Transport Exponent Research Articles

Superdiffusion on a comb structure.

We study specific properties of particles transport by exploring an exact solvable model, a so-called comb structure, where diffusive transport of particles leads to subdiffusion. A performance of the Lévy-like process enriches this transport phenomenon. It is shown that an inhomogeneous convection flow is a mechanism for the realization of the Lévy-like process. It leads to superdiffusion of particles on the comb structure. This superdiffusion is an enhanced one with an arbitrary large transport exponent, but all moments are finite. A frontier case of superdiffusion, where the transport exponent approaches infinity, is studied. The log-normal distribution with the exponentially fast superdiffusion is obtained for this case.

A characterization of the Anderson metal-insulator transport transition

We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. We introduce a local transport exponent β(E) and set the weak metallic transport region to be the part of the spectrum with nontrivial transport (i.e., β(E)>0). We prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent β(E) provides a characterization of the metal-insulator transport transition. Moreover, we show that if there is such a transition, then β(E) has to be discontinuous at a transport mobility edge. More precisely, we show that if the transport is nontrivial, then β(E)≥1/(2d), where d is the space dimension. These results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors' bootstrap multiscale analysis. We also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis, proving that the multiscale analysis is valid all the way up to a transport mobility edge.

Space-time complexity in Hamiltonian dynamics.

New notions of the complexity function C(epsilon;t,s) and entropy function S(epsilon;t,s) are introduced to describe systems with nonzero or zero Lyapunov exponents or systems that exhibit strong intermittent behavior with "flights," trappings, weak mixing, etc. The important part of the new notions is the first appearance of epsilon-separation of initially close trajectories. The complexity function is similar to the propagator p(t(0),x(0);t,x) with a replacement of x by the natural lengths s of trajectories, and its introduction does not assume of the space-time independence in the process of evolution of the system. A special stress is done on the choice of variables and the replacement t-->eta=ln t, s-->xi=ln s makes it possible to consider time-algebraic and space-algebraic complexity and some mixed cases. It is shown that for typical cases the entropy function S(epsilon;xi,eta) possesses invariants (alpha,beta) that describe the fractal dimensions of the space-time structures of trajectories. The invariants (alpha,beta) can be linked to the transport properties of the system, from one side, and to the Riemann invariants for simple waves, from the other side. This analog provides a new meaning for the transport exponent mu that can be considered as the speed of a Riemann wave in the log-phase space of the log-space-time variables. Some other applications of new notions are considered and numerical examples are presented.

A comparative study of swelling properties of hydrogels based on poly(acrylamide‐co‐methyl methacrylate) containing physical and chemical crosslinks

AbstractPoly (acrylamide‐co‐methyl methacrylate) hydrogels of different ratios were prepared by using chemical and physical crosslinks to study the effect of nature of crosslinks on swelling behavior of hydrogels. The chemically crosslinked gels were prepared by using NN′‐methylene bis acrylamide, while physically crosslinked hydrogels were prepared by precipitation polymerization method, using dioxane as solvent. Detailed swelling kinetics such as swelling ratio, transport exponent n, diffusion coefficient D and the effect of pH on equilibrium swelling studies. The study revealed that the nature of crosslinks alter the swelling characteristics of the hydrogel. In chemically crosslinked hydrogels the water transport is Fickian in nature, while in the case of the physically crosslinked hydrogels the water transport mechanism is anomalous indicating major change in relaxation mechanism due to nature of crosslinks. The results also indicate that with increasing acrylamide content the swelling ratio of the hydrogels were also increased, but the transport exponent n remains nearly constant. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 89: 779–786, 2003

Evidence of fractional transport in point vortex flow

Advection properties of passive particles in flows generated by point vortices are considered. Transport properties are anomalous with characteristic transport exponent μ∼1.5. This behavior is linked back to the presence of coherent fractal structures within the flow. A fractional kinetic analysis allows to link the characteristic transport exponent μ to the trapping time exponent γ=1+μ. The quantitative agreement is found for different systems of vortices investigated and a clear signature is obtained of the fractional nature of transport in these flows.

Jets, stickiness, and anomalous transport.

