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Bortolini, Graziéle
O oscilador harmônico e o gás de Lennard-Jones na dinâmica de Nosé-Hoover /
Graziéle Bortolini. – 2015.
70 p.: il.; 21 cm
Orientador: Edgard Pacheco Moreira Amorim
Bibliografia: p. 61-63
Dissertação (mestrado) – Universidade do Estado de Santa Catarina, Centro de
Ciências Tecnológicas, Programa de Pós-Graduação em Física, Joinville, 2015.
1. Física. 2. Dinâmica de Nosé-Hoover. 3. Oscilador Harmônico. 4. Gás de LennardJones. I. Amorim, Edgard Pacheco Moreira. II. Universidade do Estado de Santa
Catarina. Programa de Pós Graduação em Física. III. Título.
CDD: 530 - 23. ed.
Ficha catalográfica elaborada pela Biblioteca Setorial do CCT
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=M
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X mi
M 2
L6@3M
s2 Q̇2i − φ(Q) +
ṡ − (dN + 1)kB T ln s
L=
2
2
i
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O6
!"#$%&' () *+$',' ,- .'/+01''2-3
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/%0'$10'9 @ %"*!-A' % 1&' %+%07#' /!$%+*#'- ! $#/! (dN + 1)kB T ln s
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3' '" '" %21'=G%" % H1-%0?I'70'+7%(
Q̇i = Pi /(mi s2 )
Ṗi = −
∂φ
∂ Q̇i
JE9KL
ṡ = Ps /M
P˙s =
X
i
Pi 2 /(mi s3 ) − (dN + 1)kB T /s.
@" %21'=G%" % &!3#&%+$! % M!") ">! "1'3%"( %$%0&#+4"$#*'" % %
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%+21'+$! Ps *!00%"/!+ % '! &!&%+$! ! 70'1 % -#5%0 ' % %O$0' s9
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'+'
X P2
Ps2
i
H=
+
φ(Q)
+
JE9EL
+ (dN + 1)kB T ln s,
2
i
2mi s
2M
! &!&%+$! $!$'-
P=
X
Pi =
i
% &!&%+$! '+71-'0 $!$'M=
X
i
X
(mi s2 Q̇),
JE9QL
X
Qi × (mi s2 Q̇i ).
JE9RL
i
Qi × Pi =
i
M!") *!+"# %0!1 21% /'0' !5$%0 '" 21'+$# ' %" #+;&#*'"( ' 3'0#S3%-
s /! % "%0 #+$%0/0%$' ' *!&! %"*'-'&%+$! /'0' ! /'""! % $%&/! ∆t
+'" "#&1-'=G%"9 : /'""! % $%&/! 0%'- ∆t′ ) !5$# ! /!0 ∆t′ = ∆t/s(
/!0$'+$! ! *!&/0#&%+$! ! /'""! % $%&/! +' #+;&#*' % M!") )
3'0#S3%- CDEF9
KQ
!"! #$%&'$() #* %+,-
!"!"
#$%&'()*+,&( -+./- 0 12.030 3- 4052 - 0
#$%&'$ 6(+7+&,0
!"
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6)($%& 0( !"#$%& 04 0+/*)+3"+$%& ρ(Q1 , Q2 ...) /&3)4 *&0& & 4/'($& 04
!(/47
8 /+/*49( 4/*4#0+0& -&9'&/*& '4.& /+/*49( !,/+-& 04 +#*4)4//4 4 & )45
/4):(*;)+& ')&0"< -&#=6")($>4/ #& !" #$% 9+-)&-(#?#+-& -&9 dN +1
6)("/ 04 .+34)0(04@ 4 ( !"#$%& 04 0+/*)+3"+$%& ρ(P, Q, Ps , s) 2 4A')4/5
/(0( -&9& δ(H − E)@ 49 B"4 δ )4')4/4#*( ( !"#$%& 04.*( 04 C+)(-7
D&)*(#*&@ /4#0& dP = dP1 dP2 ...dPN 4 dQ = dQ1 dQ2 ...dQN @ (
!"#$%& '()*+$%& 0& /+/*49( 4/*4#0+0& '&04 /4) 4/-)+*( -&9&@
Z
Z=
Z
dPs
ds
Z
dP
Z
dQ
δ(H − E).
EF7GH
I4#0& & J(9+.*&#+(#& 0& /+/*49( 4/*4#0+0& 2 04=#+0& '&) F7F
!"#$%& '()*+$%& 0& /+/*49( 4/*4#0+0& (0B"+)4 ( !&)9(@
Z=
Z
dPs
Z
ds
Z
dP
Z
dQδ
"
X Pi 2
+ φ(Q)+
2mi s2
i
Ps 2
+ (dN + 1)kB T ln s − E .
