Non-relativistic Case Research Articles

Study of small perturbations of a stationary state in a model of upper hybrid plasma oscillations

It is shown that a constant external magnetic field, generally speaking, is not able to prevent breaking (loss of smoothness) of relativistic plasma oscillations, even if they are arbitrarily small perturbations of the zero steady-state. This result sharply differs from the non-relativistic case, for which it is possible to suppress the breaking of oscillations at any initial deviations by increasing the intensity of the magnetic field \cite {RCharx}. Nevertheless, even in the relativistic case, there are subclasses of solutions corresponding to solutions that are globally smooth in time.

Spectra of Magnetic Turbulence in a Relativistic Plasma

We present a phenomenological and numerical study of strong Alfvénic turbulence in a magnetically dominated collisionless relativistic plasma with a strong background magnetic field. In contrast with the nonrelativistic case, the energy in such turbulence is contained in magnetic and electric fluctuations. We argue that such turbulence is analogous to turbulence in a strongly magnetized nonrelativistic plasma in the regime of broken quasi-neutrality. Our 2D particle-in-cell numerical simulations of turbulence in a relativistic pair plasma find that the spectrum of the total energy has the scaling k −3/2, while the difference between the magnetic and electric energies, the so-called residual energy, has the scaling k −2.4. The electric and magnetic fluctuations at scale ℓ exhibit dynamic alignment with the alignment angle scaling close to . At scales smaller than the (relativistic) plasma inertial scale, the energy spectrum of relativistic inertial Alfvén turbulence steepens to k −3.5.

Toverline{T} deformations and the width of fundamental particles

We provide a simple geometric meaning for deformations of so-called Toverline{T} type in relativistic and non-relativistic systems. Deformations by the cross products of energy and momentum currents in integrable quantum field theories are known to modify the thermodynamic Bethe ansatz equations by a “CDD factor”. In turn, CDD factors may be interpreted as additional, fixed shifts incurred in scattering processes: a finite width added to the fundamental particles (or, if negative, to the free space between them). We suggest that this physical effect is a universal way of understanding Toverline{T} deformations, both in classical and quantum systems. We first show this in non-relativistic systems, with particle conservation and translation invariance, using the deformation formed out of the densities and currents of particles and momentum. This holds at the level of the equations of motion, and for any interaction potential, integrable or not. We then argue, and show by similar techniques in free relativistic particle systems, that Toverline{T} deformations of relativistic systems produce the equivalent phenomenon, accounting for length contractions. We also show that, in both the relativistic and non-relativistic cases, the width of particles is equivalent to a state-dependent change of metric, where the distance function discounts the particles’ widths, or counts the additional free space. This generalises and explains the known field-dependent coordinate change describing Toverline{T} deformations. The results connect such deformations with generalised hydrodynamics, where the relations between scattering shifts, widths of particles and state-dependent changes of metric have been established.

Relativistic calculation of spectral line broadening by electron collisions in plasmas: Case of hydrogenic ions

In the present work, we compute the amplitude of the real part of the impact collision operator for the electron broadening of hydrogenic ion lines in plasmas, taking into account relativistic effects in the dynamics of the perturbing electron. Specifically, two relativistic effects are included: the modification of the cross section Mott instead of the Rutherford cross section and the modification of the velocity distribution (Maxwell–Juttner). The results are compared to the non-relativistic case.

Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations

Abstract Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issue, and developing simpler but accurate approximate models is still worthwhile. Here, we propose a new family of paraxial asymptotic models that approximates the Vlasov–Maxwell system of equations. We introduce parameters in our models that allow us to handle relativistic cases, much slower beams or even non-relativistic cases. These models are derived by introducing a small parameter and provide static or quasi-static approximate equations that are 𝑛-th order accurate; 𝑛 may be chosen as required. Practically, one can select a model by determining the regime one is interested in and choosing the degree of accuracy needed.

Dark photon dark matter from a rolling inflaton

We study in detail a recently proposed mechanism for producing non-thermal dark photon dark matter at the end of inflation in the mass range μ eV≲ m ≲ 10 TeV. A tachyonic instability induced by a rolling inflaton leads to the coherent production of dark (abelian) gauge bosons with a peak in the power spectrum corresponding to the Hubble scale at the end of inflation. As the Universe expands after inflation the dark photons redshift and, at some point in their cosmic evolution, they obtain a mass. We focus in particular on the case where the dark photons are relativistic at the time their mass is generated and examine the associated cosmic evolution to compute the relic abundance today. We also examine the late time power spectrum demonstrating explicitly that it preserves the peak generated at the end of inflation. We show that the peak corresponds to small physical scales today, ℓtoday∼ cm-100 km, with large density fluctuations at ℓtoday leading to a clumpy nature for the dark photon dark matter. We also discuss potential phenomenology and future directions, briefly commenting on the non-relativistic case.

