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Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics

Timothy Halpin-Healy, Yi-Cheng Zhang

1995
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Physics reports
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Kinetic interfaces form the basis of a fascinating, interdisciplinary branch of statistical mechanics. Diverse stochastic growth processes can be unified via an intriguing nonlinear stochastic partial differential equation whose consequences and generalizations have mobilized a sizeable community of physicists concerned with a statistical description of kinetically roughened surfaces. Substantial analytical, experimental and numerical effort has already been expended. Despite impressive
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... s, however, there remain many open questions, with much richness and subtlety still to be revealed. In this review, we give an unorthodox account of this rapidly growing field, concentrating on two main lines -the interface growth equations themselves, and their directed polymer counterparts. We emphasize the intrinsic links among the topics discussed, as well as the relationships to other branches of natural science. Our aim is to persuade the reader that multidisciplinary statistical mechanics can be challenging, enjoyable pursuit of surprising depth. 218 T. Halpin-Healy, Y-C. Zhang/Physics Reports 254 (1995) 215-414 Introduction Consider a wheat field of dark golden hue and densely planted in level ground, being roughly rectangular in shape, but rather large in extent, and stretching lazily toward the distant horizon. On a cool, but calm August evening, with nary a breeze about, the edge of the field is ignited, in preparation for leaving the soil fallow the following season. The propagating fire front, initially straight by virtue of its birth along the edge, evolves in a kinetic, violent fashion and heads mercilously into the bulk of the field. Burning shafts of wheat communicate the conflagaration locally to their neighbors, and the narrow, bright, and tortuously shaped fire line, an interface separating the blackened region from the portion of the field soon to be consumed, becomes increasingly rough as random elements, such as local inhomogeneities in the moisture content or density of the wheat, begin to have a large scale cumulative effect. Rather, consider a Petri dish filled with agar, amply nutritious, upon which is very carefully placed a stretched string, infested with common bacteria. Because the nutrient is present in abundance, the growth proceeds at a furious pace, being reaction rather than diffusion limited, with the in situ population doubling every half-hour. Soon, the consequences are observable on a scale visible to the naked eye-the bacterial colony has spread laterally on the surface of the agar and exhibits a self-affine, rough growth front that has progressed several millimeters from the string. In the days that follow, the colony morphology behind the front has become extraordinarily complex, with greedy microorganisms saving themselves and going dormant because of the sudden paucity of food, while others permit self-sacrifice with the assistance of piggy-back viruses that destroy the individual bacterium, but pass on its genetic material for the greater good of the colony. Nevertheless, at the front, things are simple -it's eat, eat, eat,... the propagating edge of the colony becoming increasing rough due to local variations in the food concentration and reproduction rate. Or finally, consider a random, very dense and essentially two-dimensional packing of tiny glass beads (roughly, but not uniformly, sand grain size, though entirely transparent), sandwiched between two rectangular glass plates that are sealed along opposite long edges. Because the bead pack is geometrically and compositionally disordered, it serves quite nicely to realize a random, but porous medium. Water is introduced along the entire length of an open edge and is forced through the broad, but very narrow channel by means of a pump. As water displaces air in the pores of the random bead pack, the interface separating these two phases, easily seen from above, pushes ahead getting kinetically roughened, thanks to local inhomogeneities in the random medium. The above physical processes, apparently unrelated, nonetheless share an important common feature. Each produces a kinetically roughened edge, whose geometrical complexity can best be understood, perhaps, within the nuance-stricken discipline of nonequilibium statistical mechanics. In recent years, much analytical, numerical and experimental effort has been expended by investigators seeking a greater understanding of the statistical properties of such self-affine edges. The theoreticians pushed the field into a mature phase of sorts with the development, in the mid-eighties, of a small industry