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 non-specialists, 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.
•A review on recent developments on KPZ, intended for non-specialists, is given.•Physical implications of theoretical results are stressed.•Connections to directed polymer, a quantum many-body system, etc. are explained.•Experimental study using liquid-crystal turbulence is also explained.
A KPZ Cocktail-Shaken, not Stirred Halpin-Healy, Timothy; Takeuchi, Kazumasa A.
Journal of statistical physics,
08/2015, Letnik:
160, Številka:
4
Journal Article
Recenzirano
The stochastic partial differential equation proposed nearly three decades ago by Kardar, Parisi and Zhang (KPZ) continues to inspire, intrigue and confound its many admirers. Here, we (i) pay debts ...to heroic predecessors, (ii) highlight additional, experimentally relevant aspects of the recently solved 1+1 KPZ problem, (iii) use an expanding substrates formalism to gain access to the 3d radial KPZ equation and, lastly, (iv) examining extremal paths on disordered hierarchical lattices, set our gaze upon the fate of
d
=
∞
KPZ. Clearly, there remains ample unexplored territory within the realm of KPZ and, for the hearty, much work to be done, especially in higher dimensions, where numerical and renormalization group methods are providing a deeper understanding of this iconic equation.
We investigate growing interfaces of topological-defect turbulence in the electroconvection of nematic liquid crystals. The interfaces exhibit self-affine roughening characterized by both spatial and ...temporal scaling laws of the Kardar-Parisi-Zhang theory in 1+1 dimensions. Moreover, we reveal that the distribution and the two-point correlation of the interface fluctuations are universal ones governed by the largest eigenvalue of random matrices. This provides quantitative experimental evidence of the universality prescribing detailed information of scale-invariant fluctuations.
This Letter reports on how the interfaces in the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) class undergo, in the course of time, a transition from the flat, growing regime to the stationary one. ...Simulations of the polynuclear growth model and experiments on turbulent liquid crystal reveal universal functions of the KPZ class governing this transition, which connect the distribution and correlation functions for the growing and stationary regimes. This in particular shows how interfaces realized in experiments and simulations actually approach the stationary regime, which is never attained unless a stationary interface is artificially given as an initial condition.
Topological defects—locations of local mismatch of order—are a universal concept playing important roles in diverse systems studied in physics and beyond, including the universe, various condensed ...matter systems, and recently, even life phenomena. Among these, liquid crystal has been a platform for studying topological defects via visualization, yet it has been a challenge to resolve three-dimensional structures of dynamically evolving singular topological defects. Here, we report a direct confocal observation of nematic liquid crystalline defect lines, called disclinations, relaxing from an electrically driven turbulent state. We focus in particular on reconnections, characteristic of such line defects. We find a scaling law for in-plane reconnection events, by which the distance between reconnecting disclinations decreases by the square root of time to the reconnection. Moreover, we show that apparently asymmetric dynamics of reconnecting disclinations is actually symmetric in a comoving frame, in marked contrast to the two-dimensional counterpart whose asymmetry is established. We argue, with experimental supports, that this is because of energetically favorable symmetric twist configurations that disclinations take spontaneously, thanks to the topology that allows for rotation of the winding axis. Our work illustrates a general mechanism of such spontaneous symmetry restoring that may apply beyond liquid crystal, which can take place if topologically distinct asymmetric defects in lower dimensions become homeomorphic in higher dimensions and if the symmetric intermediate is energetically favorable.
We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they belong to the Kardar-Parisi-Zhang ...(KPZ) universality class for 1+1 dimensions Takeuchi and Sano in Phys. Rev. Lett. 104:230601,
2010
; Takeuchi et al. in Sci. Rep. 1:34,
2011
. Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but beyond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrating also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the temporal correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the interfaces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.
Memory and Universality in Interface Growth De Nardis, Jacopo; Le Doussal, Pierre; Takeuchi, Kazumasa A
Physical review letters,
2017-Mar-24, Letnik:
118, Številka:
12
Journal Article
Recenzirano
Odprti dostop
Recently, very robust universal properties have been shown to arise in one-dimensional growth processes with local stochastic rules, leading to the Kardar-Parisi-Zhang (KPZ) universality class. Yet ...it has remained essentially unknown how fluctuations in these systems correlate at different times. Here, we derive quantitative predictions for the universal form of the two-time aging dynamics of growing interfaces and we show from first principles the breaking of ergodicity that the KPZ time evolution exhibits. We provide corroborating experimental observations on a turbulent liquid crystal system, as well as a numerical simulation of the Eden model, and we demonstrate the universality of our predictions. These results may give insight into memory effects in a broader class of far-from-equilibrium systems.
Stochastic motion of a point - known as Brownian motion - has many successful applications in science, thanks to its scale invariance and consequent universal features such as Gaussian fluctuations. ...In contrast, the stochastic motion of a line, though it is also scale-invariant and arises in nature as various types of interface growth, is far less understood. The two major missing ingredients are: an experiment that allows a quantitative comparison with theory and an analytic solution of the Kardar-Parisi-Zhang (KPZ) equation, a prototypical equation for describing growing interfaces. Here we solve both problems, showing unprecedented universality beyond the scaling laws. We investigate growing interfaces of liquid-crystal turbulence and find not only universal scaling, but universal distributions of interface positions. They obey the largest-eigenvalue distributions of random matrices and depend on whether the interface is curved or flat, albeit universal in each case. Our exact solution of the KPZ equation provides theoretical explanations.
We experimentally investigate the critical behavior of a phase transition between two topologically different turbulent states of electrohydrodynamic convection in nematic liquid crystals. The ...statistical properties of the observed spatiotemporal intermittency regimes are carefully determined, yielding a complete set of static critical exponents in full agreement with those defining the directed percolation class in 2+1 dimensions. This constitutes the first clear and comprehensive experimental evidence of an absorbing phase transition in this prominent nonequilibrium universality class.
We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ) universality class with curved initial conditions. By simulations of a cluster growth model and experiments with liquid-crystal ...turbulence, we determine the universal scaling functions that describe the height distribution and the spatial correlation of the interfaces growing outward from a ring. The scaling functions, controlled by a single dimensionless time parameter, show crossover from the statistical properties of the flat interfaces to those of the circular interfaces. Moreover, employing the KPZ variational formula to describe the case of the ring initial condition, we find that the formula, which we numerically evaluate, reproduces the numerical and experimental results precisely without adjustable parameters. This demonstrates that precise numerical evaluation of the variational formula is possible at all, and underlines the practical importance of the formula, which is able to predict the one-point distribution of KPZ interfaces for general initial conditions.
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