Monte Carlo. Lecture 15 4/9/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky

Transcription

1 Monte Carlo Lecture 15 4/9/18 1

2 Sampling with dynamics In Molecular Dynamics we simulate evolution of a system over time according to Newton s equations, conserving energy Averages (thermodynamic properties) only includes phenomena that occur in the time scale of the MD simulation Sometimes excitations of the system are beyond MD time scale A i i A e i e E E i i 1 T T 0 A( t) dt Rely on ergodicity to sample the entire phase space Need to know dynamics! 2

3 Simulation boundary conditions Specify the macroscopic boundary conditions What thermodynamic variable are kept constant? Need to specify one out of each conjugate pair of variables du TdS PdV dn... 3

4 Counting excitations If excitations are easily countable can construct free energies explicitly using ensemble statistics F( V, T ) E( V ) EZP ( V ) TS( V, T ) Bose Einstein statistics V FV T EV k T, V q q,, B ln 1 exp q, 2 q, vibrational spectrum C v Ab initio [1]... Expt [2] 4

5 Sampling with dynamics Difficult to explore the entire configuration space if potential energy fluctuates and rare events need to be included Solution: bias dynamics to evenly explore (free) energy landscape along a low dimensional collective variable Umbrella sampling, metadynamics, hyperdynamics Either explicitly constrain dynamics or encourage exploration of barriers 5

6 Random sampling Configuration space (x, p) has high dimensionality ~6N Number of configurations / excitations to sample is very large Instead of following time dynamics, would it makes sense to just choose configurations randomly and compute averages? How do you properly average over such random samples? The key is to average with the correct weights, given by the probability that a system is found in particular microstate U P i E A i P A i i i i Probability distributions are given by statistical mechanics 6

7 Monte Carlo algorithm 1. Define a domain of possible inputs (groups of states) 2. Generate inputs randomly from a probability distribution over the domain 3. Compute observable quantities: Perform a deterministic computation on the inputs 4. Aggregate the results 7

8 Random variables Random numbers on computers are pseudo random, they are generated deterministically using a mathematical formula, e.g. X 0 = seed Built in library functions often have poor statistical properties and some will repeat patterns after tens of thousands of trials For 32 bit integers maximum period is 2 32, or about 4 billion Need: uncorrelated, uniform, fast, portable, reproducible True random numbers can only be obtained from physical devices sampling thermal noise, radioactive decay, etc 8

9 Testing randomness PHP rand() on Microsoft Windows 2008: Debian GNU/Linux OpenSSL vulnerability due to reduced randomness in security keys 9

10 Probability distributions A continuous random variable X is distributed according to a probability density function f X : Example: Get an exponential random variable Given a uniform random variable p on [0,1] will be exponentially distributed 10

11 Monte Carlo integration Randomly throwing darts in a square and counting fraction inside the circle can give an estimate of π 11

12 Buffon needle (1777) Needles dropped randomly. What is the probability of a needle falling on one of parallel lines? 12

13 Monte Carlo integration MC can be an efficient way to perform difficult integrations Track the fraction of times you randomly end up below the curve, and multiply by the area of the box A bit silly in 2 or 3 dimensions can use uniform grids What about 6N dimensions? Can t possibly sample everywhere Need m 6N grid points 100 atoms, 5 points = > universe 13

14 Unbiased sampling Want to include only statistically significant states The number of available states at high energy is much larger... Mainly pick states with low weights This is similar to trying to integrate a non uniform function A b a f ( x) dx 1 M M i1 f ( x i ) 14

15 Canonical ensemble A i P i A i Probability of finding a state decreases exponentially with energy 15

16 Importance sampling When most of the weight of the integral comes from a small important range of x where f(x) is large, sampling more often in this region increases accuracy Instead of uniformly random {x i } choose using probability p(x) 16 M i i i b a x p b a x p x f M x p x f dx x p x p x f dx x f A 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) (

