Brownian Surfaces Research Articles

Determination of local fractal dimension of the catalyst support MgAl2O4 surface

In this paper, a procedure for estimating local fractal dimensionality values for greyscale images was developed. This procedure was tested on synthesised fractal and multifractal Brownian surfaces. The use of the method of detrended fluctuation analysis to determine the Hurst index for samples with a size not exceeding 2000 points was analysed. The optimal values of the sample size and the scale parameter have been obtained, allowing the Hurst index to be calculated with an error of less than 20%. The procedure was also tested on greyscale images, which are 8-bit representations of a synthesised fractal surface. The peculiarities of the application of the procedure for calculating the local fractal dimension for images obtained by scanning electron microscopy have been analysed. The values of the local fractal dimension of the oxide support MgAl2O4 are calculated using SEM images at different magnifications. It has been shown that, taking into account the fractal dimensionality calculation, different morphologies of the particle surface are observed at different scales of image magnification. At the highest magnification, the particle surface is found to consist of embedded smooth micron particles. The surface morphology of the MgAl2O4 particle depends on the micro- and mesopores between the embedded particles, this surface can be characterised by a fractal surface with FD2 = 2.3–2.4.

Describing and Modeling Rough Composites Surfaces by Using Topological Data Analysis and Fractional Brownian Motion.

Many composite manufacturing processes employ the consolidation of pre-impregnated preforms. However, in order to obtain adequate performance of the formed part, intimate contact and molecular diffusion across the different composites' preform layers must be ensured. The latter takes place as soon as the intimate contact occurs and the temperature remains high enough during the molecular reptation characteristic time. The former, in turn, depends on the applied compression force, the temperature and the composite rheology, which, during the processing, induce the flow of asperities, promoting the intimate contact. Thus, the initial roughness and its evolution during the process, become critical factors in the composite consolidation. Processing optimization and control are needed for an adequate model, enabling it to infer the consolidation degree from the material and process features. The parameters associated with the process are easily identifiable and measurable (e.g., temperature, compression force, process time, ⋯). The ones concerning the materials are also accessible; however, describing the surface roughness remains an issue. Usual statistical descriptors are too poor and, moreover, they are too far from the involved physics. The present paper focuses on the use of advanced descriptors out-performing usual statistical descriptors, in particular those based on the use of homology persistence (at the heart of the so-called topological data analysis-TDA), and their connection with fractional Brownian surfaces. The latter constitutes a performance surface generator able to represent the surface evolution all along the consolidation process, as the present paper emphasizes.

Fault Roughness at Seismogenic Depths and Links to Earthquake Behavior

AbstractFault geometry affects the initiation, propagation, and cessation of earthquake rupture, as well as, potentially, the statistical behavior of earthquake sequences. We analyze 18,250 (−0.27 < M < 4.4) earthquakes of the 2016–2019 Cahuilla, California, swarm and, for the first time, use these high-resolution earthquake locations to map, in detail, the roughness across an active fault surface at depth. We find that the strike-slip fault is 50% rougher in the slip-perpendicular direction than parallel to slip. 3D mapping of fault roughness at seismogenic depths suggests that roughness varies by a factor of 8 for length scales of 1 km. We observe that the largest earthquake (M 4.4) occurred where there is significant fault complexity and the highest measured roughness. We also find that b-values are weakly positively correlated with fault roughness. Following the largest earthquake, we observe a distinct population of earthquakes with comparatively low b-values occurring in an area of high roughness within the rupture area of the M 4.4 earthquake. Finally, we measure roughness at multiple scales and find that the fault is self-affine with a Hurst exponent of 0.52, consistent with a Brownian surface.

Heat Transfer on Micro and Nanostructured Rough Surfaces Synthesized by Plasma

The review summarizes recent experimental results of studying heat transfer on rough surfaces synthesized by plasma. The plasma-surface interaction leads to the stochastic clustering of the surface roughness with a high specific area breaking the symmetry of the virgin surface of the initial crystalline materials. Such a surface is qualitatively different from the ordinary Brownian surface. The micro- and nanostructured surface consist of pores, craters, and nanofibers of size from tens of nanometers to tens of microns, which can provide new heat transfer properties related to a violation of the symmetry of the initial materials. In recent years, new results have been obtained in the study of heat transfer during phase change on plasma-modified surfaces in relation to energy, chemical, and cryogenic technologies. The objective of the review is to describe the specific structure of refractory metals after high-temperature plasma irradiation and the potential application of plasma processing of materials in order to create heat exchange surfaces that provide a significant intensification of two-phase heat transfer. Refractory metals with such a highly porous rough surface can be used as plasma-facing components for operation under extreme heat and plasma loads in thermonuclear and nuclear reactors, as catalysts for hydrogen production, as well as in biotechnology and biomedical applications.

