Conclusions

As shown in Fig. 2, lines of traffic form near the second-order phase transition line. In this region there is a cooperative effect between the fluctuations and the ordering effects. Much below the transition line the fluctuations are too great, and no cooperative structures form. However, too far above the transition line the fluctuations are suppressed, and order dominates overwhelmingly. This means that the ants are basically induced to ``turn around" with sufficient probability that only patches form. The role of such ``U-turns", has been investigated experimentally and theoretically in [21], where it was also suggested that they might play an important role in the self-organization of ant traffic.

The region in which the lines form represents a cooperative effect between the fluctuations and the ordering behavior. The formation of lines can also be understood in terms of the theory in [9,10]. In the region of the lattice where there are lines, the clumping effect is both unstable in the transverse direction and stable in the lateral direction of the particle motion, and it is this fact which gives rise to the formation of stable lines of traffic. The result is a fairly robust region of line formation. Motion in this region can perhaps be likened to motion in the liquid crystal phase of matter, or the reptation of polymer chains in polymer melts, in which the order along one axis is frozen in, while along another motion can occur freely.

Since information (in the form of the orderly movement of ants from one place to another) can be said to flow only in the region of stable line formation, which lies above the disordered region and below the very ordered region, these results might mistakenly be thought similar to the ``meta-theory" of ``complexity at the edge of chaos" which asserts that complex behavior emerges in the vicinity of a marginally stable state.[22,23,24,25] However we point out that the system, as far as we know, does not ``self-organize" to this region unless it may be said that on the evolutionary time scale the biological organisms found such behavior adaptive. In addition there is not an ``edge" but rather a large robust region where the most complex structures form. Lastly we do not believe that the reason for this type of behavior has anything to do with the very speculative ones which are sometimes suggested. It is quite natural that complex behaviors resulting from the competition between stable and unstable modes should appear near a phase transition line, since the first unstable modes appear there. Such behaviors can be incorporated as biological functionality as shown in the few examples discussed in this letter, but this functionality has nothing to do with the notion of complexity ``at the edge of chaos" and the hypothetical universal computational properties which may or may not exist there.[23,25]

ER would like to thank the the Santa Fe Institute and the NSF REU program, and the UGS program at LANL which supported parts of this research.