Non-relativistic Case Research Articles

Strong-field tunneling ionization in the relativistic regime

We study the photoelectron energy and momentum distributions of strong-field tunneling ionization in the relativistic regime. Using the relativistic strong-field approximation, we have derived the position of the tunnel exit, the initial momentum distributions at the tunnel exit, and the instantaneous ionization rate when the electron tunnels through the Coulomb barrier in the relativistic regime. Since the tunneling electron energy is of the order of 1% of the rest of its mass, those nonadiabatic tunneling coordinates are quite different from the nonrelativistic case. We further incorporate the nonadiabatic relativistic tunneling coordinates into the classical trajectory Monte Carlo model and have calculated photoelectron energy and momentum distributions by accurately considering the field distribution in the focus of relativistic femtosecond laser pulses. We show that the nonadiabatic tunneling effect and the focal field distribution are very crucial to modeling strong-field ionization in the relativistic regime because the quiver radius of electrons can be as large as the beam waist.

Pressure in warm and hot dense matter using the average-atom model

Expressions of pressure in warm and hot dense matter using the average-atom model are presented. They are based on the stress-tensor approach. Nonrelativistic and relativistic cases are considered. The obtained formulas are simple and can be easily implemented in an average-atom model code. Comparisons with experimental data and quantum molecular dynamics and path integral Monte Carlo simulations are shown. The present formalism agrees well with experimental results for a large variety of elements in the warm dense matter regime and with ab initio simulations in the warm and hot dense matter regime for aluminum.

Lie symmetries of nonrelativistic and relativistic motions

We study the Lie symmetries of non-relativistic and relativistic higher order constant motions, in $d$ spatial dimensions, like constant acceleration, constant rate-of-change -of-acceleration (constant jerk), and so on. In the non-relativistic case, these symmetries contain the $z=\frac 2N$ Galilean conformal transformations, where $N$ is the order of the differential equation that defines the constant motion. The dimension of this group grows with $N$. In the relativistic case the vanishing of the ($d+1$)-dimensional space-time relativistic acceleration, jerk, snap, ... , is equivalent, in each case, to the vanishing of a $d$-dimensional spatial vector. These vectors are the $d$-dimensional non-relativistic ones plus additional terms that guarantee the relativistic transformation properties of the corresponding $d+1$ dimensional vectors. In the case of acceleration there are no corrections, which implies that the Lie symmetries of zero acceleration motions are the same in the non-relativistic and relativistic cases. The number of Lie symmetries that are obtained in the relativistic case does not increase from the four-derivative order (zero relativistic snap) onwards. We also deduce a recurrence relation for the spatial vectors that in the relativistic case characterize the constant motions.

Atomic basis functions for molecular electronic structure calculations

Electronic structure methods for accurate calculation of molecular properties have a high cost that grows steeply with the problem size, therefore, it is helpful to have the underlying atomic basis functions that are less in number but of higher quality. Following our earlier work [Chem. Phys. Lett. 416, 116 (2005)] where general correlation-consistent basis sets are defined, for any atom, as solutions of purely atomic functional minimization problems, and which are shown to work well for chemical bonding in molecules, we take a further step here and define a new kind of atomic polarization functionals, the minimization of which yields additional sets of diffuse functions that help to calculate better molecular electron affinities, polarizabilities, and intermolecular dispersion interactions. Analytical representations by generally-contracted Gaussian functions of up to microhartree numerical accuracy grades are developed for atoms Hydrogen through Nobelium within the four-component Dirac-Coulomb theory and its scalar-relativistic approximation, and also for Hydrogen through Krypton in the two-component nonrelativistic case. The convergence of correlation energy with the basis set size is studied, and complete-basis-set extrapolation formulas are developed.

