Alignment of Rods and Partition of Integers

Abstract

We study dynamical ordering of rods. In this process, rod alignment via pairwise interactions competes with diffusive wiggling. Under strong diffusion, the system is disordered, but at weak diffusion, the system is ordered. We present an exact steady-state solution for the nonlinear and nonlocal kinetic theory of this process. We find the Fourier transform as a function of the order parameter, and show that Fourier modes decay exponentially with the wave number. We also obtain the order parameter in terms of the diffusion constant. This solution is obtained using iterated partitions of the integer numbers.

6 pages, 4 figures

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• Condensates in Driven Aggregation Processes
• Glassy dynamics and hysteresis in a linear system of orientable hard rods
• Collective dynamics of active filament complexes
• Pattern Selection and Super-patterns in the Bounded Confidence Model
• Random many-particle systems: applications from biology, and propagation of chaos in abstract models