Brownian Dynamics pada Temperatur Konstan.
Problem Molecular Dynamics = Problem mendapatkan solusi gerak partikel
yang dipengaruhi oleh persamaan gaya klasik Newton: F= ma. Dalam
simulasi, besaran waktu yang seharusnya bersifat kontinu dianggap diskrit,
dengan selang waktu konstan h.
MD: solusi dari md2
R/dt
2
= -F( R )
Konservasi energi adalah konsep penting dalam dinamika temperatur dari
sistem. Konservasi energi dalam simulasi dapat diperlihatkan, dengan syarat
dalam metode numeriknya, persamaan energi potensialnya harus dapat
dideferensialkan. Permasalahan diferensiabilitas fungsi energi potensial pada
batas kotak latis yang menyebabkan asumsi differensiability tak terpenuhi
dapat diatasi dengan menerapkan fungsi potensial Lennard-Jones dan radius
cutoff rc pada:
Vc(r ) = V(r) –V(rc).
Brownian Dynamics
-Merupakan pemodelan simulasi dinamis campuran antara deterministik dan
stokastik. Pada Brownian Dynamics beberapa derajat kebebasan hanya
direpresentasikan melalui pengaruh stokastik yang dilibatkan secara eksplisit dalam
persamaan gerak klasik dari partikel-partikel.
-Studi simulasi dilakukan dalam jerangka kerja canonical ensemble. Ide dasar dari
simulasi adalah: Pengaruh temperatur konstan pemanasan di suatu ruangan tertutup
(tangki), dengan medan gaya stokastik yang bekerja pada partikel-partikel.
-Dalam beberapa metode simulasi Dinamika gerak Brown, system digambarkan
dengan persamaan-persamaan Differensial stokastik. Contohnya, persamaan gerak
dari suatu partikel yang melakukan gerak Brown adalah:
Persamaan differensial orde 1 di atas, dikenal sebagai persamaan gerak Langevin,
yang merupakan persamaan gerak stokastik. Pemanasan tangki dapat direalisasikan
melalui gaya stokastik R(t), sehingga menimbulkan sebuah pertanyaan: Apakah sifat
yang seharusnya dimiliki R(t), agar nilainya ekwivalen dengan temperatur
pemanasan ruang T ?
- Investigasi mengenai R(t) yang akan membawa v ke distribusi klasik invarian
Maxwell-Boltzmann, yang diharapkan dalam canonical ensemble.
- Mulai dengan kasus 1D.
- Penggunaan teory proses stokastik dan rantai Markov, dapat menunjukkan
kalau hal tersebut dapat dicapai jika:
dengan; 3
-
-
-
-h : interval waktu untuk integrasi numerik dari persamaan gerak.
Pemilihan R(t) membawa pada hasil:
Algoritma untuk mensimulasikan gerak Brown Dinamis, secara umum menggunakan
algoritma berikut.
Algoritma :
1. Tetapkan posisi awal dan kecepatan awal dari partikel.
2. Bangkitkan bilangan acak dengan distribusi Gauss untuk rata-rata= 0 dan variansi
= R2
sebagaimana yang dijelaskan di atas. Proses ini akan memberikan R(t).
3. Integralkan persamaan gerak dengan nilai R yang diperoleh di langkah 2 dan
tentukan posisi baru berikut kecepatan dari partikel.
4. kembali ke langkah 2.
Implementasi langkah 3 pada program simulasi dilakukan dengan menggunakan
bentuk penjumlahan dari algoritma Verlet A3 berikut:
Algoritma A3. NVE MD Velocity Form:
(i) Tentukan posisi awal ri
1
(ii) Tentukan kecepatan awal vi
1
(iii) Hitung posisi semua parttikel pada saat n+1 dengan formula:
ri
n+1
= ri
n
+ hvi
n
+ 1/2m-1
h2
Fi
n
.
(iv) Hitung kecepatan semua parttikel pada saat n+1 dengan formula:
vi
n+1
= vi
n
+ h(Fi
n+1
+ Fi
n
)/2m.
Pada mulanya, partikel diletakkan pada permukaan-pusat dari latis kubus sehingga
mengakibatkan jumlah partikel merupakan kelipatan 4Æ 256 = 44
.
latis kubus
4
Nilai kecepatan dianggap berdistribusi normal. Besaran kecepatan partikel diatur
sedemikian rupa secara periodik agar dihasilkan level energi tertentu yang dapat
ditoleransi, yaitu setara dengan rata-rata dari temperatur sistem.
Catatan Listing Program Simulasi:
Analisis dari source code program simulasi Brownian dynamics dilakukan dengan
freeware yang diambil dari internet. Karena alasan tersebut software tidak dapat
digunakan untuk mengeksekusi kode secara keseluruhan. Pada tugas simulasi ini
kami menggunakan dua freeware: Force 2.08 (Fortran compiler and editor) dan
pemrograman basic untuk Windows, Liberty. Bas. Dari force 2.08 diperoleh analisis
source code seperti yang terungkap di bawah ini. Oleh karena permasalahan teknis
pada Force 2.08, hasil output tidak bisa ditampilkan maka kami menggunakan
Liberty.bas untuk menjalankan program dengan memperhatikan hasil analisis
dengan Force.
