Monte-Carlo Methods in Statistical Physics

౷ܭ෺ཧʹ͓͚ΔϞϯςΧϧϩ๏

ࠤʑ໦ɹࢤ߶ ∗

Monte-Carlo Methods in Statistical Physics

Munetaka SASAKI ∗

1. ^͸͡Ίʹ

ੈքʹॳΊͯిؾػցࣜܭࢉػʢίϯϐϡʔλʣ͕ݱ Εͨͷ͸

1940

೥୅લ൒Ͱ͋Δ͕ɼͦΕҎདྷίϯϐϡʔ λͷੑೳ͸ɼ·͞ʹࢦ਺ؔ਺తͳਐԽΛ਱͖͛ͯͨɽྫ

͑͹ɼूੵճ࿏্ͷτϥϯδελͷ਺͸ɼ

1960

೥୅͔Β ࠓ೔ʹࢸΔ·Ͱɼ

1.5

ʙ

2

೥͝ͱʹ

2

ഒͷϖʔεͰ૿Ճ͠

ଓ͚͍ͯΔɽ͜ͷܦݧଇ͸ʮϜʔΞͷ๏ଇʯͱݺ͹ΕΔɽ

·ͨɼ

1946

೥ʹΞϝϦΧͰ։ൃ͞Εͨ

ENIAC

͸ຖඵ໿

5,000

ճͷՃݮࢉ͕ՄೳͰ͕͋ͬͨ⁽¹⁾ɼݱࡏͷҰൠతͳՈ

ఉ޲͚ίϯϐϡʔλ͸ຖඵ

10

¹¹ճఔ౓ͷුಈখ਺఺਺ԋ

ࢉ⁽²⁾Λߦ͏͜ͱ͕Ͱ͖Δɽ͞Βʹɼ೔ຊ࠷଎ͷεʔύʔ ίϯϐϡʔλʮژʯ͸ɼͦͷ໊͕ࣔ͢௨Γɼຖඵ

10

¹⁶ճ ͷුಈখ਺఺਺ԋࢉ͕ՄೳͰ͋Δɽ

͜ͷΑ͏ͳίϯϐϡʔλͷരൃతͳൃలʹ൐͍ɼίϯ ϐϡʔλʹΑΔ਺஋γϛϡϨʔγϣϯ͸ɼՊֶʹ͓͚Δ

ॏཁͳݚڀखஈͷ

1

ͭͱͯ͠੒௕Λ਱͖͛ͯͨΘ͚ͩ

͕ɼͦͷൃలʹ͔ܽͤͳ͔ͬͨͷ͕਺஋ܭࢉख๏ʢΞϧΰ ϦζϜʣͷ։ൃͰ͋Δɽ࣮ͦͯ͠ࡍʹɼ

20

^{ੈلʹ͸༷ʑ} ͳΞϧΰϦζϜ͕։ൃ͞Εͨͷ͕ͩɼ

2000

^೥ʹ

SIAM (Society for Industrial and Applied Mathematics)

^͸ɼͦ

ͷதͷτοϓ

10

Λൃදͨ͠⁽³⁾ɽ͜ͷϦετʹ͸ɼߴ଎

ϑʔϦΤม׵΍ΫΠοΫιʔτͳͲɼඇৗʹ༗໊ͳΞϧ ΰϦζϜͷ໊લ͕ͣΒͬͱฒΜͰ͍Δ͕ɼͦΕΒͷதͰɼ

࠷΋ݹ͍ΞϧΰϦζϜͱͯ͠࠷ॳʹ঺հ͞Ε͍ͯΔͷ͕ɼ ຊߘͰ঺հ͢ΔϞϯςΧϧϩ๏Ͱ͋Δɽ

ݩʑϞϯςΧϧϩ๏͸ɼதੑࢠ͕෺࣭಺Λಈ͖ճΔ༷

ࢠΛ୳ΔͨΊʹ։ൃ͞Εͨɼཚ਺Λ༻͍ͨ਺஋ܭࢉ๏Ͱ

͋Δɽ໋໊ऀ͸։ൃऀͷҰਓͰ͋ΔϑΥϯɾϊΠϚϯͰ

͋ΓɼΧδϊͰ༗໊ͳϞφίެࠃͷϞϯςΧϧϩ

(Monte

∗।ڭतɹ෺ཧֶڭࣨ

Associate Professor, Institute of Physics

Carlo)

͕ͦͷ໊ͷ༝དྷͱͳ͍ͬͯΔɽҰൠʹϞϯςΧϧ

ϩ๏͸ɼཚ਺Λ༻͍ͨγϛϡϨʔγϣϯख๏ͷ૯শͰ͋

Γɼྫ͑͹Ұ༷ཚ਺Λ༻͍ͨԁप཰π^{ͷܭࢉ(4)ͳͲ΋ɼ} ϞϯςΧϧϩ๏ͷҰྫͰ͋Δɽ͔͠͠౷ܭ෺ཧͷ෼໺Ͱ

͸ɼ୯ʹϞϯςΧϧϩ๏ͱ͍͏ͱ΄ͱΜͲ৔߹ɼϚϧί ϑ࿈࠯ϞϯςΧϧϩ๏ͷ͜ͱΛࢦ͢ɽ͜Ε͸Ϛϧίϑ࿈

࠯Ͱ͋ΒΘ͞ΕΔμΠφϛΫεΛར༻ͯ͠ɼ༩͑ΒΕͨ

֬཰෼෍͔ΒͷαϯϓϦϯάΛಘΔํ๏Ͱ͋Δɽ౷ܭ෺

ཧͰ͸ଟ͘ͷ৔߹ɼͦͷ֬཰෼෍ΛϘϧπϚϯ෼෍ͱ͢

Δɽ͜ͷΑ͏ʹɼϚϧίϑ࿈࠯ϞϯςΧϧϩ๏͸Ϟϯς Χϧϩ๏ͷҰछͰ͋Δ͕ɼ೚ҙͷ֬཰෼෍͔Βͷαϯϓ ϦϯάΛՄೳͱ͢ΔۃΊͯ൚༻ੑͷߴ͍ख๏Ͱ͋Γɼͦ

ͷԠ༻ൣғ͸෺ཧͷΈͳΒͣɼ޻ֶɾ਺ֶɾੜ෺ֶɾۚ

༥޻ֶͳͲඇৗʹଟذʹ౉Δɽ

ຊղઆͰ͸౷ܭ෺ཧʹ͓͚ΔϞϯςΧϧϩ๏ʹ͍ͭͯ

঺հΛ͢Δɽຊߘͷߏ੒͸ҎԼͷ௨ΓͰ͋Δɽ·ͣୈ

2

અͰ͸ɼࠓ೔ඪ४తʹ༻͍ΒΕ͍ͯΔٙࣅཚ਺ੜ੒๏Ͱ

͋Δ

Mersenne Twister

๏ʹ͍ͭͯ঺հ͢Δɽͦͯ͠ୈ

3

અͰ͸Ϛϧίϑ࿈࠯ϞϯςΧϧϩ๏ʹ͍ͭͯɼୈ

4

^અͰ

͸୅දతͳϞϯςΧϧϩ๏Ͱ͋Δ

Metropolis

^ΞϧΰϦζ Ϝʹ͍ͭͯ঺հ͢Δɽୈ

5

અͰϞϯςΧϧϩ๏ͷޮ཰ͱ

؇࿨࣌ؒͷؔ܎ʹ͍ͭͯड़΂ɼҎ߱ͷୈ

6

ʙ

9

અͰ͸ۙ

೥։ൃ͞Εͨޮ཰తϞϯςΧϧϩ๏Ͱ͋ΔɼΫϥελʔ ΞϧΰϦζϜɼ֦ுΞϯαϯϒϧϞϯςΧϧϩ๏ɼ֬཰

తΧοτΦϑ๏ʹ͍ͭͯ঺հ͢ΔɽಛʹචऀΒ͕։ൃ͠

ͨ֬཰తΧοτΦϑ๏ʹ͍ͭͯ͸ɼୈ

8

અͱୈ

9

અͰৄ

͘͠঺հ͢Δɽୈ

10

અͰຊߘͷ·ͱΊΛड़΂Δɽ

2. Mersenne Twister^๏

ϞϯςΧϧϩ๏ʹΑΔίϯϐϡʔλɾγϛϡϨʔγϣ ϯͰ͸ཚ਺Λ༻͍Δ͕ɼίϯϐϡʔλ͸ຊ౰ͷҙຯͰͷ

ཚ਺Λ࡞Δ͜ͱ͕Ͱ͖ͳ͍ͨΊɼ΄ͱΜͲͷ৔߹ɼͦͷ

͔ΘΓʹٙࣅཚ਺Λ༻͍Δɽٙࣅཚ਺ͱ͸Ұݟཚ਺ͷΑ

͏ʹݟ͑Δ͕ɼ࣮ࡍʹ͸ԿΒ͔ͷϧʔϧʹैͬͯ֬ఆత ʹੜ੒͞ΕΔɼ਺ྻͷ਺ͷ͜ͱͰ͋Δɽྫ͑͹ɼ࠷΋؆

ศͳٙࣅཚ਺ੜ੒๏Ͱ͋Δઢܗ߹ಉ๏Ͱ͸ɼ࣍ͷϧʔϧ ʹैͬͯٙࣅཚ਺ྻ

{ X

n

}

^{Λੜ੒͢Δɿ}

X

_n+1=

(a × X

n+

b) mod m

,

(1)

͜͜Ͱ

p mod q

͸੔਺

p

Λ੔਺

q

Ͱׂͬͨ࣌ͷ༨ΓͰ

͋Δɽ

a

,

b

,

m

ͷ૊Έ߹Θͤʹ͍ͭͯ͸ز͔ͭͷఏҊ͕͋

Δ͕ɼద੾ͳ΋ͷΛબ΂͹ɼ਺ྻ

{ X

n

}

^͸

0

͔Β

2

³¹

− 1

ͷؒͷ੔਺Λ΄΅ۉ౳ʹऔΔͷͰɼ

X

nΛ

2

³¹ͰׂΕ͹

0

͔Β

1

ͷؒͷٙࣅҰ༷ཚ਺Λ࡞Δ͜ͱ͕Ͱ͖Δɽ͔͠͠

ઢܗ߹ಉ๏͸ʮपظ͕୹͍ʢ࠷େͰ΋

m

ʣʯɼʮཚ਺ͷ࣭͕

ѱ͍ʯͳͲͷܽ఺Λ࣋ͭͨΊɼݱࡏͰ͸·ͣ࢖ΘΕͳ͍ɽ

ͦΕʹର͠ɼຊઅͷද୊ʹ͋Δ

Mersenne Twister

๏^(5,6)

