Force-free Case Research Articles

Surprise ballistic and scaling inverted dynamics of a system coupled to a Hamiltonian thermostat

We study the induced generalized Brownian dynamics of a Hamiltonian thermostat, and specifically the diffusive properties. For a tagged particle regulated by a logarithmic-oscillator thermostat and moving in an external logarithmic potential, we reveal a distinct inversion of the scaling exponents for the mean-squared displacement ⟨Δx 2(t)⟩ ∼ t λ around the exponent λ = 2 − α covering regimes from ballistic diffusion to confinement motion (i.e. 0 ⩽ α ⩽ 2). This behavior contrasts with the expanding behavior associated with superdiffusive processes in a tilted periodic potential. The Hamiltonian thermostat is shown to maintain a fixed kinetic energy and, although this system is nonergodic in the force-free case, leads to a confined system that approaches thermal equilibrium slowly with the same temperature as the thermostat. In addition, we find its displacement has a significant dependence on λ via the amplitude of the external logarithmic potential. The continuous reduction of the scaling exponent is vital in the quantitative evaluation of all diffusive processes.

Gauge models of musical forces

Metaphors involving motion and forces are a source of inspiration for understanding tonal music and tonal harmonies since ancient times. Starting with the rise of quantum cognition, the modern interactional conception of forces as developed in gauge theory has recently entered the field of theoretical musicology. We develop a gauge model of tonal attraction based on SU(2) symmetry. This model comprises two earlier attempts, the phase model grounded on U(1) gauge symmetry, and the spatial deformation model derived from SO(2) gauge symmetry. In the neutral, force-free case both submodels agree and generate the same predictions as a simple qubit approach. However, there are several differences in the force-driven case. It is claimed that the deformation model gives a proper description of static tonal attraction. The full model combines the deformation model with the phase model through SU(2) gauge symmetry and unifies static and dynamic tonal attraction.

Analytic mean-fieldα2-dynamo with a force-free corona

Stellar dynamos are affected by boundary conditions imposed by stellar coronae. Under some approximations it is possible to find analytical solutions. Interior dynamo models often consider a current-free coronae without taking into account the constraints imposed by the presence of currents in the corona. We aim to analytically evaluate the effect of coronal currents and of an outer boundary condition on the efficiency of an $\alpha^2$ dynamo. We intend to estimate the change in geometry and dinamo excitation numbers with respect to the current-free case. We analytically solve the turbulent dynamo induction equation for a homogeneous, non-mirror symmetric turbulence, in a spherical domain surrounded by a linear force-free corona with magnetic field B satisfying curl B = \beta B. The main result is that the dynamo number is a decreasing function of \beta. Moreover, if the current is parallel to the field (\beta > 0) the dynamo number is smaller than in the force-free case. On the contrary (\beta < 0) the dynamo number is greater than the in force-free case. We conclude that the presence of currents in the corona needs to be taken into account because it affects the condition for excitation of a dynamo.

Effective Potential from the Generalized Time-Dependent Schrödinger Equation

We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.

General relativistic monopole magnetosphere of neutron stars: a pseudo-spectral discontinuous Galerkin approach

The close vicinity of neutron stars remains poorly constrained by observations. Although plenty of data are available for the peculiar class of pulsars we are still unable to deduce the underlying plasma distribution in their magnetosphere. In the present paper, we try to unravel the magnetospheric structure starting from basic physics principles and reasonable assumptions about the magnetosphere. Beginning with the monopole force-free case, we compute accurate general-relativistic solutions for the electromagnetic field around a slowly rotating magnetized neutron star. Moreover, here we address this problem by including the important effect of plasma screening. This is achieved by solving the time-dependent Maxwell equations in a curved space-time following the 3+1~formalism. We improved our previous numerical code based on pseudo-spectral methods in order to allow for possible discontinuities in the solution. Our algorithm based on a multi-domain decomposition of the simulation box belongs to the discontinuous Galerkin finite element methods. We performed several sets of simulations to look for the general-relativistic force-free monopole and split monopole solutions. Results show that our code is extremely powerful in handling extended domains of hundredth of light-cylinder radii~$\rlight$. The code has been validated against known exact analytical monopole solutions in flat space-time. We also present semi-analytical calculations for the general-relativistic vacuum monopole.