Algoritma yang digunakan dalam listing program:
(Implementasi Langkah 1)
DEFINITION OF THE SIMULATION PARAMETERS
SET THE ORDER PARAMETER
SET UP FCC LATTICE FOR THE ATOMS INSIDE THE BOX
(Implementasi Langkah 2 - 4)
ASSIGN BOLTZMANN DISTRIBUTED VELOCITIES AND CLEAR THE FORCES
START OF THE ACTUAL BROWNIAN DYNAMICS PROGRAM:
THE EQUATIONS OF MOTION ARE INTEGRATED USING THE 'SUMMED FORM'
THE STOCHASTIC PART IS FOLLOWS: AT REGULAR TIME INTERVALS THE VELOCITIES ARE
REPLACED BY VELOCITIES DRAWN FROM A BOLTZMANN DISTRIBUTION AT SPECIFIED
TEMPERATURE.
ADVANCE POSITIONS ONE BASIC TIME STEP
APPLY PERIODIC BOUNDARY CONDITIONS
COMPUTE THE PARTIAL VELOCITIES
COMPUTES THE FORCES ON THE PARTICLES
COMPUTE THE VELOCITIES
COMPUTE THE KINETIC ENERGY
COMPUTE THE AVERAGE VELOCITY
REPLACE THE VELOCITIES
Output Simulasi
COMPUTE VARIOUS QUANTITIES:
EK = Energi Kinetik
EPOT = Energi Potensial
ETOT =Energi total = EK + EPOT
TEMP = Temperatur
PRES = Tekanan
VEL = Kecepatan
RP = (COUNT / 256.0) *100.0
Hasil analisis program menggunakan Force 2.08 mengungkapkan beberapa masalah
yang terdapat di dalam program, sebagai berikut:
C:\WINDOWS\TEMP\cccWYOfb.o: In function `MAIN__':
//F/browniandynamics1.f:86: undefined reference to `ranset_' 5
C:\WINDOWS\TEMP\cccWYOfb.o: In function `mxwell_':
//F/browniandynamics1.f:307: undefined reference to `ranf_'
//F/browniandynamics1.f:308: undefined reference to `ranf_'
Hal ini berarti, subroutine RANSET(ISEED) pada baris kode call ranset (Iseed) yang
hasilnya akan digunakan pada dengan CALL MXWELL (VH,N3,H,TREF) perlu
ditambahkan.
Pembangkitan Bilangan Acak dalam FORTRAN 77
Pembangkit bilangan acak FORTRAN 77 yang diberikan oleh para vendor
menggunakan CRAY Y-MP UNICOS. Subrutin RANSET dengan argumen integer
digunakan untuk mempersiapkan dan mengubah seed; dengan fungsi presisi tunggal
yang disebut dengan RANF untuk membangkitkan bilangan acak. RANF tidak
memerlukan argumen apapun, berikut ini contoh penulisan kodenya:
INTEGER ISEED
REAL RANF, U
...
CALL RANSET(ISEED)
U = RANF()
Fungsi pada subprogram RANF mengembalikan nilai bilangan acak semu yang
terdistribusi secara seragam dalam selang (0,1), dengan membuang nilai yang
terakhir. Metode yang digunakan adalah perkalian secara kongruensi. Subrutin
subprogram RANGET membuat nilai seed terbaru dari RANF agar dapat digunakan
lebih lanjut, dan subrutin RANSET menyimpan nilai seed untuk penggunaan lanjut
oleh RANF.
struktur:
SUBROUTINE and FUNCTION subprograms
User Entry Names: RANF, DRANF, RANGET, RANSET
Penggunaan:
Dalam sembarang ekspresi aritmatika,
RANF() diatur ke sebuah nilai yang lebih besar dari nol dan kurang dari satu. RANF
bertipe REAL.
CALL RANGET(SEED)
CALL RANSET(SEED)
SEED
(REAL untuk kasus CDC, presisi ganda pada kasus lain). Pada saat keluar dari
RANGET, nilai SEED akan diatur ke suatu nilai yang menentukan posisi terbaru
dalam barisan bilangan acak. Nilai ini mungkin akan digunakan kemudian sebagai
suatu argumen dalam memanggil subrutin RANSET untuk memulai pembangkitan
bilangan acak dari titik tersebut.
Berikut ini beberapa referensi dari Intel Fortran Libraries, yang diambil dari internet: 6
RANSET
Portability Subroutine: Sets the seed for the random number generator.
Module: USE IFPORT
Syntax
CALL RANSET (seed)
Seed (Input) REAL(4). The reset value for the seed.
Compatibility
CONSOLE STANDARD GRAPHICS QUICKWIN GRAPHICS WINDOWS DLL LIB
See Also: “RANGET”
RANF
Portability Function: Generates a random number between 0.0 and RAND_MAX.
Module: USE IFPORT
Syntax
result = RANF ( )
Results:
The result type is REAL(4). The result value is a single-precision pseudo-random number
between 0.0 and RAND_MAX as defined in the C library, normally 0x7FFF 215–1. The
initial seed is set by the following:
CALL SRAND(ISEED) where ISEED is type INTEGER(4).
Compatibility
CONSOLE STANDARD GRAPHICS QUICKWIN GRAPHICS WINDOWS DLL LIB
Pada subrutin Mxwell terdapat u1=ranf() dan u2=ranf(), berdasarkan penjelasan di
atas, u1 dan u2 dapat disubstitusi dengan rnd(seed).