Dynamical and statistical properties of the vortex and passive particle advection in chaotic flows generated by 4- and 16-point vortices are investigated. General transport properties of these flows are found to be anomalous and exhibit a superdiffusive behavior with typical second moment exponent mu approximately 1.75. The origin of this anomaly is traced to the presence of coherent structures within the flow, the vortex cores, and the region far from where vortices are located. In the vicinity of these regions stickiness is observed and the motion of tracers is quasiballistic. The chaotic nature of the underlying flow dictates the choice for thorough analysis of transport properties. Passive tracer motion is analyzed by measuring the mutual relative evolution of two nearby tracers. Some tracers travel in each other's vicinity for relatively long times. This is related to a hidden order for the tracers, which we call jets. Jets are localized and found in sticky regions. Their structure is analyzed and found to be formed of a nested set of jets within jets. The analysis of the jet trapping time statistics shows a quantitative agreement with the observed transport exponent.

Quantum breaking time scaling in superdiffusive dynamics.

We show that the breaking time of quantum-classical correspondence depends on the type of kinetics and the dominant origin of stickiness. For sticky dynamics of quantum kicked rotor, when the hierarchical set of islands corresponds to the accelerator mode, we demonstrate by simulation that the breaking time scales as tau(Planck's over 2pi) approximately (1/Planck's over 2pi)(1/mu) with the transport exponent mu>1 that corresponds to superdiffusive dynamics [B. Sundaram and G. M. Zaslavsky, Phys. Rev. E 59, 7231 (1999)]. We discuss also other possibilities for the breaking time scaling and transition to the logarithmic one tau(Planck's over 2pi) approximately ln(1/Planck's over 2pi) with respect to Planck's over 2pi.

Chaotic advection near a three-vortex collapse.

Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. Tracer dynamics is analyzed by numerical construction of Poincaré sections, and is found to be strongly chaotic: advection pattern in the region around the center of vorticity is dominated by a well developed stochastic sea, which grows as the vortex system's initial conditions are set closer to those leading to the collapse of the vortices; at the same time, the islands of regular motion around vortices, known as vortex cores, shrink. An estimation of the core's radii from the minimum distance of vortex approach to each other is obtained. Tracer transport was found to be anomalous: for all of the three numerically investigated cases, the variance of the tracer distribution grows faster than a linear function of time, corresponding to a superdiffusive regime. The transport exponent varies with time decades, implying the presence of multi-fractal transport features. Yet, its value is never too far from 3/2, indicating some kind of universality. Statistics of Poincaré recurrences is non-Poissonian: distributions have long power-law tails. The anomalous properties of tracer statistics are the result of the complex structure of the advection phase space, in particular, of strong stickiness on the boundaries between the regions of chaotic and regular motion. The role of the different phase space structures involved in this phenomenon is analyzed. Based on this analysis, a kinetic description is constructed, which takes into account different time and space scalings by using a fractional equation.

Magnetic Field Line Wandering and Shock‐Front Acceleration: Application to the Radio Emission of SN 1987A

The superposition of random fluctuations δ on a main, regular magnetic field 0 has been shown by Ragot to produce, on the scales of interest, a nondiffusive random walk of the magnetic field lines in the quasi-linear regime of turbulence. The spreading Δx2 of the field lines across the direction of 0 increases as zβ, z being the distance along 0 with a transport exponent β that is not necessarily equal to 1. This puts into question the diffusivity of magnetic field lines in a non-quasi-linear regime as well and could imply an anomalous transport of the cosmic rays accelerated at a shock front if their decorrelation time from the field lines is longer than their acceleration time. Under the assumption of such a long decorrelation time, the acceleration time of cosmic rays scattered along the wandering field lines is derived here as a function of the obliquity and speed of the shock for a general transport exponent of the field lines. It is shown that for high-speed shocks, the importance of magnetic field line wandering (MFLW) is not limited to quasi-perpendicular shocks as usually believed. Moreover, the turbulence ahead of the shock is in a non-quasi-linear regime on the length scale traveled by accelerated particles whenever MFLW is important. The inclusion of MFLW can make the acceleration time depart by orders of magnitude from its value estimated in the approximation of simple scattering. This discrepancy is actually most welcome to explain the acceleration time of GeV electrons at the reappearance of SN 1987A in the radio waveband, since the inferred diffusion coefficient along the shock normal is about 5 orders of magnitude greater than the limiting Bohm value of a cross-field scattering coefficient. From the requirement of a realistic turbulence spectrum and in the approximation of quasi-linear transport coefficients, it is found that the magnetic field lines ahead of the SN 1987A shock strongly supradiffuse (β ≈ 1.8) and that the compression ratio of the shock is larger than 3. Finally, this paper argues in favor of a radio silence of SN 1987A prior to 1990 due to a lack of scattering of the GeV electrons when the shock speed exceeds about one-tenth of the speed of light.