Relativistic regularized kappa distributions

The special relativistic generalization of isotropic regularized kappa distributions is derived and compared to that of the original Olbertian (or standard) kappa distributions. It is demonstrated that for the latter, the kappa parameter is even more strongly limited than in the non-relativistic case, while for the former, all positive kappa values remain possible. After a derivation of non-relativistic limits, the pressures of the distributions are studied as a specific case of moments of both the relativistic standard and regularized kappa distributions.

Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach

<p style='text-indent:20px;'>We consider linear stability of steady states of 1<inline-formula><tex-math id="M1">\begin{document}$ \frac{1}{2} $\end{document}</tex-math></inline-formula> and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.</p>

Relativistic tunneling through two "transparent" successive barriers

In the case of tunneling of relativistic particles, differently from the nonrelativistic case, a limit of "transparent" barrier can also lead to an apparent "superluminal" behavior when considering the phase time. In this limit, the restricting condition of "opaque" barrier of the nonrelativistic case is avoided, nevertheless, the very thin width of a single barrier to obtain this ?transparent? limit can result in a problem itself, for probing the effect. A combination of two successive transparent barriers can show an apparent "superluminal" behavior along a macroscopic arbitrary distance "L". Two solutions for energy E above and below the potential square barrier V are found, for both solutions there the apparent superluminal behavior is possible above a threshold of free travelling group velocity (energy) and dependent on the ratio barriers length free path as function of the ratio group velocity - speed of light.

QuantumFDTD - A computational framework for the relativistic Schrödinger equation

We extend the publicly available quantumfdtd code. It was originally intended for solving the time-independent three-dimensional Schrödinger equation via the finite-difference time-domain (FDTD) method and for extracting the ground, first, and second excited states. We (a) include the case of the relativistic Schrödinger equation and (b) add two optimized FFT-based kinetic energy terms for the non-relativistic case. All the three new kinetic terms are computed using Fast Fourier Transform (FFT).We release the resulting code as version 3 of quantumfdtd. Finally, the code now supports arbitrary external filebased potentials and the option to project out distinct parity eigenstates from the solutions. Our goal is quark models used for phenomenological descriptions of QCD bound states, described by the three-dimensional Schrödinger equation. However, we target any field where solving either the non-relativistic or the relativistic three-dimensional Schrödinger equation is required.

The relativistic Schrödinger equation through FFTW 3: An extension of quantumfdtd

In order to solve the time-independent three-dimensional Schrödinger equation, one can transform the time-dependent Schrödinger equation to imaginary time and use a parallelized iterative method to obtain the full three-dimensional eigen-states and eigen-values on very large lattices. In the case of the non-relativistic Schrödinger equation, there exists a publicly available code called quantumfdtd which implements this algorithm. In this paper, we (a) extend the quantumfdtd code to include the case of the relativistic Schrödinger equation and (b) add two optimized Fast Fourier Transform (FFT) based kinetic energy terms for non-relativistic cases. The new kinetic energy terms (two non-relativistic and one relativistic) are computed using the parallelized FFT-algorithm provided by the FFTW 3 library. The resulting quantumfdtd v3 code, which is publicly released with this paper, is backwards compatible with version 2, supporting explicit finite-differences schemes in addition to the new FFT-based schemes. Finally, we (c) extend the original code so that it supports arbitrary external file-based potentials and the option to project out distinct parity eigen-states from the solutions. Herein, we provide details of the quantumfdtd v3 implementation, comparisons and tests of the three new kinetic energy terms, and code documentation. Program summaryProgram Title:quantumfdtd v3CPC Library link to program files:https://doi.org/10.17632/9p8zyvmdy2.1Developer's repository link:https://github.com/quantumfdtd/quantumfdtd_v3Licensing provisions: GPLv3Programming language:C++, Python, ShellJournal reference of previous version: [1–4]Does the new version supersede the previous version?: YesReasons for the new version: We extended the previous version of quantumfdtd, which solves the time-independent non-relativistic three-dimensional Schrödinger equation, to the case of the relativistic kinetic energy. Additionally, we have added the capability of using arbitrary three-dimensional potentials from external files. This functionality is required, for instance, in order to interface with lattice QCD code.Summary of revisions: (a) Include the case of the relativistic Schrödinger equation, (b) add two optimized Fast Fourier Transform (FFT) based kinetic energy terms for non-relativistic cases, and (c) support arbitrary external file-based potentials and the option to project out distinct parity eigen-states from the solutions. The new kinetic energy terms (two non-relativistic and one relativistic) are computed using the parallelized FFT-algorithm provided by the FFTW 3 library.Nature of problem: We compute the ground, first, and second excited states of the time-independent three-dimensional Schrödinger equation. As input, we accept a number of hard-coded, analytical, three-dimensional potentials, as well as arbitrary potentials via external files. The outputs of the program are the corresponding eigen-values and eigen-vectors (energies and wave-functions). We can also project out distinct parity eigen-states of the solutions.Solution method: The time-dependent Schrödinger equation is transformed to imaginary time. We use a parallelized iterative method to obtain the full three-dimensional time-independent eigen-states and eigen-values on large lattices. The new kinetic energy terms (two non-relativistic and one relativistic) are computed using the parallelized FFT-algorithm provided by the FFTW 3 library. The old non-relativistic kinetic term is computed using the finite-difference time-domain (FDTD) algorithm. Finally, we provide several Python and Shell scripts for the analysis steps. This includes a Python program for projecting out distinct parity eigen-states from the solutions.Additional comments including restrictions and unusual features: We require the number of MPI processes being a divisor of the number N of spatial grid points. The following external programs/libraries are used and thus required:•MPI library•FFTW_MPI, version 3•GNU Scientific Library (GSL), linked to CBLAS•For some of the post-processing scripts, Python 3 is required References[1]A. Dumitru, Y. Guo, M. Strickland, Phys. Rev. D 79 (2009) 114003, https://doi.org/10.1103/PhysRevD.79.114003[2]A. Dumitru, Y. Guo, A. Mocsy, M. Strickland, Phys. Rev. D 79 (2009) 054019, https://doi.org/10.1103/PhysRevD.79.054019[3]M. Strickland, and D. Yager-Elorriaga, J. Compt. Phys. 229 (2010) 6015, https://doi.org/10.1016/j.jcp.2010.04.032[4]M. Margotta, K. McCarty, C. McGahan, M. Strickland, D. Yager-Elorriaga, Phys. Rev. D 83 (2011) 105019; Erratum: Phys. Rev. D 84 (2011) 069902, https://doi.org/10.1103/PhysRevD.83.105019