Multiscale anisotropy analysis of second-harmonic generation collagen imaging of human pancreatic cancer.

Pancreatic ductal adenocarcinoma (PDAC) is one of the deadliest cancers with a minority (< 10%) of patients surviving five years past diagnosis. This could be improved with the development of new imaging modalities for early differentiation of benign and cancerous fibrosis. This study intends to explore the application of a two-photon microscopy technique known as second harmonic generation to PDAC using the 2D Wavelet Transform Modulus Maxima (WTMM) Anisotropy method to quantify collagen organization in fibrotic pancreatic tissue. Forty slides from PDAC patients were obtained and eight images were captured per each tissue category on each slide. Brownian surface motion and white noise images were generated for calibration and testing of a new variable binning approach to the 2D WTMM Anisotropy method. The variable binning method had greater resistance to wavelet scaling effects and white noise images were found to have the lowest anisotropy factor. Cancer and fibrosis had greater anisotropy factors (Fa) at small wavelet scales than normal and normal adjacent tissue. At a larger scale of 21 μm this relationship changed with normal tissue having a higher Fa than all other tissue groups. White noise is the best representative image for isotropy and the 2D WTMM anisotropy method is sensitive to changes induced in collagen by PDAC.

Small-Angle Scattering from Fractional Brownian Surfaces

Recent developments in nanotechnology have allowed the fabrication of a new generation of advanced materials with various fractal-like geometries. Fractional Brownian surfaces (fBs) are often used as models to simulate and characterize these complex geometries, such as the surface of particles in dilute particulate systems (e.g., colloids) or the interfaces in non-particulate two-phase systems (e.g., semicrystalline polymers with crystalline and amorphous phases). However, for such systems, a realistic simulation involves parameters averaged over a macroscopic volume. Here, a method based on small-angle scattering technique is proposed to extract the main structural parameters of surfaces/interfaces from experimental data. It involves the analysis of scattering intensities and the corresponding pair distance distribution functions. This allows the extraction of information with respect to the overall size, fractal dimension, Hurst and spectral exponents. The method is applied to several classes of fBs, and it is shown that the obtained numerical values of the structural parameters are in very good agreement with theoretical ones.

Scale Symmetry of Stochastic Surface Clustering under Plasma Influence in Fusion Devices

Titanium, tungsten, carbon, lithium, and beryllium surface structure were analyzed after plasma irradiation in fusion devices. Exceptional extreme high-temperature plasma load in fusion devices leads to specific surface clustering. It is strictly different from any other conditions of material’s clustering. The hierarchical granularity with cauliflower-like shape and surface self-similarity have been observed. Height’s distribution is deviated from the Gaussian function. The relief roughness differs qualitatively from the ordinary Brownian surface and from clustering under other conditions. In fusion devices, the specific conditions regulate material surface clustering faced to plasma. Ions and clusters melt on the surface and move under the effect of stochastic electromagnetic field driven by the near-wall turbulent plasma. In such a process, long-term correlations lead to the growth of surface with a self-similar structure. The multiscale synergistic effects influence the self-similarity–fractal growth from nanometers to millimeters. Experimental results illustrate universality of stochastic clustering of materials irradiated with plasma in fusion devices.

Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense

Recent works have shown that random triangulations decorated by critical (p=1∕2) Bernoulli site percolation converge in the scaling limit to a 8∕3-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE6 in two different ways: The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov–Hausdorff topology. There is a bijective encoding of the site-percolated triangulation by means of a two-dimensional random walk, and this walk converges to the correlated two-dimensional Brownian motion which encodes SLE6-decorated 8∕3-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield (2014); this is sometimes called peanosphere convergence. We prove that one in fact has joint convergence in both of these two senses simultaneously. We also improve the metric convergence result by showing that the map decorated by the full collection of percolation interfaces (rather than just a single interface) converges to 8∕3-LQG decorated by SLE6 in the metric space sense. This is the first work to prove simultaneous convergence of any random planar map model in the metric and peanosphere senses. Moreover, this work is an important step in an ongoing program to prove that random triangulations embedded into C via the so-called Cardy embedding converge to 8∕3-LQG.

