In an earlier post, I explain why I would find it useful if researches could shed some light on the history and the circumstances behind their publications. So, this post is about my paper

Christian Scholz, Gerd Schröder-Turk and Klaus Mecke

Pattern-fluid interpretation of chemical turbulence, Phys. Rev. E 91, 042907 (2015)

and why there might be an unsolved issue.

Some decades ago scientists realized that some of the patterns that we observe in nature cannot be easily explained by the basic interactions between atoms, molecules, cells etc. in the system. Instead these systems appeared to create patterns from an almost symmetric initial state. This would violate thermodynamic principles, but only, and that turned out to be very important, if the system was at equilibrium. It is obvious, that this is not the case for biological systems. But it took some time to realize this is also the case in many “dead” systems, such as granular or active matter, as long as there is a constant influx and dissipation of energy in the system.

One example for such systems are reaction-diffusion models, where chemicals diffuse and react with each other. It was found that if chemicals display a special kind of non-linear interaction (via catalysis and inhibition) and are constantly refreshed, spontaneous pattern formation can be observed.

Curiously, this was first predicted numerically, with Alan Turing being one of the first researchers who published on this (tragically, it was Turings last paper before his death). The heterogeneous patterns in reaction-diffusion systems are therefor often called Turing-Patterns. Finding a real chemical system that displayed this behavior proved to be challenging. It took quite some time after first predictions, until the 1990s, when the first stable chemical realization of a pattern forming reaction-diffusion model could be demonstrated. Typical patterns include stripe or zebra-patterns or dots with local hexagonal order.

In the chemical system, however, there were also interesting spatio-temporal chaotic patterns. This was interesting to me and I was curious if we can simulate the formation of such patterns. I tried various parameters and models, all of which have been shown to reproduce the stationary heterogeneous patterns very well. However, when it came to the chaotic patterns, I could not find a nice agreement between model and experiment. There are definitely some spatio-temporal patterns in the model as well, in fact, there is a large variety of patterns that has been found in the past in similar models. However, in all these well described models I wasn’t able to find a set of parameters that would really reproduce the experimental observations.

While playing around with the patterns I realized that you could simply overlap a fairly big amount of stationary patterns with random orientations and translations and this would nicely reproduce snapshots of the chaotic patterns. We found this really interesting and thought, if there was some possibility that chaotic patterns can be a kind of pattern fluid, where stationary patterns represent ground states and chaotic patterns are superpositions with random phases. Of course the equations of the model are non-linear partial differential equations, so this does not allow simple superpositions of single solutions to form a new one.

We could not come up with a true mathematical solutions to this, even though it is tempting to try a variational approach and from that define a sort of interaction energy between fundamental patterns. We published our results, but it was quite clear that we could not convince many referees of our results. I believe there are some common reasons for this. Referees thought we might just not have looked well enough to find the spatio-temporal chaotic patterns in standard reaction-diffusion models. I did quite some search over the years, but cannot complete rule it out. I would be extremely happy if someone would just be able to find the right model with the right set of parameters to reproduce the experimental patterns, because it would solve a very strange discrepancy between experiment and simulation. Or there is something in the experiment that the models do not quite include, maybe the actual three-dimensionality of the experimental system, boundary conditions or temporal fluctuations of input concentrations. All my attempts to include this in the model did not result in qualitatively new patterns. Nevertheless, this requires more research still. And finally, we cannot present a self-contained mathematical model for our pattern superposition model.

Nevertheless, I believe there is an unsolved problem, even though it might not be a huge issue. Keep in mind, the experimental article is cited many hundred times as an example for formation of spatio-temporal chaotic patterns in chemical systems. I think it would advance our understanding and make the literature on these type of patterns more robust, if we could find a model that fully replicates the observation.