What is the significance of the fractional derivative? Does it define any process or have any (current) uses in the real world?
fractional fourier transforms come to mind
>>7966417
oil drilling, it has to do with the porousness of the ground
>>7966417
I use it at my job at starbucks. We take pride in our lattes by integrating mocha directly into coffee with fractional derivatives
As with most obscure physics or applied mathematical techniques, it finds applications in condensed matter theory.
>>7966757
Kek
I use it at my job at starbucks. We take pride in our lattes by integrating mocha directly into coffee with fractional derivatives
>>7966757
I integrated the area of the interior of a Fleshlight just for the zozzles. Actually rigorous work depending on which model you use.
>>7966417
>implying that the average /sci/ lurker know basic calculus or what is the gamma function
>>7967652
So that means I can't pose a question?
It concernes the theory of Lévy flights (google it).
I'd say whole or integer power derivatives are special in the way random processes p with finite variance are special - for additive p-weighted observables you get a gaussian (central limit theorem), which fulfills a diffusion/heat/Schrödinger equation, and more tricky p are associated with propagations that corewapond to a deformed Laplacian, a fractional derivative.
I'm on the phone right now but I cab do a proper rant at night.
PS
https://en.m.wikipedia.org/wiki/Fractional_quantum_mechanics
>>7967701
OP here, looking forward to it. A friend of mine's professor just recently wrote a book on fractional derivatives which made me interested in the subject. It seems that there are more uses for it than i had previously thought.
>>7967721
Basically, I’ve made the point: Differential equations using fractional derivatives are associated some settings in statistical physics once you continuously deform the lower moments.
Do you know some quantum mechanics, Feynman–Kac formula or stochastic integrals?
I can try to line out QM like classical theory of diffusion (not working in terms of the time evolution <operator> but instead focusing on the time by time stochastic evolution )
Below it will help to always track the dimensions. e.g. consider the drift diffusion equation
[math] \dfrac {d} {d t} \psi = \mu \frac {d} {d x} \psi+ \kappa^2 \dfrac {d^2} {d x^2} \psi. [/math]
Dimensional analysis tells us that [math] \mu [/math] is a characteristic length per time (drift velocity) while [math] \kappa [/math] is a characteristic length per square root of time. This small factoid has curious consequences. In statistical physics, [math] \kappa^2 = 2D [/math] is the diffusion coefficient. What follows also applies to non relativistic quantum mechanics, except the diffusion coefficient is imaginary, [math] \kappa^2= \frac {i \hbar} {2m} [/math].
Firstly, the accumulation of the values of a function F along a smooth path x(t) is
[math] \int_ {t_0}^ {t_1} F(x(s)) \, dx(s) [/math],
which is
[math] \int_ {t_0}^ {t_1} F(x(s)) \, x'(s) \, ds [/math],
where
[math] x'(t) = \lim_ { \Delta t \to 0} \dfrac { x ( t + \Delta t ) - x (t) } { \Delta t } [/math].
The Itō integral is a means of computing the accumulation of a function along a path in cases where x’ isn't defined in a sensible way.
Given the value x(t) of a curve/stochastic process at time t, for any time interval [math] \Delta t > 0 [/math], we can test for [math] x( t+ \Delta t) [/math] and the increment [math] \Delta x \equiv x(t+ \Delta t) - x(t) [/math] is probabilistic and dependents on [math] \Delta t [/math] (and possibly on t or even on x(t)).
For example, in the case of a Brownian motion each new [math] \Delta x [/math] takes values according to the distribution
[math] P( \Delta x)= \dfrac {1} { \kappa \sqrt { \Delta t} \sqrt {2 \pi} } \exp \left( - \dfrac {1} {2} \dfrac { ( \Delta x)^2} { \kappa^2 \, \Delta t } \right) [/math].
(I set [math] \mu=0 [/math] and note that usually one uses a variable [math] \sigma= \kappa \sqrt { \Delta t} [/math])
The Gauss curve distribution for [math] \Delta x [/math] says that even for very small [math] \Delta t [/math], there is a non-vanishing change that [math] x( t + \Delta t ) [/math] is far away from x(t). For bigger [math] \Delta t [/math], the distribution flattens out and the chance for bigger net deviation grows.