Ergodic–Nonergodic Transition Induced by Non-Gaussian Noise

The ergodic behavior of a system subjected to both internal thermal and external non thermal noises has been studied, by using the Khinchin theorem. For the internal noise we have considered Gaussian white noise but the external noise non-Gaussian in characteristic. We discuss the process which is either free, or takes place in a bounded potential landscape. In a force-free case, the system is ergodic for q ≤ 1, whereas for q > 1 the ergodicity is breakdown. An ergodic–nonergodic transition is found at a critical value q = 1. After adding an external bounded potential, the ergodicity for q > 1 is recovered, which is the characteristic behavior of nonergodic Brownian motion of class-I. Furthermore, we compare our results with the ergodicity definition and Lee’s condition to discuss applicability of the latter within non-Gaussian Brownian dynamics.

Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise

We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion.

Generalized Klein–Kramers equation: solution and application

A generalized Klein–Kramers equation based on the continuous time random walk model is investigated. The equation generalizes the ordinary and fractional Klein–Kramers equations. Analytic solutions for the probability density and first two moments (for the force-free case) are obtained, and their dynamic behaviors are investigated in detail. The model is used to describe the cell migration of two migrating transformed renal epithelial Madin–Darby canine kidney (MDCK-F) cell strains: wild-type (NHE+) and NHE-deficient (NHE−) cells. Our theoretical predictions are in good agreement with experimental work in the paper (Dieterich et al 2008 Proc. Nat. Acad. Sci. USA 105 459).

A Numerical Solution to Fractional Diffusion Equation for Force-Free Case

A collocation finite element method for solving fractional diffusion equation for force-free case is considered. In this paper, we develop an approximation method based on collocation finite elements by cubic B-spline functions to solve fractional diffusion equation for force-free case formulated with Riemann-Liouville operator. Some numerical examples of interest are provided to show the accuracy of the method. A comparison between exact analytical solution and a numerical one has been made.

Generalized Klein-Kramers equations

A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000)]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated.

Continuous-time random walk: exact solutions for the probability density function and first two moments

We consider the decoupled continuous-time random walk model with a finite characteristic waiting time and approximate jump length variance. We take the waiting time probability density function (PDF) given by a combination of the exponential and the Mittag-Leffler function. Using this waiting time PDF, we investigate the diffusion behavior for all times. We obtain exact solutions for the first two moments and the PDF for the force-free and linear force cases. Due to the finite characteristic waiting time and jump length variance, the model presents, for the force-free case, normal diffusive behavior in the long-time limit. Further, the model can describe anomalous behavior at intermediate times.

Continuous time random walk: Galilei invariance and relation for the nth moment

We consider a decoupled continuous time random walk model with a generic waiting time probability density function (PDF). For the force-free case we derive an integro-differential diffusion equation which is related to the Galilei invariance for the probability density. We also derive a general relation which connects the nth moment in the presence of any external force to the second moment without external force, i.e. it is valid for any waiting time PDF. This general relation includes the generalized second Einstein relation, which connects the first moment in the presence of any external force to the second moment without any external force. These expressions for the first two moments are verified by using several kinds of the waiting time PDF. Moreover, we present new anomalous diffusion behaviours for a waiting time PDF given by a product of power-law and exponential function.

Magnetohydrodynamics dynamical relaxation of coronal magnetic fields

Context. For the last thirty years, most of the studies on the relaxation of stressed magnetic fields in the solar environment have onlyconsidered the Lorentz force, neglecting plasma contributions, and therefore, limiting every equilibrium to that of a force-free field. Aims. Here we begin a study of the non-resistive evolution of finite beta plasmas and their relaxation to magnetohydrostatic states, where magnetic forces are balanced by plasma-pressure gradients, by using a simple 2D scenario involving a hydromagnetic disturbance to a uniform magnetic field. The final equilibrium state is predicted as a function of the initial disturbances, with aims to demonstrate what happens to the plasma during the relaxation process and to see what effects it has on the final equilibrium state. Methods. A set of numerical experiments are run using a full MHD code, with the relaxation driven by magnetoacoustic waves damped by viscous effects. The numerical results are compared with analytical calculations made within the linear regime, in which the whole process must remain adiabatic. Particular attention is paid to the thermodynamic behaviour of the plasma during the relaxation. Results. The analytical predictions for the final non force-free equilibrium depend only on the initial perturbations and the total pressure of the system. It is found that these predictions hold surprisingly well even for amplitudes of the perturbation far outside the linear regime. Conclusions. Including the effects of a finite plasma beta in relaxation experiments leads to significant differences from the force-free case.

