2pm chemistry seminar zhang mit
TRANSCRIPT
Lecture 8 .-
Monte Carlo sampling
@2pm , chemistry seminar,
Bin Zhang , MITin 2 weeks @ 2pmZob Distasio
, Cornell
Today : Nobel Prize in Physics 2021Georgio Parisi
Reminder 40> =fdÑOWPW= { ✗ , , Xz , . . . , Xp , Pi , Pa . . . - , Psv}
☒ = { ×. . . - , Asu) F- {A, . . - Psn}
☒ ={tip}itptx
, -51=17×71757it Oli,p→1= 01×-3
@ constant N,HT
Pfxip , = e-BYKingz
E-fdxfdpe-mteiip-itttiypt-EP.IM;t Uli )
PC Iii') = eIaH
2- ice Zpe
PLÉ ) = §dp→ PCÉ,p→ ) =ftp.PCEIPcp?fdpPCp-7--I
-BUCH= C
[ position'D
Generate a set of configurations
{ It } where these are distributedaccording to PCE )
{ ✗+3 wait
Playpen, =
⇐ e- Piuij
Dilip = Ulxj ) - Uki)
Algorithm: ( want]✗c- → ✗ c- +1 St
P( ✗c- til /p(✗+, = e-PM±tyt
•Generate a Markov Chain
i. → i. → ii. →ii. - -¥"
Markovian " : no memory
P(Ñt→Ñt+, ) only depends on it
[ not É , . . - ¥?
Eg Molecular Dynamics
¥,
-
- Et ist Liii ] ii. pjm
iii. , = É+ + Est
want :-Bluth
,'t
Patti )|p(✗+,→ e
converges ? as 1-→✗
① Markovian② Detailed Balance &
D.etailedy.BE#pg A
¥:*DB:
P(A) - PCA→B) = PCB) - PCB→A)
✗t → ✗c- + ,
→ ✗c- +2
-
PCA) PEX t →✗++1) = Plitt, )P(✗c-+i>✗t)
choosing aru6J~
P ( ✗→ y)↳ Pgen ( ✗ → y) Paccept (✗→y)
Generation, make new config y
Accept /reject going to y" rate
"
PIN Pgen (✗→ y l Paul✗→ g) = r(✗→y)
Ply) Pgenly → x) Pauly→x) /Pace ( ✗→ y)=[¥,P§g¥y ] Pauly→x)- -
Metropolis Monte-Carlo 119531
MTTRR → Manhattan Pngecf• - - ' ' Lusitanos Computing Center
Pace (✗→y ) = min ( 1,rlx →y) )
Paul ✗→ y)=[¥,P§h¥f) Pauly→xD1- -
Pace (✗→yl = r(✗→y) Pace Cy→x )
Pace ( y → a) = rly -5×1Pace ( ✗→ y)
r (✗→ y ) = →×)
② r( ✗ → y) 71 st My -3×1<1
⑥rlx→ y ) < 1 St pr(y→x) > 1
Race =mn[ 1,rcx ->yl]
←
Pace (✗→y)-
= r( ✗→y )
Pauly -3×1 F)a) Pace /✗ →g) = 1
Pauly -1×1 = Yr(✗→ y) )b) Pace ( ✗ →g) = rC✗→y ) < 1
Pauly -5×1=1
Algorithm : start at config It① Propose Xttl W/ prob Pages 1×+0×+1 )-
generate rand number"a"
[0,1)fhiforn -
② if a < min [ 1 , rlx+→ ✗t.is] ←
accept,more to Xtti
else :
✗1- + i = ✗ t
③ go back to I
711×1 pl = PEN +& kid
sample
Plxl = e-PUCH
If e-pacy=
§nT¥e→±k"
"
✗¥÷
Move rule , that is symmetricpropose :
✗tti = ✗+ + az.by ←
rzt ( -1 , 1) uniform random
by biggest possible move
Xttz = ✗c-+ , Taz. §
Pga ( y→ x)
1¥>g)= I rlx-syi-e-f.tn
Pace ( ✗+→ ✗+a) = min ( 1 , e-Pd" )
Pace ( ✗+→ ✗++ 1) = min ( 1 , e-Pd" )
= " ""
how doesat
compareto last
9current posmove 1
, energy goesdown
U ( Xz ) - Ucx , I <0,e-Pd"
> 1,alwaysaccept
Move 2 Ulxz)-uh , , > 0 , e-PAUL 1
,acceptprobe-Blu
Tune our moves , here §St average acceptance rate ~
0.25-3 0.5
Tradeoff between efficiency &explorationif E is large , each accepted
more will go far,but most
moves will be rejectedby is really small , almost always accept
but stay close to starting point
to be this/i¥qDenture
simpleI 2
① pick an atom [ atom 47
② generate a random move
Ñy(1- til =IyCH + random
REC-1 , 1)
Ii + [mamarandom ☒ { ]random . {
③then calc e- Psu for whole system
Random move
① more com of molecule
✗can= ✗
can✗ it {
② rotate molecule by arandom angle
Why MC & why not :
① easy
② CI choose very smart
types of moves, jump our
energy barriers st . explorationis very fast
WH:① not real dynamics [gives
static properties]
② usually only tiny charges accepted
Do we have to use Mei nieGlauber Rule
- PAULPaulk?y) = e- ←
e- Path + etpbulz
why? All = Uly )- UCHxy
buy ✗ = UCH-Uly) = -All✗y
%¥⇒, -- e-nunsor
acceptrate Mmetrope's
i
Go%
-xtx