PACS numbers: 02.70.Ns, 34.30.+h

(1)

Energy Transfer to a Many-body System via a Periodic External Force by Using Molecular Dynamics Simulation

Jaehwa Kim ^∗

Department of Physics, Gangneung-Wonju National University, Gangneung 210-702, Korea (Received 23 July 2013 : revised 5 August 2013 : accepted 10 October 2013)

We investigated the energy transfer to a many-particle system by periodic external force acting vertically. This investigation addressed a two-dimensional system by using a molecular dynamics simulation. As expected, we found that the energy of a system with a periodic boundary conditions was not disturbed by a periodic external force. However, the energy of a system with hard-wall fixed boundary conditions converged to different values according to the angular frequencies of the external forces. Especially, an external force with a characteristic angular frequency pushes the system to have a higher energy, and this characteristic angular frequency depends on the density of the system. These results are inferred to have come from the hard-wall boundary itself and from a special interaction between the periodic external force and the hard-wall boundary.

PACS numbers: 02.70.Ns, 34.30.+h

Keywords: Molecular dynamics simulation, Energy transfer, Periodic external force, Boundary conditions

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»< é s ^∗

y

© 2 £ x" é ¶ Å Ò@ / < Æ § Ó ü t o < Æõ , y © 2 £ x 210-702

(2013¸ 7 Z 4 23{ 9 ~ Ã Î6 £ §, 2013¸ 8 Z 4 5{ 9 Ã º& ñ : r ~ Ã Î6 £ §, 2013¸ 10 Z 4 10{ 9 > F S X & ñ )

Y

U -× ¼ > r Û ¼ íJ $ ¶ û _ © ñ 6 x` ¦ ° ú H 2 " é ¶ { 9 > \ @ /K " f Ã ºf ~ ½ Ó ¾ ÓÜ ¼ Ð Å Òl & h ü @§ 4 s 6

x ½ + É M : { 9 > _ \ -t # Qb G> ì ø Í6 £ x H \ ¦ ì r 1 l x% i < Æ_ ~ ½ ÓZ O ` ¦ : x # · ú Ð ¤ . Å Òl & h

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¼ Ð \ -t \ ¦ ² ú H 1 l x Ã º > r F < Ê` ¦ Ð% i . Õ ªo ¦ s Qô Ç õ H é ß é ß ô Ç # 4 _ ¦& ñ ) a â >

¸| s Å Ò H ´ òõ ü < 8Ô ¦ # Q â > õ ü @Â Ò\ " f K ï r Å Òl & h j Ë µõ _ © ñ 6 x \ _ ô Ç ´ òõ ½ + Ë K

õ s 9 s ´ òõ H > _ x 9 ¸\ ² ú f ` ¦ Ð ¤ .

PACS numbers: 02.70.Ns, 34.30.+h

Keywords: ì r 1 l x% i < Æ, \ -t ² ú , Å Òl & h ü @§ 4 , â > ¸|

∗

E-mail: [email protected]

1128

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License

(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any

medium, provided the original work is properly cited.

(2)

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@

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s

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" f { 9 -Û ¼á Ôa A> \ @ /K " f H s ¿ º t © ì ø Í ) a ´ ò õ

& h { © ô Ç \ " f ] X Ø æ| ¨ c M : ¸ ú · ú 9 / B N & ³ © s { 9

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>

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ü @§ 4 s 6 x H ~ ½ Ó ¾ ÓÜ ¼ Ð é ß é ß ô Ç # 4 Ü ¼ Ð ¦& ñ ) a â

>

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H & ³ © s Å Òl & h ü @§ 4 õ a ?§ 4 # \ -t \ ¦ f ¨ Ã º

H כ s 0 p x ½ + É כ ? s \ @ /ô Ç ² ú ` ¦ % 3 l 0 A # ì r

1 l x% i < Æ& h ~ ½ ÓZ O [19]` ¦ 6 x # Ã ºf ~ ½ Ó ¾ ÓÜ ¼ Ð H ¦

&

ñ ) a â > ¸| ` ¦ ° ú ¦ Ã º¨ î ~ ½ Ó ¾ ÓÜ ¼ Ð H Å Òl & h â > ¸

II. ¤M X ì Äß Ã Å ß O Ë4 ºü g Å

Ä

º _ { 9 Å Òl & h ü @§ 4 \ _ # # Qb G> î

r1 l x H \ ¦ · ú Ð . s { 9 _ î r1 l x ~ ½ Ó& ñ d É r m dv

dt = F 0 cos(ωt) (1) s

9 v(t) = F 0

mω sinωt +

v(t 0 ) − F 0

mω sinωt 0

(2) Ü

¼ Ð ³ ð & ³ ) a . " f 5 Å q ¸ H y 1 l x Ã º ω 9 þ t Ã º2 ¤

É r ; ¤` ¦ t ¦ 1 l x < Ê` ¦ · ú Ã º e . (3)d _ { 9 -\ - t

& ñ o \ ¦ s 6 x W = m

2 v ² | ^r _r

0

= Z r

r

₀

F · d~ ~ r = Z t

t

₀

F vdt (3)

{ 9

ü @§ 4 \ _ K t 0 \ " f t r ç ß 1 l x î ß % 3 É r î r1 l x \ -t

H

W = F ₀ ²

2mω ² sin ² ωt + F ₀

ω

v(t 0 ) − F ₀ mω sinωt 0

sinωt

− v(t 0 ) F 0

ω sinωt 0

+ F ₀ ²

2mω ² sin ² ωt 0 (4) s

. (4)d \ " f { 9 _ î r1 l x \ -t H sinusoidal > Å Ò l

& h Ü ¼ Ð 9 ô Ç Å Òl (t ⁰ \ " f t ⁰ + ^2π _ω ) 1 l x î ß \ ü @§ 4

\

_ K ² ú ) a \ -t H % ò e ` ¦ Ð# ï r . 7 £ ¤ _ { 9

ü @§ 4 \ _ K ² ú ~ Ã Î H \ -t H ô ÇÅ Òl s © _ Ø æ ì

r ô Ç r ç ß s t ¨ î ç H& h Ü ¼ Ð % ò s ) a .

