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Monte Carlo Techniques

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Molecular Modeling and Simulation: An Interdisciplinary Guide

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 21))

Abstract

From Washington D.C. to Wall Street to Los Alamos, statistical techniques termed collectively as Monte Carlo (MC) are powerful problem solvers. Indeed, disciplines as disparate as politics, economics, biology, and high-energy physics rely on MC tools for handling daily tasks.

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Notes

  1. 1.

    Buffon used a Monte Carlo integration procedure to solve the following problem: a needle of length L is thrown at a horizontal plane ruled with parallel straight lines separated by d > L; what is the probability that the needle will intersect one of these lines? Buffon derived the probability as an integral and attempted an experimental verification by throwing the needle many times and observing the fraction of needle/line intersections. It was Laplace who in the early 1800s generalized Buffon’s probability problem and recognized it as a method for calculating π.

  2. 2.

    We use the term random for brevity in most of this chapter, though the terms pseudorandom or quasi-random are technically correct.

  3. 3.

    We say that x lies in [a, b] if a ≤ x ≤ b and that x lies in [a, b) if a ≤ x < b; similarly, x in (a, b) means a < x < b.

  4. 4.

    The random variables x 1 and x 2 are independent if the joint probability density function ρ(x 1, x 2) is equal to the product of the individual probability density functions: ρ(x 1, x 2) = ρ1(x 1)ρ2(x 2).

  5. 5.

    Though for complex systems, the state descriptors (e.g., coordinates) are unlikely to be repeated in phase with the cycle of a (short) random number generator, subtle problems may occur in some applications, making the goal of long period generally desirable.

  6. 6.

    How many pairs of rabbits can be produced in a year from one rabbit pair? Assume that every month each pair produces a new offspring couple, which from the second month also becomes productive.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences and Department of Chemistry, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Prof. Tamar Schlick

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Schlick, T. (2010). Monte Carlo Techniques. In: Molecular Modeling and Simulation: An Interdisciplinary Guide. Interdisciplinary Applied Mathematics, vol 21. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6351-2_12

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