͸ɼपظ͕

2

¹⁹⁹³⁷

− 1 ≈ 10

⁶⁰⁰²,

(2)

ͱඇৗʹ௕͘ɼ͔͠΋ߴ଎ʹ࣭ͷྑ͍ཚ਺Λ࡞Δ͜ͱ͕

Ͱ͖ΔͨΊɼݱࡏͰ͸͜ͷํ๏͕ඪ४తʹ࢖ΘΕ͍ͯΔɽ

ͪͳΈʹɼ

2

ⁿ

− 1

ʢ

n

͸੔਺ʣͷܗͷࣗવ਺͸

Mersenne

਺ɼͦͷதͰૉ਺ͷ΋ͷ͸

Mersenne

ૉ਺ͱݺ͹ΕΔͷ

͕ͩɼࣜ

(2)

ࠨลͷ਺͸

Mersenne

ૉ਺Ͱ͋Γɼ͜ͷ͜ͱ

͕

“Mersenne Twister”

ͱ͍͏໊લͷ༝དྷͱͳ͍ͬͯΔɽ

3. Ϛϧίϑ࿈࠯ϞϯςΧϧϩ๏

1

અͰड़΂ͨΑ͏ʹɼϚϧίϑ࿈࠯ϞϯςΧϧϩ๏ͱ

͸ϞϯςΧϧϩ๏ͷҰछͰ͋Δ͕ɼ౷ܭ෺ཧͷ෼໺Ͱ͸ɼ ୯ʹϞϯςΧϧϩ๏ͱ͍͏ͱ΄ͱΜͲ৔߹ɼϚϧίϑ࿈

࠯ϞϯςΧϧϩ๏ͷ͜ͱΛࢦ͢ɽҎԼɼʮϚϧίϑ࿈࠯Ϟ ϯςΧϧϩ๏ʯͷ͜ͱΛʮϞϯςΧϧϩ๏ʯͱه͢͜ͱ ͱ͢ΔɽຊઅͰ͸͜ͷʢϚϧίϑ࿈࠯ʣϞϯςΧϧϩ๏

ʹ͍ͭͯ؆୯ʹ঺հ͢Δɽ

Ұൠʹɼ

M

ݸͷঢ়ଶ͔Β੒Δܥʹ͍ͭͯߟ͑Δɽ෺ཧ ͷ৔߹ɼ͜ΕΒͷঢ়ଶ͸ඍࢹతͰ͋Γɼྫ͑͹εϐϯܥ ͷ৔߹ɼݸʑͷεϐϯ͕͋Δಛఆͷํ޲Λ޲͍ͨঢ়ଶΛ ද͢ɽ͜ͷঢ়ଶΛαͱද͢ɽϞϯςΧϧϩ๏ͷ໨త͸ɼ

͜ΕΒ

M

ݸͷঢ়ଶΛɼ͋Δ໨ඪ֬཰෼෍

{π

α

}

^ʹैͬͯ

ੜ੒͢Δ͜ͱͰ͋Δɽ

{π

α

}

͸೚ҙʹબͿ͜ͱ͕Ͱ͖Δ͕ɼ

෺ཧʹ͓͍ͯϞϯςΧϧϩ๏Λ༻͍Δ৔߹ɼͦΕΛϘϧ πϚϯ෼෍ʹબͿ৔߹͕΄ͱΜͲͰ͋Δɽͦͷ࣌

{π

α

}

^͸

࣍ࣜͰ༩͑ΒΕΔɽ

πα=

exp( − E

_α/

k

B

T)

/

Z

,

(3)

͜͜Ͱɼ

E

_α͸ঢ়ଶα^{ͷΤωϧΪʔɼ}

k

_B͸ϘϧπϚϯఆ

਺ɼ

T

͸Թ౓ɼ

Z

͸෼഑ؔ਺Ͱ͋ΔɽҎԼɼಛʹஅΒͳ

͍৔߹ɼ໨ඪ֬཰෼෍

{π

α

}

͸ϘϧπϚϯ෼෍ͱ͢Δɽ

͜ͷ໨తͷͨΊϞϯςΧϧϩ๏Ͱ͸ɼݱࡏͷঢ়ଶα^͔ Β࣍ͷεςοϓͰͷঢ়ଶβ^{ΛɼભҠ֬཰}

W

_α→βʹैͬͯ

֬཰తʹܾΊΔͱ͍͏͜ͱΛߦ͏ɽα^ͱβ^{ͷऔΓ͑Δ஋}

͸ͦΕͧΕ

M

ݸ͋ΔͷͰɼ

W

_α→β͸

M × M

ͷߦྻͰ͋

Δɽܥ͸ݱࡏͷঢ়ଶ͔Βɼඞ͍ͣͣΕ͔ͷঢ়ଶʹભҠ͢

ΔͨΊɼ

W

_α→β͸࣍ͷอଘଇΛຬͨ͢ɿ

β

W

_α→β=

1

.

(4)

͜ͷભҠ֬཰ߦྻʹΑΓɼ

n

εςοϓ໨ͱ

n

+

1

εςοϓ

໨ͷ֬཰෼෍͸࣍ࣜͰؔ܎͚ͮΒΕΔɽ

p

_α

(n

+

1)

=

γ

W

_γ→α

p

_γ

(n)

,

(5)

͜͜Ͱ

p

_γ

(n)

͸ɼ

n

εςοϓ໨ͷঢ়ଶ͕γ^{Ͱ͋Δ֬཰Λ} ද͢ɽ

͜ͷ֬཰աఔͷେ͖ͳಛ௃͸ɼ࣍ͷεςοϓͰͲͷঢ় ଶʹભҠ͢Δ͔͸ݱࡏͷঢ়ଶͷΈʹґଘ͠ɼաڈʹ͸ґ ଘ͠ͳ͍͜ͱͰ͋Δɽݴ͍׵͑Δͱɼݱࡏͷঢ়ଶα^͔Β

࣍εςοϓͰঢ়ଶβʹભҠ͢Δ֬཰͸ɼ୯७ʹߦྻཁૉ

W

_α→βͰॻ͖ද͞Εɼ͜Ε·ͰͲ͏͍͏ཤྺΛͨͲͬͯ

ঢ়ଶαʹ౸ୡ͔ͨ͠ʹશ͘ґଘ͍ͯ͠ͳ͍ɽ͜ͷΑ͏ʹɼ ະདྷͰͷڍಈ͕ݱࡏͷঢ়ଶͷΈʹґଘ͠ɼաڈʹશ͘ґ ଘ͠ͳ͍֬཰աఔͷ͜ͱΛϚϧίϑաఔɼͦͷதͰঢ়ଶ ͱ͕࣌ؒ཭ࢄతͳ΋ͷΛϚϧίϑ࿈࠯ͱ͍͏ɽ͜ͷϚϧ ίϑ࿈࠯Λ༻͍͍ͯΔ͜ͱ͕ɼʮϚϧίϑ࿈࠯ϞϯςΧϧ ϩ๏ʯͱ͍͏໊લͷ༝དྷͱͳ͍ͬͯΔɽ

֬཰෼෍

{p

α

(n)}

͕ͲͷΑ͏ʹ࣌ؒൃల͢Δ͔͸ભҠ

֬཰ߦྻ

{ W

_α→β

}

ͷ༩͑ํʹґଘ͢ΔΘ͚͕ͩɼҰൠʹ ϞϯςΧϧϩ๏Ͱ͸ɼ͜ͷભҠ֬཰ʹҎԼͷ

2

ͭͷ৚݅

Λ՝͢ɿ

(i)

Τϧΰʔυੑɿ೚ҙͷঢ়ଶ͔Βελʔτͯ͠΋ɼঢ় ଶભҠΛ܁Γฦ͢͜ͱʹΑΓɼܥ͸શͯͷঢ়ଶʹͨ

ͲΓண͘͜ͱ͕ՄೳͰ͋Δɽ

(ii)

௼Γ߹͍৚݅ɿ೚ҙͷঢ়ଶα^{ʹର͠ɼ࣍ͷ౳͕ࣜ੒}

Γཱͭɿ

γπγ

W

_γ→α=

γπα

W

_α→γ=πα,

(6)

͜͜Ͱɼୈ

2

͔ࣜΒୈ

3

ࣜΛಋ͘ࡍʹࣜ

(4)

Λ༻

͍ͨɽ͜ͷࣜ͸ɼ໨ඪ෼෍

{π

α

}

^{Ͱ͸ɼ೚ҙͷঢ়ଶ} αʹ͓͍ͯɼྲྀೖ͢Δ֬཰ͱྲྀग़͢Δ֬཰͕௼Γ߹

͏͜ͱΛද͍ͯ͠Δɽ

͜ͷ

2

ͭͷ৚͕݅ຬͨ͞ΕΔ࣌ɼ೚ҙͷॳظ෼෍

{ p

_α

(0) }

ʹର͠ɼແݶεςοϓޙͷ֬཰෼෍͕

{π

α

}

^{ʹऩଋ͢Δ͜}

ͱɼͭ·Γࣜ

lim

_n→∞

p

_α

(n)

=πα,

(7)

͕੒Γཱͭ͜ͱ͕਺ֶతʹݫີʹࣔ͞Ε͍ͯΔ⁽⁷⁾ɽ͜Ε ΛΤϧΰʔυఆཧͱ͍͏ɽ

͜ͷΑ͏ʹɼભҠ֬཰͕໨ඪ෼෍

{π

α

}

^{ʹऩଋ͢ΔͨΊ} ʹ͸௼Γ߹͍৚͕݅ຬͨ͞ΕΕ͹ྑ͍ͷ͕ͩɼࣜ

(6)

͸

੍໿͕͋·Γʹ؇͘ɼ·ͨ྆ลʹଟ਺ͷભҠ֬཰ΛؚΉ

ͨΊɼ͜ͷࣜΛຬͨ͢ҰൠతͳભҠ֬཰Λߟ͑Δ͜ͱ͸

͔ͳΓࠔ೉Ͱ͋ΔɽͦͷͨΊϞϯςΧϧϩ๏Ͱ͸ɼભҠ

֬཰Λߏ੒͢Δࡍʹɼ࣍ͷৄࡉ௼Γ߹͍৚݅Λ՝͢͜ͱ

͕ଟ͍ɽ

(ii)’

ৄࡉ௼Γ߹͍৚݅ɿ೚ҙͷ

2

^ͭͷঢ়ଶα, γ^ʹର͠ɼ

࣍ͷ౳͕ࣜ੒Γཱͭɿ

πγ

W

_γ→α=πα

W

_α→γ.