Ä

ºo _ ' a d É r ^ > \ @ /ô Ç \ -t ² ú s Ù ¼ Ð s

Qô Ç © S ! ` ¦ 2 " é ¶ ¨ î _ { 9 > \ @ / # & h 6 x # Ð

. s \ ¦ 0 A # ì r 1 l x% i < Æ& h ~ ½ ÓZ O ` ¦ 6 x ¦ ô Ç

. ü @§ 4 s 6 x H â Ä º Ø æ[ t s { 9 # Q H s $ à Ô\ ¦ Æ Ò

&

h # r Ó ý t Y Us H ~ ½ ÓZ O É r > í ß © _ # Q 9¹ ¡ § s ß ¼ .

" f r ç ß \ { 9 _ C & h ` ¦ Æ Ò& h K H ~ ½ ÓZ O ` ¦

(3)

Fig. 1. (Color online) Total Energy variation with time at various angular frequencies from ω = 0.0 to ω = 9.0.

6 x ¦ s M : { 9 ç ß _ © ñ 6 x É r (5)d _ Y U × ¼- > r Û

¼ íJ $ ¶ û [20,21]` ¦ 6 x % i .

u(r ij ) = 4

( σ

r _ij ) ¹² − ( σ r _ij ) ⁶

. (5)

#

l " f r ^ij H ¿ º { 9 s _ o s 9 É r íJ $ ¶ û _ U · s

\ @ /6 £ x ÷ & ¦ σ H ì r _ f â s . ì r 1 l x§ 4 < Æ_ ~ ½ Ó Z O

É r ô Ç { 9 \ @ / # Qt ¸ H { 9 Å Ò H j Ë µ` ¦ > í

ß ¦ s כ ` ¦ ¸ H { 9 \ @ / # Û ¼ ± p # > í ß Ù ¼

Ð s õ & ñ \ " f ´ ú § É r ( É Óh A " é ¶ s èכ ¹ ) a . Ä ºo _ 3 l q

&

h s ü @§ 4 \ _ ô Ç { 9 > _ \ -t ² ú s t > ^ _

\ P

% i < Æ& h $ í | 9 \ ' a d s e H כ É r m Ù ¼ Ð { 9 Ã º H 12 × 12, 7 £ ¤ 144 > h & ñ ¸ Ð % i . # l " f 6 x H é ß 0

A H ¸¿ º MD (Molecular Dynamics)é ß 0 A Ð [19] | 9 | ¾ Ó,

\

-t , o H y y m, , σ s é ß 0 As 9 s M : r ç ß _ é

ß 0 A H τ = q mσ

²

s . ¢ ¸ô Ç > _ 8 ú x \ -t H { 9 { ©

° ú

H 8 ú x \ -t \ ¦ l ï r Ü ¼ Ð ? /% 3 . î r1 l x ~ ½ Ó& ñ d ` ¦ :

x K 5 Å q ¸ü < 0 Au \ ¦ ½ ¨ l 0 Aô Ç Ã ºu & h ì r É r > h½ ¨o 8 A l

(leap-frog) · ú ¦o 7 £ § [22] ` ¦ 6 x % i ¦ × æ \ é ß é ß ô

Ç # 4 Ü ¼ Ð ¦& ñ ) a â > ¸| õ q §\ ¦ 0 AK $ 2 " é ¶ _ x» ¡ ¤ ~ ½ Ó ¾ Óõ y» ¡ ¤ ~ ½ Ó ¾ Ó ¸¿ º Å Òl & h â > ¸| ` ¦ ° ú H â Ä

º\ ¦ · ú Ð ¸2 ¤ % i . íl 5 Å q ¸ H ¸ H { 9 ° ú É r 5

Å

q ¸\ ¦ ° ú t ë ß ~ ½ Ó ¾ Ó É r ~ ½ ÓÜ ¼ Ð Y J ¦À Ò ì r í ¸2 ¤

%

i . x 9 ¸ 0.5 s ¦, { 9 _ íl 5 Å q ¸ 20.0 â Ä º

\

@ /K " f, ü @§ 4 s \ O H â Ä º H ω = 0.0 Ð Z ~ l Ð ô Ç .

ü

@§ 4 _ 1 l x Ã º\ ¦ ω = 0.0 \ " f ω = 9.0 t 10t _

"

f Ð É r ω \ @ / # r Ó ý t Y Us ô Ç õ Fig. 1\

e .

Figure 1` ¦ Ð ô Ç × ¦ Ð Ðs t ë ß 10> h_ " f Ð É r X

<s ' × ¦ s ô Ç × ¦ Ð ¸# " f ) a כ Ü ¼ Ð ü @§ 4 _ y 1 l x Ã º

# Q " ° ú כs ÷ &~ t ¸¿ º { 9 { © t H 8 ú x \ -t H í l

° ú כ Õ ª@ / Ð Ð > rH d` ¦ Ð# ï r . 7 £ ¤ ü @§ 4 \ _ ô Ç 6 f& Ç l

Fig. 2. Particle motion with fixed hard wall boundary in y direction.

¨ î ç H& h Ü ¼ Ð { 9 > _ \ -t \ Á º % ò ¾ Ó` ¦ Å Òt

· ú

§6 £ §` ¦ · ú Ã º e . s H (4)d \ " f ô Ç> h_ { 9 \ @ / ô

Ç Å Òl & h ü @§ 4 s Å Òl _ & ñ Ã ºC _ r ç ß s t ± ú M :

ª _ ~ ½ Ó ¾ Óõ 6 £ § _ ~ ½ Ó ¾ ÓÜ ¼ Ð ° ú î r1 l x \ -t \ ¦ 7 £ x

r ( y èr \ ¦ ì ø Í4 ¤ > H X < s Qô Ç 6 x s { 9

ç ß _ © ñ 6 x \ _ K " f % ò ¾ Ó ~ Ã Ît · ú §6 £ §` ¦ S X

#

ï r .

