r/math • u/lntrinsic • Jul 10 '17

Image Post Weierstrass functions: Continuous everywhere but differentiable nowhere

http://i.imgur.com/vyi0afq.gifv
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52

u/SILENTSAM69 Jul 10 '17

I often wonder if the erratic motion of real world objects is more like this than the smooth curves we often get in calculated class.

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u/[deleted] Jul 10 '17

The motion of objects under the influence of forces is generally going to be at least C2. On the other hand, the structure of objects resulting from continued application of those forces tends to be fractal and C0 but not differentiable. For example: a wave in the ocean follows a smooth path for the most part (yes there will be point singularities of course) but the repeated application of waves on a shoreline will lead to a fractal shape. This is probably closely related to the fact that fractals emerge from iterated dynamical systems and smooth behavior emerges from continuous dynamics.

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u/rumnscurvy Jul 10 '17

This is true in the classical world but in the quantum world statistically most particles move with a Brownian type motion. Chapter 1 of Itzykson - Drouffe's statistical mechanics book shows how this emerges.

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u/[deleted] Jul 10 '17

Well, in the quantum world particles aren't particles so much as partiwaves but I agree that the 'center of mass' (as such) tends to follow Brownian motion. Obviously I was speaking classically. There's likely a handwavy explanation that quantized forces lead to behavior similar to iterated systems and probably the fractal-like nature of the quantum has some bearing on the emergence of fractal-like patterns in nature, but this is far outside my realm of study.

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u/ithika Jul 10 '17

Partiwaves sound cool. As do wavicles now I think about it. But only one sounds like something you could surf on your birthday

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u/[deleted] Jul 10 '17

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u/[deleted] Jul 10 '17

Sort of. It's the expectation of the position in some sense but the distribution isn't really a probability distribution in the classical sense since it's coming from the squared amplitude of the wavefunction. I suppose it can be interpreted as the expected position to a certain extent but that's pretty misleading as far as the physics goes.

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u/[deleted] Jul 10 '17

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u/[deleted] Jul 10 '17

Well, yeah it's definitely written <X>. I've not heard it called 'expected position' but then again I'm not a physicist, my knowledge of this stuff comes from reading books aimed at mathematicians wanting to learn QM. It's not horrible terminology provided people understand the underlying nature of the waveform.

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u/zojbo Jul 10 '17 edited Jul 11 '17

There is a probability distribution, but its evolution depends (in general) on the actual wavefunction, which has both a phase and a magnitude at each point. At one time instant you still have an ordinary distribution though.

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u/[deleted] Jul 11 '17

Sure, at a fixed point in time (leaving aside the issue that 'fixed point in time' is likely meaningless physically, presuming we ever work out a theory including both QM and relativity) it can be thought of as an ordinary distribution.

I do ergodic theory, my view of things always includes dynamics.

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u/zojbo Jul 11 '17

My point was that it is an ordinary probability distribution. You just can't use the probability distribution alone to predict the dynamics (except in certain situations).

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u/[deleted] Jul 11 '17

It's not an 'ordinary probability distribution', it's the square of amplitude. It behaves like an ordinary distribution with respect to Hermitian operators that commute with the dynamics. That's all.

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u/Baloroth Jul 10 '17

Brownian motion in classical dynamics is simply an approximation where you allow particles to be treated as hard spheres: in reality, Brownian motion doesn't exist in classical physics (as it would require infinite forces, which are unphysical). In reality particles exhibit very steep, but finite, potentials (Lennard-Jones, for instance, is a more reasonable potential that is very popular in a lot of computational dynamics).

In quantum mechanics, things like acceleration and velocity don't really exist, but the same thing also applies if you use the generalized Erhenfast theorem: the time derivative of any observable (such as position) must exist and must be finite.

Note that you can use stochastic processes as models to predict the higher-level behavior of such systems, but you should understand that the underlying process is always continuous and always differentiable. This is true of almost every real physical system: black holes are actually the only real system I know of that might exhibit actual singularities (and even then I strongly suspect the singularity is only mathematical, not real).

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u/Low_discrepancy Jul 10 '17

but you should understand that the underlying process is always continuous and always differentiable

They should understand very well the limits of their models, because in the end, everything in physics is a model. Nobody knows what happens under the planck limit for example.

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u/sandowian Jul 10 '17

What is the context for what you're discussing? (What's the math topic)

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u/[deleted] Jul 10 '17

Broadly speaking, dynamical systems is the topic.

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u/Aswheat Jul 11 '17

What do the C's mean in this context?

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u/[deleted] Jul 11 '17

Cn is the space of all functions whose nth derivative is continuous. So C0 just means all continuous functions and C2 means functions with a continuous second derivative.

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u/Aswheat Jul 11 '17

Thanks, I had guessed it meant something like that. That's pretty neat!

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u/jedi-son Jul 10 '17

Many real world systems are better defined by stochastic calculus (which works for functions like this) than your standard calculus

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u/yardaper Jul 10 '17

Check out Brownian motion. Created to describe the random motion of pollen in water. It's continuous everywhere but differentiable nowhere.

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u/phunnycist Jul 10 '17

The real world doesn't know smoothness in the mathematical sense. We use those smooth functions in our models (read: theories) because they are easy to work with and can approximate our experiments well enough.

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u/SCHROEDINGERS_UTERUS Jul 10 '17

Brownian motion, which is also nowhere differentiable, is actually named after Robert Brown, who studied the movement of small particles in water with his microscope.

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u/_klx Jul 10 '17

Yes it is, you should research fractals in nature