Lecture RN1

  • Upload
    rover

  • View
    220

  • Download
    0

Embed Size (px)

Citation preview

  • 7/24/2019 Lecture RN1

    1/25

    !"#$%& ()&*+,-!"#$%&' )#*+,-.%"*

    .+*"-/"#% 012"/!"#$%&'#()"*#),--./ 012 0345$--6

    7%8!7#()"7"9:$#(7!7!"#$%&7;?7@A()96A>,;B*"48

  • 7/24/2019 Lecture RN1

    2/25

    !"# %&'()* '+*,-%./

    4#6A() =-3)A C%6$-/ "DE!#(%$ "6-(A!! FD#(D (-3)%#3! 6%34-.3A!!

    A*G*/ 6%4#-%(&HA 4A(%EI )DA6.%$ .-&-3

    J!#."$AK =-3)A C%6$- #3)AG6%&-3/ .9$&L4#.A3!#-3%$ #3)AG6%$!

    =%6M-H CD%#3 =-3)A C%6$-/ !#.9$%&-3 #3 C$%!!#(%$ N)%&!&(%$ =A(D%3#(!I

    O9%3)9. =A(D%3#(! A)(*

  • 7/24/2019 Lecture RN1

    3/25

    3%4 5% 6"26)2"5+ "# 1#5+7,"28 9-5"#$",$ &+5:%$; -:%,5 ,+=1+4?

    @A 1#5+7,"28

    AB$1&+#-1%#"2 1#5+7,"28

    C# 0:>-16-8

    I= dxF(x)a

    b

    "

    I#b$a

    Nw

    i

    i=0

    N

    % F(x i)

    &' b$a

    N

    (

    )

    *+

    ,

    -k

    ' N$k

    "# N$k/D

    D " number of particles " at least 102, ideally 1023

    D,,%,8

    M P -64A6 -8 )DA %$G-6#)D.

    A*G* (-."-!#)A )6%"AQ#9.I N#."!-3 MP0 -6 1; P 39.:A6 -8 !9:L#3)A6H%$!7H-$9.A!

    I= " dx(1)dx(2)dx(D )F(x(1),x(2 ),,x(D ))"

    D,,%,8

  • 7/24/2019 Lecture RN1

    4/25

    /012

    /032

    $0/0124/0322

    +E7E8 A F G

  • 7/24/2019 Lecture RN1

    5/25

    H)-5 %#+ +I"&J2+8 +#+,7> %K " 62"--16"2 7"-L21M)1$

    HR,P( ) =p2

    2m

    +V R( ) , V R( ) =i

    " v rj# rj( )i< j

    "

    R = r1,r2,,rNpart( ), P = p1,p2,,pNpart( ), Npart = number of particles

    E Npart,V,T( ) =3

    2Npart kBT+ V R( )

    V R( ) = 1Z

    d3NpartRV R( )exp #V R( ) kBT( )( )$

    Z= d3NpartRexp #V R( ) kBT( )( )$

    ! C# 5:1- 6"-+8 A F N(J",5

    O5:+, +I"&J2+-8

    -J1#B&%$+2-

  • 7/24/2019 Lecture RN1

    6/25

    C#5+7,"/%#8 96,)$+; R%#5+ T",2% "JJ,%"6:

    f =1

    Nf(xn )

    n=1

    N

    " +o N#1/ 2( )

    x1, x2, , xN

    I= f(x)dxa

    b

    " = (b # a) f f =1

    b # af(x)dx

    a

    b

    "

    ,"#$%& #)&*+,- 415: )#1K%,& $1-5,1*)/%# 1# $1-5,1*)5+$ J%1#5-

    D,,%, -6"2+- "- (B@LG"2-% 1# :17:+, $1&+#-1%#- A

    RT J+,K%,&-

  • 7/24/2019 Lecture RN1

    7/25

    [,)+ ,"#$%& #)&*+,-

    3",$ 5% %*5"1#8

    LK 4A(%E -8 6%4#-%(&HA #!-)-"A!

