Gravity's Rainbow

    Consider a system of n particles in a 2-dimensional box, repulsing each other with long range forces. No doubt, the ground state structure of such a system is very close to a perfect triangular lattice. Submerge the system within a strong, uniform gravitational field. Which is the structure of its ground state now?
    The question is by no means trivial. One may expect that locally the hexagonal symmetry of the triangular lattice should be preserved. Thus, the ground state structure should be built from hexagonal clusters - large at the top of the box, small, compressed by the gravitational field, at the bottom of the field. But, how to built a sensible wall from bricks the dimension of which changes continuously with the height? The clever way to perform the task was indicated by the nature itself. Below you will find a picture taken in a laboratory experiment with a set of magnetized (and thus repulsing) steel spheres [1].

Gravity's rainbow : the hint provided by nature itself.

    As seen in the figure, the local symmetry of the gravity's rainbow structure, as I have called it,  is everywhere hexagonal. The nearest neighbor hexagons are small at the bottom and large at the top of the sample. What is it, what we see? How to describe it? The answer, suggested independently by K.W. Wojciechowski and F. Rothen, is: the ground state structure is very close to a conformally transformed triangular lattice [2-8]. The complex transformation applied by the nature in this case seems to be ln z. Figure below presents this case.

Nearly conformal structures were observed also in other experiments. Below, I present a picture taken by Florence Elias in her experiments on the ferrofluid froth.

Conformally transformed lattice of soap bubbles (numerical, of course), and its experimental, magnetic froth realization.

<>A similar structure has been also observed  performed by physicists from the Trinity College at the University of Dublin. Have a look at the paper by Wiebke Drenkham et al: The demonstration of conformal maps with two-dimensional foams,  Eur. J. Phys. 25, 429-438 (2004) where she describes results of the experiments on 2D foams placed between two inclined glass plates.
What about the 3-D problem? Here the answer is not known yet. Figure below shows a numerically generated structure in the vicinity of which I hope to find the ground state.


[1]  P.Pieranski, Gravity's rainbow. Structure of a 2D crystal grown in a strong gravitational field, Proceedings of the NATO Advanced Study Institute, Geilo, 1989.

[2] F. Rothen, P. Pieranski, N. Rivier and A.  Joyet, Cristaux conformes, Eur.  J.  Phys. 14, 227 (1993)

[3] N. Rivier, P. Pieranski and F. Rothen, Conformal crystals and their defects,  Proceedings of the conference ‘Aperiodic 94’, Les Diablerets, 17-21, September 1994.

[4] F. Rothen, P. Pieranski, Mechanical equilibrium of conformal crystals, Phys.  Rev., E 53, 2828 (1996)

[5] A. C. Branka, P. Pieranski, K. W. Wojciechowski, Confined planar mesoscopic suspensions in external field: Brownian Dynamics simulation, Chem. Phys., 204, 51 (1996).

[6] K. W. Wojciechowski, J. Klos and A. C. Branka, Do Conformal crystals Exist in Uniform Fields, in "Monte Carlo and Molecular Dynamics in Condensed Matter", Conference Proceedings, Vol. 49, Eds. K. Binder and G. Ciccotti, SIF, Bologna, (1996).

[7] K. W. Wojciechowski and J. Klos,  Soft Condensed Matter in Strong Uniform Fields; Gravity's Rainbow Revisited, Journal of Physics A: Math. & General  29, 483 (1996).

[8] K. W. Wojciechowski, The inverse problem for conformal crystals in two- and higher-dimensions, in "Slow Dynamics in Complex Systems", Conference Proceedings, Eds. M. Tokuyama and I. Oppenheim, AIP 1999 (in print).

Why Gravity's Rainbow?

The name - gravity's rainbow  -  flushed through my mind as for the first time I have seen the conformal beauty.
It flashed through it, because a week or two before I lost a battle to read in original the Pynchon's novel.
Polish translation has just been published (2001). A new battle is in front of me.

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