III. m m X ¢ ä Ä® z º w X N Ëc Ü R ß O Ë4 ºü g Å

Ä

ºo [ þ t s ¦ H 7 H _ ü < © S ! É r ¸F K Ø Ôt ë ß Å

Òl & h ü @§ 4 \ _ K / B N & ³ © ` ¦ { 9 Ü ¼v H ¢ ¸ É r \ V

s 9 þ t Ðà Ô : r 5 Å q l [23]\ ¦ Ò q ty K Ð . s 9 þ t Ðà Ô

: r 5 Å q l H l © B5 Å q \ { 9 q Å Òl & h ü @Â Ò

l © \ _ K t 5 Å q& h Ü ¼ Ð \ -t \ ¦ % 3 H © u Ð" f s { 9

5 Å q l _ l : r É r (6) d õ ° ú É r & h ] X ô Ç 1 l x Ã º_ Å Ò l

& h ü @Â Ò l © s { 9 \ K f Ü ¼ Ð" f H / B N

& ³ © ` ¦ s 6 x ô Ç כ s .

ω = qB

m . (6)

s

M : s 9 þ t Ðà Ô : r 1 l x Ã º H { 9 _ 5 Å q ¸ü < H Á º ' a

Ù ¼ Ð © @ / : r& h ´ òõ H % ò % i t H l

© B\ @ /6 £ x ÷ & H & h ] X ô Ç 1 l x Ã º_ ü @Â Ò l © ` ¦

t 5 Å q& h Ü ¼ Ð \ -t \ ¦ % 3 ` ¦ Ã º e .

s

ü < q 5 p w > { 9 > \ Å Òl & h ü @§ 4 s \ -t \ ¦ × ¦ Ã

º e H 0 p x$ í Ü ¼ Ð Fig. 2ü < ° ú s x» ¡ ¤ ~ ½ Ó ¾ ÓÜ ¼ Ð H Å Òl

&

h â > ¸| ` ¦ ° ú t ë ß y» ¡ ¤ ~ ½ Ó ¾ ÓÜ ¼ Ð H é ß é ß ô Ç # 4 Ü ¼ Ð

¦& ñ ) a â > ¸| ` ¦ ° ú H { 9 > \ ¦ Ò q ty K Ð .

7 £ ¤ Fig. 2 ü < ° ú É r © S ! \ " f { 9 > y» ¡ ¤ ~ ½ Ó ¾ ÓÜ ¼ Ð 6

x H Å Òl & h ü @§ 4 \ _ K " f # Qb G> % ò ¾ Ó ~ Ã Î H t · ú

Ð ¸2 ¤ ô Ç . é ß í H > Ð ¦& ñ ) a â > ¸| M :ë H \ ü

@§ 4 _ 1 l x Ã º

ω = 2π v _y

L (7)

(4)

Fig. 3. (Color online) Total Energy variation with time at various angular frequencies from ω = 0.0 to ω = 9.0 for initial velocity v = 20.0 and density d = 0.1.

Fig. 4. (Color online) Total energy variation with time at various velocities without external force at density d

= 0.1.

_

¸| ` ¦ t ü @§ 4 _ 1 l x Ã ºü < ¦& ñ ) a â > ¸| s

´ ú > ÷ &# Q / B N & ³ © s 0 p x > Ð . # l " f L É r é ß é

ß > ¦& ñ ) a â > # 4 s _ o s . Õ ª Q · ú ¡\ " f

\

V Ð H s 9 þ t Ðà Ô : r 5 Å q l ü < H Ø Ô> / B N 1 l x Ã º

5 Å

q ¸\ _ > r Ù ¼ Ð ü @§ 4 \ _ K 5 Å q ¸ @ /6 £ x ÷ &

H y 1 l x Ã º ² ú t l M :ë H \ { 9 _ î r1 l x+ þ AI H 4 ¤

¸ ú

ô Ç ª © Ü ¼ Ð > ) a . s Â Òì r _ ½ ¨^ & h ? /6 x

É

r 6 £ § \ À Ò ¸2 ¤ ¦ Ä ºo H ^ > s Qô Ç 4 ¤ ¸ ú ô

Ç ª © \ @ /K # Qb G> % ò ¾ Ó ~ Ã Î H t \ -t ² ú _ ' a& h

\

" fë ß 8 ú ©& h ` ¦ ´ ú Æ Òl Ð ô Ç .

Ä

º x 9 ¸ d = 0.1 â Ä º\ Ã ºf » ¡ ¤ (y» ¡ ¤) ~ ½ Ó ¾ ÓÜ ¼ Ð é

ß é ß ô Ç # 4 Ü ¼ Ð ¦& ñ ) a â > \ ¦ t H { 9 > \ @ /K r Ó

ý

t Y Us ` ¦ % i . Å Òl & h â > ¸| ` ¦ ° ú H â Ä ºü <

1 l x{ 9 ô Ç ¸| _ q §\ ¦ 0 AK â > ¸| s ü @_ © S ! É r ¸

¿

º 1 l x{ 9 > % i . Fig. 3 É r 144 > h_ { 9 " f Ð Y U

×

¼- > r Û ¼ íJ $ ¶ û Ð Å Ò# Qt H © ñ 6 x` ¦ ¦ e H © S !

\

" f { 9 _ íl 5 Å q ¸ v = 20.0 â Ä º\ r Ó ý t Y Us ô

Ç õ _ Õ ªa Ë >Ü ¼ Ð ü @§ 4 _ 6 x s \ O H ω = 0.0 \ " f

Fig. 5. (Color online) Total energy variation with time at various velocities for ω = 1.0 and density d = 0.1.

