Monte Carlo Methods in Statistical Mechanics

1. Monte Carlo Methods in Statistical Mechanics Aziz Abdellahi CEDER group Materials Basics Lecture : 08/18/2011

2. 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.

3. 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  ∞)

4. 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)

5. 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)

6. 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

7. 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.

8. 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

9. 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

10. 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

11. 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)

12. Questions ?