Talk:Nicholas Metropolis

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Monte Carlo Method

Reading the article "Equation of State Calculations by Fast Computing Machines" by N. Metropolis, et al, I realized that it is not very accurate to describe the idea "Instead of choosing configurations randomly, then weighting them with $\textstyle e^{-{\frac {E}{kT}}}$ , we choose configurations with a probability $\textstyle e^{-{\frac {E}{kT}}}$ and weight them evenly." as a foundation in the Monte Carlo method. Actually, such idea belongs to the modified Monte Carlo method (at least, that's how the authors called it). Maybe it is useful to write down the context in which the sentence was used:

"Thus the most naive method of carrying out the integration [referring to integrate over multi-dimensional phase space] would be to put each of the N particles at a random position in the square (this defines a random point in the 2N-dimensional configuracion space), then calculate the energy of the system according to Eq. (1) [ $\textstyle E={\frac {1}{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}V(d_{ij})$ with $\textstyle i\neq j$ where $d_{ij}$ is the minimum distance between particles i and j, and V is the potential between particles] and give this configuration a weight $\textstyle e^{-{\frac {E}{kT}}}$ . This method, however, is not practical for close-packed configurations, since with high probability we choose a configuration where $\textstyle e^{-{\frac {E}{kT}}}$ is very small; hence a configuration of very low weight. So the method we employ is actually a modified Monte Carlo scheme, where, instead of choosing configurations randomly, then weighting them with $\textstyle e^{-{\frac {E}{kT}}}$ , we choose configuration with a probability $\textstyle e^{-{\frac {E}{kT}}}$ and weight them evenly."

Therefore, the Monte Carlo method is the method which consists simply "... of integrating over a random sampling of points instead of over a regular array of points" and the so called 'modified Monte Carlo method' is referred to in the quote as the new approach to follow. I don't know if for historical reasons the modified Monte Carlo method usurped the name. However, the original method is described in a footnote as "This method has been proposed independently by J. E. Mayer and by S. Ulam. Mayer suggested the method as a tool to deal with the problem of the liquid state, while Ulam proposed it as a procedure of general usefulness. B. Alder, J. Kirkwood, S. Frankel, and V. Lewinson discussed an application very similar to ours.

Thoughts or clarifications will be really appreciated.

Rsmith31415 (talk) 05:31, 4 October 2010 (UTC)[reply]

I would just suggest changing description from "to solve deterministic many-body problems" to "using a system of equations with probabilistic inputs to solve many-particle problems". Reason is that "many-body problems" sounds like the solar system (earth, moon and sun) rather than a chain reaction of neutrons expanding to thousands or millions of particles. Jim.Callahan,Orlando (talk) 23:29, 28 January 2016 (UTC)[reply]

Better yet, "using a system of equations (Markov Chain) with probabilistic inputs to solve many-particle problems". Need to get Markov Chain in there because modern usage is MCMC (Markov Chain, Monte Carlo). See also Monte Carlo Method Jim.Callahan,Orlando (talk) 23:54, 28 January 2016 (UTC)[reply]