Basic Monte Carlo (chapter 3)

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Basic Monte Carlo(chapter 3)

• Algorithm
• Detailed Balance
• Other points

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Molecular Simulations
MD

• Molecular dynamics solve equations of motion
• Monte Carlo importance sampling

r1
r2
rn
MC
r1
r2
rn
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Does the basis assumption lead to something that
is consistent with classical thermodynamics?
Systems 1 and 2 are weakly coupled such that
they can exchange energy. What will be E1?
BA each configuration is equally probable but
the number of states that give an energy E1 is
not know.
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Energy is conserved! dE1-dE2
This can be seen as an equilibrium condition
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Canonical ensemble
1/kBT
Consider a small system that can exchange heat
with a big reservoir
Hence, the probability to find Ei
Boltzmann distribution
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Thermodynamics
What is the average energy of the system?
Compare
Hence
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Statistical Thermodynamics
Partition function
Ensemble average
Probability to find a particular configuration
Free energy
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Monte Carlo simulation
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Ensemble average
Generate configuration using MC
with
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Monte Carlo simulation
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Questions

• How can we prove that this scheme generates the
  desired distribution of configurations?
• Why make a random selection of the particle to be
  displaced?
• Why do we need to take the old configuration
  again?
• How large should we take delx?

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Detailed balance
o
n
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NVT-ensemble
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Questions

• How can we prove that this scheme generates the
  desired distribution of configurations?
• Why make a random selection of the particle to be
  displaced?
• Why do we need to take the old configuration
  again?
• How large should we take delx?

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Questions

• How can we prove that this scheme generates the
  desired distribution of configurations?
• Why make a random selection of the particle to be
  displaced?
• Why do we need to take the old configuration
  again?
• How large should we take delx?

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Mathematical
Transition probability
?0
Probability to accept the old configuration
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Keeping old configuration?
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Questions

• How can we prove that this scheme generates the
  desired distribution of configurations?
• Why make a random selection of the particle to be
  displaced?
• Why do we need to take the old configuration
  again?
• How large should we take delx?

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Not too small, not too big!
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Periodic boundary conditions
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Lennard Jones potentials

• The Lennard-Jones potential

• The truncated Lennard-Jones potential

• The truncated and shifted Lennard-Jones potential

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Phase diagrams of Lennard Jones fluids
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Non-Boltzmann sampling
Why are we not using this?
T1 is arbitrary!
We only need a single simulation!
We perform a simulation at TT2 and we determine
A at TT1
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T1
T2
T3
T4
T5
Overlap becomes very small
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How to do parallel Monte Carlo

• Is it possible to do Monte Carlo in parallel
• Monte Carlo is sequential!
• We first have to know the fait of the current
  move before we can continue!

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Parallel Monte Carlo

• Algorithm (WRONG)
• Generate k trial configurations in parallel
• Select out of these the one with the lowest
  energy
• Accept and reject using normal Monte Carlo rule

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Conventional acceptance rule
Conventional acceptance rules leads to a bias
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Why this bias?
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Detailed balance
o
n
?
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Modified acceptance rule
Modified acceptance rule remove the bias exactly!