Multifractional kinetics

Multifractional generalizations are considered for kinetic description of chaotic dynamics. The multiplicity of dimensions is motivated by the multiplicity of sticky island hierarchies in phase space. The set of such islands is considered as a singular support that, in the large time asymptotics, defines the main features of the kinetics which we call “multifractional”. Finally, the moments of the displacement in phase space are not definied by the only transport exponent. Full transport asymptotically exhibits anomalous diffusion with log-periodic oscillations and with intermittent dependence of the transport exponent and period of oscillations on the order of the considered moment. We speculate on the appearance of the same features in experiments and simulations for particle dispersion in turbulent flows.

Phase space structure and anomalous diffusion in a rotational fluid experiment

The transport of passive scalars is considered in a model of rotating annulus experiments. The system has a chain of vortices and a jet, separated by a stochastic layer. For special values of the control parameters, the boundary of the stochastic layer can contain self-similar structures of islands with regular trajectories. Two such values are identified, with the structure being on the jet boundary and on the vortex boundary, respectively. The transport properties for both cases were studied by high-precision direct numerical integration of the equations of motion. The presence of such structures is found to significantly affect the statistical properties of the trajectories and the transport exponent. The results of the computations are compared with various theoretical models of anomalous diffusion. The particle behavior was found to depend significantly on the time scale, with different theories being applicable on different time intervals. Some regimes do not match any of the existing theories. (c) 2000 American Institute of Physics.

On the Quasi‐linear Transport of Magnetic Field Lines

Under the assumption that a finite correlation length, Lc, exists and is much smaller than the size of the system, the quasi-linear theory for weak magnetic turbulence predicts, beyond Lc, a linear spreading of the magnetic field lines, Δr, as a function of the elapsed distance z along the main magnetic field, B0. When the power spectrum of the turbulence flattens at low wavenumbers, as was deduced from early observations of the interplanetary turbulence and is expected for an isotropic turbulence, the correlation length can be estimated as the Alfven speed times the reciprocal of the upper frequency in the flat spectrum. However, more recent measurements of the solar wind magnetic power spectrum indicate a more complex spectrum at low frequencies. The field-lines' spreading in the quasi-linear regime of turbulence is derived here as a function of z for a general power spectrum. The transport exponent α, defined by Δr ∝ zα, can be deduced from the projection along B0 of the spectrum. Normal (α = 1), supradiffusive (α > 1), and subdiffusive (α < 1) predictions are made, depending on the shape at low wavenumbers assumed for the projected spectrum, which is related to the anisotropy and ||-form of the three-dimensional spectrum.

Transport of Dust Grains in Turbulent Molecular Clouds

Recently, F. P. Pijpers has estimated anew the average residence time of dust grains in turbulent molecular clouds, modeling the grains' dynamics as a random walk along turbulent eddies of a Kolmogorov energy spectrum, with distribution versus size similar to that of the clumps. Implicitly, he assumed that this results in a diffusive transport process (the spreading of the dust grains Δr2(t) increasing linearly with time t), and obtained a residence time scaling almost as the Reynolds number of the flow, and hence much longer than previous estimates. Following the same random walk model, the grains are shown here to be transported anomalously quickly (Δr2(t) ∝ tα with α ≥ 3), and a much shorter time is found, scaling as the Reynolds number to some negative power. This might indicate the need to take into account the organization of the turbulent structures when computing the transport exponent α.