The Tutte Embedding of the Poisson\u2013Voronoi Tessellation of the Brownian Disk Converges to sqrt{8/3}-Liouville Quantum Gravity

Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a sqrt{8/3}-Liouville quantum gravity (LQG) surface. In particular, Brownian motion on a Brownian surface is well-defined. The construction in these works is indirect, however, and leaves open a basic question: is Brownian motion on a Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on mathbb {R}^2 is the epsilon rightarrow 0 limit of simple random walk on epsilon mathbb {Z}^2? We answer this question affirmatively by showing that Brownian motion on a Brownian surface is (up to time change) the lambda rightarrow infty limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is lambda times the associated area measure. Among other things, this implies that as lambda rightarrow infty the Tutte embedding (a.k.a. harmonic embedding) of the discretized Brownian disk converges to the canonical conformal embedding of the continuum Brownian disk, which in turn corresponds to sqrt{8/3}-LQG. Along the way, we obtain other independently interesting facts about conformal embeddings of Brownian surfaces, including information about the Euclidean shapes of embedded metric balls and Voronoi cells. For example, we derive moment estimates that imply, in a certain precise sense, that these shapes are unlikely to be very long and thin.

Fractal Dimension Estimation for Color Texture Images

As an important feature of image texture, fractal dimension is widely used to index, segment or classify texture images. Previously, many methods have been proposed to estimate fractal dimension. However, most of them are for binary and gray-scale images. Only a few of them are for color images. In this paper, by extending the differential box-counting approach to five-dimensional Euclidean hyper-space, we present a new robust and simple algorithm to estimate fractal dimension for color texture images. By the differential box-counting method, a real empty box not covering any pixel even with ideal resolution is also counted as a box having pixels, resulting in errors of computations. This problem is eliminated by our method based on fractal Brownian surface model. Our experiments demonstrate that our algorithm is able to capture the complexity of color natures, and outperforms other methods in terms of color texture complexity ranking and discrimination, robustness and computational complexity.

Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces

In a recent series of works, Miller and Sheffield constructed a metric on $\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-LQG surfaces (e.g., the LQG sphere, wedge, cone and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane and disk). We identify the metric gluings of certain collections of independent $\sqrt{8/3}$-LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane along their positive boundaries is isometric to a certain LQG wedge decorated by an independent chordal $\mathrm{SLE}_{8/3}$ curve. If one identifies the two sides of the boundary of a single Brownian half-plane, one obtains a certain LQG cone decorated by an independent whole-plane $\mathrm{SLE}_{8/3}$. If one identifies the entire boundaries of two Brownian half-planes, one obtains a different LQG cone and the interface between them is a two-sided variant of whole-plane $\mathrm{SLE}_{8/3}$. Combined with another work of the authors, the present work identifies the scaling limit of self-avoiding walk on random quadrangulations with $\mathrm{SLE}_{8/3}$ on $\sqrt{8/3}$-LQG.

Procedural Method for Fast Table Mountains Modelling in Virtual Environments

Natural terrains created by long-term erosion processes can sometimes have spectacular forms and shapes. The visible form depends often upon internal geological structure and materials. One of the unique terrain artefacts occur in the form of table mountains and can be observed in the Monument Valley (Colorado Plateau, USA). In the following article a procedural method is considered for terrain modelling of structures, geometrically similar to the mesas and buttes hills. This method is not intended to simulate physically inspired erosion processes, but targets directly the generation of eroded forms. The results can be used as assets by artists and designers. The proposed terrain model is based on a height-field representation extended by materials and its hardness information. The starting point of the technique is the Poisson Faulting algorithm that was originally used to obtain fractional Brownian surfaces. In the modification, the step function as the fault line generator was replaced with a circular one. The obtained geometry was used for materials’ classification and the hardness part of the modelled terrain. The final model was achieved by the erosive modification of geometry according to the materials and its hardness data. The results are similar to the structures observed in nature and are achieved within an acceptable time for real-time interactions.

On tessellations of random maps and the $$t_g$$ t g -recurrence

We study the masses of the two cells in a Voronoi tessellation of the Brownian surface of genus $$g\ge 0$$ centered on two uniform random points. Making use of classical bijections and asymptotic estimates for maps of fixed genus, we relate the second moment of these random variables to the Painleve-I equation satisfied by the double scaling limit of the one-matrix model, or equivalently to the “ $$t_g$$ -recurrence” satisfied by the constants $$t_g$$ driving the asymptotic number of maps of genus $$g\ge 0$$ . This raises the question of giving an independent probabilistic or combinatorial derivation of this second moment, which would then lead to new proof of the $$t_g$$ -recurrence. More generally we conjecture that for any $$g\ge 0$$ and $$k\ge 2$$ , the masses of the cells in a Voronoi tessellation of the genus-g Brownian surface by k uniform points follows a Dirichlet $$(1,1,\ldots ,1)$$ distribution.

