Advanced Interacting Sequential Monte Carlo Sampling for ...
(Metropolis) Monte Carlo sampling in the canonical and ... · Monte Carlo method: importance...
Transcript of (Metropolis) Monte Carlo sampling in the canonical and ... · Monte Carlo method: importance...
(Metropolis) Monte Carlo sampling
in the canonical and other ensembles
“Classical” statistical mechanics, ensemble average
“Classical” statistical mechanics, ensemble average
Monte Carlo method
© D. Frenkel
Monte Carlo method
© D. Frenkel
Monte Carlo method
Monte Carlo method: importance sampling
I.e., generating M random values of x uniformly distributed between a and b: Highly inefficient!
Importance sampling: generating M random values of u uniformly distributed between 0 and 1
The benefit is in the variance:
Monte Carlo method
Monte Carlo method
Monte Carlo method
Monte Carlo method
Monte Carlo method
Monte Carlo method
1. Select ensemble, determine Ω
2. Bearing in mind (detailed balance): Ω(o) α(o→n) acc(o→n)
= Ω(n) α(n→o) acc(n→o)
design α()
3. Determine acc()
Monte Carlo method
The general recipe
1. Select a particle at random, and calculate its energy
2. Give the particle a random displacementand calculate its new energy
3. Accept the move from to with probability,
Monte Carlo method
The NVT recipe
Periodic Boundary Conditions
Clusters ARE different from bulk Surface! In a cube of length L, one particle per unit length : [L³(L2)³]/L³ ~ 6L²/L³
Van der Waal's EoSIdeal gas EoS
What to evaluate?
Equations of state: p = p( V , T )
Heat capacity:
Monte Carlo method: detailed balance really needed?
vs
Example: LennardJones fluid
(model for dispersionforces – e.g., van der Waals – interacting systems)
Monte Carlo method: why recounting old configuration?
Phase diagram of hard spheres
Rosenbluth and Rosenbluth (1954)Wood and Jacobson (1957)
Historical (first) application: hard spheres
Fluid
Solid
NPTMonte Carlo, NPT
Ideal gas (MN)
Monte Carlo, NPT
N
M
Partition function: product of the partition function of the subsystems
1
NPTMonte Carlo, NPT
Consider the limits:
For:For: βP
NPTMonte Carlo, NPT
NPTMonte Carlo, NPT
Alternative scheme, lnV instead of V:
NPTMonte Carlo, NPT
NPTMonte Carlo, NPT
NPTMonte Carlo, grandcanonical ensemble μVT
A zeolite in contact with a gas
A model for grandcanonical ensemble
Ideal gas (MN)
NPTMonte Carlo, grandcanonical ensemble μVT
N
M
Exchange particles
Fixing N and M
NPTMonte Carlo, grandcanonical ensemble μVT
Gas system in V' much larger than interacting system in V
Ideal gas:
NPTMonte Carlo, grandcanonical ensemble μVT
Gas system in V' much larger than interacting system in V
Ideal gas:
NPTMonte Carlo, grandcanonical ensemble μVT
Grandcanonical MC scheme:
Displacement:
Insertion/removal:
NPTMonte Carlo, grandcanonical ensemble μVT
Justification:
NPTMonte Carlo, grandcanonical ensemble μVT
NPTMonte Carlo, grandcanonical ensemble μVT
NPTApplication: adsorption isotherm in a zeolite
Adsorption isotherm(s) of methane in silicalite. Black symbols: experiment. Open symbols: grandcanonical MC
NPTReference states: this was nice, but what is μ?
For a system of N molecules, each having M atoms
for an atom
NPTReference states: this was nice, but what is μ?
For ideal (molecular) gas:
NPTReference states: this was nice, but what is μ?