\ " f  © œ„  s ‰ & ³ © œ õ

 e ” > & h \ " f_  Ó ü t o † < Æ& h  : £ ¤$ í `  ¦ ƒ  ½ ¨   H X < ×  æ כ ¹ô  Ç % i 

½

+ É`  ¦ “ ¦ e ”   [1–10]. s  ƒ  ½ ¨\ " f  H ¿ º t  ˜ Д > r — ¸+ þ A (Ising    l ^ ‰ — ¸+ þ Aõ  q ¨ î + þ A Ising     l ^ ‰“   Katz- Lebowitz-Sphon (KLS) — ¸+ þ A [11])_  | 9 " fB > h  à º_  : £ ¤

$ í

`  ¦ ƒ  ½ ¨ l  0 AK " f Monte Carlo „  í ß –r Ð 3 x`  ¦ à º' Ÿ  

%

i  . ˜ Д > r — ¸+ þ Aõ  q ˜ Д > r — ¸+ þ A_  ×  æ כ ¹ô  Ç s & h “ É r | 9 " f B

> h  à º_  ˜ Д > r$ í s  e ” Ö ¼
 \ O Ö ¼
\  e ”  . ¿ º t  ˜ Ð

”

> r — ¸+ þ A“ É r “  § 4 `  ¦  l    H þ j“  ] X   © œ  ñ Œ •6   x \  _ K " f

 

 5 Å q _  { 9  [ þ t s  S X ‰ í ß –   H — ¸+ þ As  . s  — ¸+ þ A\ " f  H { 9

 [ þ t _  ×  ¦ # Q[ þ t  
 Z þ t # Q
t  · ú §“ ¦ 8 ú x > hà º ˜ Д > r s 

 )

a  . s  Qô  Ç { 9   ˜ Д > r _  $ í | 9  M :ë  H \ , ˜ Д > r — ¸+ þ A\ " f  H

³

ðï  r& h “   | 9 " fB > h  à º  H † ½ Ó © œ { 9 & ñ ô  Ç ° ú כ`  ¦ t “ ¦   

t  · ú §Ü ¼Ù ¼– Ð | 9 " fB > h  à º– Ð" f_  % i ½ + É`  ¦ ] j@ /– Ð Ã º' Ÿ 

`

 ¦ t  3 l w ô  Ç .   " f ‘ : r ƒ  ½ ¨\ " f  H ˜ Д > r — ¸+ þ A`  ¦ ¸ ú ˜ l

Õ ü t ½ + É Ã º e ”   H | 9 " fB > h  à º Á º% Á “  \  ¦ · ú ˜ ˜ Ð 9“ ¦ ô

 Ç .



^ ‰> \  ¦ ƒ  ½ ¨   H : Ÿ x > Ó ü t o † < Æ\  e ” # Q" f, U  ´s  " é ¶ L s  Ä »ô  Çô  Ç  Œ •“ É r ½ ¨% i  (blocks)Ü ¼– Ð > \  ¦
¾ º# Q" f  Œ •

“ É

r ½ ¨% i [ þ t _  | 9 " fB > h  à º[ þ t _  ì  r Ÿ í\  ¦ s 6   x # Œ > _ 

| 9

" fB > h  à º\  ¦ ½ ¨   H ~ ½ ÓZ O “ É r V , o   6   x s  ÷ &# Q4 R M ® o

∗

E-mail: [email protected]



. s  Qô  Ç r • ¸  H Ising    Ä »^ ‰— ¸+ þ A_   © œ/ B N” > r (phase coexistence)`  ¦ s K  “ ¦ e ” > t à º\  ¦ > í ß –   H X < & h 6   x ÷ &

#

Q M ® o   [6,7]. ³ ðï  r Ising    l ^ ‰ — ¸+ þ A\ " f  H q # Qe ”   H þ

j“  ] X     – Ð_  ‡ © œØ  æ 8 Al  q  (hopping rate)  H Á º l 



© œ Ising K ‰x 9 ž Ðm î ß – (Ising Hamiltonian in zero magnetic field) \  @ /K " f  © œ[ jç  H+ þ A› ¸| `  ¦ ë ß –7 á ¤ # Œ  ô  Ç . Õ ª Q

 q ¨ î + þ A Ising    l ^ ‰ — ¸+ þ A“  , KLS— ¸+ þ A“ É r { 9  [ þ t s  Á

ºô  Çy   H ü @Â Ò Ä » „  l  © œ\  _ ô  Ç j Ë µ`  ¦ ~ à Î>   ) a  . Á ºô  Ç ô

 Ç Ä » „  l  © œõ  Å Òl & h   â > › ¸|  \ " f à º' Ÿ ÷ &  H q 

% i

 1 l x§ 4 † < Æ\  _  # Œ >   H ¨ î + þ A © œI \  • ¸² ú ˜½ + É Ã º \ O Ü ¼ 9,

± ú

“ É r “ : r • ¸\ " f > _  { 9  [ þ t“ É r { — ¸€ ª œ_   e ” D h (configu- ration)\  ¦ Ä » „  l  © œõ  ¨ î ' Ÿ s  ÷ &• ¸2 Ÿ ¤ + þ A$ í ô  Ç  [12,13].

‘

: r ƒ  ½ ¨\ " f  H ! QF K ½ ¨% i  | 9 " fB > h  à º (sub-block order parameter)\  ¦ Metropolis ü < heat-bath „  s Ö  ¦`  ¦  6   x 

#

Œ 8 £ ¤& ñ % i   [13–15]. e ” > “ : r • ¸ü < | 9 " fB > h  à º_  e ” 

>

t à º\  ¦ ! QF K ½ ¨% i  | 9 " fB > h  à ºü < ’ < H  © œ ! QF K ½ ¨% i  | 9 " f B

> h  à º (damage sub-block order parameter)_  Ä »ô  Çß ¼ l

 » ¡ ¤' ‘  (finte-size scaling)`  ¦  6   x # Œ y Œ •y Œ • > í ß – % i  .