Note that this weight also arises in the quantization of [math] L(q, { \dot q } ) \propto { \dot q}^2 [/math]:
[math] \frac { ( \Delta x)^2 } { \Delta t } = \left( \frac { \Delta x} { \Delta t} \right)^2 \Delta t \approx \int_0^ { \Delta t} \left( \frac { dx} { dt} \right)^2 dt [/math]
Now, for the above P, we have:
[math] \langle \Delta x \rangle = 0 [/math]
[math] \langle \left| \Delta x \right| \rangle = \sqrt { \tfrac {2} { \pi } } \, \kappa \, \sqrt { \Delta t}[/math]
[math] \langle ( \Delta x)^2 \rangle= \kappa^2 \, \Delta t [/math]
This says that the movement has no preferred direction, but for a finite waiting time [math] \Delta t [/math] and if x(t) is some mean path, we expect [math] x(t+ \Delta t) = x(t) + \kappa \sqrt { \Delta t} [/math], see picture.
The intuition is that for a very small waiting time, you could possibly already have a big deviation and the longer the wait the farther you get away from the center - however this movement is sub-linear because with more time, more and more cancellation occur as well. The non-differentiability of the curve manifests itself here: While we know the overall deviation goes as [math] \sqrt { \Delta t} [/math], we can't make a good estimate for the instantaneous growth, because at [math] \Delta t = 0 [/math] the slope of the square root function isn’t finite! There is no x'(t)!
An Itō process is a stochastic process [math] X_t [/math] which is the sums of a Lebesgue and an Itō integral:
[math] X_t = X_0 + \int_0^t \mu_s(X_s, s) \, ds + \int_0^t \sigma_s(X_s, s) \, dW_s [/math]
One writes
[math] dX_t = \mu_t(X_s, s) \, dt + \sigma_t (X_s, s) \, dW_t [/math]
If [math]X_t[/math] isn't known, this is called a stochastic differential equation in [math]X_t[/math].
Being an Itō process is the stochastic analog of being differentiable.
If [math] \mu_t [/math] and [math] B_t [/math] are time independent, we speak of Itō diffusion.
A geometric Brownian motion is characterized via [math] \mu_t(X_s, s) = X_s \, \mu [/math] and [math] \sigma_t(X_s, s)=X_s \, \sigma [/math], i.e. both are "just" [math] \propto X_s[/math].
Itō lemma:
[math] df(t,X_t) = \left( \dfrac { d f} { d t} + \dfrac { \sigma_t^2} {2} \dfrac { d^2f} { d x^2} \right) dt + \dfrac {df} {dx}\, dX_t [/math]
As this really is an integral relation, it corresponds to a version of the fundamental theorem of calculus. If we know how to integrate against [math] X_t [/math], we can compute [math] f(t,X_t) [/math] as such an integral (plus an ordinary integral).
The following comment is foreplay to the next part:
Note that for f(x,t)= (1/2) x^2 and [math] \frac { \sigma_t^2} {2}= \kappa^2[/math] we get
[math] d \left( \frac {m} {2} X_t^2 \right) = m \, \kappa^2 dt + X_t \,m \, dX_t [/math]
This gives the manifestation of the quantum mechanical commutation relations:
Let [math] p_ { \Delta t} (t) = m \frac { x ( t+ { \Delta t})-x(t)} { { \Delta t}} [/math]. If the limit
[math] \lim_ { \Delta t \to 0}p_ { \Delta t}(t) [/math] exists, then for auxiliary
[math] \delta[/math], we have [math] \lim_ { \Delta t \to}x(t+ \delta^2 { \Delta t})=x(t) [/math].
Hence, for ever smaller time grid size [math] \Delta t[/math], e.g. an expression like
[math] x(t+ \delta_1^2 { \Delta t}) \,x(t+ \delta_2^2 { \Delta t}) \,x(t+ \delta_3^2 { \Delta t}) [/math]
converges to [math] x(t)^3 [/math].