ENERGY RELEASE AND TRANSFER IN SOLAR FLARES: SIMULATIONS OF THREE-DIMENSIONAL RECONNECTION

Using three-dimensional magnetohydrodynamic simulations we investigate energy release and transfer in a three-dimensional extension of the standard two-ribbon flare picture. In this scenario, reconnection is initiated in a thin current sheet (suggested to form below a departing coronal mass ejection) above a bipolar magnetic field. Two cases are contrasted: an initially force-free current sheet (low beta) and a finite-pressure current sheet (high beta), where beta represents the ratio between gas (plasma) and magnetic pressure. The energy conversion process from reconnection consists of incoming Poynting flux turned into up- and downgoing Poynting flux, enthalpy flux, and bulk kinetic energy flux. In the low-beta case, the outgoing Poynting flux is the dominant contribution, whereas the outgoing enthalpy flux dominates in the high-beta case. The bulk kinetic energy flux is only a minor contribution in the downward direction. The dominance of the downgoing Poynting flux in the low-beta case is consistent with an alternative to the thick target electron beam model for solar flare energy transport, suggested recently by Fletcher & Hudson, whereas the enthalpy flux may act as an alternative transport mechanism. For plausible characteristic parameters of the reconnecting field configuration, we obtain energy release timescales and energy output rates that compare favorably with those inferred from observations for the impulsive phase of flares. Significant enthalpy flux and heating are found even in the initially force-free case with very small background beta, resulting mostly from adiabatic compression rather than Ohmic dissipation. The energy conversion mechanism is most easily understood as a two-step process (although the two steps may occur essentially simultaneously): the first step is the acceleration of the plasma by Lorentz forces in layers akin to the slow shocks in the Petschek reconnection model, involving the conversion of magnetic energy to bulk kinetic energy. However, due to pressure gradient forces that oppose the Lorentz forces in approximate, or partial force balance, the accelerated plasma becomes slowed down and compressed, whereby the bulk kinetic energy is converted to heat, either locally deposited or transported away by enthalpy flux and deposited later. This mechanism is most relevant in the downflow region, which is more strongly governed by force balance; it is less important in the outflow above the reconnection site, where more energy remains in the form of fast bulk flow.

Fluctuation-driven directed transport in the presence of Lévy flights

The role of Lévy flights on fluctuation-driven transport in time independent periodic potentials with broken spatial symmetry is studied. Two complementary approaches are followed. The first one is based on a generalized Langevin model describing overdamped dynamics in a ratchet type external potential driven by Lévy white noise with stability index α in the range 1 < α < 2 . The second approach is based on the space fractional Fokker–Planck equation describing the corresponding probability density function (PDF) of particle displacements. It is observed that, even in the absence of an external tilting force or a bias in the noise, the Lévy flights drive the system out of the thermodynamic equilibrium and generate an up-hill current (i.e., a current in the direction of the steeper side of the asymmetric potential). For small values of the noise intensity there is an optimal value of α yielding the maximum current. The direction and magnitude of the current can be manipulated by changing the Lévy noise asymmetry and the potential asymmetry. For a sharply localized initial condition, the PDF of staying at the minimum of the potential exhibits scaling behavior in time with an exponent bigger than the − 1 / α exponent corresponding to the force free case.

Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving

We investigate a subdiffusive, fractional Fokker-Planck dynamics occurring in time-varying potential landscapes and thereby disclose the failure of the fractional Fokker-Planck equation (FFPE) in its commonly used form when generalized in an ad hoc manner to time-dependent forces. A modified FFPE (MFFPE) is rigorously derived, being valid for a family of dichotomously alternating force fields. This MFFPE is numerically validated for a rectangular time-dependent force with zero average bias. For this case, subdiffusion is shown to become enhanced as compared to the force free case. We question, however, the existence of any physically valid FFPE for arbitrary varying time-dependent fields that differ from this dichotomous varying family.

