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# Monte Carlo Methods in Statistical Mechanics - PowerPoint PPT Presentation

Monte Carlo Methods in Statistical Mechanics . Aziz Abdellahi CEDER group. Materials Basics Lecture : 08/18/2011. What is Monte Carlo ?. Monte Carlo is an administrative area of the principality of Monaco. Famous for its casinos ! .

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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

Finding the value of π (“shooting darts”)

The simplest Monte Carlo method

• 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  ∞)

MC : Common features and applications

• 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)

Results 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)

How to calculate 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

Monte Carlo or “Importance Sampling”

• 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 actual probability

Monte Carlo or “Importance Sampling”

• 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 actual probability

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 ? actual probability

Metropolis algorithm (3)

• 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 actual probability

Conclusion

• 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)

Questions ? actual probability