    LK %).-!"DA6#( 3-#!A

    LK (-..A6(#%$ "6-49()!/ A*G* O

  • 7/24/2019 Lecture RN1

    8/25

    444E1$M)"#/M)+E6%&

  • 7/24/2019 Lecture RN1

    9/25

    90-+)$%; ,"#$%& #)&*+,-

    @-3G !AU9A3(A! -8 39.:A6! JVBI V0I VW X K GA3A6%)A4 :E %3 %$G-6#)D. -3 %

    (-."9)A6

    ;-) )69$E 6%34-.I (-."$A)A$E 4A)A6.#3#!&(

    "#8 ? M3-F/ VBIV0IX IV3 ? (%3 %3&(#"%)A/ V3YB

    ,A"6-49(#:$A/ %$F%E! )DA !%.A !AU9A3(A 86-. )DA !%.A !AA4 VB

    @--M Z3A%6$E 6%34-.[/ D%HA !)%&!&(%$ "6-"A6&A! !#.#$%6 )- )69A 6*3*

    A*G*/ 93#8-6. 4#!)6#:9&-3 \JVK P B #8 ] ^P V ^ BI P] -)DA6F#!A

    T%,,+2"/%#-8

    C$+"22>

  • 7/24/2019 Lecture RN1

    10/25

    \6-49(A % !AU9A3(A -8 #3)AGA6 39.:A6! 9!#3G )DA 893(&-3/

    0'1234 &0' 1 5 6 *)( *

    # SDA _6!) A$A.A3) 51 '7/A BC%D #* 3

  • 7/24/2019 Lecture RN1

    11/25

    .6:,"7+a- 5,16b JN(D6%GAI BefeK

    4A_3A )F- #3)AGA6!/ U P '77 J3-)A/ #3)AGA6 4#H#!#-3K

    & P ' .-4 7

    !- )D%)I 'P7ZY&

    L#) #! "-!!#:$A )- !D-F )D%) % G--4 GA3A6%)-6 #!/

    [+6:#16"2 J,%*2+& 415: ^T]8 %=+,c%4 %K &0'

    aYnmodm =

    a(Ynmodq)" (Y

    n/q)r if this is # 0

    a(Ynmodq)" (Y

    n/q)r +m otherwise

    $%&

    J3-)A/ DA6A #3)AGA6 -"A6%&-3!bK

  • 7/24/2019 Lecture RN1

    12/25

    [+-5- K%, 5:+ M)"215> %K )#1K%,& J-+)$%B,"#$%& #)&*+,-

    7+#+,"5%,

    gA6#8E )DA )#1K%,&15>-8 )DA 4#!)6#:9&-3/

    J= #! % $%6GA #3)AGA6K

    (%$(9$%)A )DA .-.A3)/

    #8 )DA 4#!)* #! 93#8-6./

    #8 4AH#%&-3 !(%$A! %! 892:;I )DA3 4#!)6#:9&-3 #3 ,"#$%&

  • 7/24/2019 Lecture RN1

    13/25

    [+-5- K%, 5:+ M)"215> %K )#1K%,& J-+)$%B,"#$%& #)&*+,-

    7+#+,"5%,

    CDA(M 8-6 (-66A$%&-3!/

    39.:A6! %6A M %"%6)/

    #8 39.:A6! %6A 6%34-./

    C(k) =1

    M"i=1

    Mx

    ix

    i+k

  • 7/24/2019 Lecture RN1

    14/25

    R",-"721" [:+%,+&8

    3L)9"$A! F#)D (--64#3%)A! -:)%#3A4 86-. (-3!A(9&HA9!A -8 )DA @hC GA3A6%)-6 F#$$ $#A -3 % _3#)A 39.:A6 -8

    AU9%$$E !"%(A4 DE"A6"$%3A! #3 3L4#.A3!#-3%$ !"%(A/ )D#!

    39.:A6 4A"A34! -8 )DA "%6%.%)A6! 74:4'

    u1,u2,

    ,uD( )

    u2,u

    3,,u

    D+1( )

    u3,u

    4,,u

    D+2( )!

    ^T] 1- -1&J2+P K"-5 *)5 J+,1%$ 1- #%5 2%#7 "#$ 6%,,+2"/%#

  • 7/24/2019 Lecture RN1

    15/25

    00.1

    0.20.3 0.4

    0.50.6

    0.70.8

    0.91

    00.1

    0.20.3

    0.40.5

    0.60.7

    0.80.9

    1

    0

    0.10.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    [,1J2+- K,%& C(deROf. T3OCTD

  • 7/24/2019 Lecture RN1

    16/25

    00.1

    0.20.3

    0.40.5

    0.60.7

    0.80.9

    1

    00.1

    0.20.3

    0.40.5

    0.60.7

    0.80.9

    1

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    [,1J2+- K,%& R1#1&"2 ]%2$+# .5"#$",$ ^T]

  • 7/24/2019 Lecture RN1

    17/25

    e 9*+W+,; )#1K%,& ,"#$%& #)&*+,- 7+#+,"5%,8

    ."

  • 7/24/2019 Lecture RN1

    18/25

    @%)A-%B-. )C D>))(E A.-+()9%&'()* '+*,-%. >-'-%&?)%.

    !+J+"/*1215>8 !%.A !AA4 "6-49(A! )DA !%.A !AU9A3(A

    !"#$%+--8 93#8-6.#)EI !.%$$ (-66A$%&-3!

    ^%#7 J+,1%$8 !AU9A3(A 6A"A%)! #)!A$8 %jA6 _3#)A :9) $-3G "A6#-4

    C#-+#-1/=+ 5% -++$8 "A6#-4 %34 6%34-.3A!! 4- 3-) 4A"A34 -3 #)

    0%,5"*1215>8 !%.A 6A!9$)! -3 4#`A6A3) (-."9)A6!