Lampiran:
1. Source Code Fortran untuk Simulasi Brownian Dynamics.
2. Source Code Liberty Basic untuk Penggalan Simulasi Brownian
Dynamics beserta Hasil Eksekusinya. 7
1. Listing Program Brownian Dynamics untuk temperatur konstan:
c================================================
c
c BROWNIAN DYNAMICS
c
c APPLICATION TO ARGON.THE LENNARD-JONES POTENTIAL IS TRUNCATED
c AT RCOFF AND NOT SMOOTHLY CONTINUED TO ZERO.INITIALLY THE
c NPART PARTCLES ARE PLACED ON AN FCC LATTTICE.THE VELOCITIES
c ARE DRAWN FROM A BOLTZMANN DISTRIBUTION WITH TEMPERATURE TREF.
c
c INPUT PARAMETERS ARE AS FOLLOWS
c
c NPART NUMBER OF PARTICLES (MUST BE A MULTIPLE OF four)
c SIDE SIDE LENGTH OF THE CUBICAL BOX IN SIGMA UNITS
c TREF REDUCED
c DEN REDUCED DENSITY
c RCOFF CUTOFF OF THE POTENTIAL IN SIGMA UNITS
c H BASIC TIME STEP
c IREP REPLACEMENT OF THE VELOCITIES EVERY IREP'TH
c TIME STEP
c TIMEMX NUMBER OF INTEGRATION STEPS
c ISEED SEED FOR THE RANDOM NUMBER GENERATOR
c
c VERSION 1.0 AS OF AUGUST 1985
c
c DIETER W.HEERMANN
c
c====================================================
c
REAL X(1:786),VH(1:786),F(1:786)
c
REAL FY(1:256),FZ(1:256),Y(1:256),Z(1:256)
c
REAL H,HSQ,HSQ2,TREF
c
REAL KX,KY,KZ
c
INTEGER CLOCK,TIMEMX
c
EQUIVALENCE (FY,F(257)), (FZ,F(513)),(Y,X(257)),(Z,X(513))
c
c================================================
c
c DEFINITION OF THE SIMULATION PARAMETERS
c
c================================================
c
NPART = 256
SIDE = 6.75284
TREF = 0.722
DEN = 0.83234
RCOFF = 2.5
H = 0.064
IREP = 10
TIMEMX = 3
ISEED = 4711
c
c================================================
c
c SET THE ORDER PARAMETER
c
c================================================
c
WRITE(6,*) 'BROWNIAN DYNAMICS SIMULATION PROGRAM' 8
WRITE(6,*) '--------------------------------------------------'
WRITE(6,*)
WRITE(6,*) 'NUMBER OF PARTICLES IS ',NPART
WRITE(6,*) 'SIDE LENGTH OF THE BOX IS ',SIDE
WRITE(6,*) 'CUT OFF IS ',RCOFF
WRITE(6,*) 'REDUCED TEMPERATUREE IS ',TREF
WRITE(6,*) 'BASIC TIME STEP IS ',H
WRITE(6,*)
c
A =SIDE/4.0
SIDEH =SIDE*0.5
HSQ =H*H
HSQ2 =HSQ*0.5
NPARTM =NPART-1
RCOFFS =RCOFF*RCOFF
TSCALE =16.0/NPARTM
VAVER =1.13*SQRT(TREF/24.0)
c IOF1 =NPART
IOF2 =2*NPART
N3 =3*NPART
c
CALL RANSET (ISEED)------Æsubroutine tak terdefinisi
c
c
c===================================================================
c
c THIS PART OF THE PROGRAM PREPARES THE INITIAL CONFIGURATION
c
c====================================================================
c
c SET UP FCC LATTICE FOR THE ATOMS INSIDE THE BOX
c--------------------------------------------------------------------------------------------------
c IJK =0
DO 10 LG=0,1
DO 10 I=0,3
DO 10 J=0,3
DO 10 K=0,3
IJK = IJK + 1
X(IJK )=I*A+LG*A*0.5
Y(IJK )=J*A+LG*A*0.5
Z(IJK )=K*A
10 CONTINUE
DO 15 LG=1,2
DO 15 I=0,3
DO 15 J=0,3
DO 15 K=0,3
IJK =IJK +1
X(IJK ) =I*A+(2-LG)*A*0.5
Y(IJK )=J*A+(LG-1)*A*0.5
15 CONTINUE
c
c ASSIGN BOLTZMANN DISTRIBUTED VELOCITIES AND CLEAR THE FORCES
c ===============================================================
c
CALL MXWELL (VH,N3,H,TREF) ----Æ terdapat fungsi yang tak terdefinisi
c
DO 123 I=1,N3
F(I)=0.0
123 CONTINUE
c
c======================================================================
c
c START OF THE ACTUAL BROWNIAN DYNAMICS PROGRAM.
c
c THE EQUATIONS OF MOTION ARE INTEGRATED USING THE 'SUMMED FORM'
c (G.DAHLQUIST AND A. BJOERK, NUMERICAL METHODS, PRENTICE HALL
c 1974). THE STOCHASTIC PART IS FOLLOWS: AT REGULAR TIME 9
c INTERVALS THE VELOCITIES ARE REPLACED BY VELOCITIES DRAWN FROM
c A BOLTZMANN DISTRIBUTION AT SPECIFIED TEMPERATURE.
c (H.C. ANDERSEN, J.CHEM. PHYS. 72. 2384 (1980) AND W.C. SWOPE.
c H.C. ANDERSEN, P.H. BERENS AND K.R. WILSON. J. CHEM . PHYS. 76
c 637 (1982).