+
2M
EF7KH
L!4*"(5/4 0"(/ *)&-(/ 04 :()+M:4+/@ 04 9&0& B"4 P′i = Pi /s 4
@ 04//( !&)9( & 4.494#*& 04 :&."94 dPdQ (//"94 ( !&)9(
dP dQ′ 7 D&)*(#*&@
Q′i = Q
dN
′
s
Z=
Z
dPs
Z
dQ
′
Z
′
dP
Z
ds s
dN
δ
"
X P′ 2
i
+ φ(Q′ )+
2m
i
i
Ps 2
+ (dN + 1)kB T ln s − E .
+
2M
EF7NH
O/(#0& ( )4.($%& 04 4B"+:(.1#-+( '()( ( !"#$%& 04.*( 04 C+)(-7
& <4)& 0( !"#$%& f (s)7 I4#0&
δ[f (s)] = δ[s − s0 ]/f ′ (s0 )@ -&9 s0 /4#0&
f (s) '()( & #&//& /+/*49( 04=#+0& -&9&
f (s) =
X P′ 2
Ps 2
i
+ φ(Q′ ) +
+ (dN + 1)kB T ln s − E.
2mi
2M
i
QR
EF7PH
!"#$%&' () *+$',' ,- .'/+01''2-3
!"#$ !%&'#(
s0 = exp
(
−1
(dN + 1)kB T
"
X P′ 2
Ps 2
i
−E
+ φ(Q′ ) +
2mi
2M
i
f ′ (s0 ) =
#)
(dN + 1)kB T
s
,
)*+,-.
/! "0%!120 34! 0 !3405'# *+6( 70$$0 0 $!2 !$821&0 8#"#
Z=
Z
dPs
Z
dQ
′
"
Z
× δ s − exp
′
dP
(
Z
sdN +1
×
(dN + 1)kB T
X P′ 2
−1
Ps 2
+ φ(Q ) +
−E
2mi
2M
i
(dN + 1)kB T
i
′
!)#
ds.
)*+,,.
9&1:1;0%<# 0 72#721!<0<! <0 <!:&0 δ[s − s0 ]f (s)ds = f (s0 )( 8#"
f (s) = sdN +1 ! f (s0 ) = s0 dN +1 ( &!2!"#$
R
Z=
Z
dPs
Z
dQ′
"
Z
× exp
(
dP′
1
×
(dN + 1)kB T
X P′ 2
Ps 2
i
+ φ(Q′ ) +
−E
2mi
2M
i
−1
(dN + 1)kB T
!)#dN +1
)*+,=.
P
>#%$1<!20%<# H0 (P′ , Q′ ) = i P′i 2 /2mi + φ(Q′ )( 0 ?4%5'# 702&15'#
7#<! $!2 !$821&0 8#"#
Z=
1
E
exp
×
(dN + 1)kB T
kB T
Z
Z
Z
−Ps 2
× exp
dPs dQ′ dP′
2M kB T
exp
−H0 (P′ , Q′ )
kB T
.
)*+,*.
@!%<# Zc = dQ dP′ exp[−H0 (P′ , Q′ )/kB T ] 0 ?4%5'# 702&15'#
<# $1$&!"0 %# 80%A%18# ! 2!$#:B!%<# 0 1%&!C20: !" Ps #D&!"#$
R
Z=
R
′
1
(dN + 1)
2πM
kB T
1/2
=F
exp
E
kB T
Zc
)*+,E.
.
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′
Q̇i = Pi /ms
∂φ
= sF(Qi )
Ṗi = −s
∂Qi
ṡ = sPs /M
X
P˙s =
P2i /(ms2 ) − (dN + 1)kB T.
MI7HQO
i
8' #,% #/@" # /+:514#-.% 2! G%"3( J%%)!& CHKE /&%/6" +'# #2#/B
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&1>)!1"S
qi = Qi ,
pi = mi q̇i = Pi /s,
d ln s
Ps
ξ=
=
.
dt
M
MI7HTO
R 21"0&1:+1-.% 2%" !"/#-%" 2! $#"! &!"+50#,0!" 2#" !*+#-L!" MI7HTO 3
4#,6,14#( ! 2! +'# '#,!1&# *+! !)10# % !"4#5#'!,0% 2% 0!'/%7 U!2!B
V,1'%" p = mq̇i ! "+:"010+;'%" (dN + 1) /%& dN ! %:0!'%"(
NW
!"#$%&' () *+$',' ,- .'/+01''2-3
ξ˙ =
P
q̇i = pi /mi
ṗi = F(qi ) − ξpi
i
p2i /mi − (dN )kB T
.