(8)

͜ͷࣜ͸ࣜ

(6)

྆ลͷ֤߲ʹ͓͍ͯ౳͕ࣜ੒Γཱͭ͜ͱ Λද͍ͯ͠Δɽ͜ͷ͜ͱ͔Β໌Β͔ͳΑ͏ʹɼৄࡉ௼Γ߹

͍৚݅͸௼Γ߹͍৚݅ͷे෼৚݅Ͱ͋Δɽ͜Ε·ͰϞϯ ςΧϧϩ๏͸ɼओʹৄࡉ௼Γ߹͍৚݅ͷ࿮૊Έ಺Ͱ։ൃ

͞Ε͖͕ͯͨɼ࠷ۙ௼Γ߹͍৚݅ͷΈΛຬͨ͢ϞϯςΧ ϧϩ๏͕ز͔ͭ։ൃ͞Ε͓ͯΓɼ͔͠΋ଟ͘ͷ৔߹Ͱ͜

ΕΒͷख๏͸ɼैདྷͷৄࡉ௼Γ߹͍৚݅Λຬͨ͢Ϟϯς Χϧϩ๏ΑΓγϛϡϨʔγϣϯޮ཰͕ྑ͍͜ͱ͕໌Β͔

ͱͳ͍ͬͯΔ^(8ʙ14)ɽ͔͠͠ɼҎԼͰຊߘͰ͸ɼৄࡉ௼Γ

߹͍৚݅Λຬͨ͢ϞϯςΧϧϩ๏ʹ͍ͭͯͷΈ঺հ͢Δɽ

4. Metropolis^ΞϧΰϦζϜ

Ұൠʹɼৄࡉ௼Γ߹͍ͷ৚݅ࣜ

(8)

ͱ֬཰อଘͷࣜ

(4)

͔Β͸ભҠ֬཰͸Ұҙʹఆ·ΒͣɼભҠ֬཰ͷ༩͑ํ͸ແ

਺ʹଘࡏ͢Δ͕ɼͦͷதͰ΋ͬͱ΋༗໊ͳͷ͕

Metropolis

ΞϧΰϦζϜ⁽¹⁵⁾Ͱ͋Δɽ͜ͷΞϧΰϦζϜ͸࣍ͷ

2

^ͭ ͷεςοϓ͔Β੒Γཱͭɿ

(1)

ݱࡏͷঢ়ଶα^{͔ΒભҠઌͷঢ়ଶ}β^Λɼ֬཰

G

_α→βʹ

ैͬͯੜ੒͢Δɽ͜ͷ֬཰͸α^ͱβ^{ʹରͯ͠ରশ} Ͱ͋Γɼ

G

_α→β=

G

_β→αΛຬͨ͢ɽ

(2)

α^͔Ββ΁ͷભҠΛ࣍ͷ֬཰Ͱडཧ͢Δɿ

A

_α→β=

min

1

,πβ

πα

=

min

1

,

exp

− E

_β

− E

_α

k

_B

T

,

,

(9)

͜͜Ͱ

2

ͭ໨ͷ౳ࣜΛಋ͘ࡍʹࣜ

(3)

Λ༻͍ͨɽ

͜ͷ࣌ભҠ֬཰

W

_α→β͸ɼ

G

_α→βͱ

A

_α→βΛ༻͍ͯɼ࣍

ͷΑ͏ʹද͞ΕΔ

:

W

_α→β=⎧⎪⎪⎪⎨

⎪⎪⎪⎩

G

_α→β

A

_α→β

(

αβ

)

,

1 −

γα

G

_α→γ

A

_α→γ

(

α=β

)

.

(10)

G

_α→βͷରশੑ͓Αͼࣜ

(9)

ΑΓɼࣜ

(10)

ͷભҠ֬཰͕

ৄࡉ௼Γ߹͍ͷ৚݅

(8)

Λຬͨ͢͜ͱ͸༰қʹ͔֬ΊΒ ΕΔɽ

͜͜Ͱɼडཧ֬཰ͷࣜ

(9)

ͷղऍʹ͍ͭͯɼগ͠આ໌

͢Δɽ͍·

G

_α→β=

G

_β→αͳͷͰɼडཧ֬཰͸ৄࡉ௼Γ

߹͍৚݅ͱಉ༷ͷࣜɼͭ·Γ

πβ

A

_β→α=πα

A

_α→β,

(11)

Λຬͨ͢ඞཁ͕͋ΔɽͦͷͨΊɼྫ͑͹πα> πβͷ࣌ɼ

A

_α→β<

A

_β→αͰ͋Δ͕ɼେ͖͍ํͷडཧ֬཰

A

_β→αʹ֬

཰ͷ্ݶ஋

1

Λ༩͑ɼখ͍͞ํͷडཧ֬཰

A

_α→βʹৄࡉ

௼Γ߹͍ͷ৚͔݅Βܾ·Δ஋πβ/παΛ༩͑ͨͷ͕ࣜ

(9)

ͷडཧ֬཰Ͱ͋Δɽैͬͯ͜ͷडཧ֬཰͸ɼৄࡉ௼Γ߹

͍৚݅ͷ࿮૊ΈͷதͰɼͦͷ஋Λ࠷େʹͨ͠΋ͷͰ͋Δ ͱݴ͑Δɽडཧ֬཰Λ͞Βʹ্͛ΔͨΊʹ͸ɼৄࡉ௼Γ

߹͍৚݅ͷ࿮૊ΈΛഁΔඞཁ͕͋Δ⁽⁹⁾ɽ

࣍ʹࣜ

(9)

^{ʹ͍ͭͯɼୈ}

3

^{ࣜʹ෼഑ؔ਺}

Z

^ؚ͕·Εͯ

͍ͳ͍఺ʹ஫໨ͯ͠௖͖͍ͨɽࣜ

(3)

^{͔ΒΘ͔ΔΑ͏ʹɼ}

໨ඪ෼෍

{π

α

}

^{ʹ͸෼഑ؔ਺}

Z

ؚ͕·Ε͍ͯΔ͕ɼࣜ

(9)

