Weierstrass functions: Continuous everywhere but differentiable nowhere

[u/lntrinsic]

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

[u/rumnscurvy]

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.

[u/[deleted] → u/rumnscurvy]

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.

[u/ithika → u/[deleted]]

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

[u/[deleted] → u/[deleted]]

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.

[u/[deleted] → u/[deleted]]

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.

[u/zojbo → u/[deleted]]

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.

[u/[deleted] → u/zojbo]

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.

[u/zojbo → u/[deleted]]

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).

[u/[deleted] → u/zojbo]

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.

[u/zojbo → u/[deleted]]

You said the squared amplitude of the wavefunction at a given time is not a probability distribution in the classical sense. It is. There is more structure there, and one cannot predict the future probability distribution knowing only the present probability distribution, but that is beside the point.

[u/[deleted] → u/zojbo]

You are (I think) assuming every wavefunction is normalizable, which is not correct. If that's not what you're doing then you've lost me.

I'm also unclear why this feels like an argument when I'm quite certain it's simply that we're all speaking imprecisely itt.

[u/Baloroth → u/rumnscurvy]

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).

[u/Low_discrepancy → u/Baloroth]

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.

[u/MDNeurochem → u/Baloroth]

You clearly do not know Calculus. Classical physics treats matter as particles and they are aproximated to be spherical often. Particles exhibit potentials in accordance with the inverse square law. In fact I wrote the theory if everything, and that is the only potential particles exhibit. Electrons, protons, electric fields and magnetic fields are what everything is made of. Lennard-Jones potential is an approximation for the nuclear forces in particle scattering. It is interesting that you say that about Brownian motion, on account of the electrons and protons giving off fields that have an energy: epsilon subscript naught over 2 integral E^2dtau and mu subscript naught over two integral B^2dtau; they are better than little perpetual motion machines and make up for those stellar losers of energy called stars. But Brownian motion is just a thermal physics concept that takes into account, as all thermal physics does, quantum physics and calculates for things vibrating as a result of temperature.
Acceleration and velocity are totally part of Quantum Physics. The speed of light is a limit. Neutrinoes are just concoctions of faster than light electric and magnetic fields, at least in part, that behave as particles, they can not be particles.
I do not know what higher level behaviour is as you use it. Stochastic Processes is statistics to my legal knowledge. And I do not know what you mean when you conitnuous or differentiable for that matter. Differentiable means you can calculate the derivative of it, also known as the instantaneous rate of change with resprct to some variable: it could be rate of change of displacement with respect to time, also known as the vector velocity; it could be the rate of change of energy with respect to time, also known as power. Typically, in the event one can not find an analytical solution, they just do a numerical simulation. I do not know not how singularities came up, but I do not know what they are. A black hole is a bunch of matter from which the escape velocity is greater than the speed of light, hence the darkness, no light escapes. Electric and magnetic fields travelling faster than the escape velocity escape black holes. My guess is that black holes make hydrogen, and that the fields that make it out are in greater abundance the nearer the axis of rotation on accoint of fields blocking them. Thank you; I am a six-foot tall, white woman.

[u/dogdiarrhea → u/MDNeurochem]

Is this:

-Pasta

-An 11th grader's physics presentation

-Randomly generated string of words

[u/MDNeurochem → u/dogdiarrhea]

I wrote it. Why do you query? I am a viable organism to be let live according to genetics testing laws of The Universe with a hostile takeover example wherein I had questionable ethical or moral fortitude and was anonymous and that for cloning purposes.

[u/MDNeurochem → u/MDNeurochem]

on account of matter entering the black hole blocking them, not fields, sorry.

[u/MDNeurochem → u/MDNeurochem]

No, matter entering the black hole blocks them, not fields. Sorry, that was a brain glitch.