Evaluation of relativistic transport coefficients and the generalized Spitzer function in devices with 3D geometry and finite collisionality

The relativistic generalization of the linearized drift kinetic equation solver NEO-2 is presented which is used for computation of neoclassical transport coefficients and the generalized Spitzer function in 3D toroidal fusion devices (tokamaks and stellarators). This upgrade allows computations of the Spitzer function playing the role of current drive efficiency in the whole experimentally relevant temperature range, from mild temperatures where finite plasma collisionality effects are important to high temperatures where relativistic effects should be taken into account. Within the Galerkin method used for problem discretization over energy, relativistic effects are included into a set of matrices constant on a flux surface. Those matrices determine coefficients of a coupled set of integro-differential equations with a reduced dimension which is of the same form as in the non-relativistic case. For energy discretization of the linearized relativistic Coulomb collision operator, it is presented in spherical momentum space variables in the symmetric integral form derived directly from Beliaev-Budker expressions. The cancellation problem pertinent to the fully analytical representation of Braams and Karney [Phys. Fluids B 1, 1355 (1989)] does not appear in this form. Examples of evaluation of relativistic transport coefficients and the Spitzer function are presented.

Lévy path integrals of particle on circle and some applications

Fractional quantum mechanics in the non-relativistic case is described by the space-fractional Schrödinger equation containing the fractional Riesz operator, which is a non-local operator. The non-locality makes it difficult to solve the fractional Schrödinger equation in the local potentials. In this paper, we study the solution of the fractional Schrödinger equation of a free particle moving on a circle by use of the Lévy path integrals approach. We present the Lévy path integrals propagator of a particle on circle and then use it to get the wave functions and energy eigenvalues of the free particle case. In addition, the Laplace transform, energy-time transform, and momentum representation of the free particle kernel are also obtained. The results of this paper contain the ones in standard quantum mechanics as special cases. The problem of a particle on circle is a fundamental problem of path integrals with topological constraints in quantum systems, and the method of Lévy path integrals can be generalized to study more complex local potentials.

Thermalization of electron–positron plasma with quantum degeneracy

The non-equilibrium electron–positron–photon plasma thermalization process is studied using relativistic Boltzmann solver, taking into account quantum corrections both in non-relativistic and relativistic cases. Collision integrals are computed from exact QED matrix elements for all binary and triple interactions in the plasma. It is shown that in non-relativistic case (temperatures kBT≤0.3mec2) binary interaction rates dominate over triple ones, resulting in establishment of the kinetic equilibrium prior to final relaxation towards the thermal equilibrium, in agreement with the previous studies. On the contrary, in relativistic case (final temperatures kBT≥0.3mec2) triple interaction rates are fast enough to prevent the establishment of kinetic equilibrium. It is shown that thermalization process strongly depends on quantum degeneracy in initial state, but does not depend on plasma composition.

Relativistic Barotropic Fluids: A Godunov–Boillat Formulation for their Dynamics and a Discussion of Two Special Classes

A relativistic fluid is called barotropic if its internal energy $${\rho}$$ and its pressure p are one-to-one related; it is called isentropic if $${\rho}$$ and p are both one-to-one related to the fluid’s particle number density n. Deriving Godunov variables and 4-potentials for the relativistic Euler equations of perfect barotropic fluids, this paper first pursues ideas on symmetric hyperbolicity going back to Godunov and Boillat that Ruggeri and Strumia have elaborated as their theory of convex covariant density systems. The associated additional balance law (not a conservation law in the presence of shock waves) has different interpretations for different fluids. Among all barotropic fluids, we notably distinguish those for which this extra equation coincides with the second law of thermodynamics. We characterize these ‘thermobarotropic’ fluids, a class which comprises in particular the so-called pure radiation gas used in cosmology. The paper also shows that isentropic fluids are not thermobarotropic and that, like in the non-relativistic case, they cannot simultaneously satisfy the conservation laws of mass, momentum, and energy, as soon as shock waves occur. Curiously, relativistic isentropic flows which conserve energy and momentum generate mass across shock waves; this is a counterpart of the well-known fact that across shock waves, non-relativistic isentropic flows which conserve mass and momentum dissipate energy.

Momentum in the Dynamics of Variable-Mass Systems: Classical and Relativistic Case

We discuss a role of a momentum vector in the description of dynamics of systems with variable mass, and show some ambiguity in expressing the 2nd Newtonian law of dynamics in terms of momentum change in time for variable-mass systems. A simple expression that the time-derivative of the momentum of the body with variable mass is equal to the net external force is not always true (only if a special frame of reference is assumed). In basic textbooks and multiple lecture notes the correct equation of motion for a variable-mass system (including relative velocities of the masses entering or leaving the body) is not sufficiently well discussed, leading to some problems with understanding the dynamics of these systems among students. We also show how the equation of motion in classical case (in translational motion) can be easily expanded to the relativistic case and discuss a motion of a relativistic rocket. For the non-relativistic case also a rotational motion is discussed. It is of course true, that most of the good literature treats the problem accurately, but some of the commonly used textbooks do not. The purpose of our letter is to pay attention to the problem of dynamics of variable-mass systems and show yet another perspective of the subject.

