Physica A 391 (2012) 4557–4563 Contents lists available at SciVerse ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa A fluctuating lattice Boltzmann scheme for the one-dimensional KPZ equation Vadzim Yermakou, Sauro Succi ∗ Istituto Applicazioni Calcolo, CNR, via dei Taurini 19, 00185, Roma, Italy article info Article history: Received 20 January 2012 Received in revised form 12 April 2012 Available online 22 May 2012 Keywords: Fluctuating interfaces Lattice Boltzmann fluids KPZ equation Growth phenomena Burgers fluids abstract Based on the well-known mapping between the Burgers equation with noise and the Kardar–Parisi–Zhang (KPZ) equation for fluctuating interfaces, we develop a fluctuating lattice Boltzmann (LB) scheme for growth phenomena, as described by the KPZ formalism. A very simple LB-KPZ scheme is demonstrated in 1 + 1 spacetime dimensions, and is shown to reproduce the scaling exponents characterizing the growth of one-dimensional fluctuating interfaces. © 2012 Elsevier B.V. All rights reserved. 1. Introduction The Kardar–Parisi–Zhang (KPZ) equation provides a powerful mathematical paradigm for the description of non- equilibrium growth of interfaces [1]. Historically devised to account for growth processes due to random deposition and diffusion, it was subsequently shown to describe a much broader class of non-equilibrium processes, ranging from directed polymer motion in random potentials to simple glassy phenomena, from domain walls and vortex lines, to flows in disorder media and biophysical processes [2]. Mathematically, the KPZ equation is possibly the simplest non-linear and stochastic extension of the ordinary diffusion equation, the so called Edwards-Wilkinson model [3,4]. It reads as follows: ∂ t h = λ 2 (∇h) 2 + ν ∇ 2 h + ξ(x, t ) (1) where h(x, t ) is the interface height at position x and time t . In the above, ν is the diffusivity, λ a coupling constant (with the dimension of velocity) promoting non-linear lateral growth of the interface, and ξ is a white Gaussian noise. The properties of the KPZ equation have been studied extensively for the last two decades. Owing to the non-linear and stochastic nature of the equation, these studies have been mostly conducted by numerical simulations, although a few precious analytical solutions have also become available in the recent past [5]. These studies have focussed mostly on the statistical properties of the solutions, and particularly on the dynamic scaling laws obeyed by the surface width, defined as the variance of the profile h(x; t ), i.e. w(t ) =⟨ ¯ h 2 − ¯ h 2 ⟩, where overbar indicates spatial average and brackets ensemble average over noise realizations. It is found that the interface width obeys the following asymptotic dynamic scaling law: w(t ; L) ∼ L α f (t /L z ) (2) where L is the averaging length scale, and the scaling function f (x) has the form f (x) ∼ x β for x ≪ 1, i.e. t ≪ τ L ≡ L z , and f (x) → Const . for x ≫ 1. As a result, for t ≫ τ L , w(L) ∼ L α . The coefficients α, β and z = α/β are known as ∗ Corresponding author. E-mail addresses: sauro.succi@gmail.com, s.succi@iac.cnr.it (S. Succi). 0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.05.014