CE 530 Molecular Simulation

Transcription

1 CE 530 Molecular Smulaton Lecture 22 Chan-Molecule Samplng Technques Davd A. Kofke Department of Chemcal Engneerng SUNY Buffalo

2 2 Monte Carlo Samplng MC method permts great flexblty n developng mproved samplng methods Basng methods mprove samplng thout changng the lmtng dstrbuton Modfcaton of tral probabltes compensated by changes n acceptance and reverse-tral probabltes Non-Boltzmann samplng methods modfy the lmtng dstrbuton Desred ensemble average obtaned by takng a eghted average over the non- Boltzmann sample M = Me β ( U U ) 0 β ( U U ) e 0 0 W W W W 0 0 W G

3 3 Smulatng Chan Molecules Slo to explore dfferent parts of phase space Concerted moves needed to detangle chans Algorthms based solely on sngle-atom moves may be nonergodc

4 Modelng Chan Molecules Detaled models use full array of potentals dscussed prevously LJ atoms, th torson, bend, stretch ntramolecular potentals Other models try to explan qualtatve features of polymer behavor hard- or soft-sphere atoms, only stretch bead-sprng; tangent spheres; fntely-extensble nonlnear elastc (FENE) each unt of model mght represent a mult-unt segment of the true polymer the only feasble approach for very long chans >0 3 unts Lattce models are very helpful dscretze space varous choces for lattce symmetry chan occupes contguous stes on lattce one chan unt per ste 4

5 5 Generatng Confguratons of Chans Open ensembles (grand-canoncal) often preferred nserton and removal of chans enhances samplng of confguratons Insertons and removals are dffcult! We ll examne three approaches Smple samplng Confguratonal bas Pruned-enrched samplng Consder methods n the context of a smple hard-excluson model (no attracton, no bendng energy) All non-overlappng chan confguratons are eghted unformly

6 6 Smple Samplng Molecules are nserted and deleted n an unbased fashon Stepse nserton: after j segments have been nserted, the (j +)th segment s placed at random at one of the stes adjonng the last segment Any attempt that leads to an overlap th an exstng segment causes the hole tral to be mmedately dscarded π = 3 reject π = 3 contnue tral π = 3 reject Successful nserton

7 7 Smple Samplng: Inserton Lkelhood What s the probablty that ths tral ll occur usng smple nserton? 63 stes In-class assgnment fgure t out

8 8 Smple Samplng: Inserton Lkelhood What s the probablty that ths tral ll occur usng smple nserton? 63 stes Inserton probablty for frst unt /63 Insert sx more unts, each th probablty /3 gong n the rght spot /3 6 Could begn on ether end of chan multply by 2 Total probablty s product τ j = 2= =

9 Confguratonal Bas Monte Carlo 9 Based on 955 dea of Rosenbluth & Rosenbluth Apply bas durng groth of chan, so that overlaps do not lead to rejecton of entre tral Remove bas durng acceptance of complete tral Accumulate Rosenbluth eght durng course of tral π = 2 π = 2 contnue tral π = 0

10 0 Confguratonal-Bas Inserton/Deleton Tral. Analyss of Tral Probabltes Detaled specfcaton of tral moves and and probabltes Event [reverse event] Select nserton tral [select deleton tral] Place molecule at {r} [delete molecule N+] Probablty [reverse probablty] ½ [½ ] /W({r}) [/(N+)] Forard-step tral probablty Reverse-step tral probablty mn(, χ) 2 W mn(, ) 2 N + χ Accept move [accept move] mn(,χ) [mn(,/χ)] We ll ork ths out later

11 Confguratonal-Bas Inserton/Deleton Tral. Analyss of Detaled Balance Forard-step tral probablty mn(, χ) 2 W Reverse-step tral probablty mn(, ) 2 N + χ Detaled balance π π j = π j π j Lmtng N π dstrbuton ( r ) ( ) N U qt ( ) e β r + = βµ Ξ N

12 2 Confguratonal-Bas Inserton/Deleton Tral. Analyss of Detaled Balance Forard-step tral probablty mn(, χ) 2 W Reverse-step tral probablty mn(, ) 2 N + χ e old Detaled balance βu + βµ N βu + βµ ( N+ ) Ξq π π j = π j π j ne e mn(, ) mn(, χ ) 2 W = q 2 N χ Ξ + N ( N+ ) Energy s zero n both confguratons qt ( ) χ = e W N + χ = qt ( ) We N + + βµ + βµ Acceptance probablty

13 Rosenbluth Weght What s W? /W s the probablty that the chan ould be nserted nto the gven poston 3 Each placement of a unt n the chan s selected th probablty π = j here s the number of non-overlap sblng alternatves avalable at generaton of the overall nserton Probablty of makng ths partcular nserton s τ = = W = = 252

14 4 NVT Confguraton Samplng CBMC s also used to generate ne confguratons of present molecules A B Acceptance of any move s based on Rosenbluth eght for gven move and the reverse move W A = 63 W B = 252 The move A à B s accepted th probablty The move B à A s accepted th probablty 63/252 = /4

