Deonstrate Teachng FM, Partcle Dnacs, and Sple MD Codes Usng Matlab & Matheatca Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 1 Recall fro Lecture: the Lagrangan Method 1. The choce of s generalzed coordnates (s nuber of degrees of freedo).. Dervaton of the knetc and potental energ n ters of the generalzed coordnates 3. The dfference between the knetc and potental energes gves the Lagrangan functon. 4. Substtuton of the Lagrange functon nto the Lagrangan equaton of oton and dervaton of a sste of s second-order dfferental equatons to be solved. 5. Soluton of the equatons of oton, usng a nuercal te-ntegraton algorth. 6. Post-processng and vsualzaton. Now, soe exaples Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst
Lagrange Functon: xaples Pendulu q1 = ϕ φ l L= l ϕ + gl & cos( ϕ) 1) Knetc energ; ) Potental energ due to external gravtatonal feld, where g s the acceleraton of gravt). Bouncng ball q1 = = & α L g e β 1) Knetc energ; ) Potental energ due to external gravtatonal feld; 3) Potental energ of repulson between the ball and floor. Partcle n a crcular cavt q = x, q = 1 x L ( x& & ) e β ( R x ) = + α + 1) Knetc energ; ) Potental energ of repulson between the partcle and cavt wall. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 3 Pendulu: Lagrange Functon Dervaton General for L= T( & ϕ) U( ϕ) Knetc energ tangental veloct T = vτ = l & ϕ l ϕ vτ = l = l & ϕ t 0 t l φ v τ l cosφ Potental energ U = gh= glcosϕ Potental energ depends on the heght h wth respect to a zero-energ level. Here, such a level s chosen at the suspenson pont,.e. below the suspenson level, the heght s negatve. The total Lagrangan = ϕ + L l gl & cos( ϕ) Paraeters used: g = 9.8/s, l = Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 4
Pendulu: quaton of Moton and Soluton Lagrange functon and equaton oton: d L L g = & + cos = 0 &&( ) + sn ( ( )) = 0 dt & ϕ ϕ l L l ϕ gl ϕ ϕ t ϕ t Intal condtons (radan): Paraeters: g = 9.8/s, l = ϕ.3, & ϕ ϕ(0) = 1.8, & ϕ ϕ(0) = 3.1, & ϕ Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 5 Bouncng Ball: Lagrange Functon Dervaton General for L= T( & ) U( ) Knetc energ T = & Potental energ U = g +αe β The total Lagrangan: & & Purple dashed lne: the frst ter (gravtatonal nteracton between the ball and the arth). Blue dotted lne: the second ter (repulson between the ball and the bouncng surface). Red sold lne: the total potental. L(, ) = g αe β β s a relatve scalng factor: the ball-surface repulsve potental growths n e β tes for a unt length ball/surface penetraton (fro = 0 to = 1). Paraeters used: g = 9.8/s, = 1kg, 1 α = 1J, β = 4 Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 6
Bouncng Ball: quaton of Moton and Soluton Lagrange functon and equaton oton: β d L L αβ β () t L = & g α e = 0 && ( t) + g e = 0 dt & Intal condtons: Paraeters: g = 9.8/s, = 1kg, α = 1J, β = 4 1 (0) = 5 (0) = 10 (0) = 15 & & & Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 7 Partcle n a Crcular Cavt: Lagrange Functon Dervaton General for L= T( x&, & ) U( x, ) Knetc energ Potental energ T = x& + & ( ) ( R r) ( R x ) β U = αe = αe β + R r x The potental energ grows quckl and becoes larger than the tpcal knetc energ, when the dstance r between the partcle and the center of the cavt approaches value R. R s the effectve radus of the cavt. At r < R, U does not alter the trajector. β s a relatve scalng factor: the potental energ growths n e β tes between r = R and r = R+1. The total Lagrangan ( R x ) Lxx (,, &, &) = ( x& + & ) αe β + Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 8
