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1 Course MP3 Lecture 4 13/11/2006 Monte Carlo method I An introduction to the use of the Monte Carlo method in materials modelling Dr James Elliott 4.1 Why Monte Carlo? The name derives from the association of statistical sampling methods with games of chance, such as those played in the famous casinos of Monte Carlo. Although the name Monte Carlo is relatively recent (1949) and intimately associated with the use of computers, statistical sampling methods are in fact much older than this (as we will see later). In a time when numerical calculations were done by hand, these methods were used to calculate integrals which cannot be solved exactly. Modern Monte Carlo (MC) techniques, however, are primarily designed to attack problems in physics. 1
2 4.2 Brief history of MC methods It was not until the advent of mechanical calculating machines that the full power of MC methods could be unleashed. In the early 20th century, mechanical adding machines were used to calculate firing tables for heavy artillery. The association with armed conflict continued into the Second World War when MC methods were used in the development of the hydrogen bomb at Los Alamos using the world s first supercomputer ENIAC. Although the first MC calculations on ENIAC came too late to make any significant contribution to the Manhattan Project, the techniques used were quickly taken up by the scientific research community. 4.3 Emergence of modern MC methods These developments lead to the first paper to coin the term Monte Carlo by Metropolis and Ulam in With the arrival of a newer, faster computer called MANIAC at Los Alamos, there followed a stream of seminal contributions to the field of MC simulation. In particular a 1953 paper by Nicolas Metropolis, Marshall and Arianna Rosenbluth, and Edward and Mici Teller on thermal importance sampling that introduced the famous Metropolis algorithm. 50th Anniversary in 2003! We will now discuss the theory of MC and the Metropolis algorithm in more detail. 2
3 4.4.1 The Monte Carlo algorithm The goal of a Monte Carlo simulation is the calculation of the expectation value of some mechanical quantity <Q>. Ideally, we would just calculate: Q 1 M = Q i exp( βe i ) Q = Q i Z i M i= 1 which is average of Q over all microstates of the system weighted with their Boltzmann probability. However, in our simulation we can only consider a tiny fraction of the total number of states, so we must average over states sampled in a very particular way The Monte Carlo algorithm The realisation that one should sample microstates according to their Boltzmann weights is the first step towards the celebrated Metropolis algorithm. The temperature which occurs in our expression for microstate probabilities is really a statistical temperature, but it corresponds very closely to what we understand by a thermodynamic temperature. This is only strictly true once the simulation has reached equilibrium, i.e. when the microstate occupation probabilities form a Boltzmann distribution. Before this point, it does not really make any sense to talk about the temperature of the system, although people frequently do! 3
4 4.5.1 MC as a Markov process However, we have so far said nothing about how we choose each state so it appears with its correct Boltzmann probability. To answer this question, we need to consider MC as a Markov process. A Markov process is a mechanism by which an initial state σ is transformed to a new state ν in a stochastic fashion using a set of transition probabilities P(σ ν) which satisfy certain conditions. Frenkel σ: p σ, T σ P(σ ν) ν: p ν, T ν MC as a Markov process The transition probabilities for a true Markov process should satisfy the following conditions: They should not vary with time They should only depend on the properties of the initial and final states, not on any other states the system has passed through For a given initial state, the sum of the transition probabilities over all final states must be equal to unity In a MC simulation, we repeatedly apply a Markov process to generate a Markov chain of states. The Markov process is chosen such that it produces a succession of states which appear with their Boltzmann probabilities. This is known as equilibration of the system, although it is not necessary that the system follow any well-defined kinetic pathway to equilibrium (more on this later). 4
5 4.5.3 MC as a Markov process In order to reach equilibrium, our Markov process should be ergodic, i.e. it should be possible to reach any state of the system from any other state. It should also satisfy the principle of detailed balance, i.e. the rate at which the system makes transitions into and out of any microstate must be equal: pp(σ ν) = pp(ν σ) σ The condition of detailed balance imposes time-reversal symmetry on the system, and gives us a sufficient (but not necessary) condition for the ratio of our transition probabilities to give a Boltzmann equilibrium distribution of microstate occupancies. ν MC as a Markov process For our equilibrium distribution to be Boltzmann-like, the ratio of transition probabilities should be: P(σ ν) P(ν σ) p = ν = ν pσ [ E E ] exp β( ) In practice, we tune the actual transition probabilities to make the acceptance ratio, i.e. the probability of a randomly generated transition being successful, as high as possible. The only constraints are the above equation, and that the sum over all transition probabilities is equal to unity. σ 5
6 4.6.1 The Metropolis algorithm Finally, we arrive at the Metropolis algorithm, which can be summarised as follows: 1. Start with a system in (an arbitrarily chosen) state σ and evaluate the energy E σ 2. Generate a new state ν by a small ergodic perturbation to σ, and evaluate E ν 3. If E ν E σ < 0 then accept the new state. If E ν E σ > 0 then accept the new state with probability exp[ β(e ν E σ )] 4. Return to step 2 and repeat until equilibrium is achieved The Metropolis algorithm is characterised by having a transition probability of unity if the new state has a lower energy than the initial state. This is more efficient than simply accepting states on the basis of their absolute Boltzmann factor The Metropolis algorithm The Metropolis algorithm is a general MC scheme for simulating in the NVT canonical ensemble, and it works reasonably efficiently in a wide range of situations (a notable exception is in the vicinity of a phase transition where it gives rise to large fluctuations). Frenkel
