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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

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Monte carlo methods in statistical mechanics l.jpg

Monte Carlo Methods in Statistical Mechanics

Aziz Abdellahi

CEDER group

Materials Basics Lecture : 08/18/2011


What is monte carlo l.jpg

What is Monte Carlo ?

  • 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.


The simplest monte carlo method l.jpg

1

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  ∞)


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Common features in Monte Carlo methods

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)


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Useful battery properties that can be obtained from Monte Carlo

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)


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Key physical quantity : The partition function

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


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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.


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Monte Carlo : Methodology

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


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Building the chain : The Metropolis Algorithm

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


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Why does the Metropolis algorithm work ?

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


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Monte Carlo in Statistical Mechanics

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)


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Questions ?


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