Fig. 6. (Color online) Total energy variation with time at various velocities for ω = 2.0 and density d = 0.1.

r

# ω = 9.0\ s Ø Ôl t 10t _ " f Ð É r 1

l

x Ã º_ X <s ' \ ¦ í < Ê ¦ e .

Õ

ªa Ë >` ¦ Ð Å Òl & h â > ¸| { 9 M : Fig. 1\ " f Ð

כ

% ! 3 ü @§ 4 _ 1 l x Ã º\ © ' a\ O s 8 ú x \ -t { 9 & ñ ô Ç

° ú

כ` ¦ t H כ õ H Ø Ô> 8 ú x \ -t _ o e Ü ¼ 9

1 l x Ã º\ o & ñ ¸ ß ¼> ² ú H כ ` ¦ · ú Ã º e

. Fig. 3\ " f \ -t \ ¦ ¸ ú f ¨ Ã ºô Ç í H " f@ / Ð 1 l x Ã º

\

¦ & h Ü ¼ (ω = 3.0) > (ω = 2.0) > (ω = 1.0) > (ω = 4.0) > (ω = 5.0) > · · · > (ω = 9.0) > (ω = 0.0) Ð r

ç ß s t z \ ω = 3.0 { 9 M : 8 Z } É r \ -t ² ú s

{ 9 # Qz ` ¦ · ú Ã º e . 8¹ ¡ ¤ s ω = 1.0, ω = 2.0 { 9 M :

H 8 ú x \ -t í o÷ &# Q # QÖ ¼ & ñ ¸ î ß & ñ o ÷ &# Qe t ë ß ω = 3.0 _ K { © ÷ & H 8 ú x \ -t o H { 9 & ñ ô Ç ° ú כ` ¦ t t

3 l w ¦ > 5 Å q 7 £ x ¦ e 6 £ §` ¦ Ð# ï r . s H f s

1 l x Ã º\ @ /K " f H r ç ß & h Ü ¼ Ð í o÷ & H \ -t \ ¸² ú

t 3 l wÙ þ ¡l M :ë H Ü ¼ Ð Æ Ò8 £ ¤ ) a . Õ ªa Ë >\ " f Ð1 p w s 1 l x Ã

º_ o\ É r \ -t f ¨ Ã º\ ¦ Ð ü @§ 4 _ 1 l x Ã º

7 £

x < Ê\ t 5 Å q& h Ü ¼ Ð > _ \ -t f ¨ Ã º Z } t

(5)

Fig. 7. (Color online) Total energy variation with time at various velocities for ω = 3.0 and density d = 0.1.

Fig. 8. (Color online) Total energy variation with time at various velocities for ω = 4.0 and density d = 0.1.

H כ s m þ j& h _ 1 l x Ã º > r F ¦ s \ ¦ t

r \ -t f ¨ Ã º y èK y ` ¦ ^ ¦ Ã º e .

s

\ H s X O > ª ô Ç o Å Ò H / B N: x& h ` ¦ É r y

¸\ " f Ðl 0 AK { 9 & ñ ô Ç 1 l x Ã º\ " f íl 5 Å q ¸\ 8

ú

x \ -t # Qb G> H \ ¦ ¶ ú ( R Ð ¸2 ¤ ô Ç . Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8[ þ t É r í H " f@ / Ð y y ü @§ 4 _

1 l x Ã º ω = 0.0, ω = 1.0, ω = 2.0, ω = 3.0, ω = 4.0 { 9 M

: 8 ú x \ -t o\ ¦ " f Ð É r íl 5 Å q ¸\ @ /K " f % 3 É r

õ s .

$ Á º ü @§ 4 s \ O ` ¦ M :_ Fig. 4\ ¦ Ð v = 0.0, v = 10.0 \ " f H @ /^ Ð _ íl \ -t \ ¦ Ä »t t ë

ß íl 5 Å q ¸ v = 20.0 â Ä º H \ -t y è #

H כ ` ¦ ^ ¦ Ã º e . ü @§ 4 s \ O H â Ä º\ 8 ú x \ -t y

èô Ç H כ É r í H y é ß é ß ô Ç # 4 Ü ¼ Ð ¦& ñ ) a â > ¸|

\

_ ô Ç ´ òõ e ` ¦ · ú Ã º e ¦ s ´ òõ H é ß é ß ô Ç # 4 \ _ ô Ç

â

> ì ø Í \ _ K { 9 î ß A á ¤ % ò % i Ü ¼ Ð [ þ t # QÙ ¼ Ð

É r { 9 Á ºo [ þ t \ _ ô Ç Y U -× ¼- > r Û ¼ íJ $ ¶ û _ = å J # Q [ þ t s

H ´ òõ M :ë H Ü ¼ Ð K $ 3 ½ + É Ã º e . Fig. 5_ ω = 1.0\

"

f H íl 5 Å q ¸\ © ' a\ O s 8 ú x \ -t 350 H % Ð Ã º§ 4

Fig. 9. (Color online) Total Energy variation with time at various angular frequencies from ω = 0.0 to ω = 9.0 for initial velocity v = 20.0 and density d = 0.5.

Fig. 10. (Color online) Total energy variation with time at various velocities without external force at density d = 0.5.

