CE 530 Molecular Simulation

Transcription

1 CE 530 Molecular Smulaton Lecture 3 Molecular Dynamcs n Other Ensembles Davd A. Kofke Department of Chemcal Engneerng SUY Buffalo kofke@eng.buffalo.edu

2 Revew Molecular dynamcs s a numercal ntegraton of the classcal equatons of moton Total energy s strctly conserved, so MD samples the VE ensemble Dynamcal behavors can be measured by takng approprate tme averages over the smulaton Spontaneous fluctuatons provde non-equlbrum condton for measurement of transport n equlbrum MD on-equlbrum MD can be used to get less nosy results, but requres mechansm to remove energy va heat transfer Two equvalent formalsms for EMD measurements Ensten equaton Green-Kubo relaton tme correlaton functons

3 3 Molecular Dynamcs n Other Ensembles Standard MD samples the VE ensemble There s need enable MD to operate at constant T and/or P wth standard MD t s very hard to set ntal postons and veloctes to gve a desred T or P wth any accuracy PT MD permts control over state condtons of most nterest EMD and other advanced methods requre temperature control Two general approaches stochastc couplng to a reservor feedback control Good methods ensure proper samplng of the approprate ensemble

4 Thermodynamc defnton T kt S = E What s Temperature? V, = ln Ω( EV,, ) E temperature descrbes how much more dsordered a system becomes when a gven amount of energy s added to t hgh temperature: addng energy opens up few addtonal mcrostates low temperature: addng energy opens up many addtonal mcrostates Thermal equlbrum Q umber of mcrostates havng gven E Dsordered: more ways to arrange system and have t look the same entropy s maxmzed for an solated system at equlbrum total entropy of two subsystems s sum of entropy of each: consder transfer of energy from one subsystem to another f entropy of one system goes up more than entropy of other system goes down, total entropy ncreases wth energy transfer equlbrum establshed when both rates of change are equal (T =T ) (temperature s guaranteed to ncrease as energy s added) tot 4 S = S + S

5 Momentum and Confguratonal Equlbrum 5 Momentum and confguraton coordnates are n thermal equlbrum Er (, p ) = K( p ) + Ur ( ) momentum and confguraton coordnates must be at same temperature or there wll be net energy flux from one to other An arbtrary ntal condton (p,r ) s unlkely to have equal momentum and confguratonal temperatures and once equlbrum s establshed, energy wll fluctuate back and forth between two forms...so temperatures wll fluctuate too Ether momentum or confguratonal coordnates (or both) may be thermostatted to fx temperature of both assumng they are coupled p Q r

6 6 An Expresson for the Temperature. Consder a space of two varables schematc representaton of phase space Contours show lnes of constant E standard MD smulaton moves along correspondng 3 dmensonal hypersurface Length of contour E relates to Ω(E) Whle movng along the E A contour, we d lke to see how much longer the E B contour s xe Analyss yelds kt x x E B E A x E A Relates to = E E B gradent and rate of change of gradent

7 Momentum Temperature 7 Knetc energy K( p ) = = p m Gradent Laplacan p p x y pk = eˆx + eˆ m m = d p pk = + = m m m = y d = Temperature kt kt = p p K K p = + d / m = m m = d = p m p x y The standard canoncal-ensemble equpartton result

8 8 Confguratonal Temperature Potental energy U ( r ) Gradent U U ru = eˆ + eˆ = F eˆ + F eˆ ( ) x y x x y y = rx ry = Laplacan Temperature F U x r r = + = rx kt U = U r r = F = F x + = rx F r F r y y y y Butler, B. D., G. Ayton, O. G. Jepps, and D. J. Evans Confguratonal temperature: verfcaton of Monte Carlo smulatons. J. Chem. Phys. 09, 659.

9 9 Lennard-Jones Confguratonal Temperature Sphercally-symmetrc, parwse addtve model Force Laplacan F U( r ) u ( r ) = r r j = j j = j< du dr j j j j ulj 6 σ σ () r 4ε = r r r du LJ 48ε σ σ = r r dr σ r r 4 8 F r du du α = r r r r dr r dr jα j j α j j j j j j j d du LJ 67ε σ σ = rdr r dr 4 σ r 7 r 6 0.B. Formulas not verfed

10 0 Thermostats All PT MD methods thermostat the momentum temperature Proper samplng of the canoncal ensemble requres that the momentum temperature fluctuates momentum temperature s proportonal to total knetc energy energy should fluctuate between K and U varance of momentum-temperature fluctuaton can be derved from Maxwell-Boltzmann fluctuatons vansh at large rgdly fxng K affects fluctuaton quanttes, but may not matter much to other averages σtp Tp = 3 All thermostats ntroduce unphyscal features to the dynamcs EMD transport measurements best done wth no thermostat use thermostat equlbrate r and p temperatures to desred value, then remove kt = = p = d m d p Q r K

11 Isoknetc Thermostattng. Force momentum temperature to reman constant One (bad) approach at each tme step scale momenta to force K to desred value advance postons and momenta apply p new = λp wth λ chosen to satsfy repeat ( λ ) p m equatons of moton are rreversble transton probabltes cannot satsfy detaled balance does not sample any well-defned ensemble = dkt

12 Isoknetc Thermostattng. One (good) approach modfy equatons of moton to satsfy constrant!r = p / m!p = F λp λ s a frcton term selected to force constant momentum-temperature K = dk dt = p = m = p p!p m = (F m λp ) 0 = λ = Tme-reversble equatons of moton no momentum-temperature fluctuatons confguratons properly sample VT ensemble (wth fluctuatons) temperature s not specfed n equatons of moton! m m p p F p

