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Department of Chemical Engineering
Monte Carlo in different ensemblesChapter 5
NVT ensembleNPT ensemble
Grand-canonical ensembleExotic ensembles
Department of Chemical Engineering2
Statistical Thermodynamics
NVTQF ln
NNNN
NVTNVT
UANQ
A rexprdr!
113
NNNNVT UN
Q rexpdr!
13
Partition function
Ensemble average
Free energy
3
1 1N r dr' r' r exp r' exp r
!N N N N N N
NNVT
U UQ N
Probability to find a particular configuration
Department of Chemical Engineering4
Detailed balance
acc( ) ( ) ( ) ( )
acc( ) ( ) ( ) ( )
o n N n n o N n
n o N o o n N o
( ) ( )K o n K n o ( ) ( ) ( ) acc( )K o n N o o n o n
o n
( ) ( ) ( ) acc( )K n o N n n o n o
Department of Chemical Engineering5
NVT-ensemble
acc( ) ( )
acc( ) ( )
o n N n
n o N o
( ) expN n U n
acc( )exp
acc( )
o nU n U o
n o
Department of Chemical Engineering7
NPT ensemble
We control the temperature, pressure, and number of particles.
Department of Chemical Engineering8
Scaled coordinates
/i i Ls r
NNNNVT UN
Q rexpdr!
13
Partition function
Scaled coordinates
This gives for the partition function
3
3
3
ds exp s ;!
ds exp s ;!
NN N
NVT N
NN N
N
LQ U L
N
VU L
N
The energy depends on the real coordinates
Department of Chemical Engineering9
The perfect simulation ensemble
Here they are an ideal gas
Here they interact
What is the statistical thermodynamics of this ensemble?
Department of Chemical Engineering10
The perfect simulation ensemble: partition function
3ds exp s ;
!
NN N
NVT N
VQ U L
N
0
0, , 03 3
ds exp s ;! !
ds exp s ;
M N NM N M N
MV NV T M N N
N N
V V VQ U L
M N N
U L
0
0, , 3 3
ds exp s ;! !
M N NN N
MV NV T M N N
V V VQ U L
M N N
Department of Chemical Engineering11
0
0, , 3 3
ds exp s ;! !
M N NN N
MV NV T M N N
V V VQ U L
M N N
To get the Partition Function of this system, we have to integrate over all possible volumes:
0
0, , 3 3
d ds exp s ;! !
M N NN N
MV N T M N N
V V VQ V U L
M N N
Now let us take the following limits:
0
constantM MV V
As the particles are an ideal gas in the big reservoir we have:
P
Department of Chemical Engineering12
We have
0
0, , 3 3
d ds exp s ;! !
M N NN N
MV N T M N N
V V VQ V U L
M N N
0 0 0 0 01 expM N M NM N M NV V V V V V M N V V
0 0 0exp expM N M N M NV V V V V PV
This gives:
3d exp ds exp s ;
!N N N
NPT N
PQ V PV V U L
N
To make the partition function dimension less
Department of Chemical Engineering13
NPT EnsemblePartition function:
3d exp ds exp s ;
!N N N
NPT N
PQ V PV V U L
N
Probability to find a particular configuration:
, exp exp s ;N N NNPTN V V PV U L s
Sample a particular configuration:• Change of volume • Change of reduced coordinates
Acceptance rules ??
