Fractal Trajectories Research Articles

Dichotomous dynamics of magnetic monopole fluids

A recent advance in the study of emergent magnetic monopoles was the discovery that monopole motion is restricted to dynamical fractal trajectories [J. N. Hallén et al., Science 378, 1218 (2022)], thus explaining the characteristics of magnetic monopole noise spectra [R. Dusad et al., Nature 571, 234 (2019); A. M. Samarakoon et al., Proc. Natl. Acad. Sci. U.S.A. 119, e2117453119 (2022)]. Here, we apply this novel theory to explore the dynamics of field-driven monopole currents, finding them composed of two quite distinct transport processes: initially swift fractal rearrangements of local monopole configurations followed by conventional monopole diffusion. This theory also predicts a characteristic frequency dependence of the dissipative loss angle for AC field-driven currents. To explore these novel perspectives on monopole transport, we introduce simultaneous monopole current control and measurement techniques using SQUID-based monopole current sensors. For the canonical material Dy2Ti2O7, we measure [Formula: see text], the time dependence of magnetic flux threading the sample when a net monopole current [Formula: see text] is generated by applying an external magnetic field [Formula: see text] These experiments find a sharp dichotomy of monopole currents, separated by their distinct relaxation time constants before and after t ~[Formula: see text] from monopole current initiation. Application of sinusoidal magnetic fields [Formula: see text] generates oscillating monopole currents whose loss angle [Formula: see text] exhibits a characteristic transition at frequency [Formula: see text] over the same temperature range. Finally, the magnetic noise power is also dichotomic, diminishing sharply after t ~[Formula: see text]. This complex phenomenology represents an unprecedented form of dynamical heterogeneity generated by the interplay of fractionalization and local spin configurational symmetry.

Ontogenetic Trends of Sutural Complexity in Jurassic Ammonites

Ontogenetic sequencing of suture fractal dimensions from a single ammonite specimen was not previously accessible without extreme measures. The necessity for destruction of the full conch due to whorl overlap or, alternatively, the prohibitively expensive or inaccessible imaging equipment for use by paleontologists has led to this rich source of data remaining elusive. Tracings of ontogenetic sequences of sutures have rarely been published since the middle of the last century. These studies only focused on the outermost whorl of a given specimen or composited several specimens. Oftentimes, they focused only on the umbilical and juvenile stages of shell growth. Complete or multi-whorled ontogenetic sequences, including those of subadult and mature specimens, are especially uncommon. Using methods published in 2021 to address the accessibility problem in sutural ontogenies, our paper represents the first comparisons of the fractal trajectories of ammonoids and includes several significant Jurassic ammonite families. We found that ammonite sutures did not steadily increase in complexity, as was previously suggested. Instead, juvenile ammonoids were more likely to exhibit ups and downs in their fractal ontogeny. We also found that pathologies must be considered as extremely common phenomena for ammonoids. Finally, we explored the ways that the unsteady ontogenetic trends of fractal complexity in ammonoids offer insights into the formation mechanism of ammonoid septa.

Mode I Fracture Toughness Test and Fractal Character of Fracture Trajectory of Red Sandstone under Real-Time High Temperature

To evaluate the stability and compactness of high-temperature underground construction, it is necessary to test the fracture toughness of surrounding rock (red sandstone) under real-time high temperature. In this paper, SCB specimens recommended by the International Society for Rock Mechanics are used to measure the mode I fracture toughness of red sandstone at real-time high temperatures. Also, to reveal its fracture characteristics and fracture mechanism, the fracture morphology observation (SEM experiment), XRD experiment, mercury intrusion porosimetry testing, and fractal measurement of fracture trajectory are carried out on the red sandstone specimens at various temperatures. The results show that (1) temperature may have a significant impact on the fracture toughness and fracture characteristics of red sandstone. On the whole, the fracture toughness values decrease with the increase in temperature, while the fractal dimensions of fractal trajectories increase with the increase in temperature. (2) Temperature has a significant influence on the fracture mode of red sandstone. At relatively low temperatures (20°C–400°C), the main fracture mode is transgranular failure. At relatively high temperatures (400°C–700°C), the fracture mode is mainly intergranular failure. (3) The weakening mechanism of red sandstone is mainly due to the effect of thermal dehydration when the temperature is between 100°C and 400°C. When the temperature is between 400°C and 700°C, the weakening mechanism is mainly due to thermal cracking and theα-βphase transition of quartz.