¦ ω = 2.0 Fig. 6\ " f H 7 á § 8 Z } É r \ -t 500 H

%

Ð Ã º§ 4 H d` ¦ ^ ¦ Ã º e . s Qô Ç Z } É r \ -t f ¨ Ã º H

1 l x Ã º ω = 3.0_ Fig. 7\ " f Ð H כ % ! 3 \ -t { 9

&

8 Z } É r 1 l x Ã º ω = 4.0\ " f H Fig. 8 õ ° ú s \ -t f

¨ Ã º y è ¦ Õ ª s Ê ê_ 1 l x Ã º\ " f ¸ \ -t Ä » t

÷ & × ¦ # Q× ¼ H + þ AI Ð ' ) a . : r& h Ü ¼ Ð Ä ºo

H é ß é ß ô Ç # 4 _ ¦& ñ ) a â > \ ¦ t H â Ä º\ H Å Òl & h

ü @§ 4 \ _ ô Ç \ -t ² ú s 0 p x 9 ´ òõ & h Ü ¼ Ð

² ú

s 0 p x ô Ç 1 l x Ã º > r F < Ê` ¦ · ú Ã º e .

s

\ H { 9 _ x 9 ¸\ ¦ 0.5 Ð Z } # ° ú É r r Ó ý t Y Us

`

¦ ' % i . s â Ä º\ ¸ ¦& ñ ) a íl 5 Å q ¸\ " f ü @§ 4 _

ª ô Ç 1 l x Ã º\ @ / # \ -t o\ ¦ r Ó ý t Y Us

# Ð ¤ . íl 5 Å q ¸ v = 20.0 â Ä º_ õ Fig. 9 \ " f \ -t \ ¦ ¸ ú f ¨ Ã ºô Ç í H " f@ / Ð 1 l x Ã º\ ¦ & h Ü

¼ (ω = 4.0) > (ω = 3.0) > (ω = 2.0) > (ω = 1.0) >

· · · > (ω = 0.0) Ð ü @§ 4 _ 1 l x Ã º 7 £ x < Ê\ \ - t

f ¨ Ã º 7 £ x þ j@ / Ð f ¨ Ã º÷ & H 1 l x Ã º\ ¦ t

(6)

Fig. 11. (Color online) Total energy variation with time at various velocities for ω = 1.0 and density d = 0.5.

Fig. 12. (Color online) Total energy variation with time at various velocities for ω = 3.0 and density d = 0.5.

r \ -t f ¨ Ã º y è H J É r x 9 ¸ d = 0.1 { 9 M

:ü < ° ú . ë ß ω = 3.0 \ " f þ j@ /\ ¦ Ðs ~ d = 0.1M : ü

< H Ø Ô> ´ òõ & h Ü ¼ Ð \ -t ² ú ÷ & H 1 l x Ã º H ω = 4.0 Ð Z } & Ü ¼ 9 ü @§ 4 _ 1 l x Ã º\ í o÷ & H

\

-t H ì ø Í& h Ü ¼ Ð ± ú & 6 £ §` ¦ Ð# ï r .

ð ø Ít Ð ¦& ñ ) a 1 l x Ã º\ " f " f Ð É r 5 Å q ¸\ @ / ô

Ç \ -t o\ ¦ Ð ü @§ 4 s \ O H â Ä º Fig. 10 X <

Fig. 4 _ x 9 ¸ 0.1{ 9 M : Ð 8 / å L y 8 ú x \ -t y

è H כ ` ¦ ^ ¦ Ã º e . s כ É r í H y é ß é ß ô Ç # 4 _ ¦& ñ

)

a â > _ ´ òõ s ¦ s ´ òõ H { 9 _ x 9 ¸ 9 þ t M : 8 ´ ú § s

è ß . Fig. 11, Fig. 12, Fig. 13, Fig. 14 H y y ü @

§

4 _ 1 l x Ã º ω = 1.0, ω = 3.0, ω = 4.0, ω = 5.0 { 9 M : _

õ Ð ω = 2.0\ " f H ω = 1.0{ 9 M :ü < _ q 5 p w # Õ

ªa Ë > É r Ò q t| Ä Ì % i . s M : Ã º§ 4 H \ -t H ω = 1.0 \

"

f H 48 H % , ω = 2.0\ " f H 51 H % , ω = 3.0\ " f H 90 H

%

Ð > 5 Å q 7 £ x " f ω = 4.0_ õ Fig. 13\ " f x ß

¼\ ¦ s À Ò# Q \ -t 100` ¦ Å # Q ¦ 8 Z } É r 1 l x Ã º ω = 5.0 \ " f H Fig. 14 _ + þ AI Ð, r Ã º§ 4 H \ -t

50s Ð b # Q& 6 £ §` ¦ Ð# ï r . s H x 9 ¸ d = 0.1

Fig. 13. (Color online) Total energy variation with time at various velocities for ω = 4.0 and density d = 0.5.

Fig. 14. (Color online) Total energy variation with time at various velocities for ω = 5.0 and density d = 0.5.

â

Ä ºü < ° ú s 1 l x Ã º 7 £ x " f { 9 > f ¨ Ã º H \

-t & t \ -t ² ú ´ òõ þ j@ / Ð ÷ & H 1 l x Ã

º\ ¸² ú ¦ r \ -t ² ú ´ òõ y èK H כ õ

° ú É r õ s . ë ß \ -t ² ú ´ òõ þ j@ / Ð ÷ & H

1 l x Ã º H x 9 ¸ d = 0.1 \ " f H ω max = 3.0 s % 3 Ü ¼ d = 0.5 \ " f H ω max = 4.0 Ð x 9 ¸ & t 8 H 1 l x Ã

º\ " f \ -t ² ú ´ òõ p u` ¦ · ú Ã º e .

