Abstract

In this paper we present a concise introduction to the general methodologies of disorderly surface growth. The dynamics of far from equilibrium surface growth systems can be investigated from a scaling approach making heavy use of fractal geometry. The information related to the complex behavior of these systems is therefore collapsed in a few quantities, known as critical exponents, that regulate their scaling properties. An introduction to the foundations of this theoretical framework is presented through the example of the Ballistic Deposition model and the relevance of correlation for saturation phenomena is discussed. Armed with the necessary theoretical knowledge, both the Random Deposition model and the Random Deposition with Surface Relaxation model are solved exactly, providing the values of their critical exponents. To that aim several new methods are presented and applied to the individual models, especially the devising of continuum stochastic growth equations through symmetry considerations and all the related techniques adopted to solve exactly for the critical exponents.

Introduction

Far from complying with the wishful rules of conventional geometry, nature abounds with examples of rough morphologies. Materials science offers a bewildering array of phenomena characterized by disorderly surface growth, from the everyday experience of fluid flow in porous media and flame fronts propagation - that everybody can recreate by spilling coffee on a tablecloth or setting fire to a paper sheet – to microscopic deposition processes, that play a fundamental role in today's technologies like Molecular Beam Epitaxy (MBE). Further examples can be found in the biological sciences, as in the case of bacterial growth in petri dishes or DNA sampling, and even in the study of coastlines, cloud formations and botany.
The study of the fundamental properties of the rough shapes arising from such phenomena began to be seriously tackled by mathematicians only in the 1960's, with the pioneering works of Benoit Mandelbrot [1], who laid the foundations of what later became known as fractal geometry. Advances in physics have been somewhat slower, hampered by the fact that most phenomena in disorderly surface growth operate far from the thermodynamic equilibrium (deposition processes, for example, don't allow energy nor the number of particles to be conserved at all, almost by definition), therefore the methods of equilibrium statistical mechanics and thermodynamics are inadequate and largely unsuccessful. Nowadays, however, disorderly surface growth is a thriving and expanding multidisciplinary field that can count on several different and mutually consistent investigation methodologies.
Scaling methods, building on the fractal concept of self-affinity, allow to reduce the apparently unassailable complexity of such systems by identifying a set of simple quantities, called critical exponents, that completely describe the dynamics and morphology of the resulting surface.
In addition, numerous experimental methods have been developed for the study of interface motion and deposition processes in the comfort of our laboratories, with resolutions previously inconceivable, as in the case of the Scanning Tunneling Microscope (STM).
Computers are also playing an unprecedented role, being the natural instruments for the implementation of discrete models in simulations, thus complementing and integrating the insights obtained in traditional experimental settings.
Lastly, the theoretical advances in the fields of stochastic differential equations and renormalization group theory have provided scientists with a wealth of techniques for deriving continuum growth equations displaying the same properties of most microscopic discrete models and unifying the disparate phenomena under analysis through the introduction of universality classes.
In this research paper we will touch upon all these methods, apart from the experimental approach, through the exposition of a topical class of phenomena in disorderly surface growth: deposition processes. One of the most famous models, that of Ballistic Deposition (BD), will lead us to the presentation of basic scaling concepts (Section1), like that of critical exponents. A rapid exposition of elementary notions in fractal geometry follows in Section 2, to offer a deeper theoretical understanding of the subject at hand.
Building on the knowledge exposed, in Section 3 we finally present the model of Random Deposition (RD), for which analytical solutions for the discrete system and its continuum analog are both provided.
Nevertheless, as a rule of thumb, not always it is possible to find solutions in closed form in the discrete formulation of the problem, therefore a section on the Random Deposition with Surface Relaxation (RDSR) model will follow (Section 4), exemplifying a case in which the use of stochastic growth equations, obtained through symmetry considerations, is both necessary and practical.
A brief conclusion at the end of the paper will help the reader to put the results previously exposed into the more general context of fractal surfaces and disorderly surface growth, summarizing the lessons that can be learned from this booming field and offering some suggestions on its future direction.

6.) Two identical springs of equilibrium length L and spring stiffn...