c
c VERSION 1.0 AUGUST 1985
c DIETER W. HEERMANN
c
c--------------------------------------------------------------------------------------------------
c
DO 200 CLOCK = 1, TIMEMX
c
c === ADVANCE POSITIONS ONE BASIC TIME STEP ===
DO 210 I=1, N3
X(I) = X(I) + VH(I) + F(I)
210 CONTINUE
c
c ===APPLY PERIODIC BOUNDARY CONDITIONS===
DO 215 I=1,N3
IF (X(I) .LT. 0) X(I) = X(I) + SIDE
IF (X(I) .GT. SIDE) X(I) = X(I) - SIDE
215 CONTINUE
c
c ===COMPUTE THE PARTIAL VELOCITIES===
c
DO 220 I=1,N3
VH(I) = VH(I) + F(I)
220 CONTINUE
c
c =================================================================
c = =
c = THIS PART COMPUTES THE FORCES ON THE PARTICLES =
c = =
c =================================================================
c
c
VIR = 0.0
EPOT = 0.0
DO 225 I=1,N3
F(I) = 0.0
225 CONTINUE
c
c
DO 270 I=1,NPART
XI = X(I)
YI = Y(I)
ZI = Z(I)
DO 270 J = I+1, NPART
XX = XI -X(J)
YY = YI -Y(J)
ZZ = ZI - Z(J)
c
IF (XX .LT. -SIDEH) XX = XX + SIDE
IF (XX .GT. SIDEH) XX = XX - SIDE
c
IF (YY .LT. -SIDEH) YY = YY + SIDE
IF (YY .GT. SIDEH) YY = YY - SIDE
c
IF (ZZ .LT. -SIDEH) ZZ = ZZ + SIDE
IF (ZZ .GT. SIDEH) ZZ = ZZ - SIDE
RD = XX* XX + YY * YY + ZZ* ZZ
IF (RD .GT. RCOFFS ) GOTO 270
c
EPOT = EPOT + RD ** (-6.0) - RD ** (-3.0)
R148 = RD ** (-7.0) - 0.5 * RD ** (-4.0)
VIR = VIR - RD* R148 10
KX = XX * R148
F(I) = F(I) + KX
F(J) = F(J) - KX
KY = YY * R148
FY(I) = FY(I) + KY
FY(J) = FY(J) - KY
KZ = ZZ * R148
FZ(I) = FZ(I) + KZ
FZ(J) = FZ(J) - KZ
270 CONTINUE
DO 275 I=1, N3
F(I) = F(I) * HSQ2
275 CONTINUE
c
c =================================================================
c = =
c = END OF THE FORCE CALCULATION =
c =================================================================
c
c === COMPUTE THE VELOCITIES===
DO 300 I=1, N3
VH(I) = VH(I) + F(I)
300 CONTINUE
c
c ===COMPUTE THE KINETIC ENERGY ===
EKIN = 0.0
DO 305 I= 1, N3
EKIN = EKIN + VH(I) * VH(I)
305 CONTINUE
EKIN = EKIN / HSQ
c
c ===COMPUTE THE AVERAGE VELOCITY===
VEL = 0.0
COUNT = 0.0
DO 306 I=1, NPART
VX = VH(I) * VH(I)
VY = VH(I + IOF1) * VH (I + IOF1)
VZ = VH(I + IOF2) * VH (I + IOF2)
SQ = SQRT (VX + VY + VZ )
SQT = SQ / H
IF (SQT .GT. VAVER ) COUNT = COUNT +1
VEL = VEL +SQ
306 CONTINUE
VEL = VEL / H
c
IF ((CLOCK/IREP)* IREP .EQ. CLOCK) THEN
c
c ===REPLACE THE VELOCITIES ===
WRITE(6,*) 'VELOCITY REPLACEMENT' -----------Æ tidak terdapat format statement
CALL MXWELL(VH, N3, H, TREF) ----Æ terdapat fungsi yang tak terdefinisi
EKIN = TREF / TSCALE
END IF
c
c
c ================================================================
c = =
c = COMPUTE VARIOUS QUANTITIES =
c = =
c ================================================================
c
EK = 24.0 * EKIN
EPOT = 4.0 * EPOT
ETOT = EK + EPOT
TEMP = TSCALE * EKIN
PRES = DEN * 16.0 * (EKIN - VIR) / NPART
VEL = VEL / NPART 11
RP = (COUNT / 256.0) *100.0
WRITE(6.6000) CLOCK,EK,EPOT,ETOT,TEMP,PRES,VEL,RP
6000 FORMAT (1I6, 7F15.6)
c
200 CONTINUE
c
c
STOP
END
c
c =================================================================
c
c M X W E L L RETURNS UPON CALLING A MAXWELL DISTRIBUTED DEVIATES
c FOR THE SPECIFIED TEMPERATURE TREF. ALL DEVIATES ARE SCALED BY
c A FACTOR
c
c CALLING PARAMETERS ARE AS FOLLOWS
c VH VECTOR OF LENGTH NPART (MUST BE A MULTIPLE OF 2)
c N3 VECTOR LENGTH
c H SCALING FACTOR WITH WHICH ALL ELEMENTS OF VH ARE
c MULTIPLIED.
