I have a question, I am stuck on for quite some time now. Imagine you can choose a two dimensional autocorrelation function $C_V(x,y)$. From this I can create the two dimensional random process $V(x,y)$ (using the Wiener–Khinchin theorem and phase-randomization).
So far so good.
What I want in addition, is that the $x$-Integral of the autocorrelation function of the $y$-derivative is zero. So:
$\int_{-\infty}^{\infty} C_F(x)\,$d$x = 0$.
With:
$C_F(s_x) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}F(x+s_x,y+s_y)F(x,y)\,\text{d}x\,\text{d}y \, |_{s_y=0}$
$F(x,y)=\frac{\partial V(x,y)}{\partial y}$.
So the question is, how to choose $C_V(x,y)$ to obtain this (or is it even possible?).
Also $C_V$ can be assumed to be radial symmetric.
To put this in perspective: I want to look at a particle, moving in the random (but correlated) potential V(x,y), which experiences the sideway Force F(x,y). Currently I am interested in a force with the specific setup above.
What I have done so far is calculating : $\int_{-\infty}^{\infty} \frac{\partial^n C_V(x,y)}{\partial y^n}|_{y=0}\,$d$x$ and tried to express it in terms of $C_F$ to connect these two functions. Also played around a lot with mathematica and numerics but did not find any expression linking these two. The correlation function I usually start with (because of specific reasons to my problem) is : $C_V(x,y)=(1-a(x^2+y^2)) \cdot e^{-b(x^2+y^2)}$ But I am not bound on this form.
I hope someone has an idea or knows where to read up something about this. Thanks :)