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Monte Carlo Methods in Statistical Mechanics

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Monte Carlo Methods in Statistical Mechanics

Aziz Abdellahi

CEDER group

Materials Basics Lecture : 08/18/2011

- Monte Carlo is an administrative area of the principality of Monaco.

- Famous for its casinos !

- Monte Carlo is a (large) class of numerical methods used to solve integrals and differential equations using sampling and probabilistic criteria.

1

1

Finding the value of π (“shooting darts”)

- π/4 is equal to the area of a circle of diameter 1.

- Integral solved
- with Monte Carlo

- Details of the Method

- Randomly select a large number of points inside the square

- (Exact when N_total ∞)

Common features in Monte Carlo methods

- Uses random numbers and selection criteria

- Requires the repetition of a large number of events

Monte Carlo method that will be discussed in this talk

- Monte Carlo in Statistical Mechanics : Calculating thermodynamic properties of a material from its first-principles Hamiltonian

Example of results obtained from MC : LixFePO4 (Li-ion battery cathode)

- Only consider configurational degrees of freedom (Li-Vacancy)

- The energies of all Li-Vacancy configurations are known (Hamiltonian)

Useful battery properties that can be obtained from Monte Carlo

- Phase diagram, Voltage profiles

- These properties are deduced from the μ(x,T) relation [or alternatively x(μ,T)]

Results obtained in the Ceder group (using Monte Carlo)

- LixFePO4 phase diagram

- Voltage profile (room temperature)

Key physical quantity : The partition function

- {j} : Set of all possible Li-Vacancy configurations

- Ej : Energy of configuration j

- Nj : Number of Li in configuration j

- Control parameters

- All thermodynamic properties can be computed from the partition function

- Etc.

Finding a numerical approximation to the partition function

- The partition function cannot be calculated directly because the number of configurations scales exponentially with the system size (2N_sites possible configurations … too hard even for modern computers !).

- Monte Carlo strategy : Calculate thermodynamic properties by sampling configurations according to their Boltzmann probability

Importance sampling : Sample states according to their actual probability

- Consider the following random variable x :

- Direct calculation of <x> :

- Importance sampling : Randomly pick 10 values of x out of a giant hat containing 10% 0’s, 80% 1’s and 10% 2’s.

- Possible outcome :

- The arithmetic average will not always be equal to the average.
- However, the two become equal in the limit of large “chains”.

- Importance sampling : Sample states with the correct probability. Works well for very large systems that have heavy probability discrepancies.

Monte Carlo : Methodology

- Start from an initial configuration C1

- Create a Markov chain of configurations, where each configuration is determined from the previous one using a certain probabilistic criteria

- C1 C2 … CN_max

- Choose the probabilistic criteria so that states are asympotically sampled with the equilibrium Boltzmann probability (that is the main difficulty !)

- En : Energy of configuration Cn

- Nn : Number of Li in configuration Cn

- Calculate thermodynamic averages directly through arithmetic averages over the Markov Chain

Building the chain : The Metropolis Algorithm

- Start from an initial configuration C1 :

- Change the occupation state of the first Li site :

- Calculate Ei-μNi (Before the change) and Ef -μNf (After the change)

- If Ef -μNf < Ei-μNi , accept the change

- If Ef-μNf > Ei –μNi , accept the change with the probability :

- (Ratio of Boltzmann probabilities…)

- Repeat for all other Li sites to get C2

Why does the Metropolis algorithm work ?

- The Metropolis algorithm generates a chain Markov consistent with Boltzmann probabilities sampling because the selection criteria has Boltzmann probabilities built into it.

It can be shown that all selection criteria that respect the condition of detailed balance produce correct sampling :

- Probability of generating configuration j from configuration i

- Because the most probable configurations are sampled preferentially, good approximations of thermodynamic averages can be obtained by sampling a relatively small number of configurations

Monte Carlo in Statistical Mechanics

- Method to approximate thermodynamic properties using clever sampling

- Good results can be obtained by sampling a relatively small number of configurations (relative to the total number of possible configurations) :

- LixFePO4 voltage profile : 50 000 states sampled instead of 21728

Other Monte Carlo methods in engineering

- Kinetic Monte Carlo (to calculate diffusivities)

- Quantum Monte Carlo (to solve the Schrodinger equation)

- Monte Carlo in nuclear engineering (to predict the evolution of the neutron population in a nuclear reactor)