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I am developing an essay on music perception by approaching some mathematical and physical notions. As such, the term I am addressing and interested in is the "convergence", which pairs in mathematics (as far as I understood) with the notion of "discontinuity point".

I was wondering whether we could use in physics the term "convergence" as a synonymous of "accumulation" or "concentration". For example, I was wondering whether we could use the term "convergence" in association with "explosion", for example in the case of "accumulation of energy to a certain threshold"?

I have difficulties to find specific material that can help me support this associations of terms. I have found much material, but on specific topics, like "convergence of shock waves", "mathematical modeling and of converging detonation", but I would like to understand this from a more general and simpler approach.

Can someone help me understand the correlation among the terms "convergence" and "accumulation/concentration" in such a physical perspective? And find resources on where to find more information about this view?

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  • $\begingroup$ Musicologists have been using pseudo-scientific language in an incomprehensible way for at least 60 or 70 years. I wouldn't get too hung up about doing the same. If it was good enough for people like Stockhausen and Boulez to write that way, it's probably good enough for you as well! $\endgroup$ – alephzero Apr 3 '20 at 12:26
  • $\begingroup$ Hahaha. I'll definitely follow your advice, but I would anyway try to provide some context for the use of such terms $\endgroup$ – TakeMeToTheMoon Apr 3 '20 at 15:07
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Convergence just means 'getting closer together'. If you have two lines that are getting closer together they are converging. If a large crowd of people suddenly starts moving towards a certain point you can say all the people are converging towards that point. In mathematics convergence is also commonly used for a particular kind of 'getting closer': that of getting closer to a solution.

If you have the function $f(x)=1/x+3$ you can say $f(x)$ converges towards 3 if $x$ goes to infinity. As $x$ gets large it gets closer and closer to 3 (try this for yourself if you're not convinced). Note that this is only a sloppy definition. This is an important notion in mathematics and physics because whether or not something converges often determines if you get a sensible answer (or complete trash). This is probably the definition that you found for discontinuity points.

Another example of this definition would be finding the area under a function that stretches from negative infinity to positive infinity. Like finding $\int_a^be^{-x^2}dx$ as $b$ goes to infinity and $a$ goes to negative infinity. It is not obvious whether this gives a number or infinity. (spoiler: it is just a number, so the area under the graph 'converges')

For you second paragraph: accumulation and concentration kind of fit this description. When you concentrate a solution the particles come closer together. For some amount of time they have to converge. This is not a common use however. Finally an explosion is not really a convergence of energy.

Hope this helps and is readable haha.

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  • $\begingroup$ That is absolutely helpful. Thanks! But let's think now that a certain parameter, like "pressure", is "getting closer and closer" to a "critical point". At that point, I would expect the system to present a discontinuity and start to release this pressure into the "default state", that I call entropy. Can this have sense? $\endgroup$ – TakeMeToTheMoon Apr 3 '20 at 15:04
  • $\begingroup$ @LucaDanieli Entropy is a bit more subtle than that and you have to spend quite some time on statistical physics to fully understand it. I see that you're really enthousiastic about these topics but trying to understand all these terms without studying the accompanying physics is like wanting to learn a complex piece from Beethoven when you just started to learn piano. $\endgroup$ – AccidentalTaylorExpansion Apr 3 '20 at 15:20
  • $\begingroup$ Ok, last question: is there a document you recommend on this topic? I have basics of Mathematics (Engineering studies - University Level) and some (little) thermodynamics. But from here, I need a path to arrive (I presume) to the Kullback–Leibler divergence. Otherwise, without any direction, I just get lost by searching on where to find information to comprehend this topics. $\endgroup$ – TakeMeToTheMoon Apr 3 '20 at 15:29

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