Force-free collisionless current sheet models with non-uniform temperature and density profiles

We present a class of one-dimensional, strictly neutral, Vlasov-Maxwell equilibrium distribution functions for force-free current sheets, with magnetic fields defined in terms of Jacobian elliptic functions, extending the results of Abraham-Shrauner [Phys. Plasmas 20, 102117 (2013)] to allow for non-uniform density and temperature profiles. To achieve this, we use an approach previously applied to the force-free Harris sheet by Kolotkov et al. [Phys. Plasmas 22, 112902 (2015)]. In one limit of the parameters, we recover the model of Kolotkov et al. [Phys. Plasmas 22, 112902 (2015)], while another limit gives a linear force-free field. We discuss conditions on the parameters such that the distribution functions are always positive and give expressions for the pressure, density, temperature, and bulk-flow velocities of the equilibrium, discussing the differences from previous models. We also present some illustrative plots of the distribution function in velocity space

Collisionless distribution functions for force-free current sheets: using a pressure transformation to lower the plasma beta

So far, only one distribution function giving rise to a collisionless nonlinear force-free current sheet equilibrium allowing for a plasma beta less than one is known (Allanson et al., Phys. Plasmas, vol. 22 (10), 2015, 102116; Allanson et al., J. Plasma Phys., vol. 82 (3), 2016a, 905820306). This distribution function can only be expressed as an infinite series of Hermite functions with very slow convergence and this makes its practical use cumbersome. It is the purpose of this paper to present a general method that allows us to find distribution functions consisting of a finite number of terms (therefore easier to use in practice), but which still allow for current sheet equilibria that can, in principle, have an arbitrarily low plasma beta. The method involves using known solutions and transforming them into new solutions using transformations based on taking integer powers (N) of one component of the pressure tensor. The plasma beta of the current sheet corresponding to the transformed distribution functions can then, in principle, have values as low as 1/N. We present the general form of the distribution functions for arbitrary N and then, as a specific example, discuss the case for N=2 in detail

From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials

We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a Vlasov–Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans’ theorem, the equilibrium distribution functions are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite polynomials. A sufficient condition on the pressure tensor is found which guarantees the convergence and the boundedness of the candidate solution, when satisfied. This condition is obtained by elementary means, and it is clear how to put it into practice. We also argue that for a given pressure tensor for which our method applies, there always exists a positive distribution function solution for a sufficiently magnetised plasma. Illustrative examples of the use of this method with both force-free and non-force-free macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the force-free Harris sheet (Allanson et al., Phys. Plasmas, vol. 22 (10), 2015, 102116). In the effort to model equilibria with lower values of the plasma beta, solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for beta=0.05

Collisionless current sheet equilibria

Current sheets are important for the structure and dynamics of many plasma systems. In space and astrophysical plasmas they play a crucial role in activity processes, for example by facilitating the release of magnetic energy via processes such as magnetic reconnection. In this contribution we will focus on collisionless plasma systems. A sensible first step in any investigation of physical processes involving current sheets is to find appropriate equilibrium solutions. The theory of collisionless plasma equilibria is well established, but over the past few years there has been a renewed interest in finding equilibrium distribution functions for collisionless current sheets with particular properties, for example for cases where the current density is parallel to the magnetic field (force-free current sheets). This interest is due to a combination of scientific curiosity and potential applications to space and astrophysical plasmas. In this paper we will give an overview of some of the recent developments, discuss their potential applications and address a number of open questions

An exact collisionless equilibrium for the force-free Harris sheet with low plasma beta

We present a first discussion and analysis of the physical properties of a new exact collisionless equilibrium for a one-dimensional nonlinear force-free magnetic field, namely, the force-free Harris sheet. The solution allows any value of the plasma beta, and crucially below unity, which previous nonlinear force-free collisionless equilibria could not. The distribution function involves infinite series of Hermite polynomials in the canonical momenta, of which the important mathematical properties of convergence and non-negativity have recently been proven. Plots of the distribution function are presented for the plasma beta modestly below unity, and we compare the shape of the distribution function in two of the velocity directions to a Maxwellian distribution

Fractal dimension of cortical functional connectivity networks & severity of disorders of consciousness.