ͷୈ

2

ࣜʹ͸ൺπβ/πα͔͠ݱΕͳ͍ͨΊɼ෼ࢠɾ෼฼ͷ

෼഑ؔ਺͕ޓ͍ʹΩϟϯηϧ͠ɼୈ

3

ࣜͰ͸

Z

͕ফ͑ͯ

͍Δɽࡉ͔͍࿩Ͱ͸͋Δ͕ɼ͜ͷ͜ͱ͸ϞϯςΧϧϩ๏

Λ࣮૷͢Δ্ͰɼۃΊͯॏཁͳ͜ͱͰ͋Δ⁽¹⁶⁾ɽ

5. ϞϯςΧϧϩ๏ͷޮ཰ͱ؇࿨࣌ؒ

͜ͷΑ͏ʹϞϯςΧϧϩ๏Λ༻͍Δͱɼ෼഑ؔ਺Λܭ

ࢉ͢Δ͜ͱͳ͘ɼϘϧπϚϯ෼෍ʹैͬͯঢ়ଶΛੜ੒͢

Δ͜ͱ͕ՄೳͱͳΔ͕ɼͦͷҰํͰϞϯςΧϧϩ๏ʹ͸ɼ ੜ੒͞ΕΔঢ়ଶʹ૬͕ؔੜ͡Δͱ͍͏໰୊͕͋Δɽྫ͑

͹εϐϯܥʹ͓͍ͯɼ֤εςοϓͰεϐϯͷ޲͖Λ

1

ͭ

ͣͭߋ৽͢Δɼ

1

εϐϯϑϦοϓͷϞϯςΧϧϩΛߦ͏

ͱɼ౰વͳ͕Βɼ͋Δεςοϓͷঢ়ଶͱͦͷ

1

εςοϓ ޙͷঢ়ଶ͸ඇৗʹࣅ͍ͯΔɽͭ·Γ૬͕ؔڧ͍ɽҰൠʹɼ

͋Δ࣌ࠁʢεςοϓʣ

t

₁ʹ͓͚Δঢ়ଶͱɼผͷ࣌ࠁ

t

₂ʹ

͓͚Δঢ়ଶ͸ɼͦͷ࣌ؒࠩ

| t

2

− t

1

|

^{͕େ͖͘ͳΔʹͭΕ}

ͯ૬͕ؔऑ·Δ͕ɼ͜ͷ૬͕ؔ΄΅ແ͘ͳΔͨΊʹඞཁ ͳ࣌ؒͷ͜ͱΛ؇࿨࣌ؒͱݴ͏ɽ

࣍ʹɼ͜ͷঢ়ଶؒͷ૬͕ؔ࣋ͭҙຯʹ͍ͭͯߟ͑Δɽ

ྫ͑͹ɼϞϯςΧϧϩ๏ʹ͓͍ͯঢ়ଶભҠΛ

K

εςοϓ

෼ߦ͏ͱɼͦͷ్தͰ

K

ݸͷঢ়ଶ͕࡞ΒΕΔɽͦͯ͠Ϟ

ϯςΧϧϩ๏Ͱ͸ɼ͜ΕΒͷঢ়ଶʹରͯ͠ฏۉΛऔΔ͜

ͱͰɼ༷ʑͳ෺ཧྔͷɼϘϧπϚϯ෼෍ʹ͓͚Δ೤ฏۉ

஋ΛධՁ͢Δɽ͔͠͠ɼ্Ͱड़΂ͨΑ͏ʹɼϞϯςΧϧ ϩ๏Ͱ࡞ΒΕΔঢ়ଶʹ͸ڧ͍૬͕ؔ͋ΔͨΊɼ͜ΕΒશ

ͯͷঢ়ଶ͸ޓ͍ʹಠཱͰ͸ͳ͘ɼಠཱͳঢ়ଶͷ਺͸

K

/τ ఔ౓͔͠ͳ͍ɽ͜͜Ͱτ͸؇࿨࣌ؒɽͦͯ͠ɼσʔλͷ ฏۉ஋ͷ౷ܭޡࠩ͸ɼಠཱͳσʔλ਺ͷ

1

/

2

৐ʹൺྫ͢

Δɽैͬͯɼ؇࿨࣌ؒ͸ϞϯςΧϧϩ๏ͷޮ཰Λද͢ό ϩϝʔλͰ͋Γɼ؇࿨͕࣌ؒ୹͍΄Ͳޮ཰͕ྑ͍͜ͱ͕

Θ͔Δ⁽¹⁷⁾ɽ

6. ΫϥελʔΞϧΰϦζϜ

લઅʹ͓͍ͯɼ؇࿨࣌ؒ͸ϞϯςΧϧϩ๏ͷޮ཰Λද

͢όϩϝʔλͰ͋Δͱड़΂͕ͨɼҰൠʹ؇࿨࣌ؒ͸సҠ Թ౓ۙ๣ʹ͓͍ͯ૿Ճ͢Δ܏޲͕͋Δɽྫ͑͹ڧ࣓ੑମ ͷ৔߹ɼߴԹ͔ΒసҠԹ౓ʹۙͮ͘ͱ࣍ୈʹεϐϯ૬ؔ

͕ൃୡ͠ɼεϐϯ͕ಉ͡ํ޲Λ޲͍ͨྖҬʢΫϥελʔʣ

͕େ͖͘ͳΔɽͦͯ͠Ϋϥελʔ͸ɼҰ౓Ͱ͖Δͱͳ͔

ͳ͔൓స͠ͳ͍ͨΊɼ؇࿨͕࣌ؒ௕͘ͳΔɽͦͯ͠ܥͷ αΠζ͕ແݶେͷ৔߹ɼసҠԹ౓ʹ͓͍ͯΫϥελʔα Πζ͸ແݶେͱͳΔͨΊɼ؇࿨࣌ؒ΋ແݶେʹൃࢄͯ͠

͠·͏ɽ͜Ε͸૬సҠΛࣔ͢ܥશൠͰ؍ଌ͞ΕΔݱ৅Ͱ

͋Γɼ

critical slowing down

ͱݺ͹ΕΔ⁽¹⁸⁾ɽ࣮ࡍͷγϛϡ Ϩʔγϣϯ͸αΠζ͕༗ݶͷܥͰߦ͏ͨΊɼ؇࿨͕࣌ؒ

ൃࢄ͢Δ͜ͱ͸ͳ͍͕ɼసҠԹ౓ۙ๣Ͱ؇࿨͕࣌ؒٸܹ

ʹ૿େ͢Δ͜ͱʹมΘΓ͸ͳ͘ɼͦͷͨΊ௨ৗͷϞϯς Χϧϩ๏Ͱ͸ɼసҠԹ౓ۙ๣ͰͷγϛϡϨʔγϣϯޮ཰

͕େ෯ʹ௿Լͯ͠͠·͏ɽ

͜ͷ໰୊Λղܾ͢Δ࠷ળͷํ๏͸ɼ໌Β͔ʹɼΫϥε λʔ಺ͷεϐϯΛҰ౓ʹ·ͱΊͯ൓స͢Δ͜ͱͰ͋Δ͕ɼ ڧ࣓ੑΠδϯάεϐϯϞσϧͰ͸ͦͷΑ͏ͳํ๏͕࣮ࡏ

͢Δ^(19,20)ɽҎ߱ɼεϐϯ΍ཻࢠͳͲͷঢ়ଶม਺ΛҰ౓ʹ

·ͱΊͯߋ৽͢Δख๏ΛΫϥελʔΞϧΰϦζϜɼͦͷ தͰจݙ

(19

,

20)

ͰఏҊ͞Εͨํ๏Λ

Swendsen-Wang

ΞϧΰϦζϜͱݺͿɽ

1969

೥ʹ

Fortuin

ͱ

Kasteleyn

͸ɼ ڧ࣓ੑ

q

ঢ়ଶϙοπϞσϧ⁽²¹⁾ͷ෼഑ؔ਺͕ɼ͋Δछͷ ύʔίϨʔγϣϯͷ໰୊ʹ

map

Ͱ͖Δ͜ͱΛ͓ࣔͯ͠

Γ^(22,23)ɼ͜Ε͸

Fortuin-Kasteleyn

ఆཧͱݺ͹Ε͍ͯΔ

͕ɼ

Swendsen-Wang

ΞϧΰϦζϜ͸͜ͷఆཧʹج͍ͮ

ͯઃܭ͞ΕͨϞϯςΧϧϩ๏Ͱ͋Δɽ͜ͷΞϧΰϦζϜ Λద༻͢Δ͜ͱʹΑΓɼڧ࣓ੑΠδϯάϞσϧͷ

critical

slowing down

ͷ໰୊͸ɼ׬શͰ͸ͳ͍͕ܶతʹܰݮͰ

͖Δ͜ͱ͕ࣔ͞Ε͍ͯΔɽ·ͨ

Swendsen-Wang

Ξϧΰ ϦζϜ͸ɼͪΐͬͱͨ͠޻෉Λ͢Δ͜ͱͰ⁽²⁰⁾ɼڧ࣓ੑͷ

XY

Ϟσϧ΍ϋΠθϯϕϧΫϞσϧʹରͯ͠΋ద༻Մೳ

ͱͳΔɽͦͷଞʹΫϥελʔΞϧΰϦζϜ͸ɼྔࢠεϐ

ϯܥ^(24ʙ26)΍ཻࢠܥ^(8,27)Ͱ΋ఏҊ͞Ε͓ͯΓɼͦͷద༻

ʹΑΓγϛϡϨʔγϣϯޮ཰͕େ෯ʹ޲্͢Δ͜ͱ͕֬

͔ΊΒΕ͍ͯΔɽ

ϞϯςΧϧϩ๏ͷޮ཰վળͷ؍఺͔Β͍͏ͱɼΫϥε λʔΞϧΰϦζϜ͸࠷ળͷखͰ͋Δͱड़΂ͯ΋աݴͰ͸

ͳ͍͕ɼ࢒೦ͳ͕Βݱঢ়Ͱ͸ɼΫϥελʔΞϧΰϦζϜ

͕࢖͑Δܥ͸͘͝ݶΒΕ͍ͯΔɽྫ͑͹

Swendsen-Wang

ΞϧΰϦζϜ͸ɼεϐϯάϥεʢϥϯμϜ࣓ੑମʣ⁽²⁸⁾ͷ Α͏ͳϑϥετϨʔγϣϯ͕͋Δܥʹద༻ͯ͠΋શ͘͏

·͘ಇ͔ͳ͍͜ͱ͕஌ΒΕ͍ͯΔɽεϐϯάϥεʹ͓͍

ͯ΋༗ޮͳΫϥελʔΞϧΰϦζϜͷ։ൃ͸͜Ε·Ͱʹ ز͔ͭߦΘΕ͍ͯΔ͕^(29ʙ31)ɼγϛϡϨʔγϣϯޮ཰ͷେ

෯ͳ޲্ʹ੒ޭͨ͠ྫ͸΄ͱΜͲͳ͍ɽྫ֎ͱͯ͠ɼ

2

^࣍ ݩΠδϯάεϐϯάϥεʹ͓͚ΔΫϥελʔΞϧΰϦζ Ϝ͕ڍ͛ΒΕΔ͕⁽²⁹⁾ɼಉϞσϧ͸༗ݶԹ౓Ͱεϐϯάϥ εసҠΛࣔ͞ͳ͍ͨΊɼεϐϯάϥε૬Ͱ༗ޮੑ͕ࣔ͞

Εͨ༁Ͱ͸ͳ͍ɽεϐϯάϥεʹݶΒͣɼϑϥετϨʔ γϣϯ͕͋ΔܥͰ΋༗ޮͳΫϥελʔΞϧΰϦζϜ͸ɼ ʢগͳ͘ͱ΋චऀ͕஌ΔݶΓͰ͸ʣݱ࣌఺Ͱଘࡏ͠ͳ͍ɽ

7. ֦ுΞϯαϯϒϧϞϯςΧϧϩ๏

ҰൠʹɼεϐϯάϥεͷΑ͏ʹϑϥετϨʔγϣϯ͕

͋ΔܥͰ͸ɼΤωϧΪʔฏ໘্ʹ਺ଟ͘ͷϩʔΧϧϛχ ϚϜ͕͋ΔͨΊɼ௿ԹͰͷ؇࿨͕ඇৗʹ஗͘ͳΔɽ͔͠

΋ɼલઅͷ࠷ޙʹड़΂ͨΑ͏ʹɼ͜ΕΒͷܥͰ͸༗ޮͳ ΫϥελʔΞϧΰϦζϜ͕ଘࡏ͠ͳ͍ɽ͜ͷɼϑϥετ Ϩʔγϣϯ͕͋ΔܥͰͷ஗͍؇࿨ͷ໰୊Λղܾ͢ΔͨΊ ʹఏҊ͞Εͨͷ͕ɼҎԼʹ঺հ͢Δ֦ுΞϯαϯϒϧϞ ϯςΧϧϩ๏Ͱ͋Δɽ

֦ுΞϯαϯϒϧϞϯςΧϧϩ๏ͷجຊతΞΠσΟΞ

͸ɼαϯϓϦϯά͢Δঢ়ଶۭؒΛ޿͛Δ͜ͱͰ؇࿨Λଅ ਐ͢Δɼͱ͍͏΋ͷͰ͋Δɽ෺ཧͷ໰୊ʹ͓͍ͯ௨ৗզʑ

͕ڵຯΛ࣋ͭͷ͸ɼసҠԹ౓ۙ๣΍ɼͦΕΑΓ௿ԹͰͷ ܥͷৼ෣͍͕ͩɼ͜ΕΒͷԹ౓Ͱ௨ৗͷϞϯςΧϧϩγ ϛϡϨʔγϣϯΛߦ͏ͱɼ͙͢ϩʔΧϧϛχϚϜʹଊΘ Εͯ؇࿨͕஗͘ͳͬͯ͠·͏ɽͦ͜Ͱ֦ுΞϯαϯϒϧ ϞϯςΧϧϩ๏Ͱ͸ɼ໨ඪ෼෍

{π

α

}

^{Λमਖ਼ͨ͠Γɼܥ} ͷঢ়ଶม਺Λਓҝతʹ֦ு͢Δ͜ͱʹΑΓɼ௿ԹͰͷঢ় ଶ͚ͩͰͳ͘ɼߴԹͰͷঢ়ଶ΋αϯϓϦϯά͢ΔΑ͏ʹ

ͯ͠΍Δɽ͜͏͢Δ͜ͱͰɼܥ͕Ұ౓ϩʔΧϧϛχϚϜ ʹଊΘΕͯ΋ɼߴԹͰͷঢ়ଶʹભҠ͢Δ͜ͱͰɼ͔ͦ͜

Βൈ͚ग़͢͜ͱ͕Ͱ͖Δɽͭ·Γɼʮٸ͕͹ճΕʯͷਫ਼ਆ Ͱ͋Δɽ͔͠΋ɼ௿ԹͰͷঢ়ଶ΋αϯϓϦϯά͢ΔͨΊɼ զʑͷڵຯͷର৅Ͱ͋Δɼ௿Թʹ͓͚Δܥͷৼ෣͍΋஌

ਤ1 ϨϓϦΧަ׵๏ͷུ֓ਤɽM=4ɽԹ౓TiͰͷ௨ৗͷϞ ϯςΧϧϩʢਤதͷʮ௨ৗMCʯʣͱϨϓϦΧަ׵Λ܁Γฦ͠ߦ

͏ɽϨϓϦΧަ׵ʹΑΓɼ֤ϨϓϦΧͷԹ౓͕࣌ؒͱڞʹมԽ͠

͍ͯΔɽ

Δ͜ͱ͕Ͱ͖Δɽ

֦ுΞϯαϯϒϧϞϯςΧϧϩ๏͸େ͖͘෼͚Δͱɼ

(a)

ΤωϧΪʔ্ۭؒͷϥϯμϜ΢ΥʔΫΛ࣮ݱ͢Δ

ํ๏ɽ

(b)

Թ౓্ۭؒͷϥϯμϜ΢ΥʔΫΛ࣮ݱ͢Δํ๏ɽ ͷ

2

ͭʹ෼͚ΒΕΔɽ

(a)

ʹଐ͢Δํ๏ͱͯ͠ϚϧνΧ ϊχΧϧ๏^(32,33)΍

Wang-Landau

^{๏(34,35)͕͋Δ͕ɼ͜Ε} Βͷํ๏Ͱ͸໨ඪ෼෍

{π

α

}

^{Λ࣍ͷΑ͏ʹબͿɿ}

πα=

C

Ω

(E

_α

)

,

(12)

͜͜Ͱ

C

͸ن֨ԽҼࢠɽ·ͨΩ

(E)

͸ΤωϧΪʔঢ়ଶີ

౓Ͱ͋Γɼ࣍ࣜͰఆٛ͞ΕΔɿ Ω

(E) ≡

γδ

(E − E

_γ

)

.

