Extremal-point densities of interface fluctuations

Universal distribution of the number of minima for random walks and Lévy flights

We compute exactly the full distribution of the number m of local minima in a one-dimensional landscape generated by a random walk or a Lévy flight. We consider two different ensembles of landscapes, one with a fixed number of steps N and the other till the first-passage time of the random walk to the origin. We show that the distribution of m is drastically different in the two ensembles (Gaussian in the former case, while having a power-law tail m^{-3/2} in the latter case). However, the most striking aspect of our results is that, in each case, the distribution is completely universal for all m (and not just for large m), i.e., independent of the jump distribution in the random walk. This means that the distributions are exactly identical for Lévy flights and random walks with finite jump variance. Our analytical results are in excellent agreement with our numerical simulations.

Finite-size and finite-time scaling for kinetic rough interfaces

We consider discrete models of kinetic rough interfaces that exhibit space-time scale invariance in height-height correlation. We use the generic scaling theory of Ramasco et al. [Phys. Rev. Lett. 84, 2199 (2000)0031-900710.1103/PhysRevLett.84.2199] to confirm that the dynamical structure factor of the height profile can uniquely characterize the underlying dynamics. We apply both finite-size and finite-time scaling methods that systematically allow an estimation of the critical exponents and the scaling functions, eventually establishing the universality class accurately. The finite-size scaling analysis offers an alternative way to characterize the anomalous rough interfaces. As an illustration, we investigate a class of self-organized interface models in random media with extremal dynamics. The isotropic version shows a faceted pattern and belongs to the same universality class (as shown numerically) as the Sneppen model (version A). We also examine an anisotropic version of the Sneppen model and suggest that the model belongs to the universality class of the tensionless Kardar-Parisi-Zhang (tKPZ) equation in one dimension.

Enhanced fluctuation for pinned surface nanobubbles

By employing molecular dynamics simulations we investigate the fluctuation of surface nanobubbles immersed in liquid phase. Our simulation results indicate that in comparison with the surrounding liquid and nanobubble interior, the vapor-liquid or gas-liquid interface of nanobubbles always exhibits the largest compressibility, demonstrating the enhanced fluctuation for nanobubble interfaces. We also find that vapor surface nanobubbles and gas surface nanobubbles exhibit different fluctuation behaviors. For vapor nanobubbles that appear in overheated pure liquid, both density fluctuation and interface fluctuation are independent on the external pressure since the internal pressure remains constant at a given temperature. For gas nanobubbles that appear in gas supersaturated solution, the density fluctuation monotonously decreases with the increase of gas concentration, while the interface fluctuation shows a nonmonotonic variation. Departure from the intermediate gas concentration with the minimal interface fluctuation would enhance the fluctuation, which may finally lead to nanobubble destabilization. Finally, our simulation results indicate that the complicated interface fluctuation of surface nanobubbles comprises two different modes: interface deformation and interface oscillation, both of which display similar trends as that of the combined interface fluctuation.

Extremal-point density of scaling processes: From fractional Brownian motion to turbulence in one dimension

In recent years several local extrema-based methodologies have been proposed to investigate either the nonlinear or the nonstationary time series for scaling analysis. In the present work, we study systematically the distribution of the local extrema for both synthesized scaling processes and turbulent velocity data from experiments. The results show that for the fractional Brownian motion (fBm) without intermittency correction the measured extremal-point-density (EPD) agrees well with a theoretical prediction. For a multifractal random walk (MRW) with the lognormal statistics, the measured EPD is independent of the intermittency parameter μ, suggesting that the intermittency correction does not change the distribution of extremal points but changes the amplitude. By introducing a coarse-grained operator, the power-law behavior of these scaling processes is then revealed via the measured EPD for different scales. For fBm the scaling exponent ξ(H) is found to be ξ(H)=H, where H is Hurst number, while for MRW ξ(μ) shows a linear relation with the intermittency parameter μ. Such EPD approach is further applied to the turbulent velocity data obtained from a wind tunnel flow experiment with the Taylor scale λ-based Reynolds number Re_{λ}=720, and a turbulent boundary layer with the momentum thickness θ based Reynolds number Re_{θ}=810. A scaling exponent ξ≃0.37 is retrieved for the former case. For the latter one, the measured EPD shows clearly four regimes, which agrees well with the corresponding sublayer structures inside the turbulent boundary layer.