IV. + s Ç Â ] Ø

Å

Òl & h ü @§ 4 s ^ > \ 6 x ½ + É M :_ ì ø Í6 £ x` ¦ Å Ò Ð ü

@§ 4 \ _ ô Ç \ -t ² ú _ ' a& h \ " f ¶ ú ( R Ð ¤ . s \ ¦ 0

AK íl 5 Å q ¸ü < ü @§ 4 _ 1 l x Ã º\ ¦ ª > o r &

" f r ç ß \ É r 8 ú x \ -t _ o\ ¦ ì r 1 l x% i < Æ_

~

½ ÓZ O ` ¦ : x # ¶ ú ( R Ð ¤ . : £ ¤ y é ß é ß ô Ç # 4 Ü ¼ Ð ¦& ñ ) a

â

> \ _ ô Ç ´ òõ \ ¦ × æd Ü ¼ Ð Å Òl & h ü @§ 4 \ _ ô Ç { 9

> _ \ -t ² ú 0 p x$ í \ ¦ À Ò# Q Ð ¤ . õ \

@

/ô Ç q §\ ¦ 0 A # Å Òl & h â > ¸| ` ¦ $ À Ò% 3

(7)

£

§ \ t H 8 ú x \ -t { 9 & ñ > Ä »t H d` ¦ ^ ¦ Ã º e % 3

. Õ ª Q é ß é ß ô Ç # 4 Ü ¼ Ð ¦& ñ ) a â > ` ¦ t H â Ä º

\

º§ 4 K 9 s Ã º§ 4 ° ú כ É r ü @§ 4 _ 1 l x Ã º\ " f Ø Ô

>

z ` ¦ Ð ¤ . é ß é ß > ¦& ñ ) a â > \ _ ô Ç ´ ò õ

H \ -t ² ú s H 8 £ ¤ \ " f F N& ñ & h ´ òõ ü < Â Ò

&

ñ & h ´ òõ \ ¦ 1 l x r \ t ¦ e % 3 . Â Ò& ñ & h כ É r â

>

\ _ ô Ç ì ø Í ´ òõ Ð { 9 Ø æ[ t Ê ê { 9 ´ ú § É r % ò

%

> _ íJ $ ¶ û \ -t × ¦ # Q× ¼ H ´ òõ Ð 6 x ¦ s

´

òõ H x 9 ¸ 9 þ t M : 8 ß ¼> z ` ¦ r Ó ý t Y Us õ

\

" f · ú Ã º e % 3 . F N& ñ & h ´ òõ H ¦& ñ ) a # 4 \ _ ô Ç Ø

§

\

_ ô Ç כ Ü ¼ Ð Ò q ty ) a . ¢ ¸ô Ç s ´ òõ H 1 l x Ã º\

Ø Ô> 9 s ´ òõ þ j@ / Ð ÷ & H 1 l x Ã º > r F

¦ s : £ ¤& ñ 1 l x Ã º ¸ x 9 ¸\ Ø Ô> z ` ¦

· ú

Ã º e % 3 .

P

c p 8 ý ò k >

s

" é ¶ \ _ # Ã º' ÷ &% 3 6 £ §.

REFERENCES

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E 73, 066223 (2006).

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[10] W. M. Visscher, Phys. Rev. A 36, 5031 (1987).

[11] M. Razavy, Phys. Rev. A 44, 2384 (1991).

[12] S. R. Jain, Phys. Rev. Lett. 70, 3553 (1993).

[13] A. J. Makowski and S. T. Dembinski, Phys. Lett. A 154, 217 (1991).

[14] L. D. Pustylnikov, Trans. Mosc. Math. Soc. 2, 1 (1978).

[15] D. G. Ladeira and J. K. da Silva, J. Phys. A 40, 11467 (2007).

[16] E. D. Leonel and A. L. P. Livorati, Physica A 387, 1155 (2008).

[17] S. Celaschi and R. L. Zimmerman, Phys. Lett. A 120, 447 (1987).

[18] G. R. Fowles and G. L. Cassiday, Analytical Me- chanics (Thomson Brooks/Cole, New York, 2005), p. 113.

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Wakeham, Intermolecular Forces (Clarendon Press, Oxford, 1981).

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1997), p. 711.

PACS numbers: 02.70.Ns, 34.30.+h

Energy Transfer to a Many-body System via a Periodic External Force by Using Molecular Dynamics Simulation

Jaehwa Kim ∗

Department of Physics, Gangneung-Wonju National University, Gangneung 210-702, Korea (Received 23 July 2013 : revised 5 August 2013 : accepted 10 October 2013)

PACS numbers: 02.70.Ns, 34.30.+h

Keywords: Molecular dynamics simulation, Energy transfer, Periodic external force, Boundary conditions



¤M X ì Äß Ã Å 8 ûI í Ä; c 8 ý X ¢ = k4 z º8 ý ; c .U   ¹ Å«  o: Ä Z Ø  Ò ÞW Ä] K ¡ ù p §   ÞA 0" e

 »< é s ∗

y

© 2 £ x" é ¶ Å Ò@ / < Æ § Ó ü t o  < Æõ , y © 2 £ x 210-702

(2013¸  7 Z 4 23{ 9  ~ Ã Î6 £ §, 2013¸  8 Z 4 5{ 9  Ã º& ñ  : r ~ Ã Î6 £ §, 2013¸  10 Z 4 10{ 9  > F  S X & ñ )

Y

U -× ¼ > r Û ¼  íJ $ ¶ û _   ©  ñ  6 x`  ¦ ° ú   H 2 " é ¶ { 9  > \  @ /K " f Ã ºf  ~ ½ Ó ¾ ÓÜ ¼ Ð Å Òl & h   ü @§ 4 s     6

x ½ + É M : { 9  > _  \  -t  # Qb  G>  ì ø Í6 £ x   H \  ¦ ì  r  1 l x% i  < Æ_  ~ ½ ÓZ O `  ¦ :  x #  · ú   Ð  ¤ . Å Òl & h



  â >  ¸| `  ¦ ° ú   H  â Ä º\   H > _  \  -t _    o ü @§ 4 _    1 l x Ã º\  Á º  ' a >  { 9 & ñ >  Ä »t ÷ &% 3 

. Õ ª Q  ô  Ç © _   â >  é ß é ß ô  Ç # 4 Ü ¼ Ð  ¦& ñ  ) a  â >  ¸| `  ¦ t   H  â Ä º\   H  â >   _  ~ ½ Ó ¾ ÓÜ ¼ Ð Ã º f 

Ü ¼ Ð K   Å Òl & h   ü @§ 4 _    1 l x Ã º\    " f > _  \  -t _    o Ø Ô>   z  ¤Ü ¼ 9 ´ òÖ  ¦& h Ü

¼ Ð \  -t \  ¦   ² ú    H   1 l x Ã º  > r F  < Ê`  ¦  Ð% i  . Õ ªo  ¦ s  Qô  Ç   õ   H é ß é ß ô  Ç # 4 _   ¦& ñ  ) a  â > 



¸| s  Å Ò  H ´ òõ ü <  8Ô  ¦ # Q  â >   õ  ü @Â Ò\ " f K ï  r Å Òl & h   j Ë µõ _   ©  ñ  6 x \  _ ô  Ç ´ òõ  ½ + Ë K

    õ s  9 s  ´ òõ   H > _  x 9  ¸\     ² ú  f  `  ¦  Ð  ¤ .

