Applied Statistical Mechanics Lecture Note - 13 Molecular Dynamics Simulation

Transcription

1 Appled Statstcal Mechancs Lectue Note - 3 Molecula Dynamcs Smulaton 고려대학교화공생명공학과강정원

2 Contents I. Basc Molecula Dynamcs Smulaton Method II. Popetes Calculatons n MD III. MD n Othe Ensembles

3 I. Basc MD Smulaton -MC vs. MD MC MD Pobablstc smulaton technque Lmtatons eque the knowledge of an equlbum dstbuton goous samplng of lage numbe of possble phase-space gves only confguatonal popetes (not dynamc popetes!) Detemnstc smulaton technque Fully numecal fomalsm numecal soluton of N-body system A d N A( d N N exp N { βu ( )} N { βu ( )} )exp

4 I. Basc MD Smulaton - The Idea Follow the exactly same pocedue as eal expements Pepae sample pepae N patcles solve equaton of motons Connect sample to measung nstuments (e.g. themomete, vscomete, ) afte equlbaton tme, actual measuement begns Measue the popety of nteest fo a cetan tme nteval aveage popetes Example : measuement of tempeatue α mv k B T T ( t) mv ( t) k N B f

5 I. Basc MD Smulaton - Equaton of Moton Classcal Newton s equaton of moton Thee fomulaton Newtonan Lagangan Hamltonan Hamltonan pefeed fo many-body systems soluton of N dffeental equatons t m t F p p Soluton methods : Fnte Dffeence Method j j j z y x z y x p p p ),, ( ),, ( F F p p

6 I. Basc MD Smulaton - Velet Algothm Velet (967) : Vey smple, effcent and popula algothm 3 4 m m 3! 3 4 m m 3! ( t + δt) () t + p() t δt + F() t δt + () t δt + O( δt ) ( t δt) () t p() t δt + F() t δt () t δt + O( δt ) m 4 ( t + δt) + ( t δt) ( t) + F( t) δt + O( δt ) m 4 ( t + δt) ( t) ( t δt) + F( t) δt + O( δt ) featue : update wthout calculatng momentum (p)

7 I. Basc MD Smulaton - Leapfog Algothm Hockeny (970), Potte (97) Half-step leap-fog algothm Mathematcally equvalent to Velet algothm m ( t + δt) ( t) + v( t + δt) δt v( t + δt) v( t δt) + F( t) δt ( t + δt) ( t) + ( t t) v δ + F( t) δtδt m

8 . Popetes Calculaton n MD - Eneges Potental enegy Can be calculated dung foce calculaton Knetc enegy K mv

9 . Popetes Calculaton n MD - Pessues In an MD smulaton, calculaton of pessue usng tenso notaton s not the most effcent method. Fo homogeneous systems, thee s smple way to calculate pessue (Ivng and Kkwood, 950) Knetc deal gas tem Confguatonal called Val P ρk V B T + m V v ( t) v j ( t) + j ( t) F j j ( t) j ( t) F j ( t) Calculate when foce update Calculate when velocty update

10 . Popetes Calculaton n MD - Tanspot Popetes Appoaches fo tanspot popetes Method : NEMD (Non-equlbum Molecula Dynamcs) Contnuous addton and emoval of conseved quanttes Gves hgh sgnal-to-nose ato (good statstcs) Method : Equlbum molecula dynamcs Stat wth ansotopc confguaton of mass, momentum and enegy Obseve natual fluctuatons and dsspaton of mass, momentum and enegy Poo sgnal-to-nose aton (poo statstcs) All tanspot popetes can be measued at once

11 . Popetes Calculaton n MD - Tanspot Popetes Dffeental Balance Equaton Mass Enegy Momentum c(, t) t + j 0 T (, t) Dv(, t) c p + q 0 ρ + τ 0 t Dt Consttutve Equatons Fck s Law Foue s Law Newton s Law j D c q k T τ v ρv ) xy y ( x

12 . Popetes Calculaton n MD - Tanspot Popetes Pupose : Obtan tanspot coeffcent by molecula smulaton Not that the laws ae only appoxmaton that apply not-too-lage gadents In pncple tansfe coeffcents depends on c, T and v Geen-Kubo Relaton Relaton between tanspot popetes and ntegal ove tme-coelaton functon.

13 . Popetes Calculaton n MD - Tanspot Popetes Consde self-dffuson n a pue substance Consde how molecules ae dsspated when ntal confguatons ae gven as Dac delta functon Combne mass balance eqn. Wth Fck s Law Dmensonalty of gven system c(, t) D c(, t) 0 t B.C. c(, t) δ ( ) Soluton c(, t) (πdt) / d exp( ) Dt

14 . Popetes Calculaton n MD - Tanspot Popetes We do not need concentaton tself c(,t) - just dffuson coeffcent (D) < ( t) > c(, t) d c(, t) d < ( t) t > dd c(, t) D c(, t) t 0 < ( t) > ( ( t)) ddt N

15 . Popetes Calculaton n MD - Tanspot Popetes < ( t) > ( ( t)) ddt N Slope hee gves D t Plot of t vs. squae of taveled dstance gves dffuson coeffcent In 3D space, < > s mean squae dsplacement (MSD)

