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Monte Carlo in different ensemblesChapter 5

NVT ensembleNPT ensemble

Grand-canonical ensembleExotic ensembles

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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 NN

NVT

U UQ N

Probability to find a particular configuration

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Detailed balance

acc( ) ( ) ( ) ( )acc( ) ( ) ( ) ( )

o n N n n o N nn 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

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NVT-ensemble

acc( ) ( )acc( ) ( )

o n N nn o N o

( ) expN n U n

acc( ) expacc( )

o n U n U on o

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NPT ensemble

We control the temperature, pressure, and number of particles.

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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 LN

V U LN

The energy depends on the real coordinates

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The perfect simulation ensemble

Here they are an ideal gas

Here they interact

What is the statistical thermodynamics of this ensemble?

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The perfect simulation ensemble: partition function

3 ds exp s ;!

NN N

NVT N

VQ U LN

0

0, , 03 3ds exp s ;

! !

ds exp s ;

M N NM N M N

MV NV T M N N

N N

V V VQ U LM 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 LM N N

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0

0, , 3 3 ds exp s ;

! !

M N NN N

MV NV T M N N

V V VQ U LM N N

To get the Partition Function of this system, we have to integrate over all possible volumes:

0

0, , 3 3d ds exp s ;

! !

M N NN N

MV N T M N N

V V VQ V U LM 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

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We have

0

0, , 3 3d ds exp s ;

! !

M N NN N

MV N T M N N

V V VQ V U LM 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:

3 d exp ds exp s ;!

N N NNPT N

PQ V PV V U LN

To make the partition function dimension less

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NPT EnsemblePartition function:

3 d exp ds exp s ;!

N N NNPT N

PQ V PV V U LN

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

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Detailed balance

acc( ) ( ) ( ) ( )acc( ) ( ) ( ) ( )

o n N n n o N nn 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

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NPT-ensemble

acc( ) ( )acc( ) ( )

o n N nn 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 nn o V PV U L

Nn

No

exp s ;exp

exp s ;

U LU n U o

U L

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NPT-ensemble

acc( ) ( )acc( ) ( )

o n N nn 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 nn o V PV U L

exp exp 0N

nn o

o

VP V V U n U

V

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Algorithm: NPT

• Randomly change the position of a particle• Randomly change the volume

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NPT simulations

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Grand-canonical ensemble

What are the equilibrium conditions?

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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?

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The Murfect simulation ensemble: partition function

3 ds exp s ;!

NN N

NVT N

VQ U LN

0

0, , 03 3ds exp s ;

! !

ds exp s ;

M N NM V M N

MV NV T M N N

N N

V V VQ U LM 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 LM N N

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0

0, , 3 3 ds exp s ;

! !

M N NN N

MV NV T M N N

V V VQ U LM 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 LM 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

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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

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Detailed balance

acc( ) ( ) ( ) ( )acc( ) ( ) ( ) ( )

o n N n n o N nn 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

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VT-ensemble

acc( ) ( )acc( ) ( )

o n N nn 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

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VT-ensemble

acc( ) ( )acc( ) ( )

o n N nn 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 nN Vn o

U LN

3

exp, exp s ;

!

NN N

VT N

N VN V U L

N

s

3

expexp

1V

UN

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Application: equation of state of Lennard-Jones

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Application: adsorption in zeolites

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Exotic ensemblesWhat to do with a biological membrane?

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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?

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Phase diagram: alcohol

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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.

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Constant surface tension simulation

A)]})(A'A);(U)A';(U[exp{min(1, NNaccP ss

AUF

A A’

L L’

A L = A’ L’ = V

, exp[ (U( ) A)]N NN T A r rN

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(Ao) = -0.3 +/- 0.6(Ao) = 2.5 +/- 0.3(Ao) = 2.9 +/- 0.3

Tensionless state: = 0