devoted, in part, to the explication of the noise-driven Burgers' equation [ B74], which was made popular following the work in the interface context [KPZ86]. This stochastic partial differential equation, incorporating effects of surface relaxation, locally normal growth (i.e., parallel propagation of the front), and the incessant peppering of stochastic noise, has come to represent a crucial step in eliciting Nature's secrets of kinetic roughening phenomena. This review uses the KPZ equation, as well as its immediate antecedents, as a point of departure, but soon focusses its attention on more re-T Halpin-Healy, E-C. Zhang/Physics Reports 254 (199.5) 215-414 219 cent interdisciplinary developments concerning anomalous kinetic roughening, where the phenomena, though patently governed in some measure by the essential nonlinear mechanism of parallel transport of the surface, nevertheless exhibit a self&fine roughness that exceeds naive predictions, In retrospect, it was hardly thinkable a few years back that the explication of such simple physical processes would necessitate the bulk of the material touched upon here. Nonetheless, subsequent research endeavors diffused far beyond the original intentions; e.g., recent suggestions concerning interfacial growth and the structure of the universe [BF94] ! Admittedly, such matters are more for scientific curiosity than for practical applications. But in a subtle way, they form an integrated part to our understanding of the original problem. Along the way, the reader will soon find himself amidst a new branch of statistical mechanics that involves an impressive array of concepts drawn from contemporary physics. Let us now have a glimpse at that list: stochastic differential equations, dynamic and functional renormalization groups, Langevin and Fokker-Planck equations, fluctuation-dissipation theorem, Bethe ansatz, infinite matrix products, Burgers equation for shock waves, Kuromato-Sivashinski equation for flame propagation, passive scalars in turbulent fluids, selforganized criticality, spin glasses, replica symmetry breaking, ultrametricity, intermittent biological evolution, ground-state instability, global optimization, pattern formation and directed percolation. These interdisciplinary matters may sometimes appear to be deceivingly simple and easy, for anyone can turn on his personal computer to do a little simulation. Indeed, if he plays in the right comer, he will probably discover something worth reporting to the Physical Review. Yet, very seasoned players can still find interesting challenges in the same community. Indeed, some of the finest contemporary minds still consider the problems of sufficient subtlety to merit their time. Even so, we shall see that this interdisciplinary branch is not just a theorist's paradise. Many aspects directly concern experiments: fire propagation on paper or in a forest, ink imbibition, solid state surfaces, chemical corrosion, electrochemical deposition, sputter-etching, Schwoebel instability, rock porosity, geomorphology and the erosion of earth's surface, vortices in ceramic superconductors, blown fuse networks, oil recovery through porous media, etc. In crafting this review paper, it was our main purpose to inspire the reader, assumed here to be at advanced graduate student or postdoctoral fellow, scanning the horizon for a worthy field of endeavor, whom we exhort to pause briefly and gaze upon the many intriguing avenues of research available within and beyond the realm of the kinetic roughening. The immediate goal was to furnish a primer of sorts -introducing the basic grammar, spelling out the essential details, passing on a few tricks of the trade, and calling attention to some technical features of interest. We make no pretense of completeness and though roughly equal time is accorded numerical, analytical and experimental aspects, the careful reader will rightly sense a slight propagandistic thread favoring traditional methodologies (typically involving pencil and paper...) over approaches dependent upon brute processing power. Indeed, it is our strong prejudice that members of the community redouble their efforts on the analytical front, despite much appreciated difficulties, since tiny chinks in the wall may suddenly yield large scale breakthroughs! We shall adopt the style to present the physics in simple language, using a rather liberal format. Intuitive reasoning is preferred over formal approaches. We do not pretend this field to be a well established one, but endeavor to expose wherever possible the pitfalls and cracks in theory. Many loose ends are emphasized as they may lead to new research directions. In any case, since the field of nonequilibrium kinetic roughening is undergoing rapid development, any treatment purporting to be comprehensive would become obsolete even before the ink had dried upon the pages. Rather, we suspected that a