Fractal and multifractal properties of exit times and Poincarérecurrences

Systems with chaotic dynamics possess anomalous statistical properties, and their trajectories do not correspond to the Gaussian process. This property imposes description of such time characteristics as the distribution of exit times or Poincar\'erecurrences by introducing a (multi-) fractal time scale in order to satisfy the observed powerlike tails of the distributions. We introduce a corresponding phase-space-time partitioning and spectral function for dimensions, and make a connection between dimensions and transport exponent that defines the anomalous ('strange') kinetics.

Self-similarity and transport in the standard map

Anomalous transport is investigated for the standard map. A chain of exact self-similar islands in the vicinity of the period 5 accelerator island is found for a particular value of the map parameter. The transport is found to be superdiffusive with an anomalous exponent related to the characteristic temporal and spatial scaling parameters of the island chain. The value of the transport exponent is compared to the theory. The escape time distribution and Poincar\'erecurrence distribution are found to have powerlike tails and the corresponding exponents are obtained and compared to the theory.

Fractional kinetics and accelerator modes

Kinetic properties of chaotic dynamics are studied for the web-map. It is shown that depending on the values of the perturbation parameter K, there are alternative possibilities: kinetics, of the quasilinear (normal, i.e. Gaussian) type or kinetics of the anomalous (Lévyan) type. If K is close to the values K = ℓ π when the accelerator modes occur, then there is superdiffusion with anomalous values of the transport exponent. We consider self-similar properties of trajectories near the islands in the phase space. Fractional kinetic equation is proposed and fractal exponents are obtained. Local properties of trajectories (exit time) and nonlocal ones (Poincaré cycles) are studied and compared for the normal and anomalous kinetics. It is shown that the power-like law of the trapping distribution function imposes the same kind of asymptotics for the Poincaré cycles distribution.

The transport exponent in percolation models with additional loops

Several percolation models with additional loops were studied. The transport exponents for these models were estimated numerically by means of a transfer-matrix approach. It was found that the transport exponent has a drastically changed value for some of the models. This result supports some previous numerical studies on the vibrational properties of similar models (with additional loops).

Fractional kinetic equation for Hamiltonian chaos

Hamiltonian chaotic dynamics of particles (or passive particles in fluids) can be described by a fractional generalization of the Fokker-Planck-Kolmogorov equation (FFPK) which is defined by two fractional critical exponents (α, β) responsible for the space and time derivatives of the distribution function correspondingly. A renormalization method has been proposed to determine (α, β) from the first principles (ie. from the Hamiltonian). The anomalous transport exponent μ is derived as μ = β/α or μ = β/2α for the first order mean displacement in self-similar transport.

Self-similar transport in incomplete chaos.

Particle chaotic dynamics along a stochastic web is studied for three-dimensional Hamiltonian flow with hexagonal symmetry in a plane. Two different classes of dynamical motion, obtained by different values of a control parameter, and corresponding to normal and anomalous diffusion, have been considered and compared. It is shown that the anomalous transport can be characterized by powerlike wings of the distribution function of displacement, flights which are similar to L\'evy flights, approximate trappings of orbits near the boundary layer of islands, and anomalous behavior of the moments of a distribution function considered as a function of the number of the moment. The main result is related to the self-similar properties of different topological and dynamical characteristics of the particle motion. This self-similarity appears in the Weierstrass-like random-walk process that is responsible for the anomalous transport exponent in the mean-moment dependence on t. This exponent can be expressed as a ratio of fractal dimensions of space and time sets in the Weierstrass-like process. An explicit form for the expression of the anomalous transport exponent through the local topological properties of orbits has been given.

Percolation conductivity of Penrose lattices by the transfer-matrix Monte Carlo method

A generalization of the Derrida and Vannimenus transfer-matrix Monte Carlo method has been applied to calculations of percolation conductivity in Penrose lattices. Strips with a length ∼10 4 and widths from 3 to 19 bond lengths have been used. Disregarding the differences for smaller strip widths, the results show that the percolation conductivity of a Penrose lattice is very close to that of a square lattice. The estimation of the percolation transport exponent once more confirms the universality conjecture for the 0–1 distribution of resistors. The conductivity of a four-grid based quasilattice has been numerically calculated.