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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