PACS numbers: 02.70.Ns, 34.30.+h

Keywords: ì  r  1 l x% i  < Æ, \  -t    ² ú , Å Òl & h   ü @§ 4 ,  â >  ¸|

E-mail: [email protected]

1128

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License

(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any

medium, provided the original work is properly cited.



© _ Z % ò  ¾ Ó\  _ ô  Ç : £ ¤ s ô  Ç B  m 7 £ § Ü ¼ Ð, Ä ºÅ Ò  _  5 Å q _

 ¨ î ç  H& h   ´ òõ  Ð Â Ò'   z  `  ¦ ] jî ß ô  Ç s Ê ê Ð s \ 

@

/ô  Ç  ¸4 S q_    ½ ¨ ´ ú §s  '  K &    [2].  è0 A Fermi-Ulam accelerator Model [3]  Ð ô  Ç    É r  ¦& ñ ÷ & ¦  É r ô  Ç    É r



 1 l x   H   Ü ¼ Ð ½ ¨$ í ÷ &# Qe  # Q ¿ º    s \  Z  ~ # e    H { 9

    1 l x   H   Ü ¼ Ð Â Ò'  \  -t \  ¦ % 3   H  ¸4 S qs  .

s

 Ð Â Ò'  Ò q t ) a ´ ú § É r D h Ðî  r  ¸4 S q[ þ t s   ¦î ß ÷ & ¦   ½ ¨

÷

&% 3  . dissipations  e     [4–8],  ª  % i  < Æ& h   2 [/ å L

`

 ¦ ô  Ç  ¸4 S q [9–13], ¢ ¸  H  è0 A bouncer model Ð · ú  9  ×  æ

§

4 s    6 x   H  ©  S ! \ " f { 9     1 l x   H   \  Â Òv 9 )  { 9

# Q   H  & ³ © [ þ t s    ½ ¨÷ &% 3   [14]. s M :   1 l x   H   õ

 ò ø Í$ í & h Ü ¼ Ð Ø  æ[  t   H  â Ä º [15]ü < q ò ø Í$ í & h Ü ¼ Ð Ø  æ[  t

  H  â Ä º\  @ /ô  Ç   ½ ¨ Ã º'  ÷ &% 3   [16,17].

¢

¸ô  Ç Ä ºo  ¸ ú  · ú  ¦e    H Û ¼á Ôa A\  B ² ú  2 ; { 9   Å Ò l

& h   ü @§ 4 \  @ /K " f # Qb  G>  ì ø Í6 £ x   H   H { 9 ì ø Í% i  < Æ\ 

"

f ç ß é ß ô  Ç \ V Ð À Ò# Qt  ¦ e  Ü ¼ 9 / B N   & ³ © õ  ° ú  É r < É ª p

e    H   õ \  ¦  Ð   [18].  Ð:  x { 9    H Å Òl & h   ü @§ 4

\

 @ /K   ' a$ í \  _ ô  Ç $  ½ Ó$ í `  ¦ ° ú   H . 7 £ ¤   1 l x   H ü @§ 4

\

 ì ø Í6 £ x   H { 9  _  5 Å q  ¸\  ¦  Ð  ü @§ 4 s   Ø Ô>    1 l x ½ + É Ã

º2  ¤ { 9  _  5 Å q  ¸  H   1 l x Ã º\  % i q Y V #  ×  ¦ # Q[ þ t >  ÷ &

9 Û ¼á Ôa A_   â Ä º  H ì ø Í@ / Ð ü @§ 4 _    1 l x s  9 þ t Ã º2  ¤ Û ¼á Ô a As  t   H ¹ ¡ §f  e  _  5 Å q  ¸\  ¦   or v l   H / '0 >  .



 " f { 9  -Û ¼á Ôa A> \  @ /K " f  H s  ¿ º t   © ì ø Í  ) a ´ ò õ

 & h { © ô  Ç   \ " f ] X Ø  æ| ¨ c M : ¸ ú  · ú  9  / B N   & ³ © s  { 9 

#

Qz  s  · ú  94 R e   . Õ ª Q  Û ¼á Ôa As  \ O   H í  H Ã ºô  Ç { 9  

>

\  @ /K " f  H Å Òl & h   ü @§ 4 s  # Q* ô  Ç % ò  ¾ Ó`  ¦ ×  ¦  כ  

? Ó ü t : r s  Qô  Ç >   H { 9  -Û ¼á Ôa A> ü < ° ú s  ü @§ 4 _    1

l

x \  @ /K  Û ¼á Ôa A% ! 3    1 l x`  ¦   ñ   H ì ø Í6 £ x`  ¦  Ðs   H



© u  \ O # Q \  -t \  ¦ f  ¨ Ã º   H B  m 7 £ § s  \ O t ë ß  ë ß  { 9

 ü @§ 4 s    6 x   H ~ ½ Ó ¾ ÓÜ ¼ Ð é ß é ß ô  Ç # 4 Ü ¼ Ð  ¦& ñ  ) a  â

>

\  ¦ ° ú   H { 9   > \  @ /K " f  H # 4 \ " f 4 A# Q  ¸  H { 9  

   5 Å q& h Ü ¼ Ð \  -t \  ¦ % 3 `  ¦ 0 p x$ í s  e  Ü ¼Ù ¼ Ð Õ ª    õ

 # Qb  G>      H t  Ò q ty  K  ^  ¦  9 כ ¹ e   . { 9  



¦& ñ  ) a  â >   \ " f ì ø Í ÷ &# Q ì ø Í@ / ~ ½ Ó ¾ ÓÜ ¼ Ð 4 A# Q 

Jaehwa Kim ^∗

¤M X ì Äß Ã Å 8 ûI í Ä; c 8 ý X ¢ = k4 z º8 ý ; c .U ¹ Å« o: Ä Z Ø Ò ÞW Ä] K ¡ ù p § ÞA 0" e