Partcle n a Crcular Cavt: quaton of Moton and Soluton Lagrangan functon and equatons: L= x + e ( R x ) x &&() t = αβe x()/ t r() t β ( R r( t)) β + (& & ) α β ( R r( t)) &&() t αβe ()/ t r() t = Potental barrer: α = = + 7 10 J, rt () x() t () t Paraeters: β = = = Intal condtons: 1 9 4n, R 10n, 10 kg x(0) =.5n, x&,, & (0) = 30n/s Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 9 Recall fro Lecture: Both Stable and Unstable Trajectores Possble Stable quasperodc trajector Unstable chaotc trajector Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 10
Transton fro Stable to Chaotc Moton: xaple Intal condtons l1 = l = 0.6 ϕ1(0) = 1.9rad, ϕ(0) = 1.55rad & ϕ (0) = & ϕ 1 ϕ1(0) = 1.9rad, ϕ(0) = 1.7rad & ϕ (0) = & ϕ 1 Perodc oton Chaotc oton Note the transton fro a stable to chaotc oton (sall varance of ntal condtons a lead to qualtatve change of soluton behavor of the sae non-lnear sste). Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 11 Recall fro Lecture: the Haltonan Method 1. The choce of s generalzed coordnates (s nuber of degrees of freedo).. Dervaton of the knetc and potental energ n ters of the generalzed coordnates. 3. Dervaton of the generalzed oenta. 4. xpresson of the knetc energ n ters of the generalzed oenta. 5. The su the knetc and potental energes gves the Haltonan functon. 6. Substtuton of the Haltonan functon nto the Lagrangan equaton of oton and dervaton of a sste of s frst-order dfferental equatons to be solved. 7. Soluton of the equatons of oton, usng a nuercal te-ntegraton algorth. 8. Post-processng and vsualzaton. Now, the exaples Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 1
Haltonan quatons of Moton: Pendulu Lagrange functon and the generalzed oentu: L L= l & ϕ + glcosϕ p = = l & ϕ & ϕ l 脨 Haltonan of the sste: p p T =, U = gl cosϕ H = gl cosϕ l l quatons of oton: ϕ H H p & =, p& = &, p gl sn p ϕ ϕ = & = l ϕ Converson to the Lagrangan for (elnaton of p): g & = && ϕ = snϕ && ϕ+ snϕ = 0 l p l gl Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 13 Haltonan quatons of Moton: Bouncng Ball Lagrange functon and the generalzed oentu: & L & ϕ β L= g e p = = & Haltonan of the sste: p β p T =, U = g+ e H = + g+ e quatons of oton: H H p & =, p& = & =, p& = g+ e p Converson to the Lagrangan for (elnaton of p): β β β β β p& = && = g+ βe && + g e = 0 β Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 14
Recall fro Lecture: The Phase Space Trajectores: xaples xaples of phase space trajectores: (Projecton to the plane x, p x ) Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 15 Recall fro Lecture: the MD Sulaton Procedure Model ndvdual partcles and boundares. Model nteracton between partcles and between partcles and boundares. Assgn ntal postons and veloctes. Solve the equatons of oton. Sulate the oveents of the sste. Analze the sulaton data to nvestgate collectve phenoena and behavor of acroscopc paraeters. Now, soe exaples Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 16
Perodc Boundar Condtons: xaple xaple sulaton of an atoc cluster wth perodc boundar condtons. A partcles, gong through a boundares returns to the box fro the opposte sde: Ths odel s equvalent to a larger sste, coprsed of the translaton age boxes: Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 17 Adabatc xaple: Interactve Partcles n a Crcular Chaber Repulsve nteracton between the partcles and the wall s descrbed b the wall functon, a one-bod potental that depends on r dstance between the partcle and the chaber s center): β ( R r ) Wwl ( r ) e e β Interacton between partcles s odeled wth the two-bod Lennard-Jones potental (r j dstance between partcles and j): W LJ The total potental: 1 6 σ σ ( rj ) = 4ε 1 6 rj r j U = W ( r) + W ( r ), wl LJ j j> r = x + ( R x ) = α = α + r = ( x x ) + ( ) j j j r j r r j R x Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 18