Maximal- and minimal-height distributions of fluctuating interfaces

Maximal- and minimal-height distributions (MAHD, MIHD) of two-dimensional interfaces grown with the nonlinear equations of Kardar-Parisi-Zhang (KPZ, second order) and of Villain-Lai-Das Sarma (VLDS, fourth order) are shown to be different. Two universal curves may be MAHD or MIHD of each class depending on the sign of the relevant nonlinear term, which is confirmed by results of several lattice models in the KPZ and VLDS classes. The difference between MAHD and MIDH is connected with the asymmetry of the local height distribution. A simple, exactly solvable deposition-erosion model is introduced to illustrate this feature. The average extremal heights scale with the same exponent of the average roughness. In contrast to other correlated systems, generalized Gumbel distributions do not fit those MAHD and MIHD, nor those of Edwards-Wilkinson growth.

Extreme fluctuations in noisy task-completion landscapes on scale-free networks

We study the statistics and scaling of extreme fluctuations in noisy task-completion landscapes, such as those emerging in synchronized distributed-computing networks, or generic causally constrained queuing networks, with scale-free topology. In these networks the average size of the fluctuations becomes finite (synchronized state) and the extreme fluctuations typically diverge only logarithmically in the large system-size limit ensuring synchronization in a practical sense. Provided that local fluctuations in the network are short tailed, the statistics of the extremes are governed by the Gumbel distribution. We present large-scale simulation results using the exact algorithmic rules, supported by mean-field arguments based on a coarse-grained description.

Synchronization landscapes in small-world-connected computer networks

Motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small-world networks, embedded in one and two dimensions. We find that the synchronization landscape (corresponding to the progress of the individual processors) exhibits Kardar-Parisi-Zhang-like kinetic roughening on regular networks with short-range communication links. Although the processors, on average, progress at a nonzero rate, their spread (the width of the synchronization landscape) diverges with the number of nodes (desynchronized state) hindering efficient data management. When random communication links are added on top of the one and two-dimensional regular networks (resulting in a small-world network), large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system-size limit (synchronized state). In the resulting synchronization scheme, the processors make close-to-uniform progress with a nonzero rate without global intervention. We obtain our results by "simulating the simulations," based on the exact algorithmic rules, supported by coarse-grained arguments.

Exact maximal height distribution of fluctuating interfaces

We present an exact solution for the distribution P(h(m),L) of the maximal height h(m) (measured with respect to the average spatial height) in the steady state of a fluctuating Edwards-Wilkinson interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h(m),L)=L(-1/2)f(h(m)L(-1/2)) for all L>0, where the function f(x) is the Airy distribution function that describes the probability density of the area under a Brownian excursion over a unit interval. For the free boundary case, the same scaling holds, but the scaling function is different from that of the periodic case. Numerical simulations are in excellent agreement with our analytical results. Our results provide an exactly solvable case for the distribution of extremum of a set of strongly correlated random variables.

Roughness scaling for Edwards-Wilkinson relaxation in small-world networks

Motivated by a fundamental synchronization problem in scalable parallel computing and by a recent criterion for "mean-field" synchronizability in interacting systems, we study the Edwards-Wilkinson model on two variations of a small-world network. In the first version each site has exactly one random link of strength p, while in the second one each site on average has p links of unit strength. We construct a perturbative description for the width of the stationary-state surface (a measure of synchronization), in the weak- and sparse-coupling limits, respectively, and verify the results by performing exact numerical diagonalization. The width remains finite in the limit of infinite system size for both cases, but exhibits anomalous scaling with p in the latter for d< or =2.