c TREF TEMPERATURE
c
c VERSION 1.0 AS OF AUGUST 1985
c DIETER W HEERMANN
c
c =================================================================
c
SUBROUTINE MXWELL(VH,N3,H,TREF)
c
REAL VH(1:N3)
c
REAL U1,U2,V1,V2,S,R
c
NPART = N3 / 3
IOF1 = NPART
IOF2 = 2 * NPART
TSCALE = 16.0 / (1.0 * NPART - 1.0)
c
DO 10 I=1,N3,2
c
1 U1 = RANF ()------Æ tak terdefinisi
U2 = RANF () -----Æ tak terdefinisi
c
V1 =2.0 * U1 - 1.0
V2 =2.0 * U2 - 1.0
S =V1 * V1 +V2 *V2
c
IF ( S .GE. 1.0 ) GOTO 1
R = -2.0 * ALOG (S) / S
VH(I) = V1 * SQRT (R)
VH(I+1)= V2 * SQRT (R)
10 CONTINUE
c
EKIN = 0.0
SP = 0.0
DO 20 I=1,NPART
SP =SP + VH (I)
20 CONTINUE
SP = SP / NPART
DO 21 I=1,NPART
VH (I) = VH (I) - SP
EKIN = EKIN + VH (I) * VH (I)
21 CONTINUE
SP = 0.0
DO 22 I=IOF1 + 1,IOF2 12
SP = SP + VH (I)
22 CONTINUE
SP = SP / NPART
DO 23 I=IOF1 + 1,IOF2
VH(I) = VH (I) - SP
EKIN = EKIN + VH(I) * VH(I)
23 CONTINUE
SP = 0.0
DO 24 I=IOF2 + 1,N3
SP = SP + VH (I)
24 CONTINUE
SP = SP / NPART
DO 25 I=IOF2 + 1,N3
VH(I) = VH (I) - SP
EKIN = EKIN + VH (I) * VH (I)
25 CONTINUE
TS = TSCALE * EKIN
SC = TREF / TS
SC = SQRT (SC)
SC = SC * H
DO 30 I=1,N3
VH (I) = VH (I) * SC
30 CONTINUE
END 13
2. Source codes Liberty Basic
Berikut ini source codes Lyberty Basic untuk program simulasi Brownian Dynamics,
setelah membuat beberapa perubahan dari source code aselinya dalam bahasa
pemrograman Fortran 77. Kami menjamin source code berikut dapat dijalankan pada
software Liberty Basic yang berlisensi. Hal ini dibuktikan dengan mengeksekusi
penggalan program:
-Komputasi konstanta di awal program
-Inisial konfigurasi latis dan
-Subrutin Boltzmann-Maxwell.
Karena alasan lisensi software-lah, maka program tidak dapat dikompilasi secara
utuh.
-Source codes untuk Brownian Dynamics dalam Liberty Basic secara
Utuh
DIM VH(1000):DIM F(1000)
NPART = 256
SIDE = 6.75284
TREF = 0.722
DEN = 0.83234
RCOFF = 2.5
H = 0.064
IREP = 10
TIMEMX = 3
ISEED = 4711
print "================================================"
print" SET THE ORDER PARAMETER"
print "================================================"
print "BROWNIAN DYNAMICS SIMULATION PROGRAM"
print " '--------------------------------------------------"
print" NUMBER OF PARTICLES IS "; NPART
print" SIDE LENGTH OF THE BOX IS ";SIDE
print" CUT OFF IS " ;RCOFF
print" REDUCED TEMPERATUREE IS ";TREF
print" BASIC TIME STEP IS "; H
A =SIDE/4.0
SIDEH =SIDE*0.5
HSQ =H*H
HSQ2 =HSQ*0.5
NPARTM=NPART-1
RCOFFS=RCOFF*RCOFF
TSCALE=16.0/NPARTM
VAVER =1.13*(TREF/24.0)^0.5
IOF1 =NPART
IOF2 =2*NPART
N3 =3*NPART
"========================================================
print " THIS PART OF THE PROGRAM PREPARES THE INITIAL CONFIGURATION"
"========================================================"
print " SET UP FCC LATTICE FOR THE ATOMS INSIDE THE BOX"
print "-------------------------------------------------------------------" 14
DIM X(1000):DIM Y(1000): DIM Z(1000)
IJK =0
FOR LG= 0 TO 1
FOR I= 0 TO 3
FOR J= 0 TO 3
FOR K= 0 TO 3
IJK = IJK + 1
X(IJK )=I*A+LG*A*0.5
Y(IJK )=J*A+LG*A*0.5
Z(IJK )=K*A
PRINT"X(";IJK;")=",X(IJK)
PRINT"Y(";IJK;")=",Y(IJK)
PRINT"Z(";IJK;")=",Z(IJK)
NEXT K
NEXT J
NEXT I
NEXT LG
FOR LG= 1 TO 2
FOR I= 0 TO 3
FOR J= 0 TO 3
FOR K= 0 TO 3
IJK =IJK +1
X(IJK ) =I*A+(2-LG)*A*0.5
Y(IJK )=J*A+(LG-1)*A*0.5
Z(IJK )= K*A +A*0.5
PRINT"X(";IJK;")=",X(IJK)
PRINT"Y(";IJK;")=",Y(IJK)
PRINT"Z(";IJK;")=",Z(IJK)
NEXT K
NEXT J
NEXT I
NEXT LG
print" ASSIGN BOLTZMANN DISTRIBUTED VELOCITIES AND CLEAR THE FORCES"
print" ===================================================="
GOSUB[MXWELL]
FOR I=1 TO N3
F(I)=0.0
NEXT I
print" START OF THE ACTUAL BROWNIAN DYNAMICS PROGRAM."