’

< H  © œ ! QF K ½ ¨% i  | 9 " f B > h  à º  H ’ < H  © œ ( f ”  ~ ½ ÓZ O  (dam- age spreading method)`  ¦ s 6   x # Œ ½ ¨ % i   [16]. 1 p x ~ ½ Ó Ä

»ô  Ç» ¡ ¤' ‘ `  ¦ s 6   x # Œ ½ ¨ô  Ç ³ ðï  r Ising    l ^ ‰ — ¸+ þ A_  e ”

> “ : r • ¸ü < e ” > t à º  H ¨ 8 Š í ß –„  s Ö  ¦ \  _ ” > r t  · ú §€ Œ ¤ .

q

¨ î + þ A— ¸+ þ A“   KLS— ¸+ þ A\ " f  H q 1 p x ~ ½ Ó Ä »ô  Ç» ¡ ¤' ‘ `  ¦ & h 6   x

% i  . t ë ß –, | 9 " fB > h  à º_  e ” > t à º  H ¨ 8 Š í ß –„  s  Ö

 ¦ \  _ ” > r t  · ú §€ Œ ¤t ë ß –, e ” > “ : r • ¸  H ¨ 8 Š í ß –„  s Ö  ¦ \  _ 

”

> r † < Ê`  ¦ ¹ 1 Ԁ Œ ¤ .

-406-

II. Ž ì ŏ ŒU ê s0 n É õ m Í T  ] Ø

1. { ¢] k ù

³

ðï  r Ising    l ^ ‰ — ¸+ þ A_  K ‰x 9 ž Ðm î ß –“ É r H = −φ X

<i,j>

σ

i

σ

j

(1)

–

Ð & ñ _   ) a  . # Œl " f & h Ä »  ) a    \  @ /K " f σ

i

= 1 s 

“

¦, q # Qe ”   H    \  @ /K " f σ

ⁱ

= −1 s “ ¦,   ½ + Ë\  -t   H φ s  .

KLS — ¸+ þ A“ É r ³ ðï  r    l ^ ‰ — ¸+ þ A_  é ß –í  H ô  Ç   + þ As “ ¦, s

 — ¸+ þ A\ " f { 9  [ þ t _  î  r1 l x“ É r Á ºô  Çy   H ü @Â Ò Ä » „  l 



© œ E\  _  # Œ Ä » „  l  © œ ~ ½ ӆ ¾ ÓÜ ¼– Ð  s # QÛ ¼\  ¦ ”   .

KLS — ¸+ þ A\ " f ¨ 8 Š í ß –„  s Ö  ¦“ É r w [β (∆H + lE)] s  . # Œl 

"

f H  H ³ ðï  r    l ^ ‰— ¸+ þ A_  K ‰x 9 ž Ðm î ß –s “ ¦, , á Ø  æ 8 A l

 ~ ½ ӆ ¾ Ós  Ä » „  l  © œ Eü < (° ú  “ É r, f ” y Œ •, ì ø Í@ /) ~ ½ ӆ ¾ Ós 


\

    l = (−1, 0, +1)_  ° ú כ`  ¦ y Œ •y Œ • ”   . Á ºô  Ç Ä » 

„

 l  © œõ  Å Òl & h   â > › ¸| \ " f, { 9  _  à º ˜ Д > r ÷ &  H q

% i  1 l x§ 4 † < Æ\  _ K " f  H ] X @ /– Ð >  ¨ î + þ A © œI \  • ¸

² ú

˜ t  3 l w >  ë ß –[ þ t “ ¦, { 9  [ þ t“ É r {  — ¸€ ª œ_   e ” D h\  ¦  



 = å Q\ O s  ¹ ¡ §f ” { 9  ÷  r s  .

2.  ¹ ÅT ù o Ú

Monte Carlo „  í ß –r Ð 3 x`  ¦ 0 Aô  Ç „  s Ö  ¦ – Ð" f, Metropolis

„

 s Ö  ¦ [18] õ  heat-bath „  s Ö  ¦`  ¦ ³ ðï  r Ising    l ^ ‰ — ¸ + þ

Aõ  KLS — ¸+ þ A\  & h 6   x % i  . Metropolis „  s Ö  ¦“ É r % ƒ6 £ §



© œI ü <
×  æ  © œI \  _ ” > r t ë ß –, heat-bath „  s Ö  ¦ [15]“ É r é

ß –t  þ j7 á x  © œI \ ë ß – _ ” > r ô  Ç . { 9  “ § ¨ 8 Š heat-bath „  s  Ö

 ¦ [13]`  ¦ ³ ðï  r    l ^ ‰ — ¸+ þ Aõ  KLS — ¸+ þ A\   6   x % i 



.

