Dear All,
First of all, I have to update the arxiv with the latest version of the paper, since there are quite a few changes. I've added an updated version in the attachment.
Your friction coefficients are rather high, I don't have any experience with those values, but here are some of my thoughts on why there is can be difference. There are actually many different things at play which can prevent you from obtaining the 'desired behavior' whatever that is, and it might not be what you think it is.
(i) LB does not reproduce the true HI in the first few time steps. This can be referred to as the "unphysical" regime.
(ii) the solution to the compressible Navier-Stokes equation is different from that of the incompressible Navier-Stokes equation (this is modeled by LB) for short times, which can lead to interpretation issues when it comes to what the bare and virtual mass of the particle are. For a compressible fluid there should be two decays one with the bare mass (mass you specify) followed by another with the virtual mass (mass of particle + half of displaced fluid mass).
(iii) However, as your colloid is actually porous, the virtual mass decay should not appear and some intermediate decay will be present.
(iv) But, all of the above is rather academic, since the unphysical coupling will not allow you to measure any of these aspects.
so you can realistically only consider the long-time tail. As Owen pointed out, getting the VACF to be precise enough to observe the correct decay in the bulk is tricky and takes a long time. Moreover, when you use a finite box with periodic boundary conditions, you will need to introduce a correction term based on the Hasimoto formalism to obtain the theoretical power-law decay. Even then, there can be problems depending on the type of experiment you carry out. My suggestion is to measure the particle's mobility/diffusion coefficient in a two-plate setup:
http://arxiv.org/abs/1503.02681. This will introduce a lot less artifacts and there are accurate theoretical expressions to compare to. However, as said, it takes a long time to even be able to pick up on the inconsistencies in some of the hydrodynamic experiments, so you probably won't be able to do so for your project.
Finally, you should be careful with the LB in general. It is compressible, so not just the Reynolds number is important, the sonic time is really crucial too. If you have a weird time-step or lattice-spacing combination then you will have effects of sonic back coupling, see the Problete ... Gompper reference in the attached paper.
I hope this helps untangle some of the issues you have been seeing and does not further confuse the matter.
Kind Regards,
Joost
theory_pt1.pdf