Innovative trends in development of plasma technologies

Experimental data on the novel materials generated in fusion devices under conditions of high-temperature plasma loads is reviewed. In fusion devices, due to the interaction of turbulent plasma with the surface, the stochastic clustering occurs and the fractal structure forms on the plasma-facing surfaces. Such surfaces are characterized by an extremely high specific area and statistical self-similarity of the surface roughness. The generated fractal structures are nano-sized and the surface consists from pores and craters ranging in size from tens of nanometers to tens of micrometers. The occurrence of such structures can result in the appearance of additional properties of such a multiscale surface, in particular, the specific catalytic properties. Statistical characteristics of such material surface characterized by the hierarchical granularity and scale invariance qualitatively differ from those of the ordinary rough Brownian surface. This gives basis for development of innovative plasma technologies for the synthesis of new nanostructured materials with the required roughness and porosity for nuclear, chemical, hypersonic technologies, as well for biotechnologies and biomedical applications.

Fast and exact synthesis of some operator scaling Gaussian random fields

Abstract Operator scaling Gaussian random fields, as anisotropic generalizations of self-similar fields, know an increasing interest for theoretical studies in the literature. However, up to now, they were only defined through stochastic integrals, without explicit covariance functions. In this paper we exhibit explicit covariance functions, as anisotropic generalizations of fractional Brownian fields ones, and define corresponding Operator scaling Gaussian random fields. This allows us to propose a fast and exact method of simulation in dimension 2 based on the circulant embedding matrix method, following ideas of Stein [34] for fractional Brownian surfaces syntheses. This is a first piece of work to popularize these models in anisotropic spatial data modeling.

Stochastic clustering of material surface under powerful load of high-temperature plasma

The results of a study of a surface formed by high-temperature plasma loads on refractory metals such as tungsten and stainless steel are presented. There is an inhomogeneous stochastic clustering of the surface with self-similarity properties of the granularity structure from nanoscale to macro scale, similar to the previously described fractal structure of carbon materials from fusion device. The statistical characteristics of hierarchical granularity and scale invariance are estimated that qualitatively differ from the properties of the roughness of the Brownian surface, which is possibly due to the universal mechanisms of stochastic clustering of materials under the influence of high-temperature plasma.

Innovative potential of plasma technology

The review summarizes recent experimental observations of materials exposed to extreme hot plasma loads in fusion devices and plasma facilities with high-temperature plasma. Plasma load on the material in such devices lead to the stochastic clustering and fractal growth of the surface on scales from tens of nanometers to hundreds of micrometers forming statistical self-similarity of the surface roughness with extremely high specific area. Statistical characteristics of hierarchical granularity and scale invariance of such materials surface qualitatively differ from the properties of the roughness of the ordinary Brownian surface which provides a potential of innovative plasma technologies for synthesis of new nanostructured materials with programmed roughness properties, for hypersonic technologies, for biotechnology and biomedical applications.

Stochastic clustering of material surface under high-heat plasma load

The results of a study of a surface formed by high-temperature plasma loads on various materials such as tungsten, carbon and stainless steel are presented. High-temperature plasma irradiation leads to an inhomogeneous stochastic clustering of the surface with self-similar granularity – fractality on the scale from nanoscale to macroscales. Cauliflower-like structure of tungsten and carbon materials are formed under high heat plasma load in fusion devices. The statistical characteristics of hierarchical granularity and scale invariance are estimated. They differ qualitatively from the roughness of the ordinary Brownian surface, which is possibly due to the universal mechanisms of stochastic clustering of material surface under the influence of high-temperature plasma.

Estimating the Illumination Direction From Three-Dimensional Texture of Brownian Surfaces.

We studied whether human observers can estimate the illumination direction from 3D textures of random Brownian surfaces, containing undulations over a range of scales. The locally Lambertian surfaces were illuminated with a collimated beam from random directions. The surfaces had a uniform albedo and thus texture appeared only through shading and shadowing. The data confirm earlier results with Gaussian surfaces, containing undulations of a single scale. Observers were able to accurately estimate the source azimuth. If shading dominated the images, the observers committed 180° errors. If cast shadows were present, they resolved this convex-concave-ambiguity almost completely. Thus, observers relied on second-order statistics in the shading regime and used an unidentified first-order cue in the shadow regime. The source elevations could also be estimated, which can be explained by the observers’ exploitation of the statistical homogeneity of the stimulus set. The fraction of the surface that is in shadow and the median intensity are likely cues for these elevation estimates.

Stochastic clustering of the surface at the interaction of a plasma with materials

The inhomogeneous stochastic clustering of the surface with the self-similarity of the granularity structure from nano- to macroscales is observed in various materials after the action of intense high-temperature plasma flows in nuclear fusion facilities. The spectral and statistical characteristics of hierarchical granularity and scale invariance are estimated. They qualitatively differ from the properties of the simplest Brownian surface roughness and clustering under other conditions possibly because of universal mechanisms of the formation of stochastic clustering of materials under the action of a high-temperature plasma.