The surface width, a global quantity that depends on time, is used to characterize the temporal evolution of growing surfaces. One of the most successful concepts for describing the property of the surface width is the famous Family-Vicsek scaling relation. We discuss an extended scaling relation that yields a complete description for various growth models.
For two linear Langevin equations, namely the Edwards-Wilkinson equation and the Mullins-Herring equation, we furthermore study analytically the behavior of global quantities related to the surface width or to a quantity which is conjugated to the diffusion constant. The global quantities depend in a non-trivial way on two different times. We discuss the dynamical scaling forms of global correlation and response functions.
For global functions related to the surface width, we show that the scaling behavior of the response can depend on how the system is perturbed. Different dynamic regimes, characterized by a power-law or by an exponential relaxation, are identified, and a dynamic phase diagram is constructed. We discuss global fluctuation-dissipation ratios and how to use them for the characterization of non-equilibrium growth processes. We also numerically study the same two-time quantities for the non-linear Kardar-Parisi-Zhang equation.
For global functions related to the quantity which is conjugated to the diffusion constant of the linear Langevin equations, we show that the integrated response is proportional to the correlation in the linear response regime. In the aging regime, the autocorrelation and autoresponse exponents are identical and the aging exponent for the response is equal to the aging exponent for the correlation. We investigate the non-equilibrium fluctuation-dissipation theorem for non-equilibrium states based on this quantity. In the non-linear response regime a certain dissipation-fluctuation ratio approximates unity for small waiting times but approaches the ratio of perturbed and unperturbed diffusion constants for larger waiting times.