3. ! aŒ £ ? ŒW Ä ù m Ç" e 8 04 
ì ŕ ¤

³

ðï  r Ising    l ^ ‰ — ¸+ þ Aõ  KLS — ¸+ þ Aõ  ° ú  “ É r ˜ Д > r — ¸ + þ

A\   H ³ ðï  r | 9 " fB > h  à º ” > r F  t  · ú §  H  . ‘ : r ƒ  ½ ¨

\

" f  H ˜ Д > r — ¸+ þ A`  ¦ ¸ ú ˜ l Õ ü t l  0 Aô  Ç | 9 " fB > h  à º– Ð

"

f ½ ¨% i ì  r Ÿ í† < Êà º (block distribution function)\  ¦  6   x 

#

Œ > í ß –  ) a ! QF K ½ ¨% i  | 9 " fB > h  à º\  ¦ ™ è> h½ + É  כ s  . ë ß – { 9

 Ä ºo  { 9  à º { 9 & ñ >  ˜ Д > r ÷ &  H ˜ Д > r Ising   



 — ¸+ þ A[ þ t ( ³ ðï  r Ising    l ^ ‰ — ¸+ þ Aõ  KLS — ¸+ þ A)`  ¦



Œ

•“ É r ½ ¨% i Ü ¼– Ð
è  H  €  , e ” > “ : r • ¸ T

c

s  © œ\ " f  H > \  { 9

 [ þ t s  ç  H| 9  >  ì  r Ÿ í÷ &# Q e ” l  M :ë  H \ , ² D G ™ è | 9 " fB 

>

h  à º ì  r Ÿ í† < Êà º (² D G ™ èì  r Ÿ í† < Êà º-local distribution func- tion) \   © œ ´ ú §s 
 
  H ² D G ™ èx 9 • ¸ (‚    ñ ² D G ™ èx 9 • ¸- preferred local density)  H ô  Çt  { 9   כ s  . e ” > “ : r • ¸ T

_c

s  \ " f  H, >  { 9  [ þ t s  [ þ t # Q  e ”   H ½ ¨% i õ   B~ Ã Ì ô

 Ç ½ ¨% i Ü ¼– Ð ì  r o ÷ &l  M :ë  H \ , ² D G ™ è | 9 " fB > h  à º_  ì  r

Ÿ

í † < Êà º  H ¿ º > h_  ‚    ñ ² D G ™ èx 9 • ¸ ( 
  H € ª œ_  ° ú כÜ ¼– Ð ρ

₊

,   É r 
  H 6 £ § _  ° ú כÜ ¼– Ð ρ

−

)\  ¦ ”   . ì  r Ÿ í† < Êà º  H

² D

G ™ èx 9 • ¸ 0“   & h `  ¦ @ /g AÜ ¼– Ð Gauss ì  r Ÿ í\  ¦`  ¦ t   H

 â

† ¾ Ó$ í s  e ”  . Õ ª QÙ ¼– Ð Û ¼— 2 ;{ © œ  µ 1 Ï l  o| ¾ Ó (sponta- neous magnetization) m“ É r ¿ º > h_  ‚    ñ ² D G ™ èx 9 • ¸ ρ

+

ü <

ρ

₋

_  ¨ î ç  H ° ú כõ  1 p x _  | ¾ Ós    H  כ `  ¦ · ú ˜ à º e ”  .



 " f ! QF K ½ ¨% i  | 9 " fB > h  à º\  ¦  6 £ § õ  ° ú  s  & ñ _ ô  Ç



:

m = ρ

+

− ρ

₋

2 (2)

F

½ ©   o s  : r Ü ¼– РÒ' 
“ : r ! QF K ½ ¨% i  | 9 " fB > h  à º\  ¦



6   x # Œ, Ä »ô  Çß ¼l  » ¡ ¤& h `  ¦   É r U  ´s  " é ¶`  ¦ t “ ¦ e ” 



 H > \  @ / # Œ à º' Ÿ  % i  .

4. Å X ØV ê s ( aÇ k È U ê s0 n É

s

 : rÒ q tÓ ü t † < Æ ì  r  \ " f µ 1 ÏÒ q t”   o : r`  ¦ ƒ  ½ ¨ €  " f, % ƒ6 £ § Ü

¼– Ð ’ < H  © œ S X ‰ í ß – (damage spreading) ~ ½ ÓZ O `  ¦ ™ è> h % i   [16]. ’ < H  © œ S X ‰ í ß –~ ½ ÓZ O “ É r, ¿ º > h_    É r œ íl  C \ P `  ¦ ° ú  “ É r è

ß –à º\  ¦ s 6   x # Œ „  í ß –r Ð 3 x`  ¦ €   Ñ ü t  s _  ’ < H  © œ (K b ç



o : Hamming distance)`  ¦ B  Monte Carlo é ß –>    8 £ ¤

&

ñ   H  כ s  . # Œl " f ¿ º C \ P \ " f ° ú  “ É r 0 Au \  e ”   H # Q

‹

"

 Ó ü t o | ¾ Ós  ° ú  Ü ¼€   Õ ª 0 Au _  ’ < H  © œ“ É r 0,  Ø Ô€   ’ < H  © œ“ É r 1 – Ð & ñ _   ) a  . s  " é ¶ Ising — ¸+ þ A [3,21,22],  Œ ™ " é ¶ Ising

—

¸+ þ A [23], Potts — ¸+ þ A [24]õ  Blume-Capel — ¸+ þ A [25] 1 p x \ 

’

< H  © œ S X ‰ í ß – ~ ½ ÓZ O õ  Monte Carlo „  í ß –r Ð 3 x`  ¦ : Ÿ x # Œ  © œ„   s

ü < e ” > ‰ & ³ © œ`  ¦ ´ ú §s  ƒ  ½ ¨ # Œ M ® o  . q 2 Ÿ ¤ ’ < H  © œ S X ‰ í ß – ~ ½ Ó Z O

s  0 Aü < ° ú  “ É r ¨ î + þ A— ¸+ þ A\ " f  H ´ ú §s  ƒ  ½ ¨÷ &# Q M ® o t ë ß –, q

¨ î + þ A — ¸+ þ A\ " f  H  f ”  V , o  ƒ  ½ ¨÷ &# Qt t   H · ú §€ Œ ¤ .