Anomalous Diffusion of Inertial, Weakly Damped Particles

The anomalous (i.e., non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of "random kicks" is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a new fractional equation of the Kramers-Fokker-Planck type is derived. The associated collision operator necessarily involves a fractional substantial derivative, representing important nonlocal couplings in time and space. For the force-free case, a closed solution is found and discussed.

Tearing and Kelvin-Helmholtz instabilities in the heliospheric plasma

We used 2.5D simulations to analyze the magnetohydrodynamic instabilities arising from an initial equilibrium configuration consisting of a plasma jet or wake in the presence of a magnetic field with strong transverse gradients, such as those arising in the solar wind. Our analysis extends previous results by considering both a force-free equilibrium and a pressure-balance condition for a jet in a plasma sheet, along with arbitrary angles between the magnetic field and velocity field. In the force-free case, the jet/wake does not contain a neutral sheet but the field rotates through the flow to invert its polarity. The presence of a magnetic field component aligned with the jet/wake destroys the symmetric nature of the fastest growing modes, leading to asymmetrical wake acceleration (or, equivalently, jet deceleration). In the case of a jet, the instability properties depend both on the magnetic field and flow gradients, as well as on the length of the jet. The results are applied to the post-termination shock jet recently found in 3D global heliospheric simulations, where our analysis confirms and explains the stability properties found in such simulations.

Effect of entrainment on stress and pulsar glitches in stratified neutron star crust

The buildup of the stress whose relaxation is presumed to account for pulsar frequency glitches can be attributed to various mechanisms, of which the most efficient involve differential rotation of the neutron superfluid in the inner layers of the (magnetically braked) solid crust of a rotating neutron star. In such a case it is usually supposed that the stress is attributable to pinning of superfluid vortices to crust nuclei, but it has been suggested that, even if the pinning effect is too weak, a comparably large stress might still arise just from the deficit of centrifugal buoyancy in the slowed down crust. The present work is a re-examination that investigates the way such processes may be affected by considerations that were overlooked in the previous work - notably uncertainties about the 'effective' masses that have to be attributed to the 'free' superfluid neutrons to allow for their entrainment by the ionic crust material. Though restricted to a Newtonian formulation, this analysis distinguishes more carefully than has been usual between true velocities, which are contravariantly vectorial, and so called 'superfluid velocities' that are proportional to momenta, which are essentially covectorial, a technicality that is important when more than one independent current is involved. The results include a Proudman-type theorem to the effect that the superfluid angular velocity must be constant on slightly deformed Taylor cylinders in the force free case, and it is shown how to construct a pair of integral constants of the motion that determine the solution for the pinned case assuming beta equilibrium. © 2006 RAS.

Riemannian geometry of twisted magnetic flux tubes in almost helical plasma flows

Riemannian geometry of curves applied recently by Ricca [Fluid Dyn. Res 36, 319 (2005)] in the case of inflectional disequilibrium of twisted magnetic flux tubes is used here to compute the magnetic helicity force-free field case. Here the application of Lorentz force-free to the magnetic flux tube in tokamaks allows one to obtain an equation that generalizes the cylindrical tokamak equation by a term that contains the curvature of the magnetic flux tube. Another example of the use of the magnetic flux tube is done by taking the electron magnetohydrodynamics (MHD) fluid model (EMHD) of plasma physics that allows one to compute the velocity of the fluid in helical and almost helical flows in terms of the Frenet torsion of thin magnetic flux tubes. The cases of straight and curved twisted tubes are examined. Second-order effects on the Frenet torsion arise on the poloidal component of the magnetic field, while curvature effects appear in the toroidal component. The magnetic fields are computed in terms of the penetration depth used in superconductors. The ratio between poloidal and toroidal components of the magnetic field depends on the torsion and curvature of the magnetic flux tube. It is shown that the rotation of the almost helical plasma flow contributes to the twist of the magnetic flux tube through the total Frenet torsion along the tube.