»< é s ^∗

© 2 £ x" é ¶ Å Ò@ / < Æ § Ó ü t o < Æõ , y © 2 £ x 210-702

(2013¸ 7 Z 4 23{ 9 ~ Ã Î6 £ §, 2013¸ 8 Z 4 5{ 9 Ã º& ñ : r ~ Ã Î6 £ §, 2013¸ 10 Z 4 10{ 9 > F S X & ñ )

U -× ¼ > r Û ¼ íJ $ ¶ û _ © ñ 6 x` ¦ ° ú H 2 " é ¶ { 9 > \ @ /K " f Ã ºf ~ ½ Ó ¾ ÓÜ ¼ Ð Å Òl & h ü @§ 4 s 6

x ½ + É M : { 9 > _ \ -t # Qb G> ì ø Í6 £ x H \ ¦ ì r 1 l x% i < Æ_ ~ ½ ÓZ O ` ¦ : x # · ú Ð ¤ . Å Òl & h

â > ¸| ` ¦ ° ú H â Ä º\ H > _ \ -t _ o ü @§ 4 _ 1 l x Ã º\ Á º ' a > { 9 & ñ > Ä »t ÷ &% 3

. Õ ª Q ô Ç © _ â > é ß é ß ô Ç # 4 Ü ¼ Ð ¦& ñ ) a â > ¸| ` ¦ t H â Ä º\ H â > _ ~ ½ Ó ¾ ÓÜ ¼ Ð Ã º f

Ü ¼ Ð K Å Òl & h ü @§ 4 _ 1 l x Ã º\ " f > _ \ -t _ o Ø Ô> z ¤Ü ¼ 9 ´ òÖ ¦& h Ü

¼ Ð \ -t \ ¦ ² ú H 1 l x Ã º > r F < Ê` ¦ Ð% i . Õ ªo ¦ s Qô Ç õ H é ß é ß ô Ç # 4 _ ¦& ñ ) a â >

¸| s Å Ò H ´ òõ ü < 8Ô ¦ # Q â > õ ü @Â Ò\ " f K ï r Å Òl & h j Ë µõ _ © ñ 6 x \ _ ô Ç ´ òõ ½ + Ë K

õ s 9 s ´ òõ H > _ x 9 ¸\ ² ú f ` ¦ Ð ¤ .

Keywords: ì r 1 l x% i < Æ, \ -t ² ú , Å Òl & h ü @§ 4 , â > ¸|

© _ Z % ò ¾ Ó\ _ ô Ç : £ ¤ s ô Ç B m 7 £ § Ü ¼ Ð, Ä ºÅ Ò _ 5 Å q _

¨ î ç H& h ´ òõ Ð Â Ò' z ` ¦ ] jî ß ô Ç s Ê ê Ð s \

/ô Ç ¸4 S q_ ½ ¨ ´ ú §s ' K & [2]. è0 A Fermi-Ulam accelerator Model [3] Ð ô Ç É r ¦& ñ ÷ & ¦ É r ô Ç É r

1 l x H Ü ¼ Ð ½ ¨$ í ÷ &# Qe # Q ¿ º s \ Z ~ # e H { 9

1 l x H Ü ¼ Ð Â Ò' \ -t \ ¦ % 3 H ¸4 S qs .

Ð Â Ò' Ò q t ) a ´ ú § É r D h Ðî r ¸4 S q[ þ t s ¦î ß ÷ & ¦ ½ ¨

&% 3 . dissipations e [4–8], ª % i < Æ& h 2 [/ å L

¦ ô Ç ¸4 S q [9–13], ¢ ¸ H è0 A bouncer model Ð · ú 9 × æ

4 s 6 x H © S ! \ " f { 9 1 l x H \ Â Òv 9 ) { 9

# Q H & ³ © [ þ t s ½ ¨÷ &% 3 [14]. s M : 1 l x H õ

ò ø Í$ í & h Ü ¼ Ð Ø æ[ t H â Ä º [15]ü < q ò ø Í$ í & h Ü ¼ Ð Ø æ[ t

H â Ä º\ @ /ô Ç ½ ¨ Ã º' ÷ &% 3 [16,17].

¸ô Ç Ä ºo ¸ ú · ú ¦e H Û ¼á Ôa A\ B ² ú 2 ; { 9 Å Ò l

& h ü @§ 4 \ @ /K " f # Qb G> ì ø Í6 £ x H H { 9 ì ø Í% i < Æ\

f ç ß é ß ô Ç \ V Ð À Ò# Qt ¦ e Ü ¼ 9 / B N & ³ © õ ° ú É r < É ª p

e H õ \ ¦ Ð [18]. Ð: x { 9 H Å Òl & h ü @§ 4

@ /K ' a$ í \ _ ô Ç $ ½ Ó$ í ` ¦ ° ú H . 7 £ ¤ 1 l x H ü @§ 4

ì ø Í6 £ x H { 9 _ 5 Å q ¸\ ¦ Ð ü @§ 4 s Ø Ô> 1 l x ½ + É Ã

º2 ¤ { 9 _ 5 Å q ¸ H 1 l x Ã º\ % i q Y V # × ¦ # Q[ þ t > ÷ &

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