7 4.6.3 The Metropolis algorithm Requires a source of a large number of good quality random numbers in order to decide whether a move which raises the total energy should be accepted. Note that the efficiency of the standard algorithm can almost always be improved when applied to a new problem by giving careful thought to choosing the best transition probabilities. The result is often a faster simulation, and the improvement can easily make the difference between finding an answer to a problem and not finding one. In lecture 5 we will discuss in more detail how to make such improvements. Statistical sampling: Buffon s needle Probably the most famous example of MC is known as Buffon s needle, an experiment in which the value of π is estimated by repeatedly dropping a needle onto a surface ruled with equally spaced lines. The experiment is named after Georges-Louis Leclerc, Comte de Buffon, who showed in 1777 that if a needle of length l is thrown at random onto lines of spacing d then the probability that the needle intersects a line is 2l/πd, provided d l. Laplace then pointed out in 1820 that if a needle is thrown down N times, and lands on a line M of those times, then an estimate for π is given by lim( 2Nl / Md ) N 7
8 4.7.1 The 2D Ising model We will now apply the Metropolis algorithm to the 2D Ising model to show how the MC method works in practice. The Ising model is a simple model of a magnet which consists of a number of two-state (up/down) spins on a lattice. Each spin can interact with its nearest neighbour, and also with an external magnetic field. Binder 4.2 B H = ε i,j s s i j B Study magnetisation or specific heat as a function of temperature i s i The 2D Ising model The MC scheme involves starting the lattice in a random (T = ) or completely ordered (T = 0) configuration, and then flipping single spin states at random. This guarantees ergodicity, as every microstate of the system is accessible, in principle. The new microstate is accepted according to the Metropolis scheme, i.e. with probability 1 if the energy is lower, and with Boltzmann probability if the energy is higher than the previous state. For the square lattice shown, the change in spin energy with zero external field will be a multiple of 2ε, depending on the sign of the 4 nearest neighbour spins. 8
9 4.7.3 The 2D Ising model The nice thing about the 2D Ising model is that it is exactly solvable (Onsager 1944), and the results of MC simulation can be compared against theory, which is derived by explicitly writing down and solving the partition function for this system. We can compare the mean magnetisation per spin and the specific heat capacity, which are given by: m k c = N 1 = si N β 2 B i 2 2 ( H H ) The 2D Ising model Comparison of exact results (solid lines) with MC simulations (points) for mean magnetisation (left) and specific heat capacity (right) for a model. Reduced temperature (k B T/ε) [from Newman and Barkema (Clarendon Press, 1999)] 9
10 4.8 Application to off-lattice systems Leach 8 It is fairly simple, at least conceptually, to extend the Metropolis algorithm to off-lattice systems given an appropriate Hamiltonian. For example, to simulate a monatomic fluid we simply construct trial moves by randomly displacing particles. If we are simulating molecules instead of atoms, then we need to include orientational moves for rigid molecules. For flexible molecules, such as polymers, we must also consider the internal degrees of freedom of each chain. We must ensure that the trial moves form ergodic set, and that their distribution satisfies the symmetry requirements of the underlying Markov chain (i.e. a Boltzmann distribution of microstates) Improving on the Metropolis algorithm Although the Metropolis algorithm performs quite well for the Ising model, equilibration can be greatly accelerated by introducing additional moves into the algorithm. One of the great advantages of the MC method is that these moves need not be physically realistic, they need only obey acceptance rules which generate a Boltzmannlike distribution of microstates. So, we are permitted to make any move that is thermodynamically permissible. However, we should be careful that the modified move set is still ergodic, at least in principle. 10
11 4.9.2 Improving on the Metropolis algorithm Accelerating equilibration Binder Cluster moves For example, in the Ising model, instead of swapping single spin states, we can swap clusters of spin states. Provided we generate these clusters probabilistically, the algorithm is still ergodic, and requires many less MC steps per lattice site for equilibration Improving on the Metropolis algorithm Accelerating equilibration Exchange of particle identity For example, in a binary mixture, we can exchange the identity of a pair of particles, even though they may be separated by a large distances or by other particles! Such moves can greatly facilitate mixing of species in a dense system. 11
12 4.9.4 Improving on the Metropolis algorithm The price we pay for using unphysical moves is a loss of information about the dynamics of the system. Strictly, MC methods should only be used to generate an equilibrium configuration of the system and thereby study the average thermodynamic properties of the system. In practice, when using a reasonably physical move set, MC can also yield valuable information about the equilibrium dynamics of the system. Such methods are known as kinetic Monte Carlo (KMC) techniques. In course MP5 (Mesoscale and Multiscale Modelling), we will see how it is possible to calibrate MC steps to an absolute timescale, and hence link short real time dynamic simulations with KMC simulations spanning much greater extents of space and time Summary In this lecture we introduced the Monte Carlo method and described some details of its early history. We discussed the Metropolis algorithm, the most widely used technique for importance sampling, and saw how MC can be thought of as a Markov process. Next lecture, we will demonstrate the extreme versatility of the MC method for handling various different thermodynamic ensembles (NpT, µvt). We will also look at configurational bias methods for simulating rare events, and how to calculate free energies using MC. 12
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