’

< H  © œ (damage)“ É r D(t) = 1

2N X

i

|O

i

(t) − R

i

(t)| (3)

–

Ð & ñ _   ) a  . # Œl " f {O

i

(t)}  H " é ¶ A  C \ P `  ¦ _ p  “ ¦, {R

_i

(t)}  H " é ¶ A  C \ P _  4 Ÿ ¤    ) a C \ P `  ¦ _ p ô  Ç 9, N“ É r > 

\

 e ”   H 8 ú x    à º\  ¦ _ p ô  Ç . r ç ß – t = 0 { 9 M : œ íl  ’ < H



© œ D

o

`  ¦ 4 Ÿ ¤    ) a C \ P \  ë ß –Ž  H  6 £ § Ê ê, r ç ß –  â õ \    É r

’

< H  © œ`  ¦ 8 £ ¤& ñ ô  Ç . ’ < H  © œ S X ‰ í ß –~ ½ ÓZ O \  e ” # Q" f, œ íl  ’ < H  © œ, œ í l

 C \ P , „  s Ö  ¦ s  — ¸¿ º ×  æ כ ¹ô  Ç כ ¹™ è[ þ t s  .

˜

Д > r — ¸+ þ A\ " f  H
¾ º# Q t t  · ú §“ É r ½ ¨% i _  ’ < H  © œ`  ¦ 8 £ ¤

&

ñ   H  כ “ É r Ó ü t o & h  _ p  \ O Ü ¼Ù ¼– Ð, ! QF K ½ ¨% i \ " f_ 

’

< H  © œ\  › ' a d ” `  ¦ t “ ¦ ƒ  ½ ¨\  ¦ ½ + É  כ s  . s  ƒ  ½ ¨\ " f

² D

G ™ è ‚    ñ ’ < H  © œ x 9 • ¸ ˜ Д > r — ¸+ þ A`  ¦ ¸ ú ˜ l Õ ü t   H | 9 " fB 

>

h   à ºe ” `  ¦ ˜ Ð{ 9   כ s  .

5. – ¥” X ¢ ± ŽM  •  ×Y 8 Ä

‚

 + þ AU  ´s  " é ¶ s  L“   ¨ î + þ A> \ " f,  © œ › ' aU  ´s  ξ  H  6 £ § _

 › ' a >  e ”  .

ξ ∼ t

^−ν

(4)

#

Œl " f t =   

T T_c

− 1 



s  . & ñ ~ ½ Ó    _  | 9 " fB > h  à º\  ¦ s

6   x ô  Ç Ä »ô  Ç» ¡ ¤' ‘ “ É r

m = L

^−βν

m ˜  L

^1ν

t 

(5)

–

Ð j þ t à º e ”  . # Œl " f ³ ðï  r Ising     l ^ ‰ — ¸+ þ A_  e ” >  t

à º  H β = 1/8 s “ ¦ ν = 1– Ð · ú ˜ 94 R e ”   [26,27].

ü

@ Ò\ " f y © œô  Ç Ä » „  l  © œ`  ¦ ~ à Γ ¦ e ”   H q 1 p x ~ ½ Ó > \  e ”

# Q" f  © œ › ' aU  ´s   H, „  l  © œõ  ¨ î ' Ÿ ô  Ç ~ ½ ӆ ¾ ÓÜ ¼– Ѝ  H ξ

_k

s “ ¦ Ã

ºf ” ~ ½ ӆ ¾ ÓÜ ¼– Ð Es  €  , q 1 p x ~ ½ ÓÄ »ô  Ç» ¡ ¤' ‘ “ É r  6 £ § õ  y Œ •y Œ •

° ú

 s  j þ t à º e ”  .

ξ

_k

∼ t

^−νk

, ξ

_⊥

∼ t

^−ν⊥

(6) U

 ´s  " é ¶ s  L

k

×L

_⊥

“   q 1 p x ~ ½ Ó> \ " f | 9 " fB > h  à º_  q  1

p

x ~ ½ Ó Ä »ô  Ç» ¡ ¤' ‘ “ É r  6 £ § õ  ° ú  s  Å Ò# Q”    [32].

m(T

c

, L

_k

, L

_⊥

) = L

−^β

νk

k

m ˜

 tL

1 νk

k

, S



(7)

#

Œl " f — ¸€ ª œ“    (shape factor) S  H  6 £ § õ  ° ú   .

S = L

^ν_k^⊥/νk

/L

_⊥

(8)

“

¦& ñ  ) a — ¸€ ª œ“   \  @ /K " f, | 9 " fB > h  à º_  q 1 p x ~ ½ Ó Ä » ô

 Ç» ¡ ¤' ‘ “ É r  6 £ § õ  ° ú  s  j þ t à º e ”  .

m(T

c

, L

_k

) = L

−^β

νk

k

m ˜

 tL

1 νk

k



(9)



© œs  : r (field theory) Ü ¼– РÒ'  [28, 29], KLS— ¸+ þ A“ É r β = 1/2, ν

_k

= 3/2, Õ ªo “ ¦ ν

⊥

= 1/2 _  e ” > t à º\  ¦ ”   

“

¦ · ú ˜ 94 R e ”  . ½ ¨› ¸“   \  ¦ | 9 " fB > h  à º– Ð & ñ _  “ ¦,

½

¨› ¸“   _  1 p x ~ ½ Ó Ä »ô  Ç» ¡ ¤' ‘ `  ¦ s 6   x # Œ S X ‰  © œ  ) a  © œs  : r s 

´ ú

§s  ƒ  ½ ¨ ÷ &# Q4 R M ® o   [30, 31]. õ  \  à º' Ÿ  ) a KLS

—

¸+ þ A\  @ /ô  Ç ƒ  ½ ¨  H ½ ¨› ¸“    S(1, 0)`  ¦ | 9 " fB > h  à º– Ð

&

ñ _  “ ¦ Metropolis „  s Ö  ¦`  ¦  6   x % i   [30–32].

