https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

We provide a broad review of fundamental electronic properties of two-dimensional graphene with the emphasis on density and temperature dependent carrier transport in doped or gated graphene structures. A salient feature of our review is a critical comparison between carrier transport in graphene and in two-dimensional semiconductor systems (e.g. heterostructures, quantum wells, inversion layers) so that the unique features of graphene electronic properties arising from its gap- less, massless, chiral Dirac spectrum are highlighted. Experiment and theory as well as quantum and semi-classical transport are discussed in a synergistic manner in order to provide a unified and comprehensive perspective. Although the emphasis of the review is on those aspects of graphene transport where reasonable consensus exists in the literature, open questions are discussed as well. Various physical mechanisms controlling transport are described in depth including long- range charged impurity scattering, screening, short-range defect scattering, phonon scattering, many-body effects, Klein tunneling, minimum conductivity at the Dirac point, electron-hole puddle formation, p-n junctions, localization, percolation, quantum-classical crossover, midgap states, quantum Hall effects, and other phenomena.

/pdf/electronic-transport-in-two-dimensional-graphene-32oou47sy4.pdf

Electronic transport in two-dimensional graphene

The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed The term ``Anderson transition'' is understood in a broad sense, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states The emphasis is put on recent developments, which include: multifractality of critical wave functions, criticality in the power-law random banded matrix model, symmetry classification of disordered electronic systems, mechanisms of criticality in quasi-one-dimensional and two-dimensional systems and survey of corresponding critical theories, network models, and random Dirac Hamiltonians Analytical approaches are complemented by advanced numerical simulations

Anderson Transitions

Chaos, fractional kinetics, and anomalous transport

Cosmology at low frequencies: The 21 cm transition and the high-redshift Universe

A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential $\ensuremath{\psi}(\mathrm{x})$. For rapidly decaying correlations of $\ensuremath{\psi}$, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.

/pdf/percolation-statistical-topography-and-transport-in-random-1ddemlupey.pdf

Percolation, statistical topography, and transport in random media

It is shown that electron transport due to a generic low-frequency electrostatic turbulence in tokamak geometry results in the relaxation to a peaked, self-sustained plasma density profile ${n}_{0}(r)$, rather than to a diffusion-induced flat distribution. The relaxed density profile depends on the magnetic geometry and the distribution of turbulence. The associated inward pinch velocity ${\mathbf{V}}_{r}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}D\mathbf{\ensuremath{\nabla}}\mathrm{ln}{n}_{0}$ results from the competition of the turbulent diffusion of trapped electrons over the poloidal magnetic flux coordinate $\ensuremath{\psi}$ and the collisional relaxation toward a Maxwellian distribution function.

Invariant measure and turbulent pinch in tokamaks.

Anomalous cross-field electron transport in a specified magnetic field with weakly-destroyed flux surfaces is discussed. Following the approach developed previously in the present paper the regimes of effective transverse plasma transport are studied systematically, both in the collisional and collisionless limits. The present analysis incorporates the nonstationary of magnetic perturbations, which was not included in the previous works. Part I of this study deals with the quasi-linear approximation, which may be written as b0L0/ delta > chi perpendicular to ) for time-independent magnetic perturbations. However, these regimes can exist in the presence a finite-frequency magnetic flutter. A unified classification of quasi-linear regimes of anomalous transport is introduced in order to further extend the analysis to the strong magnetic turbulence limit b0L0/ delta >>1 (see Part II), which is considered as a companion paper. (Isichenko, ibid., vol.33, no.7, p.809-26 (1991)).

Effective plasma heat conductivity in 'braided' magnetic field-I. Quasi-linear limit

For pt.I see ibid., vol.33, no.7, p.795-807 (1991). This paper is devoted to the problem of anomalous transport across a magnetic field that includes a small stochastic component delta B. The perturbation is assumed to be so strongly stretched along the background magnetic field B0 that the parameter R is large: R identical to b0L0/ delta >>1 (here b0 identical to delta Bperpendicular to /B0<<1, and L0 is the longitudinal and delta the transverse correlation length of the magnetic perturbation). This strong turbulence limit, which is opposite to the quasi-linear one (R<<1), has certain notable features. The principal result is that the main transport is concentrated in very thin regions, being fractal sets with the dimension df, which can range in value from 2 to 2.75, depending on the spectrum of the magnetic perturbation. These regions consist of a small fraction of magnetic lines that percolate, that is, walk from the nonperturbed magnetic flux surfaces to a distance large compared to the transverse correlation length delta . Due to such a strong inhomogeneity of the transport distribution, as well as the long correlations, the standard transport averaging techniques fail, and one should make use of the percolation theory methods. Thus the strong turbulence regime is referred to here as the percolation limit. In comparison with the quasi-linear limit, the percolation limit has several additional intermediate regimes and the expressions for the effective heat conductivity chi eff include the critical exponents of 2-D percolation theory. The estimates of chi eff are obtained both in the collisional and collisionless limits, including the case of nonstationary magnetic perturbations.

http://iopscience.iop.org/article/10.1088/0741-3335/33/7/005/pdf

Effective plasma heat conductivity in 'braided' magnetic field-II. Percolation limit

The long-time nonlinear evolution of generic initial perturbations in stable Vlasov plasma and two-dimensional (2D) ideal fluid is studied. Even without dissipation, these systems relax to new steady states (Landau damping). The asymptotic damping laws are found to be algebraic, such as ${t}^{\ensuremath{-}1}$ for 1D plasma potential, or ${t}^{\ensuremath{-}5/2}$ for evolving stream function in a flow with nonvanishing shear. The rate of the relaxation is fast so that phase-space/fluid-element displacement in certain directions is uniformly small, implying that decaying Vlasov and 2D fluid turbulences are not ergodic.

M. B. Isichenko

Papers

Percolation, statistical topography, and transport in random media

Invariant measure and turbulent pinch in tokamaks.

Effective plasma heat conductivity in 'braided' magnetic field-I. Quasi-linear limit

Effective plasma heat conductivity in 'braided' magnetic field-II. Percolation limit

Nonlinear landau damping in collisionless plasma and inviscid fluid