Mound formation and coarsening from a nonlinear instability in surface growth

We study spatially discretized versions of a class of one-dimensional, nonequilibrium, conserved growth equations for both nonconserved and conserved noise using numerical integration. An atomistic version of these growth equations is also studied using stochastic simulation. The models with nonconserved noise are found to exhibit mound formation and power-law coarsening with slope selection for a range of values of the model parameters. Unlike previously proposed models of mound formation, the Ehrlich-Schwoebel step-edge barrier, usually modeled as a linear instability in growth equations, is absent in our models. Mound formation in our models occurs due to a nonlinear instability in which the height (depth) of spontaneously generated pillars (grooves) increases rapidly if the initial height (depth) is sufficiently large. When this instability is controlled by the introduction of a nonlinear control function, the system exhibits a first-order dynamical phase transition from a rough self-affine phase to a mounded one as the value of the parameter that measures the effectiveness of control is decreased. We define an "order parameter" that may be used to distinguish between these two phases. In the mounded phase, the system exhibits power-law coarsening of the mounds in which a selected slope is retained at all times. The coarsening exponents for the spatially discretized continuum equation and the atomistic model are found to be different. An explanation of this difference is proposed and verified by simulations. In the spatially discretized growth equation with conserved noise, we find the curious result that the kinetically rough and mounded phases are both locally stable in a region of parameter space. In this region, the initial configuration of the system determines its steady-state behavior.

Statistics of extremal intensities for Gaussian interfaces

The extremal Fourier intensities are studied for stationary Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We calculate the probability distribution of the maximal intensity and find that, generically, it does not coincide with the distribution of the integrated power spectrum (i.e., roughness of the surface), nor does it obey any of the known extreme statistics limit distributions. The Fisher-Tippett-Gumbel limit distribution is, however, recovered in three cases: (i) in the nondispersive (white noise) limit, (ii) for high dimensions, and (iii) when only short-wavelength modes are kept. In the last two cases the limit distribution emerges in nonconventional scenarios.

Algorithmic scalability in globally constrained conservative parallel discrete event simulations of asynchronous systems

We consider parallel simulations for asynchronous systems employing L processing elements that are arranged on a ring. Processors communicate only among the nearest neighbors and advance their local simulated time only if it is guaranteed that this does not violate causality. In simulations with no constraints, in the infinite L limit the utilization scales [Korniss et al., Phys. Rev. Lett. 84, 1351 (2000)]; but, the width of the virtual time horizon diverges (i.e., the measurement phase of the algorithm does not scale). In this work, we introduce a moving Delta-window global constraint, which modifies the algorithm so that the measurement phase scales as well. We present results of systematic studies in which the system size (i.e., L and the volume load per processor) as well as the constraint are varied. The Delta constraint eliminates the extreme fluctuations in the virtual time horizon, provides a bound on its width, and controls the average progress rate. The width of the Delta window can serve as a tuning parameter that, for a given volume load per processor, could be adjusted to optimize the utilization, so as to maximize the efficiency. This result may find numerous applications in modeling the evolution of general spatially extended short-range interacting systems with asynchronous dynamics, including dynamic Monte Carlo studies.

Suppressing roughness of virtual times in parallel discrete-event simulations

In a parallel discrete-event simulation (PDES) scheme, tasks are distributed among processing elements (PEs) whose progress is controlled by a synchronization scheme. For lattice systems with short-range interactions, the progress of the conservative PDES scheme is governed by the Kardar-Parisi-Zhang equation from the theory of nonequilibrium surface growth. Although the simulated (virtual) times of the PEs progress at a nonzero rate, their standard deviation (spread) diverges with the number of PEs, hindering efficient data collection. We show that weak random interactions among the PEs can make this spread nondivergent. The PEs then progress at a nonzero, near-uniform rate without requiring global synchronizations.

