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Oftentimes, when the derivative and integral operations are introduced within the realm of physics, we are taught some physical interpretation of them:

  • Velocity is the derivative of position
  • Velocity is the integral of acceleration

Integrals can additionally be used for finding quantities such as the centroid of a mass.

Where do fractional derivatives/integrals "naturally" arise in physics? How would one go about interpreting these phenomena in a physical sense? These need not be just derivatives with respect to time (space or some other quantity might make sense as well). One application I've seen fractional derivatives used in as well is in the definition of fractional Brownian motion (which has some "memory" associated with it).

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    $\begingroup$ Here's a phys.SE post playing around with fractional derivatives. Not that much in the way of intuition, though. $\endgroup$ – ACuriousMind Mar 23 '15 at 16:46
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    $\begingroup$ More on fractional derivatives. $\endgroup$ – Qmechanic Mar 23 '15 at 16:50
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Consider the drift diffusion equation

$$\dfrac{\partial}{\partial t}\psi=\mu\dfrac{\partial}{\partial x}\psi+\kappa^2\dfrac{\partial^2}{\partial x^2}\psi.$$

Dimensional analysis tells us that $\mu$ is a characteristic length per time (drift velocity) while $\kappa$ is a characteristic length per square root of time. This small factoid has curious consequences.

In statistical physics, $\kappa^2=2D$ is the diffusion coefficient. What follows also applies to non relativistic quantum mechanics, except the diffusion coefficient is imaginary, $\kappa^2=\frac{i\hbar}{2m}$.


Given the value $x(t)$ of a curve/stochastic process at time $t$, for any time interval $\Delta t > 0$, we can test for $x(t+\Delta t)$ and the increment $\Delta x\equiv x(t+\Delta t)-x(t)$ is probabilistic and dependents on $\Delta t$ (and possibly on $t$ or even on $x(t)$). For example, in the case of a Brownian motion each new $\Delta x$ takes values according to the distribution

$P(\Delta x)=\dfrac{1}{\kappa\sqrt{\Delta t}\sqrt{2\pi}}\exp \left( -\dfrac{1}{2}\dfrac{(\Delta x)^2}{\kappa^2\,\Delta t} \right)$.

(I set $\mu=0$ and note that usually one uses a variable $\sigma=\kappa\sqrt{\Delta t}$)

The Gauss curve distribution for $\Delta x$ says that even for very small $\Delta t$, there is a non-vanishing change that $x(t+\Delta t)$ is far away from $x(t)$. For bigger $\Delta t$, the distribution flattens out and the chance for bigger net deviation grows.

(Sidenote:Note that this weight also arises in the quantization of $L(q,{\dot q})\propto {\dot q}^2$:

$\frac{(\Delta x)^2}{\Delta t}=\left(\frac{\Delta x}{\Delta t}\right)^2\Delta t\approx \int_0^{\Delta t} \left(\frac{{\mathrm d}x}{{\mathrm d}t}\right)^2{\mathrm d}t$.)

Now, for the above $P$, we have:

$\langle \Delta x\rangle=0$

$\langle \left|\Delta x\right| \rangle=\sqrt{\tfrac{2}{\pi}}\,\kappa\,\sqrt{\Delta t}$

$\langle (\Delta x)^2\rangle=\kappa^2\,\Delta t$

This says that the movement has no preferred direction, but for a finite waiting time $\Delta t$ and if $x(t)$ is some mean path, we expect $x(t+\Delta t)=x(t)+\kappa\sqrt{\Delta t}$, 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. A non-differentiability of the curve in the model manifests itself here: While we know the overall deviation goes as $\sqrt{\Delta t}$, we can't make a good estimate for the instantaneous growth, because at $\Delta t=0$ the slope of the square root function $\frac{∂}{∂\Delta t}\sqrt{\Delta t}\propto\frac{1}{\sqrt{\Delta t}}$ isn’t finite! There is no $x'(t)$!

enter image description here


The accumulation of the values of a function $F$ along a smooth path $x(t)$ is

$\int_{t_0}^{t_1} F(x(s))\, {\mathrm d}x(s)$,

which is

$\int_{t_0}^{t_1} F(x(s))\, x'(s)\, {\mathrm d}s$,

where

$x'(t)=\lim_{\Delta t\to 0}\frac{x(t+\Delta t)-x(t)}{\Delta t}$.

Stochastic integrals are a means of computing the accumulation of a function along a path in cases where the above isn't defined. An Itō process is a stochastic process $X_t$ which is the sums of a Lebesgue and an Itō integral:

$X_t = X_0 + \int_0^t \mu_s(X_s, s)\,\mathrm ds + \int_0^t \sigma_s(X_s, s) \,\mathrm dW_s$

An Itō integral is roughly a Riemann integral of random variables. The norm in which the limit of partial sums converges is not the norm on $\mathbb R$, but instead the result is defined as a random variable for which the probability of being different than the limit goes to zero.

