2pm chemistry seminar zhang mit

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

run;¥¥¥"÷,¥+

pY.bpar1ForcanonicaIensen@fpygpyy.e-p[ucy, .ua,]

e-P[HÉnp% - Hiii:p? ,

Mike generation symmetric

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