PRINT"---------------------------------------------------"
FOR T=1 TO TIMEMX
print" === ADVANCE POSITIONS ONE BASIC TIME STEP ==="
FOR I=1 TO N3
X(I) = X(I) + VH(I) + F(I)
PRINT"X(";I;")=",X(I)
NEXT I
print" ===APPLY PERIODIC BOUNDARY CONDITIONS==="
DO until I=N3
IF (X(I) < 0) then X(I) = X(I) + SIDE
IF (X(I) > SIDE) then X(I) = X(I) - SIDE
loop
print" ===COMPUTE THE PARTIAL VELOCITIES==="
FOR I= 1 TO N3
VH(I) = VH(I) + F(I)
NEXT I
print" ===================================================" 15
print" = THIS PART COMPUTES THE FORCES ON THE PARTICLES ="
print" =================================================="
VIR = 0.0
EPOT = 0.0
FOR I=1 TO N3
F(I) = 0.0
NEXT I
FOR I= 1 TO NPART
XI = X(I)
YI = Y(I)
ZI = Z(I)
FOR J = 1+I TO NPART
XX = XI -X(J)
YY = YI -Y(J)
ZZ = ZI - Z(J)
IF (XX < -1 *SIDEH) then XX = XX + SIDE
IF (XX > SIDEH) then XX = XX - SIDE
IF (YY < -1* SIDEH) then YY = YY + SIDE
IF (YY > SIDEH) then YY = YY - SIDE
IF (ZZ < -1* SIDEH) then ZZ = ZZ + SIDE
IF (ZZ > SIDEH) then ZZ = ZZ - SIDE
RD = XX* XX + YY * YY + ZZ* ZZ
IF (RD > RCOFFS ) then GOTO 270
EPOT = EPOT + RD ^(-6) - RD ^ (-3)
R148 = RD ^ (-7.0) - 0.5 * RD ^(-4.0)
VIR = VIR - RD* R148
KX = XX * R148
F(I) = F(I) + KX
F(J) = F(J) - KX
KY = YY * R148
FY(I) = FY(I) + KY
FY(J) = FY(J) - KY
KZ = ZZ * R148
FZ(I) = FZ(I) + KZ
FZ(J) =FZ(J) -KZ
270 NEXT J
NEXT I
FOR I= 1 TO N3
F(I) = F(I) * HSQ2
PRINT"F(";I;")=",F(I)
275 NEXT I
print" =================================================="
print" END OF THE FORCE CALCULATION"
print" =================================================="
print" === COMPUTE THE VELOCITIES==="
for I=1 to N3
VH(I) = VH(I) + F(I)
next I
print" ===COMPUTE THE KINETIC ENERGY ==="
EKIN = 0.0 16
for I= 1 to N3
EKIN = EKIN + VH(I) * VH(I)
305 next I
EKIN = EKIN / HSQ
print" ===COMPUTE THE AVERAGE VELOCITY==="
VEL = 0.0
COUNT = 0.0
for I= 1 to NPART
VX = VH(I) * VH(I)
VY = VH(I + IOF1) * VH (I + IOF1)
VZ = VH(I + IOF2) * VH (I + IOF2)
SQ = SQRT (VX + VY + VZ )
SQT = SQ / H
IF (SQT > VAVER ) then COUNT = COUNT +1
VEL = VEL +SQ
306 next I
VEL = VEL / H
IF ((CLOCK/IREP)* IREP = CLOCK) THEN
print" ===REPLACE THE VELOCITIES ==="
print" VELOCITY REPLACEMENT"
GOSUB [MXWELL]
EKIN = TREF / TSCALE
END IF
print" ================================================"
print" = COMPUTE VARIOUS QUANTITIES ="
print" ================================================="
EK = 24.0 * EKIN
EPOT = 4.0 * EPOT
ETOT = EK + EPOT
TEMP = TSCALE * EKIN
PRES = DEN * 16.0 * (EKIN - VIR) / NPART
VEL = VEL / NPART
RP = (COUNT / 256.0) *100.0
print "CLOCK=", T; "ENERGI KINETIK=",EK; "ENERGI POTENSIAL=", EPOT; "TOTAL
ENERGI",ETOT
PRINT "TEMPERATUR=",TEMP;"TEKANAN=", PRES;"KECEPATAN PARTIKEL=", VEL;
"RP=",RP
NEXT T
STOP
END
[MXWELL]
NPART = N3 / 3
IOF1 = NPART
IOF2 = 2 * NPART
TSCALE = 16.0 / (1.0 * NPART - 1.0)
FOR I=1 TO N3 STEP 2
1 U1 = Rnd(4711)
U2 = Rnd(4711)
V1 =2.0 * U1 - 1.0
V2 =2.0 * U2 - 1.0
S =V1 * V1 +V2 *V2
IF ( S >= 1.0 )THEN GOTO 1
R = -2.0 * log (S) / S
VH(I) = V1 * R^.5
VH(I+1)= V2 * R^.5 17
10 NEXT I
EKIN = 0.0
SP = 0.0
FOR I=1 TO NPART
SP =SP + VH (I)
20 NEXT I
SP = SP / NPART
FOR I=1 TO NPART
VH (I) = VH (I) - SP
EKIN = EKIN + VH (I) * VH (I)
21 NEXT I
SP = 0.0
FOR I=IOF1 + 1 TO IOF2
SP = SP + VH (I)
22 NEXT I
SP = SP / NPART
FOR I=IOF1 + 1 TO IOF2
VH(I) = VH (I) - SP
EKIN = EKIN + VH(I) * VH(I)
23 NEXT I
SP = 0.0
FOR I=IOF2 + 1 TO N3
SP = SP + VH (I)
24 NEXT I
SP = SP / NPART
FOR I=IOF2 + 1 TO N3
VH(I) = VH (I) - SP
EKIN = EKIN + VH (I) * VH (I)
25 NEXT I
TS = TSCALE * EKIN
SC = TREF / TS
SC = (SC)^0.5
SC = SC * H
FOR I=1 TO N3
VH (I) = VH (I) * SC
PRINT"VH(";I;")=",VH(I)
30 NEXT I
RETURN 18
-Source codes untuk Brownian Dynamics-1
print "BROWNIAN DYNAMICS"
print "APPLICATION TO ARGON.THE LENNARD-JONES POTENTIAL IS TRUNCATED"
print "AT RCOFF AND NOT SMOOTHLY CONTINUED TO ZERO.INITIALLY THE"
print "NPART PARTCLES ARE PLACED ON AN FCC LATTTICE.THE VELOCITIES"
print "ARE DRAWN FROM A BOLTZMANN DISTRIBUTION WITH TEMPERATURE TREF."