(13)

͜͏͢ΔͱΤωϧΪʔ͕

E

ͷঢ়ଶ͕αϯϓϦϯά͞ΕΔ

֬཰

P(E)

͸

P(E) ≡

απαδ

(E − E

_α

)

=

C

,

(14)

ͱͳΓɼશͯͷΤωϧΪʔ४Ґ͕౳֬཰ͰαϯϓϦϯά

͞ΕΔ͜ͱͱͳΔɽҰൠʹΩ

(E)

͸ະ஌Ͱ͋Δ͕ɼ࠷ॳ͸

ద౰ͳؔ਺͔Βελʔτͯ͠ɼγϛϡϨʔγϣϯʹΑͬͯ

͋ΔछͷֶशΛ͢Δ͜ͱʹΑΓɼۙࣅతʹධՁ͢ΔɽϚ ϧνΧϊχΧϧ๏ͱ

Wang-Landau

๏͸ɼຊ࣭తʹ͸ಉ͡

ख๏Ͱ͋Δ͕ɼΩ

(E)

ͷֶशํ๏͕ҟͳ͍ͬͯΔɽ

Wang-

Landau

๏ͰఏҊ͞Εֶͨशํ๏͸ඇৗʹ؆ศͰɼ͔͠΋

ֶशޮ཰͕ྑ͍ͨΊɼݱࡏͰ͸

Wang-Landau

๏͕༻͍

ΒΕΔ͜ͱ͕ଟ͍ɽ

࣍ʹ

(b)

ʹଐ͢Δํ๏ͱͯ͠ɼ

simulated annealing

๏^(36,37)΍ϨϓϦΧަ׵๏⁽³⁸⁾͕͋Δɽ͜ΕΒ

2

ͭͷํ๏

͸ɼԹ౓্ۭؒͷϥϯμϜ΢ΥʔΫΛ࣮ݱ͢Δ఺͸ڞ௨

͕ͩɼલऀ͕

1

ͭͷϨϓϦΧͷΈΛѻ͏ͷʹର͠ɼޙऀ

͸ෳ਺ͷϨϓϦΧΛѻ͏఺͕ҟͳΔɽҎԼͰ͸ϨϓϦΧ

ަ׵๏ʹ͍ͭͯ঺հ͢Δɽ͜ͷख๏ͷུ֓Λࣔͨ͠ͷ͕

ਤ

1

Ͱ͋ΔɽϨϓϦΧަ׵๏Ͱ͸࠷ॳʹɼΦϦδφϧͷ ܥͱશ͘ಉ͡ϋϛϧτχΞϯΛ࣋ͭίϐʔʢϨϓϦΧʣ Λ

M

ݸ༻ҙ͢ΔɽͦΕͱಉ࣌ʹɼ

M

ݸͷҟͳΔԹ౓΋

༻ҙ͢Δɽ͜ͷԹ౓ηοτ

{ T

i

} (i

=

1

,

2

,

· · ·

,

M)

͸ɼσʔ λΛऔΓ͍ͨ௿Թଆͱɼ؇࿨͕଎͘ϩʔΧϧϛχϚϜ͔

Β༰қʹ୤ग़Ͱ͖ΔߴԹଆɼ͞Βʹ͜ΕΒΛͭͳ͙தؒ

Թ౓ྖҬؚ͕·ΕΔΑ͏ʹબͿɽͦͯ͠ɼ֤ϨϓϦΧ͸

ͦΕͧΕಠཱʹɼҟͳΔԹ౓ͷ೤ཋͱ઀͍ͯ͠Δͷ͕ͩɼ

࣌ʑྡΓ߹͏Թ౓ͷϨϓϦΧΛަ׵͢Δɽ͜ͷަ׵͸ɼৄ

ࡉ௼Γ߹͍৚݅Λຬͨ͢ɼ͋Δద౰ͳ֬཰ʹैͬͯߦ͏ɽ

ͦͷ݁Ռɼ֤ϨϓϦΧͷԹ౓͸࣌ؒͱڞʹࠁҰࠁͱมԽ Λ͢ΔɽͦͷͨΊɼԾʹ͋ΔϨϓϦΧ͕௿ԹͰϩʔΧϧ ϛχϚϜʹଊΘΕͯ΋ɼߴԹʹͳͬͨ࣌ʹ͔ͦ͜Β୤ग़

͢Δ͜ͱ͕Ͱ͖ΔɽԹ౓ηοτ

{ T

i

}

^{͸ɼϨϓϦΧަ׵ͷ} डཧ֬཰ͷฏۉ஋͕ɼԹ౓ʹΑΒͣҰఆʹͳΔΑ͏ʹબ

͹ΕΔ͜ͱ͕ଟ͍ɽ۩ମతʹ͸ɼจݙ

(38)

ͰఏҊ͞Εͯ

͍Δํ๏ͳͲΛ༻͍ͯɼ֤Թ౓ͷ஋Λஞ࣍తʹमਖ਼͢Δ

͜ͱʹΑΓɼͦͷΑ͏ͳԹ౓ηοτΛ༻ҙ͢Δ͜ͱ͕Ͱ

͖Δɽ·ͨۙ೥ɼϨϓϦΧͷԹ౓্ۭؒͷӡಈͷ֦ࢄ཰

ΛଌΓɼϘτϧωοΫʹͳ͍ͬͯΔͱ͜ΖʹΑΓଟ͘ͷ Թ౓Λ഑ஔ͢Δ͜ͱͰɼԹ౓ηοτΛ࠷దԽ͢Δํ๏΋

ఏҊ͞Ε͍ͯΔ⁽³⁹⁾ɽ

8. ^{֬཰తΧοτΦϑ๏}

Ұൠʹ௕ڑ཭૬ޓ࡞༻ܥ͸ɼ਺ଟ͘ͷ૬ޓ࡞༻ΛѻΘ ͳ͚Ε͹ͳΒͳ͍ͨΊɼ਺஋γϛϡϨʔγϣϯΛߦ͏ͷ

͕ࠔ೉Ͱ͋Δɽྫ͑͹ɼΫʔϩϯ૬ޓ࡞༻΍࣓ؾ૒ۃࢠ ૬ޓ࡞༻ͳͲͷ

2

ମͷ௕ڑ཭૬ޓ࡞༻͸ɼܥͷαΠζ ʢཻࢠ਺΍εϐϯ਺ͳͲʣΛ

N

ͱ͢Δͱɼ૬ޓ࡞༻ͷ਺

͸_N

C

2

≈ O(N

²

)

͚ͩ͋ΔɽͦΕʹର͠ɼۙ͘ͷཻࢠ΍ε ϐϯͱ͔͠૬ޓ࡞༻͠ͳ͍ۙڑ཭૬ޓ࡞༻ͷ৔߹ɼ૬ޓ

࡞༻ͷ਺͸͔͔ͨͩ

O (N)

Ͱ͋ΔɽͦͷͨΊɼಛʹԿͷ

޻෉΋ͤͣʹϞϯςΧϧϩΛߦ͏ͱɼ௕ڑ཭૬ޓ࡞༻ܥ Ͱͷܭࢉ࣌ؒ͸ۙڑ཭૬ޓ࡞༻ܥͰͷͦΕͷ

O (N)

ഒͱ ͳͬͯ͠·͏ɽ͜ͷ໰୊Λࠀ෰͢ΔͨΊɼπϦʔ๏^(40,41)

΍ߴ଎ଟॏల։๏^(42ʙ45)ͳͲͷख๏͕։ൃ͞Ε͍ͯΔ͕ɼ

͜ΕΒͷํ๏Ͱ͸ΤωϧΪʔͷۙࣅධՁΛ͍ͯ͠ΔͨΊɼ

ۙࣅΛؚΉख๏ͱͳ͍ͬͯΔɽ·ͨߴ଎ϑʔϦΤม׵Λ ར༻ͨ͠ख๏΋։ൃ͞Ε͍ͯΔ͕⁽⁴⁶⁾ɼ͜ͷख๏Ͱ͸ɼ

1

εϐϯߋ৽Λߦ͏ͨͼʹຖճΤωϧΪʔΛධՁ͢Δͱ͍

͏͜ͱ͸ͤͣɼͦΕΛؒҾ͍ͯߦ͍ͬͯΔͨΊɼ΍͸Γ

ۙࣅΛؚΉख๏ͱͳ͍ͬͯΔɽ௕ڑ཭૬ޓ࡞༻ܥʹ͓͍

ͯɼۙࣅͳ͠Ͱܭࢉ࣌ؒͷେ෯ͳ࡟ݮʹ੒ޭͨ͠ख๏ͱ

ͯ͠

Luijten

ͱ

Bl¨ote

ͷख๏⁽⁴⁷⁾͕͋Δ͕ɼ͜ͷํ๏͸ڧ

࣓ੑΠδϯάϞσϧʹ͔͠ద༻Ͱ͖ͳ͍ͱ͍͏ܽ఺͕͋

ΔɽͦΕʹରͯ͠ຊઅͰ঺հ͢Δ֬཰తΧοτΦϑ๏⁽⁴⁸⁾

͸ɼҰൠͷ௕ڑ཭૬ޓ࡞༻ܥʹద༻ՄೳͰɼ͔ͭϞϯς Χϧϩ๏ͱͯۙ͠ࣅΛؚ·ͳ͍ख๏ͱͳ͍ͬͯΔɽ֬཰

తΧοτΦϑ๏ͱ΄΅ಉ࣌ظʹɼ͜Εͱࣅͨಛ௃Λ࣋ͭ

ख๏͕෱Ҫɾ౻ಊʹΑΓ։ൃ͞Ε͍ͯΔͷͰ⁽⁴⁹⁾ɼڵຯͷ

͋Δํ͸ͦͪΒ΋ޚࢀর௖͖͍ͨɽ

͜͜Ͱ֬཰తΧοτΦϑ๏ͷઆ໌Λ͢Δલʹɼಉख๏

Ͱ༻͍͍ͯΔ

Stochastic Potential Switching (SPS)

