Scaling and Localization in Multipole-Conserving Diffusion

We study diffusion in systems of classical particles whose dynamics conserves the total center of mass. This conservation law leads to several interesting consequences. In finite systems, it allows for equilibrium distributions that are exponentially localized near system boundaries. It also yields an unusual approach to equilibrium, which in $d$ dimensions exhibits scaling with dynamical exponent $z=4+d$. Similar phenomena occur for dynamics that conserves higher moments of the density, which we systematically classify using a family of nonlinear diffusion equations. In the quantum setting, analogous fermionic systems are shown to form real-space Fermi surfaces, while bosonic versions display a real-space analog of Bose-Einstein condensation.

Figure 1

(a) Schematic of our dipole-conserving hopping process, where particles hop toward opposite directions in pairs. (b) Evolution of the standard deviation of the density $\sigma (t)$ starting from initial Gaussian distribution. In both 1D (circles) and 2D (triangles), $\sigma (t)$ exhibits an excellent scaling collapse with dynamical exponent $z=4+d$ [the straight line has slope 1, demonstrating $\sigma (t)\propto t^{1/z}$]. (c) Relaxation of a large number $N_0=6000$ of particles localized at $x=0$ in the presence of a small constant-density background ${\rho }_0=10$, showing a crossover from $z=5$ at short times to $z=4$ at long times. (d) Exponentially localized equilibrium density profile for 1D dipole-conserving diffusion on the half-space $x>0$.

Figure 2

(a) A schematic illustration of a 1D quadrupole-conserving diffusion process: two particles move two sites in one direction, while another particle moves three sites in the opposite direction. (b) The time evolution for quadrupole-conserving diffusion. The density is initially given as two widely separated Gaussian packets, which merge over time (black $\to$ red $\to$ blue) into a single Gaussian profile, with the standard deviation $\sigma$ remaining conserved (solid curves are guides to the eye).