Visual voice activity detection via chaos based lip motion measure robust under illumination changes

In this paper, a vision based voice activity detection (VVAD) algorithm is proposed using chaos theory. In conventional VVAD algorithm, the movement measure of lip region is found by applying an optical flow algorithm to detect the visual speech frame using a motion based energy feature set. However, since motion based feature is unstable under illumination changes, a new form of robust feature set is desirable. It is propositioned that contextual changes such as lip opening or closing motion during speech utterances under illumination variation can be observed as chaos-like and the resultant complex fractal trajectories in phase space can be observed. The fractality is measured in phase space from two sequential video input frames and subsequently any visual speech frames are robustly detected. Representative experiments are performed in image sequence containing a driver scene undergoing illumination fluctuations in moving vehicle environment. Experimental results indicate that a substantial improvement is obtained in terms of achieving significantly lower false alarm rate over the conventional method.

The Surface Fractal Structure of Fish Scales

By atomic force, microscopy studied the morphology of the surface of the scale of the Caspian Kutum. Analysis of the structure of channels of sensory systems showed that the nanostructures are not only fractal, but also self-similar. Fractal trajectories originate from the primary nucleus and extend to the sprawling low-density structure radically outwards scales like Lichtenberg figures. However, in contrast to the figures, Lichtenberg fractal band scales of fish each year are surrounded by concentric threads. The most likely mechanism for the formation of structural self-organization on the surface of the scales can be a model Witten-Sander.

Reinterpreting aircraft measurements in anisotropic scaling turbulence

Abstract. Due to both systematic and turbulent induced vertical fluctuations, the interpretation of atmospheric aircraft measurements requires a theory of turbulence. Until now virtually all the relevant theories have been isotropic or "quasi isotropic" in the sense that their exponents are the same in all directions. However almost all the available data on the vertical structure shows that it is scaling but with exponents different from the horizontal: the turbulence is scaling but anisotropic. In this paper, we show how such turbulence can lead to spurious breaks in the scaling and to the spurious appearance of the vertical scaling exponent at large horizontal lags. We demonstrate this using 16 legs of Gulfstream 4 aircraft near the top of the troposphere following isobars each between 500 and 3200 km in length. First we show that over wide ranges of scale, the horizontal spectra of the aircraft altitude are nearly k-5/3. In addition, we show that the altitude and pressure fluctuations along these fractal trajectories have a high degree of coherence with the measured wind (especially with its longitudinal component). There is also a strong phase relation between the altitude, pressure and wind fluctuations; for scales less than ≈40 km (on average) the wind fluctuations lead the pressure and altitude, whereas for larger scales, the pressure fluctuations leads the wind. At the same transition scale, there is a break in the wind spectrum which we argue is caused by the aircraft starting to accurately follow isobars at the larger scales. In comparison, the temperature and humidity have low coherencies and phases and there are no apparent scale breaks, reinforcing the hypothesis that it is the aircraft trajectory that is causally linked to the scale breaks in the wind measurements. Using spectra and structure functions for the wind, we then estimate their exponents (β, H) at small (5/3, 1/3) and large scales (2.4, 0.73). The latter being very close to those estimated by drop sondes (2.4, 0.75) in the vertical direction. In addition, for each leg we estimate the energy flux, the sphero-scale and the critical transition scale. The latter varies quite widely from scales of kilometers to greater than several hundred kilometers. The overall conclusion is that up to the critical scale, the aircraft follows a fractal trajectory which may increase the intermittency of the measurements, but doesn't strongly affect the scaling exponents whereas for scales larger than the critical scale, the aircraft follows isobars whose exponents are different from those along isoheights (and equal to the vertical exponent perpendicular to the isoheights). We bolster this interpretation by considering the absolute slopes (|Δz/Δx|) of the aircraft as a function of lag Δx and of scale invariant lag Δx/Δz1/Hz. We then revisit four earlier aircraft campaigns including GASP and MOZAIC showing that they all have nearly identical transitions and can thus be easily explained by the proposed combination of altitude/wind in an anisotropic but scaling turbulence. Finally, we argue that this reinterpretation in terms of wide range anisotropic scaling is compatible with atmospheric phenomenology including convection.