Detailed balance
Department of Chemical Engineering14
Detailed balance
acc( ) ( ) ( ) ( )
acc( ) ( ) ( ) ( )
o n N n n o N n
n o N o o n N o
( ) ( )K o n K n o ( ) ( ) ( ) acc( )K o n N o o n o n
o n
( ) ( ) ( ) acc( )K n o N n n o n o
Department of Chemical Engineering15
NPT-ensemble
acc( ) ( )
acc( ) ( )
o n N n
n o N o
, exp exp s ;N N NNPTN V V PV U L s
Suppose we change the position of a randomly selected particle
Nn
No
exp exp s ;acc( )
acc( ) exp exp s ;
N
N
V PV U Lo n
n o V PV U L
Nn
No
exp s ;exp
exp s ;
U LU n U o
U L
Department of Chemical Engineering16
NPT-ensemble
acc( ) ( )
acc( ) ( )
o n N n
n o N o
, exp exp s ;N N NNPTN V V PV U L s
Suppose we change the volume of the system
N
N
exp exp s ;acc( )
acc( ) exp exp s ;
Nn n n
No o o
V PV U Lo n
n o V PV U L
exp exp 0
N
nn o
o
VP V V U n U
V
Department of Chemical Engineering17
Algorithm: NPT
• Randomly change the position of a particle
• Randomly change the volume
Department of Chemical Engineering23
Grand-canonical ensemble
We impose:– Temperature– Chemical potential– Volume
– But NOT pressure
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The Murfect ensemble
Here they are an ideal gas
Here they interact
What is the statistical thermodynamics of this ensemble?
Department of Chemical Engineering25
The Murfect simulation ensemble: partition function
3ds exp s ;
!
NN N
NVT N
VQ U L
N
0
0, , 03 3
ds exp s ;! !
ds exp s ;
M N NM V M N
MV NV T M N N
N N
V V VQ U L
M N N
U L
0
0, , 3 3
ds exp s ;! !
M N NN N
MV NV T M N N
V V VQ U L
M N N
Department of Chemical Engineering26
0
0, , 3 3
ds exp s ;! !
M N NN N
MV NV T M N N
V V VQ U L
M N N
To get the Partition Function of this system, we have to sum over all possible number of particles
0
0, , 3 3
0
ds exp s ;! !
M N NN MN N
MV N T M N NN
V V VQ U L
M N N
Now let us take the following limits:
0
constantM MV V
As the particles are an ideal gas in the big reservoir we have:
3lnBk T
30
expds exp s ;
!
NNN N
VT NN
N VQ U L
N
Department of Chemical Engineering27
MuVT EnsemblePartition function:
Probability to find a particular configuration:
3
exp, exp s ;
!
NN N
VT N
N VN V U L
N
s
Sample a particular configuration:• Change of the number of particles• Change of reduced coordinates
Acceptance rules ??
Detailed balance
30
expds exp s ;
!
NNN N
VT NN
N VQ U L
N
Department of Chemical Engineering28
Detailed balance
acc( ) ( ) ( ) ( )
acc( ) ( ) ( ) ( )
o n N n n o N n
n o N o o n N o
( ) ( )K o n K n o ( ) ( ) ( ) acc( )K o n N o o n o n
o n
( ) ( ) ( ) acc( )K n o N n n o n o
Department of Chemical Engineering29
VT-ensemble
acc( ) ( )
acc( ) ( )
o n N n
n o N o
Suppose we change the position of a randomly selected particle
Nn3
No3
expexp s ;acc( ) !
expacc( )exp s ;
!
N
N
N
N
N VU Lo n N
N Vn oU L
N
exp 0U n U
3
exp, exp s ;
!
NN N
VT N
N VN V U L
N
s
Department of Chemical Engineering30
VT-ensemble
acc( ) ( )
acc( ) ( )
o n N n
n o N o
Suppose we change the number of particles of the system
1N+1
3 3
N3
exp 1exp s ;
1 !acc( )
expacc( )exp s ;
!
N
nN
N
oN
N VU L
No n
N Vn oU L
N
3
exp, exp s ;
!
NN N
VT N
N VN V U L
N
s
3
expexp
1
VU
N
Department of Chemical Engineering36
Model membrane: Lipid bilayer
hydrophilic head group
two hydrophobic tails
water
water
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Questions
• What is the surface tension of this system?
• What is the surface tension of a biological membrane?
• What to do about this?
Department of Chemical Engineering40
Simulations at imposed surface tension
• Simulation to a constant surface tension– Simulation box: allow the area of the bilayer to
change in such a way that the volume is constant.
Department of Chemical Engineering41
Constant surface tension simulation
A)]})(A'A);(U)A';(U[exp{min(1, NNaccP ss
AUF
AA’
LL’
A L = A’ L’ = V
, exp[ (U( ) A)]N NN T A r rN