Ξϧ ΰϦζϜʹ͍ͭͯઆ໌͢Δ^(50,51)ɽ࠷ॳʹɼϋϛϧτχΞ ϯ͸ଟ਺ͷ૬ޓ࡞༻ʢϙςϯγϟϧʣͷ࿨Ͱද͞ΕΔͱ Ծఆ͢Δɽྫ͑͹

2

ମͷ௕ڑ཭૬ޓ࡞༻ͷ৔߹ɼϋϛϧ τχΞϯ͸࣍ͷΑ͏ʹද͞ΕΔɿ

H

=

i<j

V

_{i j}

(S

_i,S_j

)

,

(15)

͜͜ͰSi͸ɼεϐϯ΍ཻࢠҐஔͳͲɼܥͷঢ়ଶΛද͢ඍ ࢹతม਺Ͱ͋Δɽ

SPS

ΞϧΰϦζϜͰ͸ɼϙςϯγϟϧ Λ֬཰

P

i jͰ

V ˜

i jʹɼ

1 − P

i jͰ

V ¯

i jʹ੾Γସ͑Δͱ͍͏

͜ͱΛߦ͏ɽ͜͜Ͱ֬཰

P

i j͸ࣜ

P

_{i j}

(S

_i,S_j

)

=

exp[

β

(

Δ

V

_{i j}

(S

_i,S_j

) −

Δ

V

_{i j}^∗

)]

,

(16)

Ͱ༩͑ΒΕΔɽβ

≡ 1

/

k

B

T

^͸ٯԹ౓ɼ

Δ

V

_{i j}

(S

_i,S_j

) ≡ V

_{i j}

(S

_i,S_j

) − V ˜

_{i j}

(S

_i,S_j

)

,

(17)

Δ

V

_{i j}^∗͸ɼΔ

V

i j

(S

i,Sj

)

ͷ࠷େ஋ͱ౳͍͔͠ɼ͋Δ͍͸ͦΕ ΑΓେ͖͍ఆ਺Ͱ͋Δɽϙςϯγϟϧ

V ˜

i j͸೚ҙʹબͿ

͜ͱ͕Ͱ͖Δ͕ɼ΋͏ยํͷϙςϯγϟϧ

V ¯

i j͸֬཰

P

i j

Λ༻͍ͯ࣍ͷΑ͏ʹද͞ΕΔɿ

V ¯

_{i j}

(S

_i,S_j

)

=

V

_{i j}

(S

_i,S_j

) −

β⁻¹

log[1 − P

_{i j}

(S

_i,S_j

)]

.

(18)

ͦͯ͠

SPS

ΞϧΰϦζϜͰ͸ɼҎԼͷखॱΛ܁Γฦ͢͜

ͱʹΑΓɼܥͷඍࢹతঢ়ଶ

{

S_i

}

^{Λߋ৽͢Δɿ}

(A)

ϙςϯγϟϧ

V

_{i j}Λ֬཰

P

_{i j}Ͱ

V ˜

_{i j}ʹɼ

1 − P

_{i j}Ͱ

V ¯

_{i j} ʹ੾Γସ͑Δɽ

(B)

੾Γସ͑ΒΕͨϋϛϧτχΞϯ

H

=

V ˜

_{i j}

(S

_i,S_j

)

+

V ¯

_{i j}

(S

_i,S_j

)

,

(19)

Λ༻͍ͯ௨ৗͷϞϯςΧϧϩΛߦ͏ɽ͜͜Ͱ͸

V ˜

΁੾Γସ͑ΒΕͨϙςϯγϟϧʹର͢Δ࿨Λɼ

͸

V ¯

΁੾Γସ͑ΒΕͨϙςϯγϟϧʹର͢Δ࿨Λ ද͢ɽ

(C) (A)

΁໭Δɽ

͜ͷ

SPS

ΞϧΰϦζϜ͸ɼΦϦδφϧͷϋϛϧτχΞϯ

H

ʹؔ͢Δৄࡉ௼Γ߹͍৚݅Λຬͨ͢͜ͱ͕ݫີʹࣔ͞

Ε͍ͯΔ^(50,51)ɽ

ͦͯ֬͠཰తΧοτΦϑ๏Ͱ͸ɼ೚ҙʹબͿ͜ͱ͕Ͱ

͖Δϙςϯγϟϧ

V ˜

i jΛ

0

ʹ͢Δ͜ͱͰɼ੾Γସ͑ΒΕ

ͨϋϛϧτχΞϯ

H

ͷܭࢉ࣌ؒΛ࡟ݮ͢Δɼͱ͍͏͜

ͱΛߦ͏ɽ੾Γସ͑ͷ݁Ռ

V ˜

_{i j}ͱͯ͠࢒Δϙςϯγϟϧ ͷ਺͸ɼϙςϯγϟϧͷڑ཭ʹର͢Δݮਰͷ଎͞ͱۭؒ

࣍ݩʹґଘ͢Δͷ͕ͩ⁽⁴⁸⁾ɼྫ͑͹

2

࣍ݩ࣓ؾ૒ۃࢠܥͷ

৔߹͸

O (N)

ͱͳΔɽ

2

ମͷ௕ڑ཭૬ޓ࡞༻ͷ৔߹ɼݩʑ ͷϙςϯγϟϧͷ਺͸

O (N

²

)

ͳͷͰɼେ൒ͷϙςϯγϟ ϧ͕

V ˜

_{i j}=

0

ʹ੾Γସ͑ΒΕΔ͜ͱʹΑΓΧοτΦϑ͞

Ε͍ͯΔ͜ͱ͕Θ͔ΔɽҰํɼखॱ

(A)

ʹ͓͚Δϙςϯ γϟϧ੾Γସ͑͸ɼϙςϯγϟϧͷ਺͕

O (N

²

)

Ͱ͋Δͨ

Ίɼී௨ʹߦ͏ͱ͜Εͱಉ͡Φʔμʔͷܭࢉ͕͔͔࣌ؒͬ

ͯ͠·͏ɽ͔͜͠͠Ε͸ɼ੾Γସ͑ͷखॱΛ޻෉͢Δ͜

ͱʹΑΓɼ

V ¯

i jͱͯ͠ੜ͖࢒Δϙςϯγϟϧ਺ͱಉ͡Φʔ μʔ·Ͱܭࢉ࣌ؒΛ࡟ݮ͢Δ͜ͱ͕Ͱ͖Δ⁽⁴⁸⁾ɽͦͷ݁Ռɼ

ྫ͑͹

2

࣍ݩ࣓ؾ૒ۃࢠܥͷ৔߹ɼ֬཰తΧοτΦϑ๏

ͷద༻ʹΑΓɼ

1

ϞϯςΧϧϩεςοϓ⁽⁵²⁾౰ͨΓͷܭࢉ

࣌ؒ͸

O (N

²

)

͔Β

O (N)

΁࡟ݮ͞ΕΔɽ

͜͜Ͱɼ͜ͷҰݟحົͳ

SPS

ΞϧΰϦζϜΛΑΓྑ͘

ཧղ͢ΔͨΊɼจݙ

(53)

^ʹ͓͚Δ

SPS

^{ΞϧΰϦζϜͷ}

࠶ܗࣜԽʹ͍ͭͯ঺հ͢ΔɽͦͷͨΊʹ࠷ॳʹɼ࣍ࣜͰ ఆٛ͞ΕΔάϥϑม਺

{ g

i j

}

^{Λಋೖ͢Δɿ}

g

i j=⎧⎪⎪⎪⎨

⎪⎪⎪⎩

0 V

_{i j}͕

V ˜

_{i j}ʹ੾Γସ͑ΒΕͨ৔߹,

1 V

_{i j}͕

V ¯

_{i j}ʹ੾Γସ͑ΒΕͨ৔߹.

(20)

࣍ʹɼࣜ

ωi j

(S

_i,S_j

; g

_{i j}

)

≡

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

e

^{−β{V˜i j(}Si,Sj)+ΔVi j^∗}

(g

i j=

0)

,

e

^{−βV¯i j(}Si,Sj)

(g

_{i j}=

1)

,

(21)

Ͱఆٛ͞ΕΔॏΈΛಋೖ͢Δɽ͜Ε͸ɼ

Swendsen-Wang

ΞϧΰϦζϜ⁽¹⁹⁾Λ࠶ܗࣜԽ͢Δࡍʹɼ

Edwards

ͱ

Sokal

͕ಋೖͨ͠ॏΈ⁽⁵⁴⁾ͱྨࣅͷ΋ͷͰ͋Δɽ

ͦͯ͠ɼ࣮͸

SPS

ΞϧΰϦζϜ͸ɼ֦ு͞Εͨঢ়ଶۭ

ؒ

( {

Si

}, { g

i j

} )

ʹ͓͍ͯɼࣜ

P

_SPS

( {

S_i

}, { g

_{i j}

} ) ≡ Z

⁻¹_SPS

i<jωi j

(S

_i,S_j

; g

_{i j}

)

,

(22)

Ͱఆٛ͞ΕΔฏߧ෼෍Λ࣮ݱ͢ΔϞϯςΧϧϩ๏ͱͳͬ

͍ͯΔɽ͜͜Ͱɼ

Z

SPS

≡ Tr

_{Si},{gi j}

i<jωi j

(S

i,Sj

; g

i j

)

.