Fig. 1. Plots of the sub-block order parameter m for the standard Ising lattice gas using (a) the Metropolis and (b) the heat-bath rates. Plot of m for the KLS model with S = 0.25 using (c) the Metropolis and (d) the heat- bath rates, and S = 0.50 using (e) the Metropolis and (f) the heat-bath rates. Simulations have been carried out with 2 × 10

⁶

MCS after discarding 8 ∼ 10 × 10

⁶

MCS.

III. + s ÇÊ Ý õ m Í w Š²  o

1. ! aŒ £ ? ŒW Ä – ¥” X ¢ •  ×Y 8 Ä

!

QF K ½ ¨% i _  ‚    ñ² D G ™ èx 9 • ¸\  ¦ | 9 " fB > h  à º– Ð ¸ ú šÜ ¼€  ,

| 9

" fB > h  à º  H d ”  (2)ü < ° ú  s  & ñ _   ) a  . ³ ðï  r Ising — ¸+ þ A

\

 @ /K " f U  ´s  " é ¶ s  L = 144“   > \  @ /K " f „   © œr Ð 3 x

`

 ¦ à º' Ÿ  % i  . Õ ªo “ ¦, ! QF K ½ ¨% i  | 9 " f B > h  à º\  ¦ y Œ •y Œ • _

 ! QF K ½ ¨% i  ß ¼l \  @ /K " f 8 £ ¤& ñ % i  . # Œl " f  6   x ) a

!

QF K ½ ¨% i _  ß ¼l   H L

^sub

= 36, 24, 18, 16, 12, 8 1 p x s  .

³

ðï  r Ising    l ^ ‰  H e ” > “ : r • ¸ „  s Ö  ¦ s  heat-bath rate Fig. 1(a) s 
 Metropolis rate Fig. 1(b)\  _ ” > r t 

· ú

§  H    H  כ `  ¦ ¹ 1 Ô Í Ç x .

KLS — ¸+ þ A\  @ /K " f U  ´s  " é ¶ s  L

||

× L

_⊥

= 432 × 48“  

>

\  @ /K " f „   © œr Ð 3 x`  ¦ à º' Ÿ  % i  . # Œl " f  6   x 0 p x ô

 Ç — ¸€ ª œ“    S = 0.25ü < 0.50`  ¦  6   x % i  . — ¸€ ª œ“   

Fig. 2. Ln-ln plots of scaling function mL

^βν

as a function of scaling variable tL

^1ν

for the standard Ising lattice gas using (a) the Metropolis and (b) the heat-bath rates. Ln- ln plots of scaling function mL

β

νk

as a function of scaling variable tL

1

νk

for the KLS model using the Metropolis rate for shape factor (c) S = 0.25 and (e) S = 0.50, and using the heat-bath rate for (d) S = 0.25 and (f) S = 0.50, respectively. All data symbols shown in these figures are the same as in Fig. 1.

S = 0.25(Fig. 1(c) ü < Fig. 1(d)]\  @ /K " f, ! QF K ½ ¨% i  | 9 

"

f B > h  à º\  ¦ y Œ •y Œ •_  ! QF K ½ ¨% i  ß ¼l \  @ /K " f 8 £ ¤& ñ 

%

i  . # Œl " f  6   x ) a ! QF K ½ ¨% i _  ß ¼l   H L

^sub_||

× L

^sub_⊥

= 64 × 16, 27 × 12, 8 × 8 1 p x s  . — ¸€ ª œ“    S = 0.50 Fig.

1(e) ü < Fig. 1(f)\  @ /K " f,  6   x ) a ! QF K ½ ¨% i _  ß ¼l   H L

^sub_||

× L

^sub_⊥

= 216 × 12, 64 × 8, 27 × 6 1 p x s  . KLS— ¸+ þ A

\

 @ /ô  Ç „   © œr Ð 3 x   õ – РÒ'  Ä ºo   H e ” > “ : r • ¸  H „  s  Ö

 ¦ _  7 á x À Ó\  _ ” > r t ë ß – — ¸€ ª œ“   \   H _ ” > r t  · ú §  H  



 H  z  ´`  ¦ · ú ˜  ? /% 3  . Metropolis „  s Ö  ¦ Fig. 1(c)) ü <

Fig. 1(e) \  @ /K " f e ” > “ : r • ¸  H T

c

= 3.15 ± 0.05 s “ ¦ heqt-bath „  s Ö  ¦ Fig. 1(d) ü < Fig. 1(f)\  @ /K " f e ” > “ : r

•

¸  H T

c

= 3.55 ± 0.05 s  .

Fig. 2(a) ü < Fig. 2(b)  H Metropolis „  s Ö  ¦(a) õ  heat- bath „  s Ö  ¦(b)`  ¦ s 6   x # Œ x» ¡ ¤`  ¦ ln(tL

^1/ν

) – Ð y» ¡ ¤`  ¦ ln(mL

^β/ν

) – Ð Õ ª 2 ; ³ ðï  r Ising    l ^ ‰ Ä »ô  Ç» ¡ ¤' ‘  Fig.s 



. ! QF K ½ ¨% i  | 9 " fB > h  à º\  ¦ : Ÿ x # Œ, “ : r • ¸ T < T

c

{ 9  M :

% 3

“ É r e ” > t à º β = 1/8e ” `  ¦ · ú ˜ à º e ”  . s    õ   H s p 

· ú

˜ 9”     õ  [26,27]ü < ¸ ú ˜ { 9 u  % i  .   " f ³ ðï  r Ising

 

 l ^ ‰_  e ” > t à ºü < e ” > “ : r • ¸ „  s Ö  ¦ \  _ ” > r t 

· ú

§  H    H  כ `  ¦ · ú ˜ à º e ”  .