Recent evidence suggests that the quantity and quality of conscious experience may be a function of the complexity of activity in the brain and that consciousness emerges in a critical zone between low and high-entropy states. We propose fractal shapes as a measure of proximity to this critical point, as fractal dimension encodes information about complexity beyond simple entropy or randomness, and fractal structures are known to emerge in systems nearing a critical point. To validate this, we tested several measures of fractal dimension on the brain activity from healthy volunteers and patients with disorders of consciousness of varying severity. We used a Compact Box Burning algorithm to compute the fractal dimension of cortical functional connectivity networks as well as computing the fractal dimension of the associated adjacency matrices using a 2D box-counting algorithm. To test whether brain activity is fractal in time as well as space, we used the Higuchi temporal fractal dimension on BOLD time-series. We found significant decreases in the fractal dimension between healthy volunteers (n = 15), patients in a minimally conscious state (n = 10), and patients in a vegetative state (n = 8), regardless of the mechanism of injury. We also found significant decreases in adjacency matrix fractal dimension and Higuchi temporal fractal dimension, which correlated with decreasing level of consciousness. These results suggest that cortical functional connectivity networks display fractal character and that this is associated with level of consciousness in a clinically relevant population, with higher fractal dimensions (i.e. more complex) networks being associated with higher levels of consciousness. This supports the hypothesis that level of consciousness and system complexity are positively associated, and is consistent with previous EEG, MEG, and fMRI studies

Neutral and non-neutral collisionless plasma equilibria for twisted flux tubes: the Gold-Hoyle model in a background field

We calculate exact one-dimensional collisionless plasma equilibria for a continuum of flux tube models, for which the total magnetic field is made up of the “force-free” Gold-Hoyle magnetic flux tube embedded in a uniform and anti-parallel background magnetic field. For a sufficiently weak background magnetic field, the axial component of the total magnetic field reverses at some finite radius. The presence of the background magnetic field means that the total system is not exactly force-free, but by reducing its magnitude, the departure from force-free can be made as small as desired. The distribution function for each species is a function of the three constants of motion; namely, the Hamiltonian and the canonical momenta in the axial and azimuthal directions. Poisson's equation and Ampère's law are solved exactly, and the solution allows either electrically neutral or non-neutral configurations, depending on the values of the bulk ion and electron flows. These equilibria have possible applications in various solar, space, and astrophysical contexts, as well as in the laboratory

Exact Vlasov-Maxwell equilibria for asymmetric current sheets

The NASA Magnetospheric Multiscale mission has made in situ diffusion region and kinetic-scale resolution measurements of asymmetric magnetic reconnection for the first time, in the Earth's magnetopause. The principal theoretical tool currently used to model collisionless asymmetric reconnection is particle-in-cell simulations. Many particle-in-cell simulations of asymmetric collisionless reconnection start from an asymmetric Harris-type magnetic field but with distribution functions that are not exact equilibrium solutions of the Vlasov equation. We present new and exact equilibrium solutions of the Vlasov-Maxwell system that are self-consistent with one-dimensional asymmetric current sheets, with an asymmetric Harris-type magnetic field profile, plus a constant nonzero guide field. The distribution functions can be represented as a combination of four shifted Maxwellian distribution functions. This equilibrium describes a magnetic field configuration with more freedom than the previously known exact solution and has different bulk flow properties

Patient preferences for topical treatment of actinic keratoses:a discrete-choice experiment

Funding: This study was funded by the National Institute for Health Research (NIHR) Research for Patient Benefit programme (PB-PG-0110-21244), Department of Health, UK. The funder was not involved in the study design. Acknowledgments: The authors gratefully acknowledge support from the Cancer Research UK Clinical Trials Unit, the UK Dermatology Clinical Trials Network, the NIHR Clinical Studies Group, and support for investigators from the British Skin Foundation and Cancer Research UK. We would also like to thank Martin Jones, Daniel Rigby and Ariel Bergmann for constructive comments on the design of the DCE.Peer reviewedPostprin