Kardar–Parisi–Zhang equation

In mathematics, the Kardar–Parisi–Zhang (KPZ) equation is a non-linear stochastic partial differential equation, introduced by Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang in 1986. It describes the temporal change of a height field with spatial coordinate and time coordinate : Here, is white Gaussian noise with average and second moment , and are parameters of the model, and is the dimension. In one spatial dimension, the KPZ equation corresponds to a stochastic version of Burgers' equation with field via the substitution . Via the renormalization group, the KPZ equation is conjectured to be the field theory of many surface growth models, such as the Eden model, ballistic deposition, and the weakly asymmetric single step solid on solid process (SOS) model. A rigorous proof has been given by Bertini and Giacomin in the case of the SOS model. Many interacting particle systems, such as the totally asymmetric simple exclusion process, lie in the KPZ universality class. This class is characterized by the following critical exponents in one spatial dimension (1 + 1 dimension): the roughness exponent , growth exponent , and dynamic exponent . In order to check if a growth model is within the KPZ class, one can calculate the width of the surface: where is the mean surface height at time and is the size of the system. For models within the KPZ class, the main properties of the surface can be characterized by the Family–Vicsek scaling relation of the roughness with a scaling function satisfying In 2014, Hairer and Quastel showed that more generally, the following KPZ-like equations lie within the KPZ universality class: where is any even-degree polynomial. A family of processes that are conjectured to be universal limits in the (1+1) KPZ universality class and govern the long time fluctuations are the Airy processes. Due to the nonlinearity in the equation and the presence of space-time white noise, solutions to the KPZ equation are known to not be smooth or regular, but rather 'fractal' or 'rough.

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Microscopic interplay of temperature and disorder of a one-dimensional elastic interface

Elisabeth Agoritsas, Thierry Giamarchi

Elastic interfaces display scale-invariant geometrical fluctuations at sufficiently large lengthscales. Their asymptotic static roughness then follows a power-law behavior, whose associated exponent provides a robust signature of the universality class to which they belong. The associated prefactor has instead a nonuniversal amplitude fixed by the microscopic interplay between thermal fluctuations and disorder, usually hidden below experimental resolution. Here we compute numerically the roughness of a one-dimensional elastic interface subject to both thermal fluctuations and a quenched disorder with a finite correlation length. We evidence the existence of a power-law regime at short lengthscales. We determine the corresponding exponent ζdis and find compelling numerical evidence that, contrarily to available analytic predictions, one has ζdis

2022

Stochastic phenomena are often described by Langevin equations, which serve as a mesoscopic model for microscopic dynamics. It has been known since the work of Parisi and Sourlas that reversible (or equilibrium) dynamics present supersymmetries (SUSYs). These are revealed when the path-integral action is written as a function not only of the physical fields, but also of Grassmann fields representing a Jacobian arising from the noise distribution. SUSYs leave the action invariant upon a transformation of the fields that mixes the physical and the Grassmann ones. We show that contrary to common belief, it is possible to extend the known reversible construction to the case of arbitrary irreversible dynamics, for overdamped Langevin equations with additive white noise—provided their steady state is known. The construction is based on the fact that the Grassmann representation of the functional determinant is not unique, and can be chosen so as to present a generalization of the Parisi-Sourlas SUSY. We show how such SUSYs are related to time-reversal symmetries and allow one to derive modified fluctuation-dissipation relations valid in nonequilibrium. We give as a concrete example the results for the Kardar-Parisi-Zhang equation.

2021