Banach spaces of fractal functions and trajectories of Brownian motions

Banach spaces of fractal functions and trajectories of Brownian motions

Aggregation with fractal trajectories of changing dimensionality

The fractal dimension of a particle–cluster aggregate depends on the space dimensionality, the geometric characteristic of the particle trajectory and on the progress of the process. The asymptotic values of fractal dimension are characteristic for large aggregates. The influence is analyzed of the step size of random walk on the structure of aggregates built by the successive addition of monomers to a growing cluster. Increasing the step size it is possible to obtain more and more compact structures [Phys. Rev. A 36 (1987) 4518] changing from that characteristic for diffusion-limited process to that observed for ballistic particle–cluster aggregation. Moreover, the trajectory fractal dimension increases as the aggregation proceeds due to the aggregate growth and the constancy of the step size of random walk. It is shown that aggregation act equation models such aggregation processes well.

Scaling Quasi-Periodic Psychological Functions

The entropic analog of the Schwarzian derivative (ESf) is used to explore the consequences of scaling and partitioning operations on time series that are known to be edge-of-chaos, or random. The nonlinear deterministic series may be fractal trajectories, and if so self-similarity may be a property that characterises their evolution. Examples are generated, and statistical methods that follow from their generation are noted. Some invariance properties of the ESf index that are associated with fractal self-similarity but not with other series appear to be a basis for process identification on relatively short time series. Surrogate and parameter optimisation methods are illustrated in real data. A possible relationship between the difference between an ESf and the ESf on its random surrogates, and the first derivative of the local largest Lyapunov exponent, is noted.

Fractals and the Quantum Classical Boundary

Early in this century, the arrival of quantum mechanics resulted in a split in the worldviews of physicists. In classical mechanics the apparent correspondence between mathematicalmodels and their physical counterparts was often so close that little distinction needed to be madebetween the two. In contrast, quantum mechanics provides a beautifully accurate description ofNature, but itself yields little explanation of microscopic phenomena. An understanding of themathematics of quantum mechanics leads one to an ability to calculate and predict, but few wouldargue that it affords a deep understanding of the phenomena being described. In the quantum world, the Heisenberg uncertainty relations seem to put a fundamentallimit on what is observable, and until recently, strongly represented the conceptual barrierseparating the quantum and classical worlds. However, it may be shown that by extendingclassical mechanics to allow Fractal trajectories, the uncertainty relations and some of thedynamical equations of quantum mechanics appear in this extended classical domain. This shift ofthe boundary between the quantum and classical world will be discussed at a general level andillustrated by some exactly solvable statistical mechanical models.

Phase space structure and fractal trajectories in 1½ degree of freedom Hamiltonian systems whose time dependence is quasiperiodic

Abstract. We consider particle motion in nonautonomous 1 degree of freedom Hamiltonian systems for which H(p,q,t) depends on N periodic functions of t with incommensurable frequencies. It is shown that in near-integrable systems of this type, phase space is partitioned into nonintersecting regular and chaotic regions. In this respect there is no different between the N = 1 (periodic time dependence) and the N = 2, 3, ... (quasi-periodic time dependence) problems. An important consequence of this phase space structure is that the mechanism that leads to fractal properties of chaotic trajectories in systems with N = 1 also applies to the larger class of problems treated here. Implications of the results presented to studies of ray dynamics in two-dimensional incompressible fluid flows are discussed.