16 . Popetes Calculaton n MD - Tanspot Popetes An altenatve fomulaton usng velocty nstead of patcle poston t ( t) v( τ ) dτ 0 t () t v( τ ) dτ v( τ ) dτ t 0 0 dτ dτ v( τ ) v( τ ) 0 0 t τ dτ dτ v( τ ) v( τ ) 0 0 t τ dτ dτ v(0) v( τ τ ) 0 0 t t dτ dτ v(0) v( τ) 0 0 t ddt t dτ v(0) v( τ) 0 t t

17 . Popetes Calculaton n MD - Tanspot Popetes t ddt t dτ < v(0) v( τ ) > 0 t D d 0 dτ < v(0) v( τ ) > < v( 0) v( τ ) > < v( 0) v( τ ) >< v( t') v( t'' ) > Autocoelaton functon : popety dffeence between two adjacent tme steps Aea unde the cuve gves the value of self-dffuson coeffcent τ

18 . Popetes Calculaton n MD - Evaluaton of tme coelaton functons Tme consumng and eque a lot of stoage Altenatve method : FFT (Fast Foue Tansfom), Coase Ganng method A(t 0 ) A(t ) A(t ) A(t 3 ) A(t 4 ) A(t 5 ) A(t 6 ) A(t 7 ) A(t 8 ) A(t 9 ) A( Δt) A(0) A 0 A + A A + A A 3 + A 3 A 4 + A 4 A 5 + A 5 A 6 + A 6 A 7 + A 7 A 8 + A 8 A A(t 0 ) A(t ) A(t ) A(t 3 ) A(t 4 ) A(t 5 ) A(t 6 ) A(t 7 ) A(t 8 ) A(t 9 ) A( Δt) A(0) A 0 A + A A 3 + A A 4 + A 3 A 5 + A 4 A 6 + A 5 A 7 + A 6 A 8 + A 7 A A(t 0 ) A(t ) A(t ) A(t 3 ) A(t 4 ) A(t 5 ) A(t 6 ) A(t 7 ) A(t 8 ) A(t 9 ) A(5 Δ t) A(0) A 0 A 5 + A A 6 + A A 7 + A 3 A 8 + A 4 A

19 . Popetes Calculaton n MD - Tanspot Popetes Zeo-shea vscosty η VkT 0 dτ < σ xy (0) σ xy ( τ ) > σ xy x y m v v + x j f y ( j ) j Themal Conductvty d λt > q mv + u( j ) VkT dt j dτ < q(0) q( τ ) 0

20 . Popetes Calculaton n MD - Radal Dstbuton Functon Tme aveaged value of numbe densty Ensemble aveaged numbe densty g ( ) ρ ( ) ρ V g ( ) < δ( j ) N j > Just count the numbe of molecules wthn a ange

21 3. MD n Othe Ensembles - Constants Wth pope choce of g(), we can calculate useful themodynamc popetes Intenal enegy Pessue Chemcal Potental U c P 0 πnρ φ( ) g( ) d ρkt πρ 3 0 dφ( ) 3 g( ) d d 3 ρλ π dv ' μ kt ln N ' q nt 3 N V V 0 dφ( ) d g( ) 3 d

22 3. MD n Othe Ensembles Constants Hamltonan fomulaton Consevaton of knetc + potental enegy H K + U (N,V,E) ensemble Cannot be appled to othe ensemble constant T, constant P, fo example we can keep const T whle H s constant dstbuton of K and U Two types of constants Holonomc constants : may be ntegated out of equaton of moton Nonholonomc constants : non-ntegable (nvolves veloctes) Tempeatue, pessue, stess,

23 3. MD n Othe Ensembles Constants Foce momentum tempeatue to eman constant One (bad) appoach at each tme step scale momenta to foce K to desed value advance postons and momenta apply p new λp wth λ chosen to satsfy epeat equatons of moton ae evesble tanston pobabltes cannot satsfy detaled balance does not sample any well-defned ensemble

24 3. MD n Othe Ensembles Constants Gauss pncple of least constants Gaussan constants : petubatve foce ntoduced nto the equaton of moton mnmzes the devaton to classcal tajectoes of patcles fom the unpetubed tajectoes Consde a functon f, a functon of patcle acceleaton f ( ) m F m f0 : nomal Newtonan equaton of moton othewse, constaned non-newtonan equaton of moton Gauss pncple : physcal acceleaton f to be mnmum ( f ( ) ζ g( )) 0 ζ : Lagangn (Gauss) Multple

25 3. MD n Othe Ensembles Constants Constant Tempeatue constants 0 ),, ( m dt dg t g 0 3 ), ( NkT m t G ( ) 0 ) ( ) ( g f ζ 0 j j j j j j j m m m F m m F ζ Newtonan Constant foce

26 3. MD n Othe Ensembles Constants Modfed equaton of moton m m F p p ζ m F ζ one of good appoach, but tempeatue s not specfed!

27 3. MD n Othe Ensembles Nose Themostat Extended Lagangan Equaton of Moton N m( s ) N Q LNose U( ) + s gkt lns p p L ms s L s Qs s Ks Qs U gktln s

28 3. MD n Othe Ensembles Nose-Hoove Themostat Equatons of moton p m p F ξp s ξ s N p ξ gkt Q m Integaton schemes pedcto-coecto algothm s staghtfowad Velet algothm s feasble, but tcky to mplement v F t-δt t t+δt At ths step, update of ξ depends on p; update of p depends on ξ p F ξp N p ξ Q m gkt