KLS — ¸+ þ A\  @ /K " f x» ¡ ¤`  ¦ ln(tL

^1/ν_|| ^||

) – Ð y» ¡ ¤`  ¦ ln(mL

^β/ν_|| ^||

) – Ð — ¸€ ª œ“    S = 0.25 Fig. 2(c) ü < Fig.

2(d) ü < — ¸€ ª œ“    S = 0.50 Fig. 2(e)ü < Fig. 2(f)\  @ / ô

 Ç q 1 p x ~ ½ Ó Ä »ô  Ç» ¡ ¤' ‘ `  ¦ z  ´' Ÿ  % i  . Õ ªo “ ¦, Ä »ô  Ç» ¡ ¤' ‘ _ 

 

õ – РÒ'  e ” > t à º β = 1/2`  ¦ % 3 % 3  . s    õ   H „  \ 

ƒ

 ½ ¨÷ &% 3 ~     õ [ þ t [28–32] õ  ¸ ú ˜ { 9 u  % i “ ¦ e ” > t à º

³

ðï  r Ising    l ^ ‰ — ¸+ þ Aõ  ° ú  s  KLS — ¸+ þ A• ¸ „  s Ö  ¦ \  _

” > r t  · ú §  H    H  z  ´`  ¦ ˜ Ð# Œï  r  .

!

QF K Ä »ô  Ç» ¡ ¤' ‘   õ \  ¦ : Ÿ x K " f, ³ ðï  r Ising    l ^ ‰ — ¸ + þ

A“ É r e ” > “ : r • ¸ü < e ” > t à º — ¸¿ º „  s Ö  ¦ \  _ ” > r t  · ú §

€

Œ

¤ . t ë ß –, KLS — ¸+ þ A\ " f  H e ” > t à º  H „  s Ö  ¦ \  _ 

”

> r t  · ú §t ë ß –, e ” > “ : r • ¸  H „  s Ö  ¦ \  _ ” > r    H  z  ´

`

 ¦ · ú ˜ à º e ”  .

2. ! aŒ £ ? ŒW Ä Å X ØV ê s – ¥” X ¢ •  ×Y 8 Ä



6 £ §“ É r ! QF K ½ ¨% i  ’ < H  © œ Ä »ô  Ç» ¡ ¤' ‘  (sub-block damage finite-size scaling) _    õ \  @ /K " f [ O " î ½ + É  כ s  . „  



© œr Ð 3 x \ " f ¿ º 7 á x À Ó_  œ íl  ’ < H  © œ`  ¦ Metropolis ü < heat- bath „  s Ö  ¦ õ  † < Êa   6   x % i  .

7 á

x À Ó I. " é ¶‘ : r Û ¼— 2 ;C \ P õ  4 Ÿ ¤   Û ¼— 2 ;C \ P s  ô  Ç t & h ë ß – ]

jü @ “ ¦ ° ú  “ É r  â Ä º. s M : œ íl  ’ < H  © œ ° ú כ D

^o

= 1/N `  ¦ Metropolis „  s Ö  ¦ õ  heat-bath „  s Ö  ¦ \  @ / # Œ  6   x 

% i  .

7 á

x À Ó II. " é ¶‘ : r Û ¼— 2 ;C \ P õ  4 Ÿ ¤   Û ¼— 2 ;C \ P s  ì ø Í@ /s “ ¦,

œ

íl  ’ < H  © œ ° ú כ D

^o

' 1`  ¦ Metropolis „  s Ö  ¦ \   6   x % i 

“

¦, œ íl  ’ < H  © œ ° ú כ D

o

= 1`  ¦ heat-bath „  s Ö  ¦ \  @ / # Œ



6   x % i  .

Fig. 3“ É r ³ ðï  r Ising    l ^ ‰ — ¸+ þ Aõ  KLS — ¸+ þ A\  @ / ô

 Ç ‚    ñ ² D G ™ è’ < H  © œx 9 • ¸ (preferred local damage densities) ρ

+

ü < ρ

−

\  ¦ ˜ Ð# Œ ï  r  . # Œl " f ρ

+

  H 7 á x À Ó II_  œ íl ’ < H  © œ Ü

¼– РÒ'  ½ ¨ô  Ç ‚    ñ ² D G ™ è’ < H  © œx 9 • ¸s “ ¦, ρ

−

  H 7 á x À Ó I_ 

œ

íl ’ < H  © œÜ ¼– РÒ'  ½ ¨ô  Ç ‚    ñ ² D G ™ è’ < H  © œx 9 • ¸ s  . ³ ðï  r Ising    l ^ ‰ — ¸+ þ A\  @ /K " f  H  © œ• ¸³ ð % ò x 9 • ¸ (zero density) \  @ /K " f @ /g As  . t ë ß – KLS — ¸+ þ A\  @ /K " f



 H  © œx 9 • ¸ % ò x 9 • ¸\  @ /K " f @ /g As   _ ” `  ¦ · ú ˜ à º e ” 



. e ” > “ : r • ¸ s  © œ\ " f  H ρ

+

ü < ρ

−

 ° ú  “ É r ° ú כ`  ¦ t “ ¦ e ”

“ ¦, e ” > “ : r • ¸ s  \ " f  H ρ

+

ü < ρ

−

  H ¿ º ° ú כs  ° ú  t  · ú §

>

  ) a  .