One writes

${\mathrm d}X_t = \mu_t(X_s, s) \, {\mathrm d}t + \sigma_t(X_s, s) \, {\mathrm d}W_t$

for the integral above. If $X_t$ isn't known, this is called a stochastic differential equation in $X_t$. Being an Itō process is the stochastic analog of being differentiable. If $\mu_t$ and $B_t$ are time independent, we speak of Itō diffusion. A geometric Brownian motion is characterized via $\mu_t(X_s, s)=X_s\,\mu$ and $\sigma_t(X_s, s)=X_s\,\sigma$, i.e. both are "just" $\propto X_s$.

The famous Itō lemma is

${\mathrm d}f(t,X_t) = \left(\dfrac{\partial f}{\partial t} + \dfrac{\sigma_t^2}{2}\dfrac{\partial^2f}{\partial x^2}\right){\mathrm d}t + \dfrac{\partial f}{\partial x}\,{\mathrm d}X_t$

The second derivative term comes from stochastic diffusion, a non-local flavor if you will.

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 $X_t$, we can compute $f(t,X_t)$ as such an integral (plus an ordinary integral).

Note that for $f(x,t)=\frac{1}{2}x^2$ and $\frac{\sigma_t^2}{2}=\kappa^2$ we get

${\mathrm d}\left(\frac{m}{2}X_t^2\right) = m\,\kappa^2{\mathrm d}t + X_t\,m\,{\mathrm d}X_t$

The next part is on the commutation relations $xp$ minus $px$, a version of the last equation which characterizes $x$ in the second term $px$ as detecting a diffusion effect. That's basically part of what Maimon writes about on the Wikipedia page on the path integral formulation of (quantum) mechanics:

Let

$p_{\Delta t}(t)=m\frac{x(t+{\Delta t})-x(t)}{{\Delta t}}$

If the limit $\lim_{\Delta t\to 0}p_{\Delta t}(t)$ exists, then for axillary $\delta$, we have $\lim_{\Delta t\to}x(t+\delta^2{\Delta t})=x(t)$.

Hence, for ever smaller time grid size ${\Delta t}$, e.g. an expression like

$x(t+\delta_1^2{\Delta t})\,x(t+\delta_2^2{\Delta t})\,x(t+\delta_3^2{\Delta t})$

converges to $x(t)^3$.

However, for

$x(t+\Delta t)\approx x(t)+\kappa{\sqrt{\Delta t}}$

with the square root, we find

$x(t+\delta^2{\Delta t})\,p_{\Delta t}(t)=\delta^2\,m\,\kappa^2+x(t)\,p_{\Delta t}(t)$.

The result says that two naively equivalent approximation schemes (e.g. $\delta=0$ vs. $\delta=1$) systematically differ by an additive diffusion term (e.g. $m\kappa^2$ here). In quantum mechanics, that's $m\kappa^2=m\frac{i\hbar}{2m}=\frac{i\hbar}{2}$.


So we had

$\frac{\partial}{\partial t} \psi = \kappa^2 \frac{\partial^2}{\partial x^2} \psi$

(note the imbalance of dimensions, $t$ vs. $x^2$) and in turn

$P(\Delta x) \propto \exp\left(c\frac{(\Delta x)^2}{\Delta t}\right)$

as next-step distribution, and then

$\langle |\Delta x|\rangle\propto (\Delta t)^{1/2}$

gives the non-smooth curve.

You may want to look at other next-step distributions, effectively giving the theories with $\langle |\Delta x|\rangle\propto (\Delta t)^{1/\alpha}$. We can see that the proportionality coefficient (the analog of $\kappa$ or the velocity) must be fractional and in turn we expect a fractional differential operator $\frac{\partial^\alpha}{\partial x^\alpha}$ in the corresponding diffusion equation. You get a Laplacian to the power of $\frac{\alpha}{2}$. The $\alpha\neq 2$-deformed theory with complex $\kappa$ is what's termed „fractional quantum mechanics“, though I don't know of really notable results, besides better understanding of the known cases.

To answer your question, associate the power of the operators to the above expectation value $\langle |\Delta x|\rangle$. The $\alpha$ it characterizes fast the system parameter in your model stochastically moves away from the center. So called Levy-flights might propagate in a rough way, the current/velocity becomes something more complicated (not proportional to the momentum, i.e. the thing that gets multiplied by $x$ in the plane wave solution). The Brownian normal-derivative case is friendliest diffusion. You wouldn't gain that intuition from that other question on fractional derivatives, because there the guy cooks up an equation with a fractional derivative of time and we're used to see all of space but only one time (the now).

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