15 5 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) Before, ths as a sum of terms ether zero or one And probablty of selectng ste j s π j = e βu j e βu τ = 2 = 2 (for example)

16 6 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) And probablty of selectng ste j s π j = e βu j 2 2 e βu 2 = 2 τ = W = 63

17 7 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) And probablty of selectng ste j s π j = e βu j 2 e βu 2 = 2 τ = W = 63 2

18 8 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) And probablty of selectng ste j s π j = e βu j 2 2 e βu 2 = 2 τ = W =

19 9 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) And probablty of selectng ste j s π j = e βu j e βu 2 = τ = W =

20 20 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) And probablty of selectng ste j s π j = e βu j e βu 2 = 2 τ = 4 2 4? W = ? In-class assgnment 2 Get the next term

21 2 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) And probablty of selectng ste j s π j = e βu j 2 e βu 2 = τ = W =

22 22 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) And probablty of selectng ste j s π j = e βu j e βu 2 = 2 τ = W =

23 23 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) And probablty of selectng ste j s π j = e βu j e βu 2 = τ = W =

24 24 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) And probablty of selectng ste j s π j = e βu j e βu 2 = 2 τ = = W = = 36288

25 Attractve Interactons Molecules th attracton Generalzaton uses Boltzmann factor to formulate Rosenbluth eght At each step eght s k u e β j= = ( j) 25 And probablty of selectng ste j s π j = e βu j e βu 2 = 2 τ = = W = = W s used just as before: accept th proby mn[,w ne /W old ] energy contrbuton s bult-n: τ βu e = = βu e W

26 26 A General Result for Markov Processes. Consder a process n hch there are several ays to generate each tral à j ( a) τ j ( b) τ j ( c) τ j To enforce detaled balance, all routes should be consdered n formulatng acceptance probablty ( a) ( b) ( c) ( a) ( b) ( c) π τj + τj + τ j mn[, χ] = π j τ j + τ j + τ j mn[,/ χ] If there are many ays to generate the tral, ths can pose dffcultes

27 27 A General Result for Markov Processes. Consder the follong recpe for a sngle-step tral Generate the tral à j va route (a), th probablty τ j (a) Choose a reverse tral j à va one of the routes, say (b) Choose t th probablty that t ould occur as the j à route Probablty = τ j (b) /τ j Accept the (forard) tral as f (a) and (b) ere the only routes ( a) ab ( b) j = j j πτ mn[, χ ] π τ mn[,/ χ ] Ths recpe satsfes detaled balance for the overall transton à j ab

28 Off the Lattce CBMC can be extended to off-lattce models Choose a set of tral orentatons at random for each atom nserton 28 et cetera Once a chan s generated n ne poston, perform same operaton tracng out ts orgnal locaton Comple Rosenbluth eght for ne and orgnal chans to use n acceptance W e βu j atoms trals j Note that each nserton may be accomplshed va multple routes, dfferng n the dscarded atom trals = ( )

29 29 CBMC General Comments Method begns to fal for suffcently long chans maybe as fe as 0 atoms Extensons of method Gbbs ensemble Branched polymers Partal chan regroth Chemcal-potental calculaton General dea can be appled n other ays Mult-step tral broken nto smaller decsons, th acceptance ncludng consderaton of the choces not taken

30 Parallel Temperng. 30 At hgh temperature a broader range of confguratons s sampled Barrers to transtons are loered Lo temperature Ensemble eght, e -u/kt escape trapped Hgh temperature Phase space, Γ Ho to smulate a lo-temperature system th hghtemperature barrer removal?

31 Parallel Temperng 2. Smulate loosely coupled hgh- and lo-temperature systems n parallel Lo temperature Hgh temperature 3 Perform moves n hch to systems sap confguratons Accept based on H( 2 ) L( 2) ( βh βl)( U2 U) e e = e = e β U U β U U ΔβΔU

32 32 Parallel Temperng 3. To get reasonable acceptance rate, temperatures should not be too dfferent Can be extended to nclude any number of systems smulated n parallel Can be extended to do temperng n other varables, such as the chemcal potental Very ell suted for use n conjuncton th hstogram reeghtng

33 33 Pruned-Enrched Rosenbluth Method At some pont along the groth process t may become clear that the chan s doomed, or the chan s really dong ell We d lke to enrch the presence of the good ones, hle prunng out the ones that look bad Use a crteron based on partal Rosenbluth eght

34 34 Pruned-Enrched Rosenbluth Method Not so good. Tme to prune W = 3/2 W = 3 W = 3 W = 9 2 W = 3 So far, so good. Let s make another W = 3/2 W = 3 W = 9 Make another W = 9 2 Other branches W = 2 Prune W = 4

35 35 Pruned-Enrched Rosenbluth Method Set cutoffs for ntermedate Rosenbluth eghts duplcate any confguraton havng W > W >, halvng eghts of ne duplcates prune confguratons havng W < W <, takng every-other such confguraton, and doublng the eght of those not taken