Time-integral iteration method for two-dimensional anomalous transport.

A methodology is developed to describe time-dependent phenomena associated with nonlocal transport in complex, two-dimensional geometries. It is an extension of the ''iterative method" introduced previously to solve steady-state transport problems [Maggs and Morales, Phys. Rev. E 99, 013307 (2019)10.1103/PhysRevE.99.013307], and it is based on the ''jumping particle" concepts associated with the continuous-time random walk (CTRW) model. The method presented explicitly evaluates the time integral contained in the CTRW master equation. A modified version of the Mittag-Leffler function is used for the waiting-time probability distributions to incorporate memory effects. Calculations of the propagation of ''anomalous transport waves" in various systems, with and without memory, illustrate the technique.

Two-dimensional chaotic thermostat and behavior of a thermalized charge in a weak magnetic field.

A two-dimensional version of a chaotic thermostat is investigated. Its structure follows the concept previously introduced by the author [G. J. Morales, Phys. Rev. E 97, 032203 (2018)2470-004510.1103/PhysRevE.97.032203] to generate a one-dimensional chaotic thermostat, namely, the usual friction force of a deterministic thermostat is supplemented with a self-consistent fluctuating force that depends on the drag coefficient associated with coupling to the heat bath. Azimuthal symmetry requires the thermostat to have two internal degrees of freedom, thus the Martyna-Klein-Tuckerman [G. J. Martyna et al., J. Chem. Phys. 97, 2635 (1992)JCPSA60021-960610.1063/1.463940] model is chosen for the heat bath. The unmagnetized system exhibits two-dimensional diffusive behavior, achieves symmetric Maxwellian velocity distributions in the absence of an external potential, and satisfies the Einstein relation when an external force is applied. The velocity fluctuations display the characteristic exponential frequency spectrum associated with chaotic systems. The model is used to explore the diffusive motion of a thermalized charge in a weak magnetic field and the associated Hall and Pedersen mobilities. Over a range of magnetic field strengths the charge exhibits absolute negative mobility.

Nonlocal transport in bounded two-dimensional systems: An iterative method.

The concept of transport mediated through the dynamics of "jumping" particles is used to develop an iterative method for obtaining steady-state solutions to the nonlocal transport equation in two dimensions. The technique is self-adjoint and capable of correctly treating spatially nonuniform, asymmetric systems. An appropriate reduced version of the iteration method is used to compare with results obtained with a self-adjoint one-dimensional transport matrix approach [Maggs and Morales, Phys. Rev. E 94, 053302 (2016)10.1103/PhysRevE.94.053302]. The transport "jump" probability distribution functions are based on Lévy α-stable distributions. The technique can handle the entire Lévy α-parameter range from one (Lorentz distributions) to two (Gaussian distributions). Cases with α=2 (standard diffusion) are used to establish the validity of the iterative method. The capabilities of the iterative method are demonstrated by presenting examples from systems with various source configurations, boundary shapes, boundary conditions, and spatial variations in parameters.

Properties of a sinusoidally driven thermostat.

This analytical and numerical study explores, from a deterministic perspective, the response of a particle in contact with a heat bath to an external force that varies sinusoidally in time. The heat bath is represented as a deterministic Nosé-Hoover thermostat. Such an idealized model encompasses features found in a variety of physical problems in which the coherent influence of an external agent competes against the thermalization tendencies of the surrounding medium, e.g., a charged particle in a thermal plasma that is acted on by a powerful electromagnetic wave. It is found that, independently of the coupling strength to the thermostat, average power can only be extracted from the sinusoidal force when the oscillatory velocity exceeds the thermal velocity. It follows from this property that a candidate heat bath whose temperature is raised solely through the application of a sinusoidal force (e.g., radio-frequency heating), reaches, in a finite time, a unique final temperature determined by the force parameters. The transition boundary separating the two domains of power transfer is shown to be dominated by chaotic behavior. The combined response to a sinusoidal force and a dc force are also considered and the relevant regions of power transfer are delineated.

Investigation of a chaotic thermostat.

A numerical study is presented of a free particle interacting with a deterministic thermostat in which the usual friction force is supplemented with a fluctuating force that depends on the self-consistent damping coefficient associated with coupling to the heat bath. It is found that this addition results in a chaotic environment in which a particle self-heats from rest and moves in positive and negative directions, exhibiting a characteristic diffusive behavior. The frequency power spectrum of the dynamical quantities displays the exponential frequency dependence ubiquitous to chaotic dynamics. The velocity distribution function approximates a Maxwellian distribution, but it does show departures from perfect thermal equilibrium, while the distribution function for the damping coefficient shows a closer fit. The behavior for the classic Nosé-Hoover (NH) thermostat is compared to that of the enlarged Martyna-Klein-Tuckerman (MKT) model. Over a narrow amplitude range, the application of a constant external force results quantitatively in the Einstein relation for the NH thermostat, and for the MKT model it differs by a factor of 2.

Self-adjoint integral operator for bounded nonlocal transport.

An integral operator is developed to describe nonlocal transport in a one-dimensional system bounded on both ends by material walls. The "jump" distributions associated with nonlocal transport are taken to be Lévy α-stable distributions, which become naturally truncated by the bounding walls. The truncation process results in the operator containing a self-consistent, convective inward transport term (pinch). The properties of the integral operator as functions of the Lévy distribution parameter set [α,γ] and the wall conductivity are presented. The integral operator continuously recovers the features of local transport when α=2. The self-adjoint formulation allows for an accurate description of spatial variation in the Lévy parameters in the nonlocal system. Spatial variation in the Lévy parameters is shown to result in internally generated flows. Examples of cold-pulse propagation in nonlocal systems illustrate the capabilities of the methodology.

