Monte Carlo Simulation

Question 1

Define the terms configuration, configurations space, and trajectory

Answer 1

For a system we know we have N atoms in d dimensions

• CONFIGURATION: corresponds to the unique vector (r1, r2… rN)
• CONFIGURATION SPACE: dN-dimensional space of all configurations
• TRAJECTORY: path in the CONFIGURATIONAL SPACE (for MD it is the time evolution)

Question 2

Name the Boltzmann factor and describe its meaning

Answer 2

• Boltzmann’s factor is exp(-E/kT)
• expresses the “probability” of a state of energy E relative to the probability of a state of zero energy.
• used introduce the temperature

p = exp(-E/kT)

Question 3

Explain ensemble averages and time averages. How are they related?

Answer 3

• Observable values estimated through numerical averaging, which can be performed via ensemble or time averaging.
• For ensemble averages, estimated by averaging over all particles (N),
• For time averages, estimated over time steps (t).
• At thermodynamic equilibrium, averages are equivalent regardless of how it was obtained as stated by “ERGOCITY theorem”

Question 4

Describe the underlying idea of a MonteCarlo algorithm

Answer 4

• Monte Carlo algorithm is based on the concept of “statistical sampling”, it relies on repeated random (stochastic) sampling to obtain a numerical result.
• For a simulation, this means we focus on the “statistical’ average instead “time evolution of a system” (deterministic)

Question 5

Explain the motivation behind the detailed balance condition

Answer 5

• There are some variables that cannot be justified by performing random sampling thus an improvement in sampling was made via the formulation of “detailed balance condition”
• For a probability to be valid it has to lie within the limits set by the system’s configuration at equilibrium

Piπ ( i → j ) = Pjπ ( j → i )

Question 6

Describe the Metropolis Criterion

Answer 6

Transition from i → j has two values

π ( i → j ) for Ej >= Ei is exp ( -β [Ej - Ei] )

else

π ( i → j ) for Ej < Ei is 1

• this simply tells that highest probability = 1 will be obtained if Ej < Ei
• logical since we would like a transition π ( i → j ), so j has to be stable than i

Question 7

How to choose the step size?

Answer 7

• maximizes performance (accuracy and speed)
• acceptance probability of ~ 50%

NOTE: optimal step size depends on the system and condition (e.g, smaller step size for denser system)

Question 8

Describe the “Improved Monte Carlo”

Answer 8

• doesn’t rely on random sampling

* includes Metropolis Criterion