A method that finds answers by repeating probabilistic trials using random numbers. In materials it includes equilibrium calculations by the Metropolis method and kinetic Monte Carlo for following diffusion.
Japanese name: モンテカルロ法
Monte Carlo method appears in 2 practice questions across 1 topic, including Computational materials science.
The Monte Carlo method is a general name for methods that repeat many trials using random numbers and estimate an answer from their average; the name comes from Monte Carlo in Monaco, known for its casino. For example, scattering points uniformly in a square and counting the fraction that fall inside the inscribed circle gives an estimate of π. The error is inversely proportional to the square root of the number of trials N, so making the result ten times more precise takes 100 times as many trials. This rate of error reduction does not depend on the number of variables, so the method is especially powerful for integrals in statistical mechanics, which have a huge number of variables, such as atomic positions.
The method most often used in materials calculations is the one presented by Metropolis and colleagues in 1953. When trials that change the state slightly are accepted or rejected with probabilities based on the Boltzmann factor, the frequency of the states visited follows the equilibrium distribution at that temperature. In alloys, repeated trials that swap atoms on lattice sites give order-disorder transition temperatures, short-range order, solubility limits and so on. The energy change from a swap is often given by a cluster expansion based on first-principles calculations. Another use is the Potts model, in which numbers distinguishing grains are placed on lattice sites to calculate grain growth.
The number of trials in the Metropolis method (Monte Carlo steps) is not real time. The sequence of states is a route for sampling the equilibrium distribution and does not represent the order of physical changes, so it normally cannot be used to discuss rates of change. Even when steps are used in place of time, as in Potts-model grain growth, their correspondence to real time is set by comparison with experiment. The part still influenced by the initial state is excluded from averages, and errors are estimated with the similarity between successive states (correlation) taken into account. When time is needed, the kinetic Monte Carlo method, which advances time by giving a rate for each event, is used.
The frequency with which an atom in a solid jumps to a neighboring site is ν exp(−E_a/kT); with a vibration frequency ν ≈ 10¹³ s⁻¹, E_a = 1.0 eV and 600 K, it is about 4×10⁴ times per second. An atom vibrates about 2.5×10⁸ times before making one jump, so molecular dynamics, which follows every vibration, can hardly follow diffusion. Kinetic Monte Carlo skips the vibrations and chooses the next event from a list of events and their rates, so it can follow precipitation and the evolution of irradiation defects on scales of seconds or hours. Events not on the list, however, never occur, so the choice of events prepared determines the result.
Answer: C. About 0.31
In the Metropolis method, moves that lower the energy are always accepted, and moves that raise it are accepted with probability exp(−ΔE/kT). Since kT = 8.617×10⁻⁵ × 1000 ≈ 0.0862 eV, ΔE/kT ≈ 1.16, and the probability is exp(−1.16) ≈ 0.31. In practice a random number between 0 and 1 is generated, and the move is accepted if it is smaller than this probability. By accepting energy-raising moves with a certain probability, the system visits states following the Boltzmann distribution, and averages give the equilibrium properties at finite temperature. If energy-raising moves were never accepted, the system would be trapped in a local energy minimum, the same as optimization at 0 K. About 0.69 mistakes the rejection probability 1 − 0.31 for the acceptance probability.
Answer: A. It assigns rates of the form ν exp(−E_a/kT) to events such as atomic jumps, selects events with probability proportional to their rates and advances time, so it can follow long-time diffusion and precipitation
In diffusion in solids, an atom vibrates a very large number of times before occasionally jumping into a neighboring vacancy. Molecular dynamics follows all these vibrations, so phenomena longer than microseconds are hard to handle. kMC skips the vibrations: it prepares a list of events such as vacancy jumps with their activation energies, selects the next event with probability proportional to the rate ν exp(−E_a/kT), and advances time according to the inverse of the sum of all event rates. Activation energies calculated by DFT are often used. This makes it possible to follow the growth of precipitates during aging and the evolution of irradiation defects in real time. Integrating equations of motion is a feature of molecular dynamics, and having no time while finding equilibrium averages is a feature of the Metropolis method.
The statistical error falls as 1/√N with the number of independent trials N. Halving the error takes four times as many trials. If successive states are correlated, the actual number of independent samples is smaller, so errors are estimated by methods such as averaging over blocks.
The Metropolis method decides whether to accept a trial from the energy difference alone and finds equilibrium averages. Kinetic Monte Carlo uses event rates obtained from activation energies and advances time each time an event is chosen, following changes on the way to equilibrium in real time.
Yes. If the random numbers have periodicity or correlations, averages can be biased or only some states may be visited. Today it is usual to use pseudo-random number generators with long periods that pass statistical tests, such as the Mersenne Twister. In parallel calculations it is also important to use an independent random-number stream for each calculation.
In the Metropolis method, moves that lower the energy are always accepted, and moves that raise it are accepted with probability exp(−ΔE/kT).
In a Monte Carlo calculation by the Metropolis method, a tr…Integrating equations of motion is a feature of molecular dynamics, and having no time while finding equilibrium averages is a feature of the Metropolis method.
Which statement about the kinetic Monte Carlo (kMC) method …Vacancy Molecular dynamics Density functional theory CALPHAD method Finite element method Interatomic potential Materials informatics Phase-field method