Relativistic Ion-Acoustic Solitary Waves in a Magnetized Pair Ion Dense Plasma with Nuclei of Heavy Elements

The propagation of ion-acoustic solitary waves (IASWs) in a magnetized collisionless degenerate plasma system for describing collective plasma oscillations in dense quantum plasmas with relativistically degenerate electrons, oppositely charged inertial ions, and positively charged immobile heavy elements is investigated theoretically. The perturbations of the magnetized quantum plasma are studied employing the reductive perturbation technique to derive the Korteweg–de Vries (KdV) and the modified KdV (mKdV) equations that admit solitary wave solutions. Chandrasekhar limits are used to investigate the degeneracy effects of interstellar compact objects through the equation of state for degenerate electrons in nonrelativistic and ultrarelativistic cases. The basic properties of small but finite-amplitude IASWs are modified significantly by the combined effects of the degenerate electron number density, pair ion number density, static heavy element number density, and magnetic field. It is found that the obliqueness affects both the amplitude and width of the solitary waves, whereas the other parameters mainly influence the width of the solitons. The results presented in this paper can be useful for future investigations of astrophysical multi-ion plasmas.

Solution of Partial Quasipotential Equations in the Relativistic Configuration Representation

Within the framework of the quasipotential approach, some variants of the interaction of two relativistic spinless particles are considered in the relativistic configuration representation. Numerical solutions of the problem of bound states and scattering states of two-particle systems with zero and nonzero orbital angular momentum are presented. The coupling between resonance and bound states is investigated. A comparison of the results with the nonrelativistic case is provided.

Nonlinear Landau damping in a relativistic electron-ion plasma–non-local nonlinear Schrödinger-equation and Krylov Bogoliubov Mitropolsky method

A nonlinear theory of amplitude modulation of electrostatic envelopes in an electron-ion plasma is developed taking care of relativistic effect, using the Vlasov-Poisson equation. By using the multiple scale reductive perturbation, it is shown that such evolution is characterized by the non-local, nonlinear Schrödinger equation. This non-local term is known to take care of the resonant particles having group velocity = ∂ω/∂k on a finite amplitude plasma wave. On the other hand it is well-known that waves in plasmas can undergo collision-less damping when they resonantly interact with trapped/free particles. Such collision-less damping was termed Landau damping. Our intention in this analysis is to analyze the effect of relativistic electron motion on such collision-less damping in detail. But since the corresponding effect of phase velocity resonance in this classical case is not so prominent, we have not attempted to treat it here and, as assumed, the amplitude is not very large. This gives rise to some new estimates of the frequency shift and energy transfer rate, in both relativistic and non-relativistic cases. In the next phase, we have analyzed the condition of a modulational instability in the non-local, nonlinear Schrödinger equation. Then, the Bogoliubov-Mitropolsky approach is used to study the change in the solitary wave profile due to the presence of non-local nonlinearity in the equation, which is being treated as a perturbation. Last but not the least, the magnitude of the Landau damping is also computed. In the concluding remarks we have compared the kinetic and the usual reductive perturbation approach to the nonlinear Schrödinger equation.

Galilean field theories and conformal structure

We perform a detailed analysis of Galilean field theories, starting with free theories and then interacting theories. We consider non-relativistic versions of massless scalar and Dirac field theories before we go on to review our previous construction of Galilean Electrodynamics and Galilean Yang-Mills theory. We show that in all these cases, the field theories exhibit non-relativistic conformal structure (in appropriate dimensions). The surprising aspect of the analysis is that the non-relativistic conformal structure exhibited by these theories, unlike relativistic conformal invariance, becomes infinite dimensional even in spacetime dimensions greater than two. We then couple matter with Galilean gauge theories and show that there is a myriad of different sectors that arise in the non-relativistic limit from the parent relativistic theories. In every case, if the parent relativistic theory exhibited conformal invariance, we find an infinitely enhanced Galilean conformal invariance in the non-relativistic case. This leads us to suggest that infinite enhancement of symmetries in the non-relativistic limit is a generic feature of conformal field theories in any dimension.