Topological classification of binary trees using the Horton-Strahler index

The Horton-Strahler (HS) index r=max(i,j)+delta(i,j) has been shown to be relevant to a number of physical (such as diffusion limited aggregation) geological (river networks), biological (pulmonary arteries, blood vessels, various species of trees), and computational (use of registers) applications. Here we revisit the enumeration problem of the HS index on the rooted, unlabeled, plane binary set of trees, and enumerate the same index on the ambilateral set of rooted, plane binary set of trees of n leaves. The ambilateral set is a set of trees whose elements cannot be obtained from each other via an arbitrary number of reflections with respect to vertical axes passing through any of the nodes on the tree. For the unlabeled set we give an alternate derivation to the existing exact solution. Extending this technique for the ambilateral set, which is described by an infinite series of nonlinear functional equations, we are able to give a double exponentially converging approximant to the generating functions in a neighborhood of their convergence circle, and derive an explicit asymptotic form for the number of such trees.

Roughness scaling in cyclical surface growth

The scaling behavior of cyclical growth (e.g., cycles of alternating deposition and desorption primary processes) is investigated theoretically and probed experimentally. The scaling approach to kinetic roughening is generalized to cyclical processes by substituting the number of cycles n for the time. The roughness is predicted to grow as n(beta) where beta is the cyclical growth exponent. The roughness saturates to a value that scales with the system size L as L(alpha), where alpha is the cyclical roughness exponent. The relations between the cyclical exponents and the corresponding exponents of the primary processes are studied. Exact relations are found for cycles composed of primary linear processes. An approximate renormalization group approach is introduced to analyze nonlinear effects in the primary processes. The analytical results are backed by extensive numerical simulations of different pairs of primary processes, both linear and nonlinear. Experimentally, silver surfaces are grown by a cyclical process composed of electrodeposition followed by 50% electrodissolution. The roughness is found to increase as a power law of n, consistent with the scaling behavior anticipated theoretically. Potential applications of cyclical scaling include accelerated testing of rechargeable batteries and improved chemotherapeutic treatment of cancerous tumors.

Comment on "extremal-point densities of interface fluctuations in a quenched random medium"

Lam and Tan [Phys. Rev. E 62, 6246 (2000)] recently studied the extremal-point densities of interface fluctuations in a quenched random medium. In this Comment we show that their results for systems on a lattice contain algebraic errors leading to invalid conclusions. Further, while most of their calculations for the continuum case are correct, they misinterpret the result to come to an agreement with the (erroneous) lattice calculations. We derive the correct expressions for the lattice, which agree with the correct interpretation of the continuum case.

Maximal height scaling of kinetically growing surfaces

The scaling properties of the maximal height of a growing self-affine surface with a lateral extent L are considered. In the late-time regime its value measured relative to the evolving average height scales like the roughness: h*(L) approximately L alpha. For large values its distribution obeys logP(h*(L)) approximately (-)A(h*(L)/L(alpha))(a). In the early-time regime where the roughness grows as t(beta), we find h*(L) approximately t(beta)[lnL-(beta/alpha)lnt+C](1/b), where either b = a or b is the corresponding exponent of the velocity distribution. These properties are derived from scaling and extreme-value arguments. They are corroborated by numerical simulations and supported by exact results for surfaces in 1D with the asymptotic behavior of a Brownian path.

Spatial persistence of fluctuating interfaces

We show that the probability, P0(l), that the height of a fluctuating (d+1)-dimensional interface in its steady state stays above its initial value up to a distance l, along any linear cut in the d-dimensional space, decays as P0(l) approximately l(theta). Here straight theta is a "spatial" persistence exponent, and takes different values, straight theta(s) or straight theta(0), depending on how the point from which l is measured is specified. These exponents are shown to map onto corresponding temporal persistence exponents for a generalized d = 1 random-walk equation. The exponent straight theta(0) is nontrivial even for Gaussian interfaces.

Extremal-point densities of interface fluctuations in a quenched random medium

We give a number of exact, analytical results for the stochastic dynamics of the density of local extrema (minima and maxima) of linear Langevin equations and solid-on-solid lattice growth models driven by spatially quenched random noise. Such models can describe nonequilibrium surface fluctuations in a spatially quenched random medium, diffusion in a random catalytic environment, and polymers in a random medium. In spite of the nonuniversal character for the quantities studied, their behavior against the variation of the microscopic length scale can present generic features, characteristic of the macroscopic observables of the system.