The decay of a crystalline cone below the roughening transition is studied. We consider local mass transport through surface diffusion, focusing on the two cases of diffusion limited and attachment-detachment limited step kinetics. In both cases, we describe the decay kinetics in terms of step flow models. Numerical simulations of the models indicate that in the attachmentdetachment limited case the system undergoes a step bunching instability if the repulsive interactions between steps are weak. Such an instability does not occur in the diffusion limited case. In stable cases the height profile, h(r, t), is flat at radii r < R(t) ∼ t 1/4 . Outside this flat region the height profile obeys the scaling scenario ∂h/∂r = F(rt −1/4 ). A scaling ansatz for the timedependent profile of the cone yields analytical values for the scaling exponents and a differential equation for the scaling function. In the long time limit this equation provides an exact description of the discrete step dynamics. It admits a family of solutions and the mechanism responsible for the selection of a unique scaling function is discussed in detail. Finally we generalize the model and consider permeable steps by allowing direct adatom hops between neighboring terraces. We argue that step permeability does not change the scaling behavior of the system, and its only effect is a renormalization of some of the parameters. 68.35.Bs, 68.55.Jk

We study the predictability of emergent phenomena in complex systems. Using nearest neighbor, one-dimensional Cellular Automata (CA) as an example, we show how to construct local coarsegrained descriptions of CA in all classes of Wolfram's classification. The resulting coarse-grained CA that we construct are capable of emulating the large-scale behavior of the original systems without accounting for small-scale details. Several CA that can be coarse-grained by this construction are known to be universal Turing machines; they can emulate any CA or other computing devices and are therefore undecidable. We thus show that because in practice one only seeks coarse-grained information, complex physical systems can be predictable and even decidable at some level of description. The renormalization group flows that we construct induce a hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity and is therefore a good candidate for a complexity measure and a classification method. Finally we argue that the large scale dynamics of CA can be very simple, at least when measured by the Kolmogorov complexity of the large scale update rule, and moreover exhibits a novel scaling law. We show that because of this large-scale simplicity, the probability of finding a coarse-grained description of CA approaches unity as one goes to increasingly coarser scales. We interpret this large scale simplicity as a pattern formation mechanism in which large scale patterns are forced upon the system by the simplicity of the rules that govern the large scale dynamics.

The relaxation process of one dimensional surface modulations is reexamined. Surface evolution is described in terms of a standard step flow model. Numerical evidence that the surface slope, D(x, t), obeys the scaling ansatz D(x, t) = α(t)F (x) is provided. We use the scaling ansatz to transform the discrete step model into a continuum model for surface dynamics. The model consists of differential equations for the functions α(t) and F (x). The solutions of these equations agree with simulation results of the discrete step model. We identify two types of possible scaling solutions. Solutions of the first type have facets at the extremum points, while in solutions of the second type the facets are replaced by cusps. Interactions between steps of opposite signs determine whether a system is of the first or second type. Finally, we relate our model to an actual experiment and find good agreement between a measured AFM snapshot and a solution of our continuum model. 68.35.Bs

Using elementary cellular automata (CA) as an example, we show how to coarse-grain CA in all classes of Wolfram's classification. We find that computationally irreducible (CIR) physical processes can be predictable and even computationally reducible at a coarse-grained level of description. The resulting coarse-grained CA which we construct emulate the large-scale behavior of the original systems without accounting for small-scale details. At least one of the CA that can be coarsegrained is irreducible and known to be a universal Turing machine.PACS numbers: 05.45. Ra, 05.10.Cc, 47.54.+r Can one predict the future evolution of a physical process which is described or modeled by a computationally irreducible (CIR) mathematical algorithm? For such systems, in order to know the system's state after (e.g.) one million time steps, there is no faster algorithm than to solve the equation of motion a million time steps into the future. Wolfram has suggested that the existence of CIR systems in nature is at the root of our apparent inability to model and understand complex systems [1, 2,3,4].Complex physical systems that are CIR might therefore seem to be inherently unpredictable. It is tempting to conclude from this that the enterprise of physics itself is doomed from the outset; rather than attempting to construct solvable mathematical models of physical processes, computational models should be built, explored and empirically analyzed. This argument, however, assumes that infinite precision is required for the prediction of future evolution. Usually coarse-grained or even statistical information is sufficient: indeed, a physical model is usually correct only to a certain level of resolution, so that there is little interest in predictions from such a model on a scale outside its regime of validity.In this Letter, we report on experiments with nearest neighbour one-dimensional cellular automata, which show that because in practice one only seeks coarsegrained information, complex physical systems can be predictable and even computationally reducible at some level of description. The implication of these results is that, at least for systems whose complexity is the outcome of very simple rules, useful approximations can be made that enable predictions about future behavior.

The flattening of a crystal cone below its roughening transition is studied by means of a step flow model. Numerical and analytical analyses show that the height profile, h(r, t), obeys the scaling scenario ∂h/∂r = F(rt −1/4 ). The scaling function is flat at radii r < R(t) ∼ t 1/4 . We find a one parameter family of solutions for the scaling function, and propose a selection criterion for the unique solution the system reaches. 68.35.Bs, 68.55.Jk In recent years it has become technologically possible to design and manufacture crystalline nanostructures, which are of tremendous importance for the fabrication of electronic devices. In many cases, these nanostructures are thermodynamically unstable, and tend to decay with time. This phenomenon has triggered experimental and theoretical efforts to try and understand the decay process [1][2][3][4][5][6][7][8][9][10][11][12][13]. Under fairly robust conditions, the decay of a nanostructure at low temperatures (below the roughening temperature, T R ) is dominated by the motion of atomic steps on the surface. Hence, attempts have been made to understand and predict the relaxation dynamics of simple step configurations.In this work we analyze, numerically as well as analytically, the time evolution of a crystalline cone formed out of circular concentric steps. The decay of other types of nanostructures such as bi-periodic surface modulations has been studied experimentally on Si(001) by Tanaka et al. [1]. Rettori and Villain [2] studied this problem theoretically in the case of small amplitude modulations. Our study, on the other hand, is relevant to large amplitude modulations and in this sense is complimentary to their work. We find that the height of the cone decays with time as h(0) − h(t) ∼ t 1/4 and the radius of the plateau at the top of the cone grows with time as R(t) ∼ t 1/4 . Consider the surface of an infinite crystalline cone, made out of circular concentric steps of radii r i (t), separated by flat terraces. The step index i grows in the direction away from the center of the cone. We assume no deposition of any new material, no evaporation and no transport of atoms through the bulk. To calculate the time dependence of the radii, we have to solve the diffusion equation for adatoms on the terraces with boundary conditions at the step edges, taking into account the repulsive interactions (of the form G(r i+1 − r i ) −2 ) between steps. Using a standard approach to do this [2,14], we arrived at a set of equations of motion for the step radii. It is convenient to present these equations in terms of dimensionless radii, ρ i , and dimensionless time τ :eq is the equilibrium concentration of a straight isolated step, T is the temperature, Γ is the step line tension, Ω is the atomic area of the solid and D s is the diffusion constant of adatoms on the terraces. k is a kinetic coefficient associated with attachment and detachment of adatoms to and from steps.The equations of motion in terms of these variables take the forṁwhere the velocities of the first and se...