(23)

ͦͯ͠ɼεςοϓ

(A)

͸ঢ়ଶม਺

{

S_i

}

^{Λݻఆ͠ͳ͕Βά} ϥϑม਺

{ g

_{i j}

}

͚ͩΛߋ৽͢Δϓϩηεʹɼεςοϓ

(B)

͸άϥϑม਺

{ g

_{i j}

}

Λݻఆ͠ͳ͕Βঢ়ଶม਺

{

S_i

}

^͚ͩΛߋ

৽͢ΔϓϩηεʹରԠ͍ͯ͠Δɽ࣮ࡍʹɼεςοϓ

(A)

ͷભҠ֬཰Λ

W

A

( { g

i j

} → { g

_{i j}

}|{

Si

} )

ɼεςοϓ

(B)

ͷભ Ҡ֬཰Λ

W

B

( {

Si

} → {

S_i

}|{ g

i j

} )

ͱ͢Δͱɼ͜ΕΒ͸࣍ͷ

ৄࡉ௼Γ߹͍৚݅Λຬͨ͢͜ͱ͕ࣔ͞ΕΔ⁽⁵³⁾ɿ

P

SPS

( {

Si

}, { g

i j

} )W

A

( { g

i j

} → { g

_{i j}

}|{

Si

} )

=

P

_SPS

( {

S_i

}, { g

_{i j}

} )W

_A

( { g

_{i j}

} → { g

_{i j}

}|{

S_i

} )

,

(24) P

SPS

( {

Si

}, { g

i j

} )W

B

( {

Si

} → {

S_i

}|{ g

i j

} )

=

P

SPS

( {

S_i

}, { g

i j

} )W

B

( {

S_i

} → {

Si

}|{ g

i j

} )

.

(25)

͜ͷ͜ͱ͸ɼ

SPS

ΞϧΰϦζϜͷฏߧ෼෍͕ࣜ

(22)

Ͱ༩

͑ΒΕΔ͜ͱΛ͍ࣔͯ͠Δɽ

࣍ʹɼ

SPS

ΞϧΰϦζϜͷฏߧ෼෍͕ࣜ

(22)

^Ͱ͋Δ

͜ͱͷҙຯʹ͍ͭͯઆ໌͢Δɽࣜ

(16)

^ɼ

(17)

^ɼ

(18)

^ɼ

(21)

͔Β༰қʹࣔͤΔΑ͏ʹɼॏΈωi j

(S

i,Sj

; g

i j

)

^͸࣍ͷੑ

࣭Λຬͨ͢ɿ

Tr

_{gi j=0,1}ωi j

(S

_i,S_j

; g

_{i j}

)

=

exp[ −β V

_{i j}

(S

_i,S_j

)]

.

(26)

͜ͷ͔ࣜΒɼ෼഑ؔ਺

Z(

β

)

ʹର͢Δɼ࣍ͷ৽͍͠දࣜΛ ಘΔɿ

Z(

β

)

=

Z

SPS

(

β

)

=

Tr

_{g

i j},{Si}

i<jωi j

(S

i,Sj

; g

i j

)

.

(27)

·ͨࣜ

(26)

͸ɼ

SPS

ΞϧΰϦζϜʹ͓͍ͯ͋Δঢ়ଶ

{

Si

}

͕࣮ݱ͢Δ֬཰͸ɼࣜ

P( {

Si

} )

=

Tr

_{gi j}

P

SPS

( {

Si

}, { g

i j

} )

=

P

B

( {

Si

} )

,

(28)

Ͱ༩͑ΒΕΔ͜ͱΛҙຯ͢Δɽ͜͜Ͱ

P

B͸ɼࣜ

P

_B

( {

S_i

} )

=

Z(

β

)

⁻¹

exp −β

i<j

V

_{i j}

(S

_i,S_j

)

,

(29)

Ͱఆٛ͞ΕΔϘϧπϚϯ෼෍Ͱ͋Δɽ͜Ε͕ɼ

SPS

Ξϧ ΰϦζϜʹΑͬͯϘϧπϚϯ෼෍ʹैͬͨঢ়ଶαϯϓϦ ϯά͕Մೳͳཧ༝Ͱ͋Δɽ

͜ͷΑ͏ʹɼݩʑͷঢ়ଶม਺

{

Si

}

^{ʹάϥϑม਺}

{ g

i j

}

^Λ Ճ͑ͨɼ֦ு͞Εͨঢ়ଶۭؒ

( {

Si

}, { g

i j

} )

ʹ͓͍ͯɼঢ়ଶ ม਺ͱάϥϑม਺Λަޓʹߋ৽͢ΔϞϯςΧϧϩ๏ͷ͜ͱ Λɼ

dual

ϞϯςΧϧϩ๏ͱݴ͏^(55,56)ɽલड़ͷڧ࣓ੑΠδ ϯάεϐϯϞσϧʹ͓͚ΔΫϥελʔΞϧΰϦζϜ^(19,20)

΍ྔࢠϞϯςΧϧϩ๏ʹ͓͚Δ

loop

ΞϧΰϦζϜ⁽²⁴⁾ͳ Ͳ΋ɼ

dual

ϞϯςΧϧϩ๏ͷҰछͰ͋Δɽ

࣍ʹࣜ

(27)

ʹΑΔ෼഑ؔ਺ͷදݱʹ͍ͭͯઆ໌͢

Δɽ͜Ε͸ɼڧ࣓ੑϙοπϞσϧͷ෼഑ؔ਺ͷ

Fourtuin- Kasteleyn

දݱ^(22,23)ͷҰൠԽͱͳ͍ͬͯΔɽ࣮ࡍɼࣜ

(27)

Λڧ࣓ੑϙοπϞσϧʹద༻͢Δ͜ͱʹΑΓɼݩʑ ͷ

Fourtuin-Kasteleyn

දݱΛಋग़Ͱ͖Δ⁽⁵³⁾ɽࣜ

(27)

ʹΑ Δදݱ͸ɼݩʑͷ

Fourtuin-Kasteleyn

දݱͱൺ΂ͯɼҎ Լͷ

2

఺ʹ͓͍ͯΑΓҰൠతͰ͋Δɿ

1)

ࣜ

(27)

ʹΑΔදݱͰ͸ϙςϯγϟϧ

V ˜

_{i j}Λ೚ҙʹબ Ϳ͜ͱ͕Ͱ͖Δɽଞํɼݩʑͷ

Fourtuin-Kasteleyn

දݱ͸

V ˜

i j=

0

ͱ͍͏ಛผͳ৔߹ʹ૬౰͍ͯ͠Δɽ

2)

ݩʑͷ

Fourtuin-Kasteleyn

දݱ͸ڧ࣓ੑϙοπϞσ ϧͰͷΈ༗ޮͳදݱͳͷʹର͠ɼࣜ

(27)

ʹΑΔද ݱ͸೚ҙͷϙςϯγϟϧ

V

i j

(S

i,Sj

)

ʹର͠ద༻Մೳ

Ͱ͋Δɽ

͜ͷҰൠԽ͞Εͨ

Fourtuin-Kasteleyn

දݱΛ༻͍Δ͜

ͱͰɼ֬཰తΧοτΦϑ๏Ͱར༻Մೳͳɼز͔ͭͷ༗ӹ ͳදࣜΛಘΔ͜ͱ͕Ͱ͖Δɽࣜ

(27)

ͷ෼഑ؔ਺Λʢٯʣ Թ౓Ͱඍ෼͢Δ͜ͱʹΑΓɼ೤ฏۉΤωϧΪʔ΍ൺ೤ʹ ର͢Δද͕ࣜಘΒΕΔ͕⁽⁵³⁾ɼ͜ΕΒͷදࣜ͸

V ¯

i jͱͯ͠

ੜ͖࢒߲ͬͨʹର͢Δ࿨ͷܗͰॻ͖ද͞ΕΔɽͦͯ͠ɼ֬

཰తΧοτΦϑ๏ʹ͓͚Δϙςϯγϟϧ੾Γସ͑Λߦ͏

ͱɼେ෦෼ͷ߲͸

V ˜

i j=

0

ʹ੾ΓସΘΔ͜ͱʹΑΓΧο τΦϑ͞ΕΔͷͰɼ೤ฏۉΤωϧΪʔ΍ൺ೤ΛධՁ͢Δ

ͨΊͷܭࢉ࣌ؒΛେ෯ʹ࡟ݮ͢Δ͜ͱ͕Ͱ͖Δɽ·ͨจ ݙ

(53)

Ͱ͸ɼϨϓϦΧަ׵๏ͷަ׵֬཰ʹର͢Δදࣜ΋

ಋग़͍ͯ͠Δͷ͕ͩɼͦΕΛ༻͍Δ͜ͱʹΑΓɼަ׵֬

཰ͷܭࢉ࣌ؒΛେ෯ʹ࡟ݮ͢Δ͜ͱ͕Ͱ͖Δɽ

9. ֬཰తΧοτΦϑ๏ͷద༻ྫ

࠷ॳʹɼ֬཰తΧοτΦϑ๏ͷ

2

࣍ݩ࣓ؾ૒ۃࢠܥ΁

ͷద༻݁Ռ⁽⁴⁸⁾Λ঺հ͢Δɽ͜ͷܥͷϋϛϧτχΞϯ͸ࣜ

H

=

− J

i jSi

·

Sj

+

D

i<j

⎡⎢⎢⎢⎢⎢

⎣Si

·

Sj

r

_{i j}³

− 3 (S

i

·

ri j

)(S

j

·

ri j

) r

⁵_{i j}

⎤⎥⎥⎥⎥⎥

⎦,

(30)

Ͱ༩͑ΒΕΔɽ͜͜ͰSi͸େ͖͞

1

ͷݹయϋΠθϯϕ ϧΫεϐϯΛɼri j͸֨ࢠ఺

i

͔Β֨ࢠ఺

j

΁޲͔͏ϕΫ τϧΛɼ

r

i j=

|

ri j

|

͸ͦͷେ͖͞Λද͢ɽ֨ࢠ͸

2

࣍ݩਖ਼

ํ֨ࢠͰ͋Δɽӈลͷୈ

1

߲͸ڧ࣓ੑతަ׵૬ޓ࡞༻Λɼ

ୈ

2

߲͸࣓ؾ૒ۃࢠ૬ޓ࡞༻Λද͢ɽ·ͨɼ܎਺

C

,

D

͸

ਤ2 2࣍ݩ࣓ؾ૒ۃࢠܥʹ͓͚Δ1ϞϯςΧϧϩεςοϓ౰ͨ

Γͷܭࢉ࣌ؒͷαΠζґଘੑ⁽⁴⁸⁾ɽԣ࣠N͸εϐϯ਺ɽ࢛͕֯֬

཰తΧοτΦϑ๏ͷɼؙ͕௨ৗͷϞϯςΧϧϩ๏ͷ݁Ռɽ

ਤ3 ֬཰తΧοτΦϑ๏ʹ͓͍ͯV¯i jͱͯ͠ੜ͖࢒ͬͨɼ1^α Πτ౰ͨΓͷϙςϯγϟϧ਺ͷԹ౓ґଘੑ⁽⁴⁸⁾ɽ

ͦΕͧΕͷ૬ޓ࡞༻ͷڧ͞Λද͓ͯ͠Γɼ

D

/

J

=

0

.