Laboratory study of avalanches in magnetized plasmas.

It is demonstrated that a novel heating configuration applied to a large and cold magnetized plasma allows the study of avalanche phenomena under controlled conditions. Intermittent collapses of the plasma pressure profile, associated with unstable drift-Alfvén waves, exhibit a two-slope power-law spectrum with exponents near -1 at lower frequencies and in the range of -2 to -4 at higher frequencies. A detailed mapping of the spatiotemporal evolution of a single avalanche event is presented.

Isotropic model of fractional transport in two-dimensional bounded domains.

A two-dimensional fractional Laplacian operator is derived and used to model nonlocal, nondiffusive transport. This integro-differential operator appears in the long-wavelength, fluid description of quantities undergoing non-Brownian random walks without characteristic length scale. To study bounded domains, a mask function is introduced that modifies the kernel in the fractional Laplacian and removes singularities at the boundary. Green's function solutions to the fractional diffusion equation are presented for the unbounded domain and compared to the one-dimensional Cartesian approximations. A time-implicit numerical integration scheme is presented to study fractional diffusion in a circular disk with azimuthal symmetry. Numerical studies of steady-state reveal temperature profiles in which the heat flux and temperature gradient are in the same direction, i.e., uphill transport. The response to off-axis heating, scaling of confinement time with system size, and propagation of cold pulses are investigated.

Origin of Lorentzian pulses in deterministic chaos.

Pulses having a temporal Lorentzian shape arise naturally from topological changes in flow trajectories or phase-space orbits associated with deterministic chaos. The pulses can appear as random intermittent events in the time series of observable quantities, and they are the cause of exponential frequency power spectra previously observed in magnetically confined plasmas and various nonlinear systems.

Generality of deterministic chaos, exponential spectra, and Lorentzian pulses in magnetically confined plasmas.

The dynamics of transport at the edge of magnetized plasmas is deterministic chaos. The connection is made by a previous survey [M. A. Pedrosa et al., Phys. Rev. Lett. 82, 3621 (1999)] of measurements of fluctuations that is shown to exhibit power spectra with exponential frequency dependence over a broad range, which is the signature of deterministic chaos. The exponential character arises from Lorentzian pulses. The results suggest that the generalization to complex times used in studies of deterministic chaos is a representation of Lorentzian pulses emerging from the chaotic dynamics.

Spontaneous thermal waves in a magnetized plasma.

Coherent temperature oscillations corresponding to thermal (diffusion) waves are observed to be spontaneously excited in a narrow temperature filament embedded in a large, but colder, magnetized plasma. The parallel and transverse propagation properties of the waves satisfy the predictions of the classical transport theory based on Coulomb collisions. The frequency of the oscillations meets the conditions for a quarter-wave thermal resonator. This is the plasma version of thermal resonators used in the study of other states of matter.

Exponential frequency spectrum in magnetized plasmas.

Measurements of a magnetized plasma with a controlled electron temperature gradient show the development of a broadband spectrum of density and temperature fluctuations having an exponential frequency dependence at frequencies below the ion cyclotron frequency. The origin of the exponential frequency behavior is traced to temporal pulses of Lorentzian shape. Similar exponential frequency spectra are also found in limiter-edge plasma turbulence associated with blob transport. This finding suggests a universal feature of magnetized plasma turbulence leading to nondiffusive, cross-field transport, namely, the presence of Lorentzian shaped pulses.

Laboratory realization of an Alfvén wave maser.

It is demonstrated that a frequency selective Alfvén-wave resonator can be realized by applying a nonuniform magnetic field to a plasma region bounded between a cathode and a semitransparent mesh anode. When a current threshold is exceeded, selective amplification results in a highly coherent, large amplitude wave that propagates into an adjacent plasma column.

Dynamics of a supersonic plume moving along a magnetized plasma.

A particle-in-cell code is used to investigate the evolution of a density plume moving through a background plasma with supersonic speed directed along the confinement magnetic field. For scale lengths representative of laboratory and auroral phenomena, the major nonlinear effects identified by the present simulations are the formation of a bipolar current system from the ballistic electrons, the appearance of transient potential layers, and the carving of deep density cavities. A 3D magnetic topology is generated by the self-consistent ballistic and diamagnetic currents that accompany highly localized potential layers.

Krook collisional models of the kinetic susceptibility of plasmas.

An assessment is made of Krook collisional models used to describe the kinetic behavior of collective oscillations, i.e., when Landau damping and collisions must be considered, as is often the case for low-frequency waves. The study focuses on an early energy-conserving model [B. D. Fried, A. N. Kaufman, and D. L. Sachs, Phys. Fluids 9, 292 (1966)] that is shown to be identical to a more modern version used in drift-wave stability studies [G. Rewoldt, W. M. Tang, and R. J. Hastie, Phys. Fluids 29, 2893 (1986)]. The inadequacy of the simpler, and often used, nonconserving model is illustrated. Comparisons are established with recent collisional studies of ion acoustic waves [V. Yu. Bychenkov, J. Myatt, W. Rozmus, and V. T. Tikhonchuk, Phys. Plasmas 1, 2419 (1994)] and electron plasma waves [C. S. Ng, A. Bhattacharjee, and F. Skiff, Phys. Rev. Lett. 83, 1974 (1999)]. A connection is also established with contemporary studies of condensed matter and quantum liquids [K. Morawetz and U. Fuhrmann, Phys. Rev. E 61, 2272 (2000); 62, 4382 (2000)]. A useful empirical fit is found that corrects the Braginskii susceptibility to incorporate the kinetic behavior associated with the Krook kinetic susceptibility.