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I must admit that I never got well how physicists handle infinitesimal quantities, mainly because of my education as a mathematician. So the following lines (taken from the preface of Berezin and Shubin's The Schrödinger Equation) destabilize me somewhat:

It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs. To appreciate what I'm hinting at consider [...] electronics, [...] transistors, [...] TV[...].

Can you provide me with some simple example of such a phenomenon? In my opinion this could not be possible, because as soon as you start introducing infinitesimal quantities, all subsequent equations need be truncated to first-order. So an equation like

$$\text{macroscopic output}=F(\text{macroscopic input} + \text{infinitesimal variation}) $$

cannot be correct. I must be wrong, but why?

Thank you.

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Consider the freezing of water, i.e., a first order phase transition. There is a discontinuity in the order parameter and you cannot possibly Taylor-expand around that. –  Lagerbaer Dec 1 '11 at 1:48
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up vote 4 down vote accepted

At some point, whatever quantities you're talking about in "experimental sciences" refer directly or indirectly to something measurable with a precision limited to a certain number of siginificant digits. Anything smaller than the quantities in the frame of reference of the discussion by more orders of magnitude than this number of significant digits may be considered as "infinitesimal", and is normally treated as such (in the mathematical sense) when theoretically describing the situation. However, in the real world out there this "ininitesimal" quantity is not necessarily an actual "mathematical infinetisimal", it may just be something happening at a different order of magnitude, or a "very small" number.

Now, the quote mentions nonlinear systems, where mathematical chaos behaviour may appear, where "very small" differences in initial conditions may yield significant variations in the behaviour of the system at later times, so that may be the meaning for the author of this quote.

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The difference is in terms. "Infinitesimal" means something different for y'all than it means for us.

I see macroscopic output all the time from otherwise undetectable input when dealing with a Foucault tester used to probe the surface of telescope optics; and this is not some exotic device, but something you could build in your garage. If someone is merely walking in another room, the output at the tester changes visibly. If there's so much as a slight breeze in the room, I see it in the output.

I would call those inputs "infinitesimal", but I studied Physics, not Math.

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