The role of the non-linearity in controlling the surface roughness in the one-dimensional Kardar--Parisi--Zhang growth process

TL;DR

This paper investigates how linear control of a non-linear stochastic growth process affects surface roughness and dynamics, revealing conditions under which the controlled system exhibits Edwards--Wilkinson or KPZ universality class behavior.

Contribution

It introduces a control protocol applied to Fourier modes of the KPZ equation, analytically and numerically analyzing the transition between Edwards--Wilkinson and KPZ regimes under control.

Findings

Controlled growth can exhibit Edwards--Wilkinson dynamics in weak non-linearity.

Strong non-linearity under control can still show KPZ scaling.

Crossover time between KPZ and dispersive regimes scales with the number of controlled modes.

Abstract

We explore linear control of the one-dimensional non-linear Kardar--Parisi--Zhang (KPZ) equation with the goal to understand the effects the control process has on the dynamics and on the stationary state of the resulting stochastic growth kinetics. In linear control, the intrinsic non-linearity of the system is maintained at all times. In our protocol, the control is applied to only a small number $n_c$ of Fourier modes. The stationary-state roughness is obtained analytically in the small-$n_c$ regime with weak non-linear coupling wherein the controlled growth process is found to result in Edwards--Wilkinson dynamics. Furthermore, when the non-linear KPZ coupling is strong, we discern a regime where the controlled dynamics shows scaling in accordance to the KPZ universality class. We perform a detailed numerical analysis to investigate the controlled dynamics subject to weak as well as strong non-linearity. A first-order perturbation theory calculation supports the simulation results in the weak non-linear regime. For strong non-linearity, we find a temporal crossover between KPZ and dispersive growth regimes, with the crossover time scaling with the number $n_c$ of controlled Fourier modes. We observe that the height distribution is positively skewed, indicating that as a consequence of the linear control, the surface morphology displays fewer and smaller hills than in the uncontrolled growth process, and that the inherent size-dependent stationary-state roughness provides an upper limit for the roughness of the controlled system.