Alignment of rods and partition of integers

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.

Figure 1

Illustrating of the alignment process.

Figure 2

The iterated integer partitions. Illustrated is the partition corresponding to $p_{3,1}=2G_{-1,4}{G}_{2,2}{G}_{1,1}^2$. Each partition $k=i+j$ with $i\ne 0$ and $j\ne 0$ generates a factor $G_{i,j}$. The factor 2 accounts for the two equivalent partitions.

Figure 3

(Color online) The order parameter versus the diffusion coefficient. The order parameter was obtained by solving polynomials of increasing order.

Figure 4

(Color online) The angular distribution for various values of $D$. The angle distribution was obtained from the first 15 Fourier modes.