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Molecular Dynamics
Basics (4.1, 4.2, 4.3)
Liouville formulation (4.3.3)
Multiple timesteps (15.3)
Molecular Dynamics
• Theory:
2
d r
Fm 2
dt
• Compute the forces on the particles
• Solve the equations of motion
• Sample after some timesteps
2
3
4
5
6
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
7
Periodic boundary conditions
8
Lennard Jones potentials
•The Lennard-Jones potential
12
6

 
  
LJ
u r   4      
 r  
 r 
•The truncated Lennard-Jones potential
u LJ r  r  rc
ur   
r  rc
 0
•The truncated and shifted Lennard-Jones potential
u LJ r   u LJ rc  r  rc
ur   
0
r  rc

9
Phase diagrams of Lennard Jones fluids
10
Saving cpu time
Cell list
Verlet-list
11
Equations of motion
t 2
t 3
rt    t 4
rt  t   rt   vt t 
f t  
2m
3!
t 2
t 3
rt    t 4
rt  t   rt   vt t 
f t  
2m
3!
t 2
rt  t   rt  t   2rt  
f t    t 4
m
 
 
 
Verlet algorithm
t 2
rt  t   2rt   r t  t  
f t 
m
Velocity Verlet algorithm
t 2
r t  t   r t   vt t 
f t 
2m
t
f t  t   f t 
vt  t   vt  
2m
12
r  0 , p  0
r  0 , p 0 ,
N
Lyaponov instability
N
N
1
, p j  0    , pi  0   
, p N 0

  1010
13

f pN ,rN

Liouville formulation
f  r f  p f
r
p


iL  r  p
r
p
df
 iLf
dt
Solution
Depends implicitly on t
Beware: this solution
Liouville
operatoruseless as
is equally
the differential
equation!
f t   exp iLt  f  0
14


iL  iLr  iL p  r  p
r
p
 
f t   exp iLr t  f 0

 
 

 exp  r 0 t  f 0 
 exp  p 0t  f 0
r 
p 


n
n
n
n








r 0t  
p
0
t

Taylorexpansion



f
0
f 0

n
n
n! r
n! p
n 0
n 0
N
N
 f p N 0, r 0  r 0t 
 f p0  p 0t  , r N 0
Let us look at them
f t   exp
iL p t f 0
separately



Shift of coordinates
Shift of momenta
r0  r0  r 0t
p0  p0  p 0t

15
iLr  r0  r0  r 0t
iL p  p0  p0  p 0t


e e

f p t , rA B t  A Be
NWe
e
Trotter identity iLr t iL p t 
e A B

 
f p t , r t   e
N
N

N

f p 0P, r 0
 lim P  e A 2 P eiLB / tPe A 2 P 
iLr t 
N
N
p


0

e
e
f
p
0
,
r
A 2P B / P A 2P P
e
e
 e e e
B iLr t
A iL p t


P
P
iLPt 2  P
A B

f p 0, r 0
haveN noncommuting
iLt  operators!
N
p
e
N
N


iLr t 
e

t
t 
iL t 2 PP
p
 f p
N
0, r 0
N
16

iLr t  r  r  rt
iLp t  p  p  pt
e
iL p t
iLp t 2 iLr t  iLp t 2
e
e
e
N



t


2
N
N
N
f p 0, r 0  f  p0  p 0 , r 0


2





N
N
N







t

t

t





Velocity
iLr t   
N Verlet!

e
f p0  p 0 , r 0  f p0  p 0 , r0  tr   



2
2
2  







1
2
N
N








r
t


t

r
t

v
t

t

F
(
t
)

t
t
t 
 
iL t 2   
     ....................
e p
f p0   p 0  , r 0  2trm

2
 
 2  

t

vt  t   v t  
F (t )  FN(t  t )
N

t 2m t
t  




f p0   p 0   p t  , r 0   tr  
 

2
2

t
p0   p0   p 0   p t 
2



 2  
t 2
r 0   r 0   tr t 2   r 0   tr 0  
F(0)
2m
17

P
Velocity Verlet:
iLp t 2 iLr t 
e
e
iLp t 2
e
Call force(fx)
Do while (t<tmax)
e
iLp t 2
 iLr t 
e
e
t
 t 
: v  t    v t  
f t 
2
2m

vx=vx+delt*fx/2
: r  t  t   r  t   tv  t  t 2 
iLp t 2
x=x+delt*vx
Call force(fx)
t
: v  t  t   v  t  t 2  
f  t  t 
2m
vx=vx+delt*fx/2
enddo
18
Liouville Formulation
Velocity Verlet algorithm:
t
p t   p t  t 
2
t 2
r t  t   r t   tr t  
F(t )
2m
pt  t   pt  
Three subsequent coordinate transformations in either r or p of
which the Jacobian is one: Area preserving
t
t
pt  t 2   pt   Fr  pt  t 2   pt  t 2 




p
t


t

p
t


t
2

Fr t 
2
2
t
r t  t   r t   pt  t 2
r t   r t 
r t   r t 
m
t Fr 
t Fr 
1
J1  det
2
0
r
1
1
1
0
J 2  det
1
t m 1
J 3  det
1
0
2
r
1
1
Other Trotter decompositions are possible!
19
Multiple time steps
• What to use for stiff potentials:
– Fixed bond-length: constraints (Shake)
– Very small time step
20
F  Fshort  Flong
Multiple
Time steps
 F 
iL  iLr  iL p  v 
r m v
iL p  iLshort  iLlong
iLshort
Fshort 

m v
Trotter expansion:


i Llong Lshort Lr t
e
Flong 
iLlong 
m v
iLlongt 2 i  Lshort Lr t iLlongt 2
e
e
e
Introduce: δt=Δt/n


i Llong Lshort Lr t
e
iLlongt 2
e
e
e
iLshortt 2 iLrt iLshortt 2 n iLlongt 2
e
e
21
e


i Llong Lshort Lr t
e
iLlongt 2
e
e
iLshortt 2 iLr t iLshortt 2 n
e
e
iLlongt 2
iLlong t 2  v  v  Flong t 2m
iLshort  t 2  v  v  Fshort  t 2m
iLr t  r  r  v t
First
e
iLlongt 2


f r 0, v(0)  f r 0, v0  Flong 0 t 2m
Now n times:
e
 f r0, v0  F
iLshortt 2 iLrt iLshortt 2 n
e
e
long
0t

2m
22
e
t
 t 
: v  t    v t  
flong  t 
2 
2m

 iLpLong t 2
iLlong t 2
iLshort  t 2 iLr  t iLshort  t 2
e
Call force(fx_long,f_short)
e
vx=vx+delt*fx_long/2
Do ddt=1,n
e
iLpShort t 2

e
e
n
 e
iLlong t 2
iLlong t 2  v  v  Flong t 2m
iLshort  t 2  v  v  Fshort  t 2m
 t  t 
 t   t
: v  t     v  t iLr t  r frShort
 vtt
2 2
2  2m


vx=vx+ddelt*fx_short/2
 iLr t 
e
: r  t   t   r  t    tv  t  t 2   t 2 
x=x+ddelt*vx
iLlong t 2
iLshort  t 2 iLr  t iLshort  t
Call force_short(fx_short)

e
e
e e
e
iLpShort
n
iLlong t 2

e


 t 2
 t

 t  t   t
: v  t    t   ivLlong
t t 2 v
 vF
fShort
t 2mt 
long 

2 2  2m

iLshort  t 2  v  v  Fshort  t 2m
vx=vx+ddelt*fx_short/2
23
enddo
iLr t  r  r  v t
2

2
24