Fig. 3. Plots of the phase diagram of the preferred local densities ρ of damage in sub-blocks for the standard lat- tice gas using (a) the Metropolis and (b) the heat-bath rates. Plots of the phase diagram for the KLS model with S = 0.25 using (c) the Metropolis and (d) the heat-bath rates, and with S = 0.50 using (e) the Metropolis and (f) the heat-bath rates, respectively. The upper branch uses D

o

' 1 for the Metropolis and D

o

= 1 for the heat- bath rates. The lower branch uses D

o

= 1/N for the Metropolis and the heat-bath rates.

Fig. 3(a) ü < Fig. 3(b)  H Ising    l ^ ‰ — ¸+ þ A\  @ /K 

"

f e ” > “ : r • ¸ T

c

 Metropolis „  s Ö  ¦ s 
 heat-bath „  s  Ö

 ¦ \  _ ” > r t  · ú §“ ¦ ° ú     H  כ `  ¦ ˜ Ð# Œï  r  . t ë ß –, KLS

—

¸+ þ A\  @ /K " f  H, „  s “ : r • ¸ — ¸€ ª œ“   \   H _ ” > r t  · ú § t

ë ß – „  s Ö  ¦ \   H _ ” > r ô  Ç   H  כ `  ¦ ˜ Ð# Œï  r  . Fig. 3(c)ü <

Fig. 3(c) \ " f % 3 “ É r Metropolis „  s Ö  ¦ \  @ /ô  Ç e ” > “ : r • ¸



 H T

c

= 3.05 ± 0.05 s “ ¦ Fig. 3(d)ü < Fig. 3(f)\ " f % 3 “ É r Metropolis „  s Ö  ¦ \  @ /ô  Ç e ” > “ : r • ¸  H T

c

= 3.45±0.05 s 



.

¿

º> h_  ‚    ñ ² D G ™ è’ < H  © œx 9 • ¸ ρ

+

ü < ρ

−

\  ¦  6   x # Œ, ’ < H  © œ S X

‰ í ß –~ ½ ÓZ O `  ¦ s 6   x ô  Ç ! QF K ½ ¨% i  | 9 " fB > h  à º\  ¦, ! QF K ½ ¨% i 

| 9

" fB > h  à ºü < ° ú  “ É r ~ ½ ÓZ O  d ”  (2)Ü ¼– Ð ½ ¨ % i  . # Œl " f

½

¨ô  Ç | 9 " fB > h  à º_  Ä »ô  Ç» ¡ ¤' ‘ `  ¦ s 6   x # Œ Metropolis Fig. 4(a) ü < heat-bath Fig. 4(b) „  s Ö  ¦ \  @ /ô  Ç ³ ðï  r    

Fig. 4. Ln-ln plots of scaling function mL

^βν

as a func- tion of scaling variable tL

^1ν

for the standard Ising lat- tice gas using (a) the Metropolis and (b) the heat-bath rates where m is the sub-block order parameter of dam- age. Ln-ln plots of scaling function mL

β

νk

as a function of scaling variable tL

1

νk

for the KLS model using the Metropolis rate for shape factor (c) S = 0.25 and (e) S = 0.50, and using the heat-bath rate for (d) S = 0.25 and (f) S = 0.50, respectively. All data symbols shown in these figures are the same as in Fig. 3.

l

^ ‰_  „  í ß –r Ð 3 x   õ \  ¦ % 3 % 3  . # Œl " f % 3 “ É r β = 1/8`  ¦

% 3

% 3 “ ¦, s p  · ú ¡\ " f % 3 “ É r ! QF K ½ ¨% i  | 9 " fB > h  à º\  ¦ s  6

 

x ô  Ç   õ ü < ° ú  s , ’ < H  © œ S X ‰ í ß –~ ½ ÓZ O `  ¦ s 6   x ô  Ç ! QF K ½ ¨% i  | 9 

"

fB > h  à º• ¸ „  s “ : r • ¸ „  s Ö  ¦ \   H _ ” > r t  · ú §  H  



 H  כ `  ¦ ˜ Ð# Œï  r  .

Fig. 4 _ 
 Qt  Õ ªa Ë >[ þ t“ É r KLS — ¸+ þ A_  „  í ß –r Ð 3 x    õ

\  ¦ ˜ Ð# Œï  r  , # Œl " f % 3 “ É r ¸ ú ˜ · ú ˜ 9”   β = 1/2`  ¦ % 3 % 3 

“

¦ [28–32], — ¸€ ª œ“    S = 0.25 Fig. 4(c)ü < Fig. 4(d)ü <

S = 0.50 Fig. 4(e) ü < Fig. 4(f){ 9  M :, s p  · ú ¡\ " f % 3 “ É r

!

QF K ½ ¨% i  | 9 " fB > h  à º\  ¦ s 6   x ô  Ç   õ ü < ° ú  s , ’ < H  © œ S X ‰ í

ß –~ ½ ÓZ O `  ¦ s 6   x ô  Ç ! QF K ½ ¨% i  | 9 " fB > h  à º• ¸ „  s “ : r • ¸

„

 s Ö  ¦ \   H _ ” > r t  · ú §  H    H  כ `  ¦ ˜ Ð# Œï  r  .

