Molecular Dynamics Basics (4.1, 4.2, 4.3) Liouville formulation (4.3.3) Multiple timesteps (15.3)

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Molecular Dynamics

Basics (4.1, 4.2, 4.3)

Liouville formulation (4.3.3)

Multiple timesteps (15.3)

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Molecular Dynamics

• Theory:

• Compute the forces on the particles• Solve the equations of motion• Sample after some timesteps

2

2

dt

dm

rF

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Molecular Dynamics• Initialization

– Total momentum should be zero (no external forces)– Temperature rescaling to desired temperature– Particles start on a lattice

• Force calculations– Periodic boundary conditions– Order NxN algorithm,– Order N: neighbor lists, linked cell– Truncation and shift of the potential

• Integrating the equations of motion– Velocity Verlet– Kinetic energy

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Periodic boundary conditions

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Lennard Jones potentials

612

4rr

ru LJ

c

cLJ

rr

rrruru

0

c

ccLJLJ

rr

rrrururu

0

•The truncated and shifted Lennard-Jones potential

•The truncated Lennard-Jones potential

•The Lennard-Jones potential

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Phase diagrams of Lennard Jones fluids

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Saving cpu time

Cell list Verlet-list

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Equations of motion 4

32

!32tt

tt

m

tttttt

rfvrr

42

2 ttm

tttttt

frrr

tm

tttttt frrr

2

2

Verlet algorithm

Velocity Verlet algorithm

tttm

tttt

tm

tttttt

ffvv

fvrr

2

2

2

432

!32tt

tt

m

tttttt

rfvrr

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Lyaponov instability 1

10

0 , 0

0 , 0 , , 0 , 0 , 0

10

N N

Nj i N

r p

r p p p p

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NNf rp ,

pp

rr

ff

f

iL

r pr p

exp i 0f t Lt f

Liouville formulationDepends implicitly on t

Liouville operatordi

d

fLf

t

Solution

Beware: this solution is equally useless as

the differential equation!

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pp

rr

pr LLL iii

0iexp ftLtf r

00exp ft

r

r

0!

0

0

fn

t

nn

nn

r

r

NN tf 00,0 rrp

0iexp ftLtf p

00exp ft

p

p

0!

0

0

fn

t

nn

nn

p

p

0,00 NNtf rpp

Shift of coordinates Shift of momenta

t000 rrr t000 ppp

Taylor expansion

Let us look at them separately

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0,0, 2ii2i NNPtLtLtLNN feeettf prp rprp

0,0, i NNLtNN fettf rprp

tLr 000i rrr tLp 000i ppp

0,0ii NNtLtL fe pr rp 0,0ii NNtLtL fee pr rp

We have noncommuting operators!

Trotter identityBABA eee

PPAPBPAP

BA eeee 2/2lim

PPAPBPABA eeee 2/2

P

tL

P

A pi

P

tL

P

B ri

P

tt

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i rL t t r r r

i pL t t p p p

0,02

00,02i NN

NNtL tffe p rpprp

NN

NN

tL tt

tf

tfe r

20,0

200,0

20i rrpprpp

NN

NNtL

ttt

ttf

tt

tfe p

20,

20

20

....................2

0,02

02i

rrppp

rrpp

)0(2

00200

02

00

2

Frrrrr

pppp

m

tttt

tt

Velocity Verlet!

)()(2

)(2

1 2

tttm

tttt

ttm

ttttt

FFvv

Fvrr

i 2 i 2ip pr

PL t L tL te e e

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i 2 i 2ip prL t L tL te e e

Velocity Verlet:

i 2: 2

2pL t t

e t t t t t tm

v v f

i 2:

2 2pL t t t

e t t tm

v v f

i : 2rL te t t t t t t r r vvx=vx+delt*fx/2

x=x+delt*vxCall force(fx)

vx=vx+delt*fx/2

Call force(fx)

Do while (t<tmax)

enddo

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Liouville FormulationVelocity Verlet algorithm:

)(2

22

tm

tttttt

tttt

ttt

Frrr

pppp

Three subsequent coordinate transformations in either r or p of which the Jacobian is one: Area preserving

Other Trotter decompositions are possible!

1

102

1det

22

1

r

rF

rr

rFpp

tJ

tt

tttt

11

01det

2

22

2

mtJ

ttm

tttt

tttt

prr

pp

1

102

1det

22

3

r

rF

rr

rFpp

tJ

tt

tt

tttt

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Multiple time steps• What to use for stiff potentials:

– Fixed bond-length: constraints (Shake)– Very small time step

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Multiple Time steps

v

F

rv

m

LLL pr iii

longshort FFF

v

F

m

L shortshorti

v

F

m

L longlongi

longshort iii LLLp

Trotter expansion: 2ii2ii longshortlongshortlong eeee tLtLLtLtLLL

rr Introduce: δt=Δt/n

2in2ii2i2ii longshortshortlongshortlong eeeeee tLtLtLtLtLtLLLrr

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mtFvrfvrftL 200,0)0(,0e long2i long

2in2ii2i2ii longshortshortlongshortlong eeeeee tLtLtLtLtLtLLLrr

long long

short short

i 2 2

i 2 2

i r

L t v v F t m

L t v v F t m

L t r r v t

Now n times:

mtFvrftLtLtL r 200,0eee long

n2ii2i shortshort

First

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Longi 2

long:2 2

pL t t te t t t

m

v v f

Shorti 2

Short:2 2 2 2

pL t t t t te t t t t t

m

v v f

Shorti 2

Short:2 2 2 2

pL t t t t te t t t

m

v v f

i : 2 2rL te t t t t t t t r r v

vx=vx+ddelt*fx_short/2

x=x+ddelt*vxCall force_short(fx_short)

vx=vx+ddelt*fx_short/2

Call force(fx_long,f_short)

Do ddt=1,n

enddo

vx=vx+delt*fx_long/2

long long

short short

i 2 2

i 2 2

i r

L t v v F t m

L t v v F t m

L t r r v t

long longshort shortni 2 i 2i 2 i 2ie e e e er

L t L tL t L tL t

long long

short short

i 2 2

i 2 2

i r

L t v v F t m

L t v v F t m

L t r r v t

long longshort shortni 2 i 2i 2 i 2ie e e e er

L t L tL t L tL t

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