However, for [math] x(t+ \Delta t) \approx x(t)+ \kappa { \sqrt { \Delta t}}[/math]. We find
[math] x(t+ \delta^2 { \Delta t}) \,p_ { \Delta t}(t)= \delta^4 \,m \, \kappa^2+x(t) \,p_ { \Delta t}(t)[/math].
The result says that two naively equivalent approximation schemes (e.g. [math] \delta=0 [/math] vs. [math] \delta=1 [/math]) systematically differ by an additive diffusion term (e.g. [math]m \kappa^2 [/math] here).
I.e. the Itō integral is defined with most left of the grid cells as in the explicit Euler-method numerical approximation scheme. The implicit Euler-method simply corresponds to a different notion of integral here and would give a different result.
In quantum mechanics, the difference of the product above is [math] m \, \kappa^2=m \frac {i \hbar} {2m}= \frac {i \hbar} {2} [/math].
Finally, coming to fractional quantum mechanics… we had
[math] \frac { d} { d t} \psi = \kappa^2 \frac { d^2} { d x^2} \psi [/math]
(note the imbalance of dimensions, t vs. [math]x^2 [/math]) and in turn
[math] P( \Delta x) \propto \exp \left(c \frac {( \Delta x)^2} { \Delta t} \right) [/math]
as next-step distribution, and then
[math] \langle |x| \rangle \propto t^ {1/2} [/math]
gives the non-smooth curve.
You may want to look at other next-step distributions,
(effectively giving the theories with [math] \langle |x| \rangle \propto t^ {1/ \alpha} [/math])
and this is all there is to „fractional quantum mechanics“.
The imbalance t vs. [math] x^\alpha [/math] for non-integer alphas forced fractional derivatives on you.
Sorry for the essay.
>>7968257
Not OP, but do you happen to know physical applications of stochastic integration with respect to "infinite dimensional Brownian motion"? (I know the terminology is not quite correct, but I can't be arsed to write out the full definition.
>>7968302
I don't know the notion, does this require an Ito lemma with infinite directions?
>>7968245
>Δ
kill yourself
>>7968313
You have a separable Hilbert space and take a positive definite trace-class operator [math]D[/math] (+additional techniqual requirements) and consider [math]X(t) = \sum_1^\infty \sqrt{\lambda_i} B_i(t)[/math] where the [math]\lambda_i[/math] are the eigenvalues of [math]D[/math] and [math]B_i[/math] are independent BMs. At least that's the most concise definition I remember. There are many equivalent ones. So actually what you get is not an "infinite dimensional BM" but rather a Gaussian process with independent increments and covariance operators [math]tD[/math] since the standard BM doesn't generalize to infinite dimensions. The Ito formula itself works pretty much the same then, except the quadratic term becomes [math]\frac{1}{2}\int_0^t \sum_1^\infty \lambda_i \cdots[/math] something with second derivatives in each direction.
>>7968409
kk, why do you ask?
Btw. do you happen to know some code packages for stochastic integration? (I take Python, Haskell (lel), MatLab (if nothing else),...)
Just asking because I saw a script here and know that there are a great bunch of quantum so and so packages done in Python
https://en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method
>>7968443
Sorry, can't help with that. Still looking for good python libraries for stochastic simulation myself. Only thing I have so far are some very simple simulation methods to implement on your own (pic related).
>>7967701
I made a separate thread for this but no one posted. You might be able to help me.
What is the physical significance to eigenvectors and eigenvalues of the fractional Schrödinger equation?
For example, I can solve the Schrödinger equation for the hydrogen atom and explain observed line spectra. What does solving the fractional Schrödinger equation for hydrogen get me?
>>7967652
>tfw I am the beta function
>>7968943
Per definition they tell you the time evolution.
But I don't know an application - if there is none in the wiki refs, there might be none yet
>>7969230
There is a fractional time-independent Schrodinger equation as well though. I'm saying if I have a hydrogenic potential, what do my eigenstates and eigenvalues represent? They obviously aren't the states of real hydrogen in the way the non-fractional Schrodinger equation solutions are, since the line spectra would be all different, and not match experimental spectra.