M
!"#$%
&'((' )*+* , -,)./0*1.,1, 23*2*(0, 2*3 -**4'3 ,+56.3' , 7*3),
H=
Z t
X p2
M ξ2
i
.
+ φ(q) + (dN )kB T
ξ(t′ )dt′ +
2mi
2
0
i
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>8
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UW
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1'*45.''63! "#$%& '#( m = k = 1& kB T = 1 ! (dN + 1) = dN & % !)*+ # '#(#&
HN =
P2
P2
Q2
+ α s + ln s.
+
2
2s
2
2
,-./0
1#234*3#& 324345$! ! 6( $+$3!(4 ! 76432# 842+98!+$ 3!(:#24+$ +* !5
:!* !*3!$; 4 '##2 !*4 4 Q& # (#(!*3# P & '##2 !*4 4 s ! # (#(!*3#
Ps . <=$!2845$! 76! α = 1/M . >4 4 6(4 !$$4$ 76432# 842+98!+$ #=!5
!'! 4 6(4 !764?@# +A!2!*'+4B ,:2+(!+24 #2 !(0 ! (#8+(!*3#& $@#
!B4$;
Q̇ = P/s2 ,
Ṗ = −Q,
,-.C0
ṡ = αPs
1 P2
˙
Ps =
−
1
.
s s2
D!!$'4B4* # # 3!(:# :!B4 842+98!B s& ! (# # 76! dtnovo = dtantigo /s&
4$ !764?E!$ ! (#8+(!*3# ! "#$% :424 # $+$3!(4 !( 76!$3@# 4 76+2!(
4 A#2(4&
Q̇ = P/s,
Ṗ = −sQ,
,-.F0
ṡ = sαPs
2
P
P˙s =
−
1
.
s2
G4H!* # 4 $!I6+*3! (6 4*?4 ! 842+98!B; q = Q& p = q̇ = P/s !
ξ = d ln s/dt = αPs & #=3!(#$ # J4(+B3#*+4*# ! "#$%5K##8!2&
p2
q2
H=
+
+
2
2
Z
t
ξ(t′ )dt′ +
0
ξ2
,
2α
,-.-0
#* ! 3!(#$ 32L$ 842+98!+$ 3!(:#24+$ +* !:!* !*3!$; 4 '##2 !*4 4
q & # (#(!*3# p ! # '#!)'+!*3! ! A2+'?@# ξ 76! #=! !'!( 4$ $!I6+*3!$
!764?E!$ ! (#8+(!*3#;
q̇ = p
FM
!"! #$%&'& '( )*%(+,-./& *0#$,)1-
ṗ = −q − ξp,
!"#$
ξ˙ = α(p2 − 1)
%&'()*( +,( '(
7-)812-9 '(
ξ>0
ξ <0
- -'./012-) .-34-)516'( .-3- ,3 -'./012-)
- -'./012-) .-34-)516'( .-3- ,3 -'./012-) 13-)5(6
./2-"
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∆t
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3-*/3(>5-H
q(t + ∆t) = q(t) + q̇(t)∆t + [−q(t) − ξ(t)q̇(t)]
∆t2
2
∆t
∆t
= q̇(t) + [−q(t) − ξ(t)q̇(t)]
q̇ t +
2
2
∆t
∆t
= s(t) + s(t)ξ(t)
s t+
2
2
∆t
∆t
ξ t+
= ξ(t) + α[q̇ 2 (t) − 1]
2
2
∆t
∆t ∆t
∆t
+s t+
ξ t+
s(t + ∆t) = s t +
2
2
2
2
∆t
∆t
∆t
ξ(t + ∆t) = ξ t +
+ α q̇ 2 t +
−1
2
2
2
%
− q(t + ∆t) ∆t
q̇ t + ∆t
2
2
q̇(t + ∆t) =
1 + ξ(t + ∆t) ∆t
2
%&'()*( +,( .->'/2()1>2-
αPs 9
q = Q9 p = q̇ = P/s
(
!"I$
ξ = d ln s/dt =
4(0- 10@-)/53- 1./31 .->'(@,/3-' (*-0,/) 5-21' 1' *1)/J*(/' 2(
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1, 2%,1,) q0 = Q0 = 0; ξ0 = Ps0 = 0 + s0 = 1'
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,
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,
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2
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ζj+1
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,
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Mj+1
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2
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˙
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45!8"( η = 2, ( )$)%#'! #)%#8"$"( &# #L# '!$) 5' %#&'()%!%(, # (81
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q̇ = p;
ṗ = −q − α1 ξp;
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O oscilador harmônico e o gás de Lennard-Jones na

Transcrição

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