Statistics of zero crossings in rough interfaces with fractional elasticity
Résumé
We study numerically the distribution of zero crossings in one-dimensional elastic interfaces described by an overdamped Langevin dynamics with periodic boundary conditions. We model the %restoring elastic forces with a Riesz-Feller fractional Laplacian of order $z = 1 + 2\zeta$, such that the interfaces spontaneously relax, with a dynamical exponent $z$, to a self-affine geometry with roughness exponent $\zeta$. By continuously increasing from $\zeta=-1/2$ (macroscopically flat interface described by independent Ornstein--Uhlenbeck processes) to $\zeta=3/2$ (super-rough Mullins--Herring interface), three different regimes are identified: (I) $-1/2<\zeta<0$, (II) $0<\zeta<1$, and (III) $1<\zeta<3/2$. Starting from a flat initial condition, the mean number of zeros of the discretized interface (I) decays exponentially in time and reaches an extensive value in the system size, or decays as a power-law towards (II) a sub-extensive or (III) an intensive value. In the steady-state, the distribution of intervals between zeros changes from an exponential decay in (I) to a power-law decay $P(\ell) \sim \ell^{-\gamma}$ in (II) and (III). While in (II) $\gamma=1-\theta$ with $\theta=1-\zeta$ the steady-state persistence exponent, in (III) we obtain $\gamma = 3-2\zeta$, different from the exponent $\gamma=1$ expected from the prediction $\theta=0$ for infinite super-rough interfaces with $\zeta>1$. The effect on $P(\ell)$ of short-scale smoothening is also analyzed numerically and analytically. A tight relation between the mean interval, the mean width of the interface and the density of zeros is also reported. The results drawn from our analysis of rough interfaces subject to particular boundary conditions or constraints, along with discretization effects, are relevant for the practical analysis of zeros in interface imaging experiments or in numerical analysis.