Feedback control of surface roughness in a one-dimensional Kardar-Parisi-Zhang growth process

Abstract

Control of generically scale-invariant systems, i.e., targeting specific cooperative features in nonlinear stochastic interacting systems with many degrees of freedom subject to strong fluctuations and correlations that are characterized by power laws, remains an important open problem. We study the control of surface roughness during a growth process described by the Kardar-Parisi-Zhang (KPZ) equation in (1 + 1) dimensions. We achieve the saturation of the mean surface roughness to a prescribed value using non-linear feedback control. Numerical integration is performed by means of the pseudospectral method, and the results are used to investigate the coupling effects of controlled (linear) and uncontrolled (non-linear) KPZ dynamics during the control process. While the intermediate time kinetics is governed by KPZ scaling, at later times a linear regime prevails, namely the relaxation towards the desired surface roughness. The temporal crossover region between these two distinct regimes displays intriguing scaling behavior that is characterized by non-trivial exponents and involves the number of controlled Fourier modes. Due to the control, the height probability distribution becomes negatively skewed, which affects the value of the saturation width. *

Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation

“…Regarding experimental studies of the KPZ equation via liquid crystal turbulence, we refer to [24][25][26]. Recent numerical treatment of the KPZ equation is shown in, e.g., [27][28][29][30][31]. From a mathematical point of view, in [32] a spacetime discretization scheme for the equivalent Burgers equation has been proposed and its convergence, albeit in a weak distributional sense (which seems to be the best to be expected), has been rigorously proven.…”

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

“…We choose b n (x) = exp (−i2πnx/L), as in our previous study (see Ref. [24] for details); the number of actuators is 2n c . The controlled equation ( 3) is solved with periodic boundary conditions,…”

“…In our previous work [24], we have shown that removing the non-linearity with different cutoff Fourier modes n c will enable the system to saturate at the desired roughness, but this significantly alters the growth dynamics. However, in order to implement a control scheme that does not change the intrinsic dynamics, it is essential to consider the effect of the non-linearity at all times.…”

“…Here, the eigenvalue λ c is obtained from the the stationary roughness of the controlled KPZ equation (3), see Ref. [24],…”

“…Thus, a detailed understanding of the effects of the non-linearity during the control process on both the dynamical and steady-state properties is needed. In our previous work [24], employing a non-linear control scheme, we demonstrated nontrivial scaling related to the KPZ universality class, which determines the time scale over which the controlled stochastic growth process is driven away from its intrinsic scale-free growth. In contrast, here we present a linear control protocol that maintains the inherent growth dynamics identical to that of the uncontrolled system, while also enforcing a desired saturation width.…”