Perhaps you could embed non-linear parameters in a smooth potential and choose the parameters at random. 

Dear Ajit, thanks a lot for this idea. I wonder whether one could use an integral transform in a similar way. Best regards Herbert

For example, we used radial potentials of the form V(\vec(r)) = k r^\alpha in a study (see below) to generate a range of related potentials by varying \alpha. 

https://www.researchgate.net/publication/229176469_Non-local_Wigner-like_correlation_energy_density_functional_Parameterization_and_tests_on_two-electron_systems

Article Non-local Wigner-like correlation energy density functional:...

How do you propose to get a random potential from these alpha-dependent potentials? You think of alpha values as obtained from some probability distribution p(alpha), right? Then,

< V(x) V(y) > =k^2 int p(alpha) (xy)^alpha d alpha ...

Even such a one-parameter modeling already yields quite a large class of random potentials since one can choose p(alpha) rather freely.

My suggestion is that Read the work of LN Trefethen in MATLAB. Use data fitting 

method .Then construction will be easy . Follow the steps 

(i) select one experimental data .

(ii) Guess a function with variables (fixed)

(iii) check theoretical with experimental 

(iv) vary the parameter in guess function  till your calculated value give nice agreement with expt.data.

(v) Select another expt data of same system and follow the above procedure

in this way you can generate new potential  on subject of your interest .

Prof Rath

This probably won't be of much help, but you can randomly distribute spheres in a volume(like in molecular dynamics), assign a smooth potential to each(centered at each), remove the spheres and you have a random potential.

Herbert,

Yes, choosing the non-linear parameters randomly from  some distribution was precisely the thought behind my suggestion. 

Cheers

Ajit

The most simple approach always worked for me: prescribe the values of the random potential as random numbers on a suitably chosen lattice of points and get a smooth connecting curve by a suitable interpolation method. Of course V(x) and V(y) are strongly correlated if x and y are close together but they are strictly independent when separated by more than n sampling points, where n depends on the interpolation scheme.

Ups..I see the question was more specific and considered the point to point correlation function as given. Then my method does not work.

https://en.wikipedia.org/wiki/Poisson%27s_equation#Potential_of_a_Gaussian_charge_density

Dear Herbert, Here   https://arxiv.org/abs/1509.05737 a distribution of random potentials is introduced and I guess they are "smooth".

Dear Bartosz, one could choose the centers randomly and sum all the smooth potentials at the centers - is it that what you mean? Or do you suggest using the spheres to avoid centers that are "too close" together? One could use the same potential at each of the centers or even choose a ramdomized potential as suggested by Ajit at each center. I think that such a "linear combination of potentials at random centers" offers a lot of possibilities for smooth random potential construction.

Dear Ulrich, in one dimension, interpolating the random values at the grid points smoothly is relatively easy. But how would you proceed in more than one spactial dimensions?

Dear Hans, do you propose to use a Poisson solver with random boundary conditions to obtain a smooth random potential?

Dear Alexander, thanks for the pointer. It is very interesting since it shows that also smooth  random vector potentials may be important, and you provided an example how to construct these.

Location density distributions describe the distribution of locations of discrete point-like artifacts that usually represent objects that act as discontinuities in the field that describes them. In that case the location density distribution is a mostly continuous function. If the distance between neighbor locations is small with respect to the average width of the Green's function of the field, then the location density distribution approaches a continuous function and is part of a continuum that represents a field. It is known that a Gaussian distribution of locations generates a field that can be described by the function ERF(r)/r, where r is the center of the location swarm. If the location swarm is generated by a spatial Poisson point process, then one of the possibilities is that the generated location density distribution is a Gaussian location distribution and the potential which is generated is the above error function divided by its argument. A stochastic process generates a coherent location swarm when the location density distribution is a continuous function. The swarm can be considered as to move as one unit if the swarm contains a displacement generator and that is the case when at every progression step the location density distribution owns a Fourier transform. The generated objects are point-like objects and represent something that hops along a hopping path and the landing locations form a location swarm. The swarm and the hopping path represent the described object. The location density distribution corresponds to the squared modulus of the wave function of the object. The Fourier transform of the location density distribution represents the characteristic function of the stochastic process that generates the hopping locations.

The question is now what mechanism ensures that the location density distribution is sufficiently continuous and owns a Fourier transform.

http://vixra.org/abs/1606.0028

Dear Hans, yet another point of view on the question raised, and what a fine one. It gets my mind spinning , since this a quite a long step to go for me :). Do you have by chance some reference to the approach that you described? Or is the topic fully described in the preprint that you cited?