Three Partcles: quaton of Moton and Soluton The total potental: U = W ( r) + W ( r ) + W ( r ) wl 1 wl wl 3 + W ( r ) + W ( r ) + W ( r ), LJ 1 LJ 13 LJ 3 r = x + r = ( x x ) + ( ) j j j quatons of oton: U U x && =, && =, = 1,,3 x Paraeters: β = = = 1 9 4n, R 10n, 10 kg Intal condtons (n, /s): x, x& (0) = 5, (0) = 3.0, & 1 1 1 1 x, x& (0) = 30, (0) = 0.5, & x, x&, (0) =.5, & 3 3 3 3 Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 19 Post-Processng: nerg = + tot kn pot ( x ) 3 kn = & + & = 1 pot U Knetc, potental and total energ vs. te (three partcles) The total energ (sold red lne): 9.65x10-5 J. Ths value does not var n te Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 0
Post-Processng: Knetc nerg, Teperature and Pressure Te averaged knetc energ of partcles s approachng the value = 3.057 10 J kn 5 whch corresponds to teperature 5 1 kn 3.057 10 T = = 0.0K 3 k 1.38 10 Note: a low teperature sste was chosen n order to observe the real-te atoc oton. Pressure n the sste s due to the radal coponents of veloctes: Averaged knetc energ vs. te (three partcles) N 9 P = vrad, vrad = vx cosγ + v snγ P= 3.3 10 Pa V Knetc energ, and therefore teperature and pressure are due to oton of the partcles. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 1 Post-Processng: Internal nerg Averaged potental energ vs. te (three partcles) Te averaged potental energ of the sste s approachng the value U = 5 0.480 10 J that gves the nternal energ of the sste. Internal energ s due to nteracton of partcles wth each other and wth external constranng felds. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst
Fve Partcles: quatons of Moton and Soluton The total potental: U = W ( r) + W ( r ) + W ( r) + W ( r ) + W ( r ) wl 1 wl wl 3 wl 4 wl 5 + W ( r ) + W ( r ) + W ( r ) + W ( r ) LJ 1 LJ 13 LJ 14 LJ 15 + W ( r ) + W ( r ) + W ( r ) LJ 3 LJ 4 LJ 5 + W ( r ) + W ( r ) + W LJ LJ 34 LJ 35 ( r ) 45 quatons of oton: U U x && =, && =, = 1,,3, 4,5 x Paraeters: 1 β = 4n, R = 10n, 9 = 10 kg Intal condtons (n, n/s): x, x& (0) = 5, (0) = 5.6, & 1 1 1 1 x, x& (0) = 30, (0) = 3.0, & x, x&, (0) = 0.5, & 3 3 3 3 x, x& (0) = 4, (0) =.5, & 4 4 4 4 x, x& (0) =, (0) = 4.9, & 5 5 5 5 Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 3 Post-Processng: nerg tot kn pot = = U kn + 5 = 1 ( x& pot + & ) Knetc, potental and total energ vs. te (fve partcles) The total energ (sold red lne): 3.66x10-5 J. Ths value does not var n te Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 4
Post-Processng: Knetc nerg, Teperature and Pressure Te averaged knetc energ of partcles s approachng the value =.847 10 kn 5 whch corresponds to teperature 1 T = k kn Note: a low teperature sste was chosen n order to observe the real-te atoc oton. J = 0.041K Averaged knetc energ vs. te (fve partcles) Knetc energ, and therefore teperature and pressure are due to oton of the partcles. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 5 Post-Processng: Internal nerg Averaged potental energ vs. te (fve partcles) Te averaged potental energ of the sste s approachng the value U = 0.813 10 5 that gves the nternal energ of the sste. J Internal energ s due to nteracton of partcles wth each other and wth external constranng felds. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 6
Fve Partcles n a Rough Wall: quatons of Moton and Soluton The total potental: U = WLJ, wall + WLJ, partcles + W quatons of oton: Paraeters: 1 β = 4n, R = 10n, 9 = 10 kg Intal condtons (n, n/s): wall The sste potental s the su of the L.J. nteractons between the partcles, the partcles and the wall and a crcular wall potental U U && x =, & =, = 1,,3,... x x1, x& 1(0) = 5, 1(0) = 5.6, & 1 x, x& (0) = 30, (0) = 3.0, & x3, x& 3, 3(0) = 0.5, & 3 x4, x& 4(0) = 4, 4(0) =.5, & 4 x, x& (0) =, (0) = 4.9, & 5 5 5 5 + 14 statc partcles representng the rough wall! Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 7 Fve Partcles n a Rough Wall: Matheatca Code Integrate the equatons of oton n te to obtan the trajectores of the partcles Defne Sste Potental, ncludng L.J. Potentals and wall potental, along wth sulaton paraeters Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 8