Holographic Fisher information metric in Schrödinger spacetime

In this paper, we study the Fisher information metric on the space of the coupling constants on both sides of the duality between non-relativistic dipole field theories and string theory in Schrödinger spacetime. We consider the following setup. In the gauge theory side one can deform a given conformal field theory by a proper scalar operator and compute the quantum information metric via the two-point correlation function between two such operators. On the string side the deformation corresponds to a scalar field probing the background. In the large N limit of the theory the probing can be done without backreaction on the original spacetime, thus one can construct a perturbative scheme for the calculation of the dual holographic Fisher information metric as shown by [1]. Considering the asymptotic behaviour of the holographic Fisher information metric close to the boundary of the Schrödinger spacetime we show that its divergence structure exactly matches its dual quantum counterpart up to the leading order, thus extending the holographic setup up to the non-relativistic case. One should note that the existence of other terms is not seen from the boundary theory to this level of approximation. Their behaviour near the boundary, however, is pointing what kind of information from the boundary theory is missing to be able to reconstruct the bulk. Obviously, more work is needed to refine and elucidate their meaning and interrelations in holographic setup.

Contact interactions II; Gross-Pitaewskii equation, Bose-Einstein condensate, Fermi sea

In Dell’Antonio (Eur Phys J Plus 13:1–20, 2021), we explored the possibility to analyse contact interaction in Quantum Mechanics using a variational tool, Gamma Convergence. Here, we extend the analysis in Dell’Antonio (Eur Phys J Plus 13:1–20, 2021) of joint weak contact of three particles to the non-relativistic case in which the free one particle Hamiltonian is H_0 = - frac{Delta }{2M} . We derive the Gross–Pitaevskii equation for a system of three particles in joint weak contact. We then define and study strong contact and show that the Gross–Pitaevskii equation is also the variational equation for the energy of the Bose–Einstein condensate (strong contact in a four-particle system). We add some comments on Bogoliubov’s theory. In the second part, we use the non-relativistic Pauli equation and weak contact to derive the spectrum of the conduction electrons in an infinite crystal. We prove that the spectrum is pure point with multiplicity two and eigenvalues that scale as frac{1}{log {n}}.

Fluid Dynamics in Curvilinear Coordinates without Fictitious Forces

The use of curvilinear coordinates is sometimes indicated by the inherent geometry of a fluid dynamics problem, but this introduces fictitious forces into the momentum equations that spoil the strict conservative form. If one is willing to work in three dimensions, these fictitious forces can be eliminated by solving for rectangular (Cartesian) momentum components on a curvilinear mesh. A thoroughly geometric approach to fluid dynamics on spacetime demonstrates this transparently, while also giving insight into a greater unity of the relativistic and nonrelativistic cases than is usually appreciated.