1

^ͱ

͍ͯ͠Δɽ

ਤ

2

^͸

1

ϞϯςΧϧϩεςοϓ౰ͨΓͷܭࢉ࣌ؒͷα ΠζґଘੑΛ͍ࣔͯ͠Δɽ࢛͕֯֬཰తΧοτΦϑ๏ͷɼ

ؙ͕௨ৗͷϞϯςΧϧϩ๏ͷ݁ՌͰ͋Δɽ௨ৗͷϞϯς Χϧϩ๏Ͱ͸ܭࢉ͕࣌ؒεϐϯ਺

N

ͷ

2

৐ʹൺྫͯ͠

͍Δͷʹର͠ɼ֬཰తΧοτΦϑ๏Ͱ͸

N

ʹൺྫ͓ͯ͠

Γɼܭࢉ͕࣌ؒେ෯ʹ୹ॖ͞Ε͍ͯΔɽ࣍ʹɼ֬཰తΧο τΦϑ๏ʹ͓͍ͯ

V ¯

_{i j}ͱͯ͠ੜ͖࢒ͬͨɼ

1

αΠτ౰ͨ

Γͷϙςϯγϟϧ਺ͷԹ౓ґଘੑΛࣔͨ͠ͷ͕ਤ

3

Ͱ͋

Δɽ֬཰తΧοτΦϑ๏͸࣓ؾ૒ۃࢠ૬ޓ࡞༻ʹରͯ͠

ͷΈద༻͍ͯ͠ΔɽҰൠʹ֬཰తΧοτΦϑ๏Ͱ͸ɼ௿

Թ΄Ͳ

V ¯

i jͱͯ͠ੜ͖࢒Δϙςϯγϟϧͷ਺͕૿͑Δ܏

޲͕͋Δͷ͕ͩɼਤ

3

͔Β΋ͦͷ܏޲͕ಡΈऔΕΔɽ·

ͨɼαΠζ͕૿͑Δͱϙςϯγϟϧͷ਺΋Θ͔ͣʹ૿͑

͍ͯΔɽ͔͠͠ɼͲͷԹ౓ɾαΠζʹ͓͍ͯ΋ɼϙςϯ

ਤ4 ^{ബບ࣓ੑମʹ͓͚Δɼ}2ͭͷ࣓Խͷؔ਺ͱͯ͠ͷࣗ༝Τω ϧΪʔF(m_⊥,m)ͷଌఆ݁Ռ⁽⁵⁷⁾ɽεϐϯ࠶഑ྻసҠԹ౓T_SRT͸ 0.33Jɼ͜͜ͰJ͸ަ׵૬ޓ࡞༻ΤωϧΪʔɽࠨ͸T<TSRTɼӈ͸

T>T_SRTͷ݁ՌɽΧϥʔϓϩοτ͍ͯ͠Δྔ͸F_diff≡F−F_minɼ

͜͜ͰF_min≡min(m⊥,m)F(m_⊥,m)ɽ࣮ઢ͸F_diff ≤TΛຬͨ͢

௿ࣗ༝ΤωϧΪʔྖҬΛࣔ͢ɽ

γϟϧ਺͸͔͕ͨͩ

25

ఔ౓Ͱ͋ΔɽͦΕʹର͠ɼྫ͑͹

L

=

256

ʢ

L

͸

1

ล౰ͨΓͷ֨ࢠ఺ͷ਺ʣͷ৔߹ɼݩʑ ͷϙςϯγϟϧ਺͸

1

αΠτ౰ͨΓ

256

²

− 1

=

65

,

535

Ͱ͋Γɼ͜ͷ͜ͱ͸ϙςϯγϟϧ੾Γସ͑ʹΑΓେ෦෼

ͷ૬ޓ࡞༻͕ΧοτΦϑ͞Ε͍ͯΔ͜ͱΛ͍ࣔͯ͠Δɽ

࣍ʹɼڧ࣓ੑബບͷݚڀʹ֬཰తΧοτΦϑ๏Λద༻

ͨ݁͠Ռʹ͍ͭͯ঺հ͢Δ⁽⁵⁷⁾ɽਨ௚࣓ؾҟํੑΛ༗͢Δ ڧ࣓ੑബບͰ͸ɼਨ௚࣓ؾҟํੑͱ࣓ؾ૒ۃࢠ૬ޓ࡞༻

ͷڝ߹ʹΑΓɼԹ౓ͷ௿Լͱڞʹ࣓Խͷ޲͖͕໘಺ํ޲

͔Β໘௚ํ޲ʹมԽ͢Δɼεϐϯ࠶഑ྻసҠ͕͠͹͠͹

؍ଌ͞ΕΔ^(58,59)ɽਤ

4

͸ɼͦͷڧ࣓ੑബບʹ͓͚Δɼ

2

ͭͷ࣓Խͷؔ਺ͱͯ͠ͷࣗ༝ΤωϧΪʔ

F (m

_⊥,

m

)

^ͷଌ ఆ݁ՌͰ͋Δ⁽⁵⁷⁾ɽ͜͜Ͱ

m

_⊥^{͸࣓Խͷ໘௚੒෼ɼ}

m

^͸໘

಺੒෼Ͱ͋Δɽզʑ͸ɼࣗ༝ΤωϧΪʔʹؔ͢ΔϚϧν ΧϊχΧϧ๏⁽⁶⁰⁾ͱ֬཰తΧοτΦϑ๏Λ૊Έ߹ΘͤΔ͜

ͱͰɼ௕ڑ཭૬ޓ࡞༻ܥʹ͓͍ͯࣗ༝ΤωϧΪʔΛޮ཰

తʹଌఆ͢Δख๏Λ։ൃ͍ͯ͠Δͷ͕ͩ⁽⁶¹⁾ɼຊଌఆͰ͸

͜ͷख๏Λ༻͍͍ͯΔɽࣗ༝ΤωϧΪʔ͕௿͍࣓Խ΄Ͳ

࣮ݱ֬཰͕ߴ͍ͷ͕ͩɼਤΑΓɼ

T

>

T

_SRTʢ

T

_SRT͸ε ϐϯ࠶഑ྻసҠԹ౓ʣͰ͸

(m

_⊥,

m

) ≈ (0

,

0

.

6)

ɼ

T

<

T

_SRT Ͱ͸

(m

_⊥,

m

) ≈ (0

.

75

,

0

.

15)

ʹࣗ༝ΤωϧΪʔͷ࠷খ఺

͕͋ΓɼసҠԹ౓Λڥʹ࠷খ఺ͷҐஔ͕ෆ࿈ଓʹมԽ͠

͍ͯΔ༷ࢠ͕ݟͯऔΕΔɽ

10. ^·ͱΊ

Ҏ্ۦ͚଍Ͱ͕͋ͬͨɼ౷ܭ෺ཧʹ͓͚ΔϞϯςΧϧ ϩ๏ʹ͍ͭͯ֓આͨ͠ɽຊߘͰ঺հͨ͠ͷ͸ϞϯςΧϧ ϩ๏ͷҰ෦Ͱ͋Γɼଞʹ΋ྔࢠܥʹ͓͚ΔϞϯςΧϧϩ

๏ɼཻࢠܥʹ͓͚ΔϞϯςΧϧϩ๏ɼΠϕϯτυϦϒϯ ܕϞϯςΧϧϩ๏ͳͲɼࢴ໘ͷؔ܎্આ໌ΛׂѪͨ͠ख

๏͕਺ଟ͋͘ΔɽϞϯςΧϧϩ๏ʹ͍ͭͯΑΓৄ͘͠஌

Γ͍ͨํ͸ɼจݙ

(62

,

63)

ͳͲΛࢀߟʹͯ͠௖͖͍ͨɽ

ୈ

1

અͰ΋ड़΂ͨΑ͏ʹɼϞϯςΧϧϩ๏͸೚ҙͷ֬

཰෼෍Ͱͷঢ়ଶੜ੒ΛՄೳͱ͢ΔۃΊͯ൚༻తͳΞϧΰ ϦζϜͰ͋Γɼͦͷద༻ൣғ͸෺ཧɾ޻ֶɾ਺ֶɾੜ෺

ֶɾۚ༥޻ֶͳͲɼඇৗʹଟذʹ౉Δɽ΋͠ຊߘʹΑΓ ϞϯςΧϧϩ๏ʹڵຯΛ࣋ͬͯ௖͚ͨͳΒɼචऀͱͯ͠

޾ਙͷࢸΓͰ͋Δɽ

ँࣙ

ຊߘͷݚڀ੒ՌͷҰ෦͸

JSPS

Պݚඅएखݚڀ

(B)

21740279

ͷॿ੒Λड͚ͨ΋ͷͰ͢ɽ

ࢀߟจݙ

(1) ৐ࢉͷܭࢉ଎౓͸ຖඵ385ճɼআࢉͷͦΕ͸40ճͰ͋Γɼ Ճݮࢉͱൺ΂͔ͯͳΓ஗͘ͳΔɽͦΕͰ΋౰࣌ͷిؾػցࣜ

ܭࢉػͱൺ΂ͯ໿ઍഒͷܭࢉ଎౓ΛތΓɼใಓͰ͸ʮGiant Brainʯͱশ͞Εͨɽ

(2) খ਺఺ͷҐஔ͕ݻఆ͞Εͳ͍ුಈখ਺఺਺Λ༻͍ͯߦ͏࢛ଇ ԋࢉɽ

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(4) 0͔Β1ͷؒʹ෼෍͢Δ2ͭͷҰ༷ཚ਺(x,y)Λੜ੒͠ɼ

r=

x²+y²≤1Λຬͨ͢఺ͷׂ߹PpiΛଌఆ͢Δͱɼ4Ppi

͕π^{ͷۙࣅ஋ͱͳΔɽ}

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΂͖Ͱͳ͍ɽͦ΋ͦ΋෼഑ؔ਺ͷ࿨ΛܭࢉͰ͖ΔͷͰ͋Ε

͹ɼͦΕͱಉఔ౓ͷܭࢉ࣌ؒͰ೚ҙͷ෺ཧྔͷ೤ฏۉ஋Λධ ՁͰ͖Δͷ͔ͩΒɼγϛϡϨʔγϣϯΛߦΘͣ௚઀೤ฏۉ஋

ͷܭࢉΛ͢Ε͹ྑ͍ɽ

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ؒͰଌΒΕͨ؇࿨࣌ؒͰͳͯ͘͸ͳΒͳ͍ɽݴ͍͔͑Δͱɼ

͍͘Βεςοϓ਺ͰΈͨ؇࿨͕࣌ؒ୹ͯ͘΋ɼͦͷ෼1ε ςοϓΛ࣮ߦ͢ΔͨΊʹඞཁͳCPU͕࣌ؒ௕͚Ε͹ɼҙຯ

͕ͳ͍ɽ

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