Recent developments on the KPZ surface-growth equation

H. S. WIO, J. A. REVELLI; C. ESCUDERO; R.R. DEZA; M.S. DE LA LAMA

PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES

ROYAL SOC

Lugar: Londres; Año: 2011 vol. 369 p. 396 - 411

1364-503X

The stochastic nonlinear partial differential equation known as the Kardar–Parisi–Zhang(KPZ) equation is a highly successful phenomenological mesoscopic model of surface andinterface growth processes. Its suitability for analytical work, its explicit symmetries andits prediction of an exact dynamic scaling relation for a one-dimensional substratumled people to adopt it as a ‘standard’ model in the field during the last quarter of acentury. At the same time, several conjectures deserving closer scrutiny were establishedas dogmas throughout the community. Among these, we find the beliefs that ‘genuine’non-equilibrium processes are non-variational in essence, and that the exactness ofthe dynamic scaling relation owes its existence to a Galilean symmetry. Additionally,the equivalence among planar and radial interface profiles has been generally assumedin the literature throughout the years. Here—among other topics—we introduce avariational formulation of the KPZ equation, remark on the importance of consistencyin discretization and challenge the mainstream view on the necessity for scaling ofboth Galilean symmetry and the one-dimensional fluctuation–dissipation theorem. Wealso derive the KPZ equation on a growing domain as a first approximation to radialgrowth, and outline the differences with respect to the classical case that arises in this