Post-Processng: nerg tot kn pot = = U kn + 5 = 1 ( x& pot + & ) Knetc, potental and total energ vs. te (fve partcles) The total energ (sold red lne): 3.66x10-5 J. Ths value does not var n te Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 9 Post-Processng: Knetc nerg, Teperature and Pressure Te averaged knetc energ of partcles s approachng the value =.680 10 kn 5 whch corresponds to teperature 1 T = k kn Note: a low teperature sste was chosen n order to observe the real-te atoc oton. J = 0.039K Averaged knetc energ vs. te (fve partcles) Knetc energ, and therefore teperature and pressure are due to oton of the partcles. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 30
Post-Processng: Internal nerg Averaged potental energ vs. te (fve partcles) Te averaged potental energ of the sste s approachng the value U = 0.645 10 5 J that gves the nternal energ of the sste. Internal energ s due to nteracton of partcles wth each other and wth external constranng felds. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 31 Isotheral xaple: 1D Lattce wth a Cold Regon Cold regon Total nuber of partcles s large. Interacton between partcles s odeled wth the two-bod haronc potental: 1 W ( r ) = kr h j j here, r j = j s the relatve dstance between partcles and j n the vertcal (-axs) drecton, k lnear nteracton coeffcent (slar to sprng stffness). Several atos n the ddle of the chan (between the ellow dashed lnes) represent the ntall cold regon. Intal veloctes and dsplaceents for these atos are zero. The reanng atos have randol dstrbuted ntal veloctes and dsplaceents. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 3
1D Lattce: quatons of Moton and Soluton Nuber of partcles sulated: 50. Boundar condtons are perodc, so that the couplng between the 50 th and 1 st partcles s establshed. For a ore setrc vew, the sulaton shows the 1 st partcle at both ends of the lattce. The total potental (the last ter s due to perodc boundar condtons):): 49 1 U = Wh( + 1) + Wh( 1 50), Wh( j) = k( j) = 1 quatons of oton: U && =, = 1,,...50 Paraeters: Intal condtons: = = 1 1 10 kg, k 10 N/ ddle atos..7 : (0) = &, other atos : (0), & (0) rando wthn [ 1,1] Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 33 Post-Processng: Knetc nerg and Teperature Te averaged knetc energ of partcles n the cold subsste s approachng the value 0.4 Averaged knetc energ per partcle vs. te (for the ntall cold subsste) = 0.86 10 J kn 1 whch corresponds to teperature kn T = k 1 0.86 10 = 3 1.38 10 41.4K Knetc e nerg, 1 0-1 J 0.3 0. 0.1 Te averaged qulbru value 0 100 00 300 400 500 Te, s In contrast to the adabatc sste exaple, the knetc energ for the open sotheral subsste both fluctuates and approaches asptotcall the statstcal average. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 34
Post-Processng: Internal nerg Averaged potental energ vs. te ( cold subsste) Te averaged potental energ of the cold subsste s approachng the value U = 5 0.480 10 J that gves the nternal energ of the subsste. Potental e nerg, 1 0-1 J 1.5 1 0.5 Te averaged Internal energ asp 0 100 00 300 400 500 Te, s In contrast to the adabatc sste exaple, the nternal energ for the open sotheral subsste both fluctuates and approaches asptotcall the statstcal average. Wng Ka Lu, duard G. Karpov,Harold Park, Davd Farrell, Dept. of Mechancal ngneerng, Northwestern Unverst 35