An appetizer to modern developments on the Kardar–Parisi–Zhang universality class

Abstract

The Kardar-Parisi-Zhang (KPZ) universality class describes a broad range of non-equilibrium fluctuations, including those of growing interfaces, directed polymers and particle transport, to name but a few. Since the year 2000, our understanding of the one-dimensional KPZ class has been completely renewed by mathematical physics approaches based on exact solutions. Mathematical physics has played a central role since then, leading to a myriad of new developments, but their implications are clearly not limited to mathematics -as a matter of fact, it can also be studied experimentally. The aim of these lecture notes is to provide an introduction to the field that is accessible to nonspecialists, reviewing basic properties of the KPZ class and highlighting main physical outcomes of mathematical developments since the year 2000. It is written in a brief and self-contained manner, with emphasis put on physical intuitions and implications, while only a small (and mostly not the latest) fraction of mathematical developments could be covered. Liquid-crystal experiments by the author and coworkers are also reviewed.

Kardar–Parisi–Zhang universality class for the critical dynamics of reaction–diffusion fronts

“…Actually, for systems in the 1D KPZ class, the dynamical behavior is known to be particularly rich and complex, including a number of additional, interesting properties, such as ergodicity loss and aging, non-trivial persistence, peculiar fluctuation properties around steady steady state, etc. [35,36]. Such properties might warrant further detailed study in future for reactiondiffusion systems of the type that we have addressed here.…”

“…Such properties might warrant further detailed study in future for reactiondiffusion systems of the type that we have addressed here. In our present work, we have considered exponents and one-and two-point statistics as the main traits characterizing the universality class, as is currently being done in the context of kinetic roughening [35,36]. Indeed, identification of surface kinetic roughening universality classes, taking into account additional properties beyond exponent values, is becoming increasingly pertinent in view of potential ambiguities [55] and, more generally, because it provides an improved understanding of scale invariance far-from-equilibrium, not only in the KPZ case, but in other universality classes as well [46,63,64].…”

“…Sparked by exact solutions of this equation and other models in the same universality class for the one-dimensional (d = 1) case, the KPZ universality class is quite recently focusing a large attention, see e.g. [34][35][36] for recent reviews. Indeed, these results have shown that KPZ universality goes much beyond the values of the critical exponents.…”

Strong Coupling in Conserved Surface Roughening: A New Universality Class?

“…Kardar, Parisi, and Zhang (KPZ) [6] discovered an important universality class for growing rough interfaces, by adding the lowest-order nonlinearity to the continuum Edwards-Wilkinson (EW) model, in which height fluctuations are driven by nonconserved noise and relax diffusively [7]. The KPZ equation inspired many analytic, numerical, and experimental studies [8][9][10] and continues to surprise researchers [9,[11][12][13][14][15], not least of all because of a strong-coupling fixed point not accessible perturbatively [6]. Several experiments have been performed [16] to confirm the KPZ universality class and recently gained sufficient statistics to show universal properties beyond scaling laws [17][18][19].…”

Analytic iteration procedure for solitons and traveling wavefronts with sources

“…In the theory of superconductivity, we envisage applying the method to the sine-Gordon equation, which is used to describe the physics of fluxons in long Josephson junctions under the influence of a driving force [22,23]. In the area of interface growth, the Kardar-Parisi-Zhang equation and the related linear stochastic heat equation come to mind as candidates for application [24]. One can think of more examples of nonlinear DEs with sources that can be investigated with the proposed method.…”