Spontaneous Symmetry Breaking and Higgs Mode: Comparing Gross-Pitaevskii and Nonlinear Klein-Gordon Equations

We discuss the mechanism of spontaneous symmetry breaking and the elementary excitations for a weakly-interacting Bose gas at a finite temperature. We consider both the non-relativistic case, described by the Gross-Pitaevskii equation, and the relativistic one, described by the cubic nonlinear Klein-Gordon equation. We analyze similarities and differences in the two equations and, in particular, in the phase and amplitude modes (i.e., Goldstone and Higgs modes) of the bosonic matter field. We show that the coupling between phase and amplitude modes gives rise to a single gapless Bogoliubov spectrum in the non-relativistic case. Instead, in the relativistic case the spectrum has two branches: one is gapless and the other is gapped. In the non-relativistic limit we find that the relativistic spectrum reduces to the Bogoliubov one. Finally, as an application of the above analysis, we consider the Bose-Hubbard model close to the superfluid-Mott quantum phase transition and we investigate the elementary excitations of its effective action, which contains both non-relativistic and relativistic terms.

Evolution of three-dimensional relativistic current sheets and development of self-generated turbulence

In this paper, the temporal evolution of 3-dimensional relativistic current sheets in Poynting-dominated plasma is studied for the first time. Over the past few decades, a lot of efforts have been conducted on studying the evolution of current sheets in 2-dimensional space, and concluded that sufficiently long current sheets always evolves into the so-called "plasmoid-chain", which provides fast reconnection rate independent of its resistivity. However, it is suspected that plasmoid-chain can exist only in the case of 2-dimensional approximation, and would show transition to turbulence in 3-dimensional space. We performed 3-dimensional numerical simulation of relativistic current sheet using resistive relativistic magnetohydrodynamic approximation. The results showed that the 3-dimensional current sheet evolve not into plasmoid-chain but turbulence. The resulting reconnection rate is $0.004$ which is much smaller than that of plasmoid-chain. The energy conversion from magnetic field to kinetic energy of turbulence is just 0.01\% which is much smaller than typical non-relativistic cases. Using the energy principle, we also showed that the plasmoid is always unstable for a displacement in opposite direction to its acceleration, probably interchange-type instability, and this always results in seeds of turbulence behind the plasmoids. Finally, the temperature distribution along the sheet is discussed, and it is found that the sheet is less active than plasmoid-chain. Our finding can be applied for many high energy astrophysical phenomena, and can provide a basic model of the general current sheet in Poynting-dominated plasma.

Transition from convective to absolute Raman instability via the longitudinal relativistic effect by using Vlasov-Maxwell simulations

The longitudinal relativistic effect on stimulated Raman backscattering (SRBS) is investigated by using one-dimensional (1D) Vlasov-Maxwell simulations. Using a short backscattered light seed pulse with a very small amplitude, the linear gain spectra of SRBS in the strongly convective regime is presented by combining the relativistic and non-relativistic 1D Vlasov-Maxwell simulations, which is in agreement with the steady-state linear theory. More interestingly, by considering transition from convective to absolute instability due to electron trapping, we successfully predict the critical duration of the seed which can just trigger the kinetic inflation of the excited SRBS after the seed leaves the simulation box. The critical duration in the relativistic case is much shorter than that in the nonrelativistic case, which indicates that the kinetic inflation more easily occurs in the relativistic case than in the nonrelativistic case. In the weakly convective regime, the transition from convective to absolute instability for SRBS can directly occur in the linear regime due to the longitudinal relativistic modification. For the same pump, our simulations first demonstrate that the SRBS excited by a short and small seed pulse is a convective instability in the nonrelativistic case but becomes an absolute instability due to the decrease of the linear Landau damping from the longitudinal relativistic modification in the relativistic case. In more detail, the growth rate of the backscattered light is also in excellent agreement with theoretical prediction.

