Monte Carlo in different ensembles Chapter 5

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Presentation transcript:

Monte Carlo in different ensembles Chapter 5 NVT ensemble NPT ensemble Grand-canonical ensemble Exotic ensembles

Statistical Thermodynamics Partition function Ensemble average Probability to find a particular configuration Free energy

Ensemble average Generate configuration using MC: with

Detailed balance o n

NVT-ensemble

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

The energy depends on the real coordinates Scaled coordinates Partition function Scaled coordinates The energy depends on the real coordinates This gives for the partition function

The perfect simulation ensemble Here they are an ideal gas Here they interact What is the statistical thermodynamics of this ensemble?

The perfect simulation ensemble: partition function

To get the Partition Function of this system, we have to integrate over all possible volumes: Now let us take the following limits: As the particles are an ideal gas in the big reservoir we have:

To make the partition function dimension less We have To make the partition function dimension less This gives:

NPT Ensemble Detailed balance Partition function: Probability to find a particular configuration: Detailed balance Sample a particular configuration: Change of volume Change of reduced coordinates Acceptance rules ??

Detailed balance o n

NPT-ensemble Suppose we change the position of a randomly selected particle

NPT-ensemble Suppose we change the volume of the system

Algorithm: NPT Randomly change the position of a particle Randomly change the volume

NPT simulations

Grand-canonical ensemble What are the equilibrium conditions?

Grand-canonical ensemble We impose: Temperature Chemical potential Volume But NOT pressure

Here they are an ideal gas The Murfect ensemble Here they are an ideal gas Here they interact What is the statistical thermodynamics of this ensemble?

The Murfect simulation ensemble: partition function

To get the Partition Function of this system, we have to sum over all possible number of particles Now let us take the following limits: As the particles are an ideal gas in the big reservoir we have:

MuVT Ensemble Detailed balance Partition function: Probability to find a particular configuration: Detailed balance Sample a particular configuration: Change of the number of particles Change of reduced coordinates Acceptance rules ??

Detailed balance o n

mVT-ensemble Suppose we change the position of a randomly selected particle

mVT-ensemble Suppose we change the number of particles of the system

Application: equation of state of Lennard-Jones

Application: adsorption in zeolites

Exotic ensembles What to do with a biological membrane?

Model membrane: Lipid bilayer hydrophilic head group two hydrophobic tails water water

Questions What is the surface tension of this system? What is the surface tension of a biological membrane? What to do about this?

Phase diagram: alcohol

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.

Constant surface tension simulation A L = A’ L’ = V

Tensionless state: g = 0 g(Ao) = 2.5 +/- 0.3 g(Ao) = 2.9 +/- 0.3