Bose–Einstein condensation and non-extensive statistics for finite systems

We study the Bose–Einstein condensation in non-extensive statistics for a free gas of bosons, and extend the results to the non-relativistic case as well. We present results for the dependence of the critical temperature and the condensate fraction on the entropic index, q, and show that the condensate can exist only for a limited range of q in both relativistic and non-relativistic systems. We provide numerical results for other thermodynamics quantities like the internal energy, specific heat and number fluctuations. We discuss the implications for high energy physics and hadron physics. The results for the non-relativistic case can be of interest in cold-atom systems.

Interaction of ion acoustic solitons for Zakharov Kuznetsov equation in relativistically degenerate quantum magnetoplasmas

Nonlinear electrostatic waves on the ion time scale have been studied in a quantum magnetoplasma in the presence of relativistically degenerate electrons. In this regard, Zakharov Kuznetsov (ZK) equation has been derived using the reductive perturbation technique. The single and two soliton solutions of the ZK equation have been obtained by employing Hirota formalism. The nonrelativistic and ultrarelativistic limits of the relativistically degenerate electrons have been studied and discussed in detail for important plasma parameters both for the single and two soliton solutions of the ZK equation. We have applied our study for the plasma parameters that are found in the white dwarfs. It has been noticed that the interaction of the ZK solitons depends on the propagation vectors in the predominant direction of propagation. Importantly, it has been found that the spatial and temporal scales over which the interaction of solitons occur are different for the nonrelativistic and ultrarelativistic cases.

Effect of relativistically degenerate electrons on linear and nonlinear structures in ion temperature gradient driven pure drift mode

Low frequency, electrostatic, pure drift mode is investigated in dense magnetized electron-ion plasma in the presence of relativistically degenerate electrons using quantum magnetohydrodynamic model (QMHD). Ions are considered as warm classical particles. Inhomogeneities in background ion density, ion temperature and external magnetic field are taken into account. In the linear regime, dispersion relation is obtained and plotted for both non-relativistic and ultra-relativistic regimes. Parametric study shows that for both non-relativistic and ultra-relativistic cases, growth rate of the wave increases by increasing, , which is the ratio of background ion temperature to density. In the nonlinear case, solitary solutions are obtained by using the functional variable method. Graphical illustrations are used to show the effect of relativistically degenerate electrons and other parameters like electron and ion number densities, magnetic field and ion temperature on nonlinear structures in both non relativistic and relativistic regimes. This work may be helpful to understand low frequency phenomena in dense inhomogeneous plasmas like neutron stars, white dwarfs and next generation lasers.

Relation between the Density of States in Different Cases and the Corresponding Examples

The study of particle system is inevitable to study its statistical law, and understanding the distribution of particles is fundamental and important. In this paper, the author points out the relation between the number of states of free particles in the relativistic and non-relativistic cases, and gives formulae for each case. Furthermore, the author enumerates some popular and basic particle systems, as well as classifies them in terms of relativistic and non-relativistic cases, also their basic formulae are deduced and enumerated in this paper. These particle systems are applicable to different formulae for the density of states for relativistic or non-relativistic cases. It is concluded that the different densities of states in different cases are related to each other through the de Broglie relation and the energy formula.

Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation

The super-Macdonald polynomials, introduced by Sergeev and Veselov (Commun Math Phys 288: 653–675, 2009), generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald–Ruijsenaars operators introduced by the same authors in Sergeev and Veselov (Commun Math Phys 245: 249–278, 2004). We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super-Macdonald polynomials and deformed Macdonald–Ruijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformed Macdonald–Ruijsenaars operators. Motivated by recent results in the nonrelativistic (qrightarrow 1) case, we propose that these models describe the particles and anti-particles of an underlying relativistic quantum field theory, thus providing a natural generalisation of the trigonometric Ruijsenaars model.

A new series solution associated with the quantum harmonic oscillator

In the non-relativistic case, the problem of quantum harmonic oscillator is solved by the aid of the Schrödinger equation. An additional term of potential energy that is related with the square of the spatial coordinate is also included in the equation. A second order linear differential equation is obtained in the stationary state. The direct solution gives the wave function in terms of the parabolic cylinder functions. This solution leads to the Hermite polynomials when the quantization of energy is considered. However, the derived expression of the wave function does not yield the correct functions for the limiting case, in which the potential energy is zero. We put forth an alternative solution for the second order linear differential equation. First of all, a new series expression is suggested as an ansatz for the power series solution of differential equations. The coefficients of the series are obtained in terms of recurrence relations. Some computational simulations are given.