Transcript Incorporating Hydrodynamics into Monte Carlo Simulations R C Ball, Physics Theory Group and Centre for Complexity Science University of Warwick assoc.

Incorporating Hydrodynamics into
Monte Carlo Simulations
R C Ball,
Physics Theory Group and
Centre for Complexity Science
University of Warwick
assoc. member
Centre for Scientific Computing
Outline
• Monte Carlo simulation and Diffusion
• Challenge of long range hydrodynamic
coupling – important in many soft matter
systems.
• Fourier method: order N^2 per unit time
– application to polymer dynamics
• Wavelet adaptation approaching order N
MC of particle systems
Configuration x
"Move"
n=3N cpts for N particles in 3D
 xi   Kij   j
i j   ij
Metropolis Paccept  min 1, eU / kBT 
for small enough  :
k  0
 correct eq'm
Paccept  1 and
 x  x   2 K  KT
identifiable with D  K  K T and  2  2 t
Equilibration purposes: keep K simple, accepting simple D.
General: K ( D) by Cholesky decomposition, cost ~ n3 .
More practical interpretation
different directions for  as different types of move:
 x ( p )   w( p )
scalar  ,
 2 =1
p  chosen move type (most simply, which atom to move)
2Dij  t  
2
 P( p) w
( p)
p
( p)
i
w
j
Particles in a fluid
diagonal blocks
off-diagonal
k BT
D pp 
I
6 a p
k BT
D pq 
I  rˆpq rˆpq 

8 rpq
p
 O(rpq )
a
rpq
q
long range form & full matrix
valid when diffusion rate of fluid momentum fastest
k BT

v

,
r2
r
r3
advection rate configurational relax'n rate
Micro Hydrodynamics
• Macromolecules
• Bacteria
• Colloids
Some studies just use Cholesky …
Banchio & Brady: long range part slow to change – update less often (JCP 2003)
Present approach: (i) Fourier (ii) Wavelets
Fourier Approach
O(r  r ') 

k  kmax
kmax 3

6 2


d 3 k k BT
ˆ ˆ cos k .(r  r ')
I

kk
3
2

k
 2 
k BT
2 2
eˆ cos  k .r    eˆ cos  k .r '  
2
k
 2kmax 3kBT 

 wkeˆ (r ) wkeˆ (r ') keˆ
 3 
eˆ
wkeˆ (r )  cos  k .r   
k
kmax 

2a
phase

pol'n
eˆ  k
k  kmax
unit strain transverse wave
Limitation to timestep
2kmax 3k BT
actual strain  =
3
estimated energy change U  VG  Nk BT 
2
2
required  k BT
 t    N
2

1
3
kmax k BT
computational cost N
2
costing order N
per natural unit of time
a
3
kBT
Polymer chain
N=1000
monomers
Using ‘phantom’ chains so
that eq’m initial configurations available
Polymer diffusion
ln d (t )
2
6
CoM
Monomers rel to CoM
4
4
6
monomer motion
10
2
0
Centre of Mass motion
2
12
ln t
N=100 monomer chains
7
6
4
6
2
Out[131]=
Out[125]=
5
4
6
2
11
12
13
14
4
3
N=400
N=1000
8
10
12
Hydrodynamic scaling
ln d (t )
N=1000
monomers
d t
abs
rel CoM
2
4
1/3
2
d  t1/4
2
4
monomers
6
10
d  t1/2
2
4
Centre of Mass
ln t
Wavelet version (untested)
Fundamental wavelet identity (1 dimension):
db da  x  a 
 b2  b f  b  f
 x ' a 

  N[ f ] ( x  x ')
 b 
 f (u)du  0
Generalise and adapt in 3D 
db d 3a d 2u  r  a   r ' a 
O (r  r ')   4  3 
bv 
, u  bv 
,u 
b
b
2
 b
  b

.v=0,
v  1 & cts, v  0
ra
 1,
b
v  0
rewrite O as
Move limits and costs
 k BT db 3 d 2u
O(r  r ') 
d a
4 


b
2
Nk BT

wbau (r ) wbau (r ')
3
 bmin
r a 
v
,u 

 b3  b
b
 r ' a 
v
,u 

 b3  b
b
b , a ,u
3
NkBT
1 bmin
timestep set by 
1  t  N
3
bmin
kBT
2
L
av cost per step c  bmin
3

bmin
(per Fourier)
db
L
3
c(b, a, u )  b   ln
4 
b
b
L
 computational cost N log
per natural unit of time.
b
Limitations and Prospects
• Imposing relative motion and strain: applicable to SOFT
matter only.
• Not capturing close-to-contact “lubrication friction”
• Potential to beat conventional MC at equilibration
– hydrodynamics accelerates large scale relaxation
• E.g. polymer
D ~ 1/N → 1/R
– motion more concerted
• Opens up hydrodynamic coupling
• Background flow (e.g. simple shear) can be added
• Walls easy to add by Fourier, challenging by Wavelets
Acknowledgements
• Line of enquiry prompted by collaboration
on translocation with D Panja and G
Berkema: hydrodynamics crucial.
• CSC CoW
• ITS desktop
• CSC seminar