17 Biased sampling Brute force unbiased sampling of the free energy landscape is difficult if it is not smooth and has deep minima and barriers Probability of sampling a 1eV barrier is ~ at 300K Can bias the sampling in order to explore high energy regions with more probability True potential Ideal biasing potential No barrier, perfect sampling 17

18 Biased sampling Perform a simulation in a modified ensemble with probabilities where W(r N ) is a biasing weight function biasing function of r N The effect of introducing the weight function W is equivalent to adding a biasing potential to the Hamiltonian We can use the bias to increase probability of sampling unlikely regions of phase space Need to correct the calculated result to remove the bias 18

19 Biased sampling The average of a quantity A in the canonical ensemble is Unbiased free energy profile can be calculated by subtracting the biasing potential from the biased free energy profile 19

20 Estimators and variance p(x) is chosen to bias the random sample to reduce variance A f f f / p f / p p M M p Choose p(x) to resemble f(x) as much as possible Variance can be made much smaller Need fewer samples to get an accurate estimate of the integral 20

21 Importance sampling If p(x)=cf(x) then variance is zero Perfect estimation with one sample A b a f ( x) dx 1 M M i i1 p( xi ) f ( x ) 21

22 Sampling the canonical ensemble Can we pick states from the ensemble with a probability proportional to Boltzmann weights? How to construct probability weighted sample? Catch: can t sample p B (R), since there are many configurations and normalization Z (partition function) is unknown 22

23 Importance sampling Calculating the average depth of a river with a random walk Only accept moves that are in the river uniform sampling importance sampling D. Frenkel and B. Smit, Understanding Molecular Simulations 23

24 Markov chain Monte Carlo (MCMC) Goal: generate a collection of states according to a desired distribution P B (R), which cannot be sampled directly Hint: we know the relative probabilities of states because we know the ratio of the numerators We will try to indirectly sample the distribution Do this by executing a sequence of random steps through configuration space transition probabilities 24

25 Markov chain Random walk across a set of states with given transition probabilities The steady state probability distribution is encoded in the transition matrix This amounts to finding eigenvector with eigenvalue 1 Example: Google's PageRank algorithm is essentially a Markov chain over the graph of the Web 25

26 Metropolis formalism Probability of finding the system at a later time in another state: This is a matrix equation in the space of probability distributions Markov chain should be ergodic: any state can reach any state We want to reach the steady state equilibrium distribution 26

27 Detailed balance Global balance (necessary at equilibrium) Detailed balance (sufficient but not necessary) no Z We have freedom to choose transition probabilities selection probability acceptance ratio 27

28 Metropolis algorithm 1. Start from initial (random) configuration 2. Randomly perturb the system (e.g. displace particles) 3. Compute energy difference between two states 4. Make a transition with probability use random numbers here ΔE 0 : always accept new configuration ΔE > 0 : accept with probability 5. Repeat steps 2 4, accumulating average 28

29 Model systems Markov chain Monte Carlo (MCMC) algorithms are generally used for sampling from multi dimensional distributions, especially when the number of dimensions is high Atomic liquids Real space moves with small displacements x( t dt) x( t) x (rand[0,1] ) x( t dt ) x( t) x (rand[-0.5, not detail balanced 0.5] ) Smaller moves: better acceptance rate, slower sampling (correlation) Bigger moves: faster sampling, lower acceptance rate Can do one particle or many particle moves (with care) Energies from pair potentials 29

30 Simulation of hard shapes No need to define physical dynamics for problems of packing Joshua A. Anderson, M. Eric Irrgang, Sharon C. Glotzer, Computer Physics Communications,

31 Acceptance ratio Is there a single optimal acceptance ratio? 50% is often cited as a target acceptance ratio. But it really depends on the specifics of your system. Does computation time to test whether a trial move is accepted depend on the magnitude of the move? Not for continuous potentials, but it does for hard spheres a move can be rejected as soon as neighbor overlap is detected. Thus, for hard spheres, rejection is cheap, and we can accommodate lower acceptance ratios (20%). 31