13 Thermostattng va Wall Collsons 3 Wall collson mparts random velocty to molecule selecton consstent wth (canoncal-ensemble) Maxwell-Boltzmann dstrbuton at desred temperature Gaussan p π ( p) = exp d / ( π mkt ) mkt random p Advantages realstc model of actual process of heat transfer correctly samples canoncal ensemble Dsadvantages can t use perodc boundares wall may gve rse to unacceptable fnte-sze effects not a problem f desrng to smulate a system n confned space not well suted for soft potentals Wall can be made as realstc as desred

14 4 Andersen Thermostat Wall thermostat wthout the wall Each molecule undergoes mpulsve collsons wth a heat bath at random ntervals Collson frequency ν descrbes strength of couplng Probablty of collson over tme dt s νdt Posson process governs collsons Smulaton becomes a Markov process Π= νδt Π + νδt Π ( ) ( ) VT VE Π VE s a determnstc TPM t s not ergodc for VT, but Π s Pt (; ν) = νe νt Clck here to see the Andersen thermostat n acton random p

15 5 osé Thermostat. Modfcaton of equatons of moton lke soknetc algorthm (dfferental feedback control) but permts fluctuatons n the momentum temperature ntegral feedback control Extended Lagrangan equatons of moton ntroduce a new degree of freedom, s, representng reservor assocate knetc and potental energy wth s L ose = momenta = m (s!r ) p L!r = m s!r p s L s = Q!s U (r ) + Q!s gkt lns U s = gkt lns K s = Q!s effectve mass

16 7 osé Thermostat. Extended-system Hamltonan s conserved H ose = p = m s +U (r ) + p s + gkt lns Q Thus the probablty dstrbuton can be wrtten π(r,p,s, p s ) = δ (H ose E) What does ths mean for the samplng of coordnates and momenta? How does ths ensure a canoncal dstrbuton?

17 8 osé Thermostat 3. Q osé = dp! s dsdp dr δ (H osé E) = p dp! s dsd p dr s 3 δ +U (r ) + p s + gkt ln s E m Q p = p s δ h(s) = δ (s s ) 0 h (s 0 ) Get canoncal ensemble for s, p' f g = 3(+) s can be nterpreted as a tme-scalng factor Δt true = Δt sm /s s vares durng smulaton, so true tme step s of varyng length

18 9 osé Thermostat 3. Q osé = dp! s dsdp dr δ (H osé E) = p dp! s dsd p dr s 3 δ +U (r ) + p s + gkt lns E m Q = 3 + s dp! s dsd p dr gkt δ s exp gkt H( p,r ) + p s Q E =! = C! gkt e E ( 3 +) gkt Get canoncal ensemble for s, p' f g = 3(+) s can be nterpreted as a tme-scalng factor Δt true = Δt sm /s dp s e ( 3 +) gkt d p dr exp p s Q d p dr exp 3( +) gkt H( p,r ) 3( +) gkt H( p,r ) p = p s δ h(s) = δ (s s ) 0 h (s 0 ) s vares durng smulaton, so true tme step s of varyng length

19 H ose = p = m s +U (r ) + p s + gkt lns Q osé-hoover Thermostat. Advantageous to work wth non-fluctuatng tme step r = r p = p / s s = s 0 Δ t = Δt / s Scaled-varables equatons of moton constant smulaton Δt fluctuatng real Δt!r = H p = p m s!p = H r = F!s = H = p s p s Q!p s = H s = s = p m s gkt d r = s dr d t dt = s p ms = p ms = p m Real-varables (' removed) equaton of moton (sp s / Q) t!r = p m!p = F sp s Q p!s s = sp s Q = Q = p m gkt

20 osé-hoover Thermostat. Real-varable equatons are of the form!r = p m!p = F ξp!s s = ξ!ξ = Q = p m gkt Compare to soknetc equatons!r = p / m!p = F λp (redundant; s s not present n other equatons) λ = m m p p Dfference s n the treatment of the frcton coeffcent osé-hoover correctly samples VT ensemble for both momentum and confguratons; soknetc does VT properly only for confguratons F p

21 osé-hoover Thermostat 3. Equatons of moton!r = p m!p = F ξp!s s = ξ!ξ = Q = p m gkt Integraton schemes predctor-corrector algorthm s straghtforward Verlet algorthm s feasble, but trcky to mplement r v F t-dt t t+dt At ths step, update of ξ depends on p; update of p depends on ξ!p = F ξp!ξ = Q = p m gkt

22 Barostats 3 Approaches smlar to that seen n thermostats constrant methods stochastc couplng to a pressure bath extended Lagrangan equatons of moton Instantaneous vral takes the role of the momentum temperature P(r,p ) = kt p (p ) V + 3V! r j f! j pars,j Scalng of the system volume s performed to control pressure Example: Equatons of moton for constrant method!r = p / m + χ(r,p )r!p = F χ(r,p )p χ(t) s set to ensure dp/dt = 0! V = 3V χ(r,p )

23 4 Summary Standard MD smulatons are performed n the VE ensemble ntal momenta can be set to desred temperature, but very hard to set confguraton to have same temperature momentum and confguraton coordnates go nto thermal equlbrum at temperature that s hard to predct eed ablty to thermostat MD smulatons ad ntalzaton requred to do EMD smulatons Desrable to have thermostat generate canoncal ensemble Several approaches are possble stochastc couplng wth temperature bath constrant methods more rgorous extended Lagrangan technques Barostats and other constrants can be mposed n smlar ways