    Di61+#6>8 !.%$$ 39.:A6 -8 T-%&3G "-#3) -"* %34 $-F .A.-6E

  • 7/24/2019 Lecture RN1

    19/25

    !(] J,%=1$+$ *> T%&J12+,-8

    C#5+2 K%,5,"#8 4-9:$A @Ch J"A6#-4 -8 %""6-V#.%)A$E B]kBlK

    7K%,5,"#8 hA-6GA =%6!%G$#%m! n?NN JnAA" ?) N#."$A N)9"#4KSDA (-3G69A3&%$ GA3A6%)-6 VJ3K P oe]oe p(4-) VJ3LBK Y BW0f0Bfll2 F#)D % "A6#-4 -8 0kW0I

    R WL!D#j !D#jL6AG#!)A6 GA3A6%)-6 F#)D % "A6#-4 -8 0kW0 L BI

    SF- BoL:#) .9$&"$ELF#)DL(%66E GA3A6%)-6! F#)D % "A6#-4 -8 2ef0fWBl0eo1l101ef q 0k2e*

    SDA -HA6%$$ "A6#-4 AV(AA4! 0kB0W*

    0]C d%,5,"# 1o :#) $%GGA4 _:-3%((# "!A94-L6%34-. !AU9A3(A F#)D %!D-6) $%G -8 2 %34 % $-3G $%G -8 Bf

    \A6#-4 J0kBf L BK r 0k12

    \%,2"#$ TLTjj @Ch/ .P0kW0 %P00oe21ff (PB

  • 7/24/2019 Lecture RN1

    20/25

    3%4 5% 6:%-+ 5:+ )#1K%,& J-+)$% ,E#E 7+#+,"5%,8

    C5 $+J+#$- %# 5:+ J,%*2+&VVV A1k+,+#5 ->-5+&- ",+ -+#-1*2+ 5% $1k+,+#5

    -5"/-/6"2 J,%J+,/+-

    f-)"22>8 -:1lB,+71-5+, 1- *+W+, 5:"# ^T]

    0,%7,"&&1#7 2"#7)"7+-L6%&J12+,- J,%=1$+ %#+8 #%5 "24">- 7%%$V

    T:+6b -5"/-/6"2 J,%J+,/+-P &"b+ " NA J2%5

    5,> &%,+ 5:"# %#+P 6%&J",+ ,+-)25-

    .%J:1-/6"5+$ %#+- ",+ "="12"*2+P +E7E K,%& 5:+ *%%b8

    DF+*-%

  • 7/24/2019 Lecture RN1

    21/25

    U,15+ " J,%7,"& -)6: 5:"58

    LK )DA Z.%#3[ "6-G6%. #3#&%$#QA! )DA !AA4/ 51 9 13

  • 7/24/2019 Lecture RN1

    22/25

    ]*]]eo2ff1 ]*WBfoW]2o

    ]*1Boeofo] ]*ef10o0ff]*1W1ff]ef ]*Be2oW2eW

    ]*]2W]2o1B ]*fBe]o]o]]*02Bfo1l] ]*1BB]WWo]

    ]*01B2oWo0 ]*e2el]WB]

    ]*1BB0Wf]e ]*ooBel1Wl]*efBo2foW ]*o2]]Wlo]

    ]*BelWWee2 ]*1ee2oBfe]*BW1eBW]W ]*1lWW0BB]

    X

    X

    DI"&J2+ %K :%4 5:+ %)5J)5 S2+ -:%)2$ 2%%b 21b+8

  • 7/24/2019 Lecture RN1

    23/25

    3%4 5% )-+ 7#)J2%58

    u G39"$-)

    G39"$-)q "$-) Z3%.A>-8>-9)"9)>_$A[ 9!#3G B/0 F#)D "-#3)!

    G39"$-)q !A) )A6.#3%$ "-!)!(6#") (-$-6

    G39"$-)q !A) -9)"9) Z3%.A>-8>"-!)!(6#")>_$A*"!v

    G39"$-)q 6A"$-)G39"$-)q U9#)

    uGH 3%.A>-8>"-!)!(6#")>_$A*"! w

  • 7/24/2019 Lecture RN1

    24/25

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

    random_couples.out u 1:2

  • 7/24/2019 Lecture RN1

    25/25

    !+K+,+#6+-8

    !E 3E ^"#$")P RE HE 0"+XP TE TE \%,$+1"#)P

    9T%&J)5"/%#"2 0:>-16- o .%2=1#7 0,%*2+&- 415: T%&J)5+,-;P UC^DpBmT3

    HE RE [:1q--+#P 9T%&J)5"/%#" 0:>16-;P TeR\!CA]D f(CmD!.C[p 0!D..

    ()&+,16"2 !+61J+-8 5:+ e,5 %K .61+#/S6 T%&J)/#7

    444E#,E6%&