A particle tracking model for non-Fickian subsurface diffusion

In this paper, a new particle tracking technique is described which can simulate non-Fickian diffusion within porous media. The technique employs fractional Brownian motions (fBms), a generalization of regular Brownian motion. These random fractal functions allow both super- and subdiffusive particle paths to be produced and hence non-Fickian diffusion of the resulting panicle clouds can be modeled. In recent years, fBm trace functions have been used by many authors to reproduce self-affine random fields to simulate various porous media properties. In contrast, a method is detailed herein which uses self-similar spatial fBm trajectories to simulate directly non-Fickian behavior of the particle clouds. Although fractal trajectories have been previously suggested as the basis for possible methods of modeling non-Fickian diffusion, the authors believe that this paper contains the first algorithm to be presented which does not require an a priori knowledge of the end condition of the random walk and, more importantly, allows both a definable scaling exponent and (fractal) diffusion coefficient to be specified. The resulting non-Fickian diffusion using the new algorithm is illustrated and some applications are discussed. The purpose of this paper is to bring the potential usefulness of fBm trajectories in simulating non-Fickian processes within homogeneous media to the attention of numerical modelers active in the simulation of subsurface diffusive processes. The method has a particular environmental application in the simulation of the non-Fickian dispersion of groundwater contaminants through porous media.

Fractal space-time and the statistical mechanics of random walks

We illustrate some of the ideas involved in fractal space-time using familiar deterministic fractals. Starting with the objective of reproducing the Heisenburg uncertainty principle for point particles, we use the Peano-Moore curve to help visualize the qualitative behaviour of particles moving on fractal trajectories in space and time. With this qualitative picture in mind we then explore exactly solvable models to verify that our ideas are mathematically consistent. We find that the Schrödinger equation describes ensembles of classical particles moving on fractal random walk trajectories. This shows that the Schrödinger equation has a straightforward microscopic model which is not, however, appropriate for quantum mechanics. The free particle Dirac equation is also derivable in terms of ensembles of classical particles and this unites the two equations conceptually in a very direct way. In both cases what we discover is a many-particle simulation of quantum mechanics and this confirms in a graphic way that the mysteries surrounding quantum mechanics lie not in the equations, but in interpretation and the theory of measurement. Finally, we discuss an exactly solvable model which incorporates fractal time. The calculation produces the Dirac equation in 1 + 1 dimensions and because of intrinsic space-time loops, constitutes a model with the potential to exhibit the wave-particle duality found in nature.

Fractal trajectories in a numerical model of the upper Indian Ocean

Abstract. A wind-driven numerical model of the Indian Ocean is used to examine the horizontal statistics of hundreds of passive tracers spread evenly over the model domain. The distribution covers several dynamically distinct regions, revealing a variety of Lagrangian behaviours associated with different geographic locations. In particular, a cluster of trajectories with scaling dimension as large as 1.3 exists throughout the equatorial zone. Spectral analysis of trajectory displacements indicates mixed Rossby-gravity waves are involved in the production of some fractal trajectories.

Fractal drifter trajectories in the Kuroshio extension

Fractal drifter trajectories in the Kuroshio extension

Fractal space-time: a geometric analogue of relativistic quantum mechanics

Considers a 'thought experiment' in which particles are confined to move on fractal trajectories in both space and time, treating the case of a Peano-Moore trajectory in detail. Generalising these results, the author uses the classical principle of relativity and a correspondence principle to show that fractal trajectories in space with Hausdorff dimension D=2 exhibit both an uncertainty principle and a de Broglie relation. The incorporation of fractal time with D=2 places an upper bound on the macroscopic velocities of 'fractalons', which in turn requires that the macroscopic physics be Lorentz covariant. On a microscopic scale, the presence of fractal time is interpreted in terms of the appearance of particle-antiparticle pairs when observation energies become of the order of mc2. The author proposes two field equation descriptions of fractalons based on random walk space-time trajectories and subsequently relates these equations to the free particle Klein-Gordon and Dirac equations respectively.

Fractal and lacunary stochastic processes

Discrete-time random walks simulate diffusion if the single-step probability density function (jump distribution) generating the walk is sufficiently shortranged. In contrast, walks with long-ranged jump distributions considered in this paper simulate Levy or stable processes. A one-dimensional walk with a selfsimilar jump distribution (the Weierstrass random walk) and its higherdimensional generalizations generate fractal trajectories if certain transience criteria are met and lead to simple analogs of deep results on the Hausdorff-Besicovitch dimension of stable processes. The Weierstrass random walk is lacunary (has gaps in the set of allowed steps) and its characteristic function is Weierstrass' non-differentiable function. Other lacunary random walks with characteristic functions related to Riemann's zeta function and certain numbertheoretic functions have very interesting analytic structure.