print "INPUT PARAMETERS ARE AS FOLLOWS"
print" NPART = NUMBER OF PARTICLES (MUST BE A MULTIPLE OF four)"
print " SIDE = SIDE LENGTH OF THE CUBICAL BOX IN SIGMA UNITS"
print "TREF = REDUCED"
print " DEN = REDUCED DENSITY"
print " RCOFF = CUTOFF OF THE POTENTIAL IN SIGMA UNITS"
print" H = BASIC TIME STEP"
print" IREP = REPLACEMENT OF THE VELOCITIES EVERY IREP'TH"
print" TIME STEP"
print " TIMEMX = NUMBER OF INTEGRATION STEPS"
print "ISEED = SEED FOR THE RANDOM NUMBER GENERATOR"
print "VERSION 1.0 AS OF AUGUST 1985"
print " DIETER W.HEERMANN"
print "===================================================="
print "================================================"
print" DEFINITION OF THE SIMULATION PARAMETERS"
print "================================================"
NPART = 256
SIDE = 6.75284
TREF = 0.722
DEN = 0.83234:PRINT"REDUCED DENSITY=",DEN
RCOFF = 2.5:PRINT "RCOFF=",RCOFF
H = 0.064
IREP = 10:PRINT "REPLACEMENT VELOCITIES EVERY=",IREP;"th-STEP"
TIMEMX = 3:PRINT"TIME MAX=",TIMEMX
ISEED = 4711:PRINT"ISEED=",ISEED
print "================================================"
print" SET THE ORDER PARAMETER"
print "================================================"
print "BROWNIAN DYNAMICS SIMULATION PROGRAM"
print " '--------------------------------------------------"
print" NUMBER OF PARTICLES IS ", NPART
print" SIDE LENGTH OF THE BOX IS ",SIDE
print" CUT OFF IS " ,RCOFF
print" REDUCED TEMPERATUREE IS ",TREF
print" BASIC TIME STEP IS ", H
A =SIDE/4.0
SIDEH =SIDE*0.5
HSQ =H*H
HSQ2 =HSQ*0.5
NPARTM =NPART-1
RCOFFS =RCOFF*RCOFF
TSCALE =16.0/NPARTM
VAVER =1.13*(TREF/24)^0.5
IOF1 =NPART
IOF2 =2*NPART
N3 =3*NPART
PRINT"A=",A,"SIDEH =",SIDEH,"HSQ=",HSQ,"HSQ2=",HSQ2
PRINT"NPARTM=",NPARTM,"RCOFFS=",RCOFFS,"TSCLAE=",TSCALE
PRINT"VAVER=",VAVER,"IOF1=",IOF1,"IOF2=",IOF2,"N3=",N3 19
print "CALL RANSET (ISEED)"
-Hasil Eksekusi dari program Brownian Dynamics-1
BROWNIAN DYNAMICS
APPLICATION TO ARGON.THE LENNARD-JONES POTENTIAL IS TRUNCATED
AT RCOFF AND NOT SMOOTHLY CONTINUED TO ZERO.INITIALLY THE
NPART PARTCLES ARE PLACED ON AN FCC LATTTICE.THE VELOCITIES
ARE DRAWN FROM A BOLTZMANN DISTRIBUTION WITH TEMPERATURE TREF.
INPUT PARAMETERS ARE AS FOLLOWS
NPART = NUMBER OF PARTICLES (MUST BE A MULTIPLE OF four)
SIDE = SIDE LENGTH OF THE CUBICAL BOX IN SIGMA UNITS
TREF = REDUCED
DEN = REDUCED DENSITY
RCOFF = CUTOFF OF THE POTENTIAL IN SIGMA UNITS
H = BASIC TIME STEP
IREP = REPLACEMENT OF THE VELOCITIES EVERY IREP'TH
TIME STEP
TIMEMX = NUMBER OF INTEGRATION STEPS
ISEED = SEED FOR THE RANDOM NUMBER GENERATOR
VERSION 1.0 AS OF AUGUST 1985
DIETER W.HEERMANN
===============================================
DEFINITION OF THE SIMULATION PARAMETERS
================================================
REDUCED DENSITY= 0.83234
RCOFF= 2.5
REPLACEMENT VELOCITIES EVERY= 10th-STEP
TIME MAX= 3
ISEED= 4711
================================================
SET THE ORDER PARAMETER
================================================
BROWNIAN DYNAMICS SIMULATION PROGRAM
'--------------------------------------------------
NUMBER OF PARTICLES IS 256
SIDE LENGTH OF THE BOX IS 6.75284
CUT OFF IS 2.5
REDUCED TEMPERATUREE IS 0.722
BASIC TIME STEP IS 0.064
A= 1.68821 SIDEH = 3.37642 HSQ= 0.004096 HSQ2=
0.002048
NPARTM= 255 RCOFFS= 6.25 TSCLAE= 0.62745098e-1
VAVER= 0.19599339 IOF1= 256 IOF2= 512 N3= 768
CALL RANSET (ISEED)