IV. + s Ç Â ] Ø

¨ î

+ þ A ˜ Д > r — ¸+ þ A (³ ðï  r Ising    l ^ ‰ — ¸+ þ A)õ  q ¨ î + þ A

˜

Д > r — ¸+ þ A (KLS — ¸+ þ A)_  e ” >  : £ ¤$ í “ É r ! QF K ½ ¨% i õ  ’ < H  © œ S X

‰ í ß –`  ¦ s 6   x ô  Ç ! QF K ½ ¨% i  Ä »ô  Ç» ¡ ¤' ‘ `  ¦ Metropolis ü < heat- bath S X ‰Ò  ¦„  s \  ¦ s 6   x # Œ ƒ  ½ ¨\  ¦ % i  . ! QF K ½ ¨% i  | 9 

"

fB > h  à ºü < ’ < H  © œ S X ‰ í ß –`  ¦ s 6   x ô  Ç ! QF K ½ ¨% i  | 9 " fB > h   Ã

º\  ¦ ¿ º — ¸+ þ A\  & h 6   x`  ¦ % i “ ¦, # Œl " f ! QF K ½ ¨% i `  ¦ s 6   x ô

 Ç | 9 " fB > h  à º ¿ º — ¸+ þ A (¨ î + þ A— ¸+ þ Aõ  q ¨ î + þ A— ¸+ þ A)\ 

"

f ¸ ú ˜ & h 6   x ) a    H  כ `  ¦ S X ‰ “   % i  .

KLS — ¸+ þ A\  & h 6   x ô  Ç ! QF K ½ ¨% i õ  ’ < H  © œ S X ‰ í ß –`  ¦ s 6   x ô  Ç ! Q F

K ½ ¨% i  Ä »ô  Ç» ¡ ¤' ‘ `  ¦ : Ÿ x K  β = 1/2`  ¦ % 3 % 3 “ ¦, s    õ   H

¸ ú

˜ · ú ˜ 9”    © œs  : r`  ¦ : Ÿ x ô  Ç   õ  [28–32]ü < ¸ ú ˜ { 9 u  ÷ &% 3 



. q ¨ î + þ A — ¸+ þ A\ " f• ¸ q 2 Ÿ ¤ e ” > “ : r • ¸  H „  s Ö  ¦ \  _ ” > r

t ë ß –, e ” > t à º  H „  s Ö  ¦ \  _ ” > r t  · ú §  H    H  כ `  ¦

· ú

˜ à º e ”  . Õ ªo “ ¦ e ” > “ : r • ¸ Metropolis „  s Ö  ¦`  ¦ s  6

 

x €  , heat-bath „  s Ö  ¦`  ¦  6   x ô  Ç „  í ß –r Ð 3 x ˜ Ð  Z  }  



 H  כ `  ¦ · ú ˜ Í Ç x . » · ¡ ­ # Œ s  Qô  Ç " é ¶ “  “ É r ü @Â Ò Ä »  „   l

 © œ\  _ ô  Ç q 1 p x ~ ½ Ó  © œ › ' a \     KLS — ¸+ þ A_  e ” > “ : r • ¸

    l ^ ‰ — ¸+ þ A_  e ” > “ : r • ¸˜ Ð  Z  }    H  כ `  ¦ ˜ Ð# Œï  r



 [11,12]. # Œl " f % 3 “ É r e ” > “ : r • ¸ T

^c

= 3.15 ± 0.05  H · ú ¡

"

f ™ è> h  ) a 1 l x& h   © œs  : r   õ  [12]ü < ¸ ú ˜ { 9 u ô  Ç .

כ

¹€  • €  , ½ ¨% i  ì  r Ÿ í † < Êà º[ þ t`  ¦ s 6   x ô  Ç KLS — ¸+ þ A_  ! Q F K | 9 " f B > h  à º\  ¦ ƒ  ½ ¨ % i  . # Œl " f ƒ  ½ ¨ô  Ç ! QF K | 9 

"

f B > h  à ºü < ’ < H  © œ S X ‰ í ß –`  ¦ s 6   x ô  Ç ! QF K | 9 " f B > h  à º



 H ¨ î + þ A— ¸+ þ A ÷  r  m   q ¨ î + þ A — ¸+ þ A\ • ¸ | 9 " f B > h  à º

–

Ð & h 6   x s  ÷ &# Q | 9  à º e ”    H  כ `  ¦ ¹ 1 Ô ? /% 3  .

P

c p 8 ý ò k >

‘

: r ƒ  ½ ¨_  ? /6   x“ É r The University of Georgia _  ~ à Ì † < Æ 0

A  7 Hë  H x 9  Physical Review E\  > F   ) a  7 Hë  H \  Ÿ í† < ÊH † d

`

 ¦ µ 1 ߘ 2 ³ .

Y

c p w Š à U Ø ”  ô

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Order Parameter in the KLS Model by Using Block Distribution Functions and Damage Spreading Methods

Jae-Suk Yang

Department of Physics, Korea University, Seoul 136-713

Wooseop Kwak

^∗

Department of Physics, Chosun University, Gwangju 501-759 (Received 13 February 2008)

We study the proper order parameters for conserved systems, the standard Ising lattice gas and the Katz-Lebowitz Sphon (KLS) model. In this paper, we introduce the sub-block order parameter by using the block distribution. We also find the damage spreading method with the block distribution function to be a good order parameter for conserved models. These order parameters work well for equilibrium, as well as nonequilibrium, models. We obtain the critical exponents of the order parameter for the standard Ising lattice gas and for the KLS model by using a heat bath and the Metropolis transition probabilities.

PACS numbers: 05.20.-y, 05.70.Ln, 64.60.Cn, 66.30.Dn Keywords: Lattice gas, Order parameter, Damage spreading

∗

E-mail: [email protected]