Thursday, May 26, 2016

The joy and mystery of discovery

My wife and I went to see the movie, The Man Who Knew Infinity, which chronicles the relationship between the legendary mathematicians Srinivasa Ramanujan and G.H. Hardy. I knew little about the story or the maths and so learnt a lot. I think one thing it does particularly well is capturing the passion that many scientists and mathematicians have about their research, including both the beauty of the truth we discover and the rich enjoyment of the finding it.

The movie obviously highlights the unique, weird, and intuitive way that Ramanujan was able surmise extremely complex formula without proof.

I subsequently read a little more. There is a nice piece on The Conversation, praising the movie's portrayal of mathematics. A post on the American Mathematical Society blog discusses the making of the movie including a discussion with the mathematician Ken Ono, who was a consultant. Stephen Wolfram also has a long blog post about Ramanujan.

I enjoyed reading the 1993 article Ramanujan for lowbrows that considers some "simple" results that most of us can understand, such as the taxi cab number 1729.

The formula below features in the movie. It is an asymptotic formula for number of partitions of an integer n.

It is interesting that this is useful in the statistical mechanics of non-interacting fermions in a set of equally spaced energy levels, in the micro canonical ensemble.
Indeed it is directly related to the linear in temperature dependence of the specific heat and the (pi^2)/3 pre factor!

This features in Problem 7.27 in the text by Schroeder, based on an article in the American Journal of Physics. See particularly the discussion around equation 8, with W(r) replaced by p(n).

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