pedagogical, though entirely informal Z Halpin-Healy, E-C. Zhang/Physics Reports 254 (1995) 215-414 221 developed by Derrida and Spohn and coworkers, merits, perhaps, a mini-review in its own right. Even so, it has already been nicely touched upon in the past by Krug and Spohn [ KrS9lr], as has been the topic of driven lattice-gases [SZ95r], a favorite of the more mathematically-minded members of the statistical mechanics community. The wonderful connections between driven lattice-gases, kinetically roughened surfaces and the DPRM are fertile, indeed, for the resourceful researcher. Much work can be done via these ties. As an indication, the reader need only glance at the recent efforts of Tang and Lyuksyutov [ TLv93], who investigate the DPRM delocalization from extended defects, a matter relevant to the wandering of elastic vortex lines in disordered superconductors plagued with columnar defects or grain boundaries, by simulating inhomogeneous KPZ stochastic growth! Similarly, Krug and Tang [ KrT94] have uncovered novel shock phenomena of open boundary driven lattice-gases by studying the DPRM analog: disorder-induced unbinding in confined geometries. Unfortunately, we had insufficient stamina to present subtle issues of replica symmetry breaking for directed polymers and manifolds in random media, as elucidated by Mtzard and Parisi within their variational approach [Pa90,MP91]. The reader is encouraged to scout out these matters directly; likewise, the dynamics of driven lines [ Hwa92,EK92], the importance of rare versus typical DPRM fluctuations [ MCz90,HF94], as well as the difficult subject of kinetic roughening within the MBE context with its associated instabilities [ KrPS93], pattern formation and coarsening phenomena [ John94,Ern94,SP94]. Hwa and Fisher [ HF94] have reformulated the DPRM using rigorous functional methods, their work is a minireview per se and gives particular insight into the ground state instability problem. Note, too, the recent renormalization group study [ Mi94] concerning overlap distributions for the many-dimensional DPRM. Again, for a systematic discussion of the huge body of simulation work done in the past few years, we defer to Meakin [ M88r,M93r]. After submission of this manuscript, we learned that Barabasi and Stanley [ BS94] have nearly completed an inspired, useful elementary book on fractal interface growth, while Maritan et al. [Ma951 are preparing a comprehensive review on similar matters. The reader is urged to consult all for completeness. 2. Early models of local stochastic growth 2.1. Eden model [E58,E61] The classic model of stochastic growth was proposed by Eden, several decades back, as a gross simplification of the biological growth process exhibited by cancerous cells [ E%,E61,WB72]. Its essential ingredients are easily appreciated -take a sheet of graph paper and shade in the small central square. Now, with equal probability, darken one of this seed's 4 immediate neighbors. The resulting two-particle cluster possesses 6 perimeter squares. Again, with equal probability, one of these is chosen as the growth site and occupied, yielding a three particle cluster, and so on, see Fig. 2 .la. This very simple stochastic growth algorithm, trivially implemented on the computer, is iterated repeatedly giving rise to an Eden cluster that is compact, nonfractal in the bulk, but with intriguing surface roughness properties - Fig. 2 .lb. Surprisingly, even for such a straightforward model, it proves highly nontrivial to gain substantial analytical insights. Some limited progress was achieved early on by exactly solving the model in infinitely large dimensionality [ PZ84] . The idea is to count exactly all the possible cluster configurations and then to take the ensemble average of the physical quantity of interest -in our case, the gyration radius, 224 7: Halpin-Healy, E-C. ZhanglPhysics Reports 254 (1995) 215-414 6(x, t) = vV2h(x, t) + 7(x, t) + c (2.4) where h( x, t) is the surface's position, x is a coordinate in the (d -1)-dimensional base space, r] the space and time-dependent stochastic noise, while c is the constant average velocity of the propagating surface. In the simplest scenario, the noise is assumed spatially and temporally uncorrelated and gaussian, with variance (7(x, t)r](x', t')) =2&7(x -x/)&t -t'). Being linear, the EW equation is readily solved via Fourier methods. A central quantity of interest is the width w of the fluctuating interface, given by w*( L, t) = (JC-(~-') Jt dd-'x[ h( X, t) -&I') where the angular brackets denote averaging over samples and h is the mean height. Direct integration shows that the surface roughness w has the following scaling behavior w2(L t) -$rxf&@) (2.5) where x = (3 -d)/2 is the EW saturation-width exponent, while the universal scaling function fEw for the surface width, Kd fEw(X) = (3 _ d) (zT)3-d

doi:10.1016/0370-1573(94)00087-j
fatcat:kwz2xwvx2zaxdbawslxuq5jij4