Alignment of Rods and Partition of Integers
E. Ben-Naim and P.L. Krapivsky
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.
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