@Herbert,

a reasonably simple and universal interpolation scheme for scattered data in arbitrary dimension is among the urgent needs for many programmers. My solution is in

www.ulrichmutze.de

Scientific Software by Ulrich Mutze

C+- code and Linux makefiles

first bulleted line: the link 'here'

cpm1/include/cpmalgorithms.h

cpm1/source/cpmalgorithms.cpp

class AveragingInterpolator

Dear Dr. Hommeler,

                                  I have done some research on random potentials in designing Stock Market Systems and working on E-Commerce nano systems. You can take a look at my paper on The String Theory of Stock Market Systems on my RG site which has been published in the European Journal of Physics as Archived paper in 2015. You will see that a way, which uses both General Relativity and String Theory differentiation and systems integration to multiply the effects of quantum information and protocols is to look at the equilibrium (Quatum Mechanics) of the systems which are using the random potentials after statistically minimising the white noise in such systems by using some optimisation like Reservoir Engineering Optimisation and analysing the zeros of the system dynamically. Of course stock exchange data could be a good source of such smooth random potential data for simulation exercises. SKM, NH, SM

Dear Dr Mallick,

could you please summarize how your random potentials are constructed? This would be very helpful for the readers of this thread in my opinion.

Thanks a lot in advance and best regards

Dear Prof. Homeier,

                                   I am copy pasting from a lecture we are preparing (in progress) which in our opinion aptly summarizes the construction of the random potentials

"8.     Renormalisation Group (SO(32) Heterotic String Group Computer System) for dimensional cooling and conducting properties :

 The infinity cancellation and compactification of the system by the objectives of Physical (Systems) Technological Efficiency*{Economic Market Efficiency*Financial Market} Efficiency coupled e-commerce network (closure in the consumption action) has a equilibrium equation system of 23 (Mallick (2015)) + 6 algorithm steps + 3 BLUE Steps of Gauss-Markov Econometric Error (System Noise) Minimisation to be classified in the String Theoretic SO(32) Heterotic String with Higgs-Englert-Bosonic Mean Field electron flow. The velocity for the Phosphorus -Ceramic Dielectric Material Group has been computed after implementing the E-Commerce Network and successful periodic equilibrium system in Lagrangian commercial market actions using the experimental equilibrium prices for that time period in the Indian markets. This is a lower dimensional space than the E8xE8 Heterotic String Group and by Newtons Law of Cooling can be cooled for superconducting properties of the E-Commerce (also genetic i.e.growing within the Edgeworth Box by collective Lagrangian genetic market actions) satellite system (see also Sullivan, Kolokythas, Raychaudhury, Vrtilek, Kantharia (2013) for radiative cooling in Astrophysics for lowering of dimensions)). It satisfies the Boolean Field property because of the linearity of the Gauss Markov Equilibrium solution space. Hence satisfies systems closure for information and energy coupled networking and Econophysics Arrow of Time, Symmetric and normalized Equilibrium Actions."

Soumitra K. Mallick, Nick Hamburger, Sandipan Mallick

Ref. Mallick (2015) is on the RG website of the first author. The Sullivan et. al. (2013) paper is from the  Bulletin of the Astrophysical Society of India

Also, . ______, Nick Hamburger & Sandipan Mallick, Design of gated switches using HAAG Theorem bypass technology, Nick Hamburger website, January 10, 2016.

Soumitra K. Mallick, Nick Hamburger, Sandipan Mallick

Dear Prof. Homeier,

                                  We have already preliminarily assigned the lecture as mentioned above for the consideration of learned scientific societies and as such the contents are confidential. So without meaning to sound alarmist the discussion in its present state is purely for academic purposes. I hope the Research Gate community understands. Thanks. Soumitra K. Mallick, Nick Hamburger, Sandipan Mallick

@Soumitra K Mallick

If I would get this letter my hypothesis would be that the authors try to surpass Sokal's hoax. I would be sure that they can't be serious.

May be another approach may help: The signature of the foundation of physical reality is visible in all aspects of the universe. See the link below.

Hans van Leunen, retired physicist

http://vixra.org/abs/1607.0146

Dear Ulrich, hoax is probably the right estimate. :) I definitely would reject refereeing that contribution.

Best regards

Herbert