-Source codes untuk Konfigurasi Awal
print "==============================================="
print " THIS PART OF THE PROGRAM PREPARES THE INITIAL CONFIGURATION"
print ================================================"
print " SET UP FCC LATTICE FOR THE ATOMS INSIDE THE BOX"
print "-------------------------------------------------------------------"
A= 1.68821
DIM X(1000):DIM Y(1000): DIM Z(1000)
IJK =0
FOR LG= 0 TO 1 20
FOR I= 0 TO 3
FOR J= 0 TO 3
FOR K= 0 TO 3
IJK = IJK + 1
X(IJK )=I*A+LG*A*0.5
Y(IJK )=J*A+LG*A*0.5
Z(IJK )=K*A
PRINT"X(";IJK;")=",X(IJK)
PRINT"Y(";IJK;")=",Y(IJK)
PRINT"Z(";IJK;")=",Z(IJK)
NEXT K
NEXT J
NEXT I
NEXT LG
FOR LG= 1 TO 2
FOR I= 0 TO 3
FOR J= 0 TO 3
FOR K= 0 TO 3
IJK =IJK +1
X(IJK ) =I*A+(2-LG)*A*0.5
Y(IJK )=J*A+(LG-1)*A*0.5
Z(IJK )= K*A +A*0.5
PRINT"X(";IJK;")=",X(IJK)
PRINT"Y(";IJK;")=",Y(IJK)
PRINT"Z(";IJK;")=",Z(IJK)
NEXT K
NEXT J
NEXT I
NEXT LG
-Hasil Eksekusi dari program untuk Konfigurasi Awal:
======================================================
THIS PART OF THE PROGRAM PREPARES THE INITIAL CONFIGURATION
======================================================
SET UP FCC LATTICE FOR THE ATOMS INSIDE THE BOX
-------------------------------------------------------------------
X(1)= 0
Y(1)= 0
Z(1)= 0
X(2)= 0
Y(2)= 0
Z(2)= 1.68821
X(3)= 0
Y(3)= 0
Z(3)= 3.37642
X(4)= 0
Y(4)= 0
Z(4)= 5.06463
X(5)= 0
Y(5)= 1.68821
Z(5)= 0
X(6)= 0
Y(6)= 1.68821
Z(6)= 1.68821
……………………….
Y(252)= 4.220525 21
Z(252)= 5.908735
X(253)= 5.06463
Y(253)= 5.908735
Z(253)= 0.844105
X(254)= 5.06463
Y(254)= 5.908735
Z(254)= 2.532315
X(255)= 5.06463
Y(255)= 5.908735
Z(255)= 4.220525
X(256)= 5.06463
Y(256)= 5.908735
Z(256)= 5.908735
-Source codes untuk Maxwell-Boltzmann-1
DIM VH(1000)
H= 6.75284
TREF= 0.722
H= 0.064
N3= 768
print"CALL RANSET (ISEED)"
print" ASSIGN BOLTZMANN DISTRIBUTED VELOCITIES AND CLEAR THE FORCES"
print" ==================================================="
print" CALL MXWELL (VH,N3,H,TREF)"
PRINT" SUBROUTINE MXWELL(VH,N3,H,TREF)"
NPART = N3 / 3
IOF1 = NPART
IOF2 = 2 * NPART
TSCALE = 16.0 / (1.0 * NPART - 1.0)
FOR I=1 TO N3 STEP 2
1 U1 = Rnd(4711)
U2 = Rnd(4711)
V1 =2.0 * U1 - 1.0
V2 =2.0 * U2 - 1.0
S =V1 * V1 +V2 *V2
IF ( S >= 1.0 )THEN GOTO 1
R = -2.0 * log (S) / S
VH(I) = V1 * R^.5
VH(I+1)= V2 * R^.5
10 NEXT I
EKIN = 0.0
SP = 0.0
FOR I=1 TO NPART
SP =SP + VH (I)
20 NEXT I
SP = SP / NPART
FOR I=1 TO NPART
VH (I) = VH (I) - SP
EKIN = EKIN + VH (I) * VH (I)
21 NEXT I 22
SP = 0.0
FOR I=IOF1 + 1 TO IOF2
SP = SP + VH (I)
22 NEXT I
SP = SP / NPART
FOR I=IOF1 + 1 TO IOF2
VH(I) = VH (I) - SP
EKIN = EKIN + VH(I) * VH(I)
23 NEXT I
SP = 0.0
FOR I=IOF2 + 1 TO N3
SP = SP + VH (I)
24 NEXT I
SP = SP / NPART
FOR I=IOF2 + 1 TO N3
VH(I) = VH (I) - SP
EKIN = EKIN + VH (I) * VH (I)
25 NEXT I
TS = TSCALE * EKIN
SC = TREF / TS
SC = (SC)^0.5
SC = SC * H
FOR I=1 TO N3
VH (I) = VH (I) * SC
PRINT"VH(";I;")=",VH(I)
30 NEXT I
END
DO until I=N3
F(I)=0.0
loop
-Hasil Eksekusi dari program Maxwell-Boltzmann-1
:
VH(1)= 0.20989832e-2
VH(2)= -0.12618094e-1
VH(3)= -0.82723833e-2
VH(4)= -0.10924963e-2
VH(5)= 0.93732508e-2
………………………………..
VH(761)= -0.99742066e-2
VH(762)= -0.77111357e-2
VH(763)= -0.49706784e-2
VH(764)= 0.35381167e-3
VH(765)= -0.13525497e-1
VH(766)= 0.15105383e-1
VH(767)= 0.10939923e-1
VH(768)= -0.71259251e-2
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