The current matrix elements from HAL QCD method

HAL QCD method is a method to construct a potential (HAL QCD potential) that reproduces the NN scattering phase shift faithful to the QCD. The HAL QCD potential is obtained from QCD by eliminating the degrees of freedom of quarks and gluons and leaving only two particular hadrons. Therefor, in the effective quantum mechanics of two nucleons defined by HAL QCD potential, the conserved current consists not only of the nucleon current but also an extra current originating from the potential (two-body current). Though the form of the two-body current is closely related to the potential, it is not straight forward to extract the former from the latter. In this work, we derive the the current matrix element formula in the quantum mechanics defined by the HAL QCD potential. As a first step, we focus on the non-relativistic case. To give an explicit example, we consider a second quantized non-relativistic two-channel coupling model which we refer to as the original model. From the original model, the HAL QCD potential for the open channel is constructed by eliminating the closed channel in the elastic two-particle scattering region. The current matrix element formula is derived by demanding the effective quantum mechanics defined by the HAL QCD potential to respond to the external field in the same way as the original two-channel coupling model.

Constraining external reverse shock physics of gamma-ray bursts from ROTSE-III limits

Assuming that the early optical emission is dominated by the external reverse shock (RS) in the standard model of gamma-ray bursts (GRBs), we intend to constrain RS models with the initial Lorentz factor $\Gamma_0$ of the outflows based on the ROTSE-III observations. We consider two cases of the RS behavior: the relativistic shock and the non-relativistic shock. For homogeneous interstellar medium (ISM) and wind circum-burst environment, the constraints can be achieved by the fact that the peak flux $F_{\rm \nu}$ at the RS crossing time should be lower than the observed upper limit $F_{\rm \nu,limit}$. We consider the different spectral regimes that the observed optical frequency $\nu_{\rm opt}$ may locate in, which are divided by the orders for the minimum synchrotron frequency $\nu_{\rm m}$ and the cooling frequency $\nu_{\rm c}$. {\bf Considering the homogeneous and wind environment around GRBs, we find that the relativistic RS case can be constrained by the (upper and lower) limits of $\Gamma_0$ in a large range from about hundreds to thousands for 36 GRBs reported by ROTSE-III. The constraints on the non-relativistic RS case are achieved with limits of $\Gamma_0$ for 26 bursts ranging from $\sim 30$ to $\sim 350$.} The lower limits of $\Gamma_0$ achieved for the relativistic RS model is disfavored based on the previously discovered correlation between the initial Lorentz factor $\Gamma_0$ and the isotropic gamma-ray energy $E_{\rm \gamma, iso}$ released in prompt phase.

Nuclear structure of bound states of asymmetric dark matter

Models of Asymmetric Dark Matter (ADM) with a sufficiently attractive and long-range force gives rise to stable bound objects, analogous to nuclei in the Standard Model, called nuggets. We study the properties of these nuggets and compute their profiles and binding energies. Our approach, applicable to both elementary and composite fermionic ADM, utilizes relativistic mean field theory, and allows a more systematic computation of nugget properties, over a wider range of sizes and force mediator masses, compared to previous literature. We identify three separate regimes of nugget property behavior corresponding to (1) non-relativistic and (2) relativistic constituents in a Coulomb-like limit, and (3) saturation in an anti-Coulomb limit when the nuggets are large compared to the force range. We provide analytical descriptions for nuggets in each regime. Through numerical calculations, we are able to confirm our analytic descriptions and also obtain smooth transitions for the nugget profiles between all three regimes. We also find that over a wide range of parameter space, the binding energy in the saturation limit is an ${\cal O}(1)$ fraction of the constituent's mass, significantly larger than expectations in the non-relativistic case. In a companion paper, we apply our results to synthesis of ADM nuggets in the early Universe.

Noncommutative space–time and Hausdorff dimension

We study the Hausdorff dimension of the path of a quantum particle in noncommutative space–time. We show that the Hausdorff dimension depends on the deformation parameter [Formula: see text] and the resolution [Formula: see text] for both nonrelativistic and relativistic quantum particle. For the nonrelativistic case, it is seen that Hausdorff dimension is always less than 2 in the noncommutative space–time. For relativistic quantum particle, we find the Hausdorff dimension increases with the noncommutative parameter, in contrast to the commutative space–time. We show that noncommutative correction to Dirac equation brings in the spinorial nature of the relativistic wave function into play, unlike in the commutative space–time. By imposing self-similarity condition on the path of nonrelativistic and relativistic quantum particle in noncommutative space–time, we derive the corresponding generalized uncertainty relation.