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I am talking about higher probability viz. Brownian Motion, Ergodic Theory, Concentration, Percolation, Random Graphs, Random Matrix, etc. Going through books, I find that somehow or the other, many theorems in probability come from motivation from physics: they observe a physics phenomenon, then they think about it mathematically, and out comes a theorem.

Probably I'll give an example. Dynamical systems are, in my opinion, an offshoot of physics. The very concept of ergodicity i.e. measure-theoretic non-decomposability of the system into two non-trivial components can be best motivated using the free flow of gas molecules in a room, rather than just writing some dry $T^{-1}A=A\implies \mu(A)\mu(A^c)=0$.

The latter is of course fine, and rigorous and precise, but does not really tell the story.

I want to do research in probability. I have a firm understanding of the mathematics. However, I have not done any undergraduate or graduate physics course. So I am afraid that despite knowing the maths, my knowledge will be dry and if I wish to devise a theorem, I have to heavily rely on measures and stuff, on purely mathematical inequalities, although if I have an understanding of statistical mechanics, then at least I would be able to "guess" what my bounds can be, say for inequalities, or what I really want to say.

Your opinion is highly solicited. To be a good researcher in probability, would it be better for me to learn physics? If so, which part? Any book recommendation? Particularly any book that "connects" probability with physics?

Thanks a lot!

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  • $\begingroup$ So you want to somehow tie physics to the study of probability for the sole purpose of gaining some intuition about it? $\endgroup$ – Ant Jul 3 '16 at 6:47
  • $\begingroup$ If you go through Grimmet's Percolation, you'll see how they start talking about physics from the start. Concentration inequalities, I've read, come from physical interpretations. I can of course use maths to answer questions but say you are trying to find a new result. How would you proceed? Instead of trying to "find" a bound, say, which somehow happens to fit the bill, what if you can visualize what's happening in physics, and already have an "idea" of what you want? So in that case, tying physics to probability wouldn't be a bad idea I guess! $\endgroup$ – Landon Carter Jul 3 '16 at 6:52
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    $\begingroup$ But this line of reasoning would apply to mathematical finance too? $\endgroup$ – BCLC Jul 3 '16 at 13:58
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    $\begingroup$ I think a prior idea of finance is good if you wish to read math finance, isn't it? $\endgroup$ – Landon Carter Jul 4 '16 at 4:07
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    $\begingroup$ "Your opinion is highly solicited. To be a good researcher in probability, would it be better for me to learn physics?" Better than what exactly? It is clearly good to learn physics and it is clearly better than some things (such as spending your time playing video games). Whether or not it is better than some other things (such as learning R programming) is less clear and doesn't really have an answer which is independent of either your background or your interests. $\endgroup$ – John Coleman Jul 7 '16 at 23:14
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Of course you don't need to learn statistical physics to do probability theory and there are many areas of probability that are unrelated to physics. However, it certainly wouldn't hurt to at least learn some of the basics. For a very mathematical approach to the subject, I highly recommend the ongoing book project Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction by Friedli and Velenik.

Besides reasons of general interest, another important reason for learning about statistical physics is that there are methods and techniques in the subject that are non-probabilistic, but that a probabilist may nevertheless want/need to employ. An example of this is the cluster expansion technique, which is discussed in Friedli-Velenik as well as in many other places. In the opposite direction from expansion methods, which are approximate, you may want to take a look at Baxter's Exactly Solved Models in Statistical Mechanics, for techniques to deal with models that have exact solutions.

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    $\begingroup$ Thanks for the nice words about our book. You might be interested in knowing that the final draft (the version sent to its publisher, Cambridge University Press) has been uploaded on our page. It should more or less coincide with the final version (apart from some modifications that might be made on the galley proofs). The book, we hope, should appear mid-2017. $\endgroup$ – Yvan Velenik Aug 26 '16 at 7:17
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    $\begingroup$ Good news! Thanks for letting me know. $\endgroup$ – Ben Aug 26 '16 at 17:36
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Besides physics, there is also statistics. Many researchers in probability theory know nothing of statistics beyond freshman-level stuff and have no idea why it's a whole separate community and a body of theory other than what they know as probabilists, and why it interacts with philosophy of science in ways that their own work can avoid.

However, most mathematicians would probably be better off if they knew a whole lot more about physics.

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