Steady-state Energy Distribution Research Articles

Non-Thermal Nitric Oxide Formation in the Earth’s Polar Atmosphere

Auroral events are the prominent manifestation of solar/stellar forcing on planetary atmospheres because they are closely related to the stellar energy deposition by and evolution of planetary atmospheres. A numerical kinetic Monte Carlo model was developed with the aim to calculate the steady-state energy distribution functions of suprathermal N(4S) atoms in the polar upper atmosphere formed due to the precipitation of high-energy auroral electrons in the N2-O2 atmospheres of rocky planets in solar and exosolar planetary systems. This model describes on the molecular level the collisions of suprathermal N(4S) atoms and atmospheric gas taking into account the stochastic nature of collisional scattering at high kinetic energies. It was found that the electron impact dissociation of N2 is an important source of suprathermal N atoms, significantly increasing the non-thermal production of nitric oxide in the auroral regions of the N2-O2 atmospheres of terrestrial-type planets.

Particle heating and acceleration by reconnecting and nonreconnecting current sheets

In this article, we study the physics of charged particle energization inside a strongly turbulent plasma, where current sheets naturally appear in evolving large-scale magnetic topologies, but they are split into two populations of fractally distributed reconnecting and nonreconnecting current sheets (CS). In particular, we implemented a Monte Carlo simulation to analyze the effects of the fractality and we study how the synergy of energization at reconnecting CSs and at nonreconnecting CSs affects the heating, the power-law high energy tail, the escape time, and the acceleration time of electrons and ions. The reconnecting current sheets systematically accelerate particles and play a key role in the formation of the power-law tail in energy distributions. On the other hand, the stochastic energization of particles through their interaction with nonreconnecting CSs can account for the heating of the solar corona and the impulsive heating during solar flares. The combination of the two acceleration mechanisms (stochastic and systematic), commonly present in many explosive events of various sizes, influences the steady-state energy distribution, as well as the transport properties of the particles in position- and energy-space. Our results also suggest that the heating and acceleration characteristics of ions and electrons are similar, the only difference being the time scales required to reach a steady state.

Energy distributions of an ion in a radio-frequency trap immersed in a buffer gas under the influence of additional external forces

An ion held in a radiofrequency trap interacting with a uniform buffer gas of neutral atoms develops a steady-state energy distribution characterised by a power-law tail at high energies instead of the exponential decay characteristic of thermal equilibrium. We have previously shown that the Tsallis statistics frequently used as an empirical model for this distribution is a good approximation when the ion is heated due to a combination of micromotion interruption and exchange of kinetic energy with the buffer gas [I. Rouse and S. Willitsch, Phys. Rev. Lett. 118, 143401 (2017)]. Here, we extend our treatment to include the heating due to additional motion of the ion caused by external forces, including the "excess micromotion" induced by uniform electric fields and rf phase offsets. We show that this also leads to a Tsallis distribution with a potentially different power-law exponent from that observed in the absence of this additional forced motion, with the difference increasing as the ratio of the mass of the neutral atoms to that of the ion decreases. Our results indicate that unless the excess micromotion is minimised to a very high degree, then even a system with very light neutrals and a heavy ion does not exhibit a thermal distribution.

Sensitivity of internal wave energy distribution over seabed corrugations to adjacent seabed features

Here we show that the distribution of energy of internal gravity waves over a patch of seabed corrugations strongly depends on the distance of the patch to adjacent seafloor features located downstream of the patch. Specifically, we consider the steady state energy distribution due to an incident internal wave arriving at a patch of seabed ripples neighbouring (i) another patch of ripples (i.e. a second patch) and (ii) a vertical wall. Seabed undulations with dominant wavenumber twice as large as overpassing internal waves reflect back part of the energy of the incident internal waves (Bragg reflection) and allow the rest of the energy to transmit downstream. In the presence of a neighbouring topography on the downstream side, the transmitted energy from the patch may reflect back; partially if the downstream topography is another set of seabed ripples or fully if it is a vertical wall. The reflected wave from the downstream topography is again reflected back by the patch of ripples through the same mechanism. This consecutive reflection goes on indefinitely, leading to a complex interaction pattern including constructive and destructive interference of multiply reflected waves as well as an interplay between higher mode internal waves resonated over the topography. We show here that when steady state is reached both the qualitative and quantitative behaviour of the energy distribution over the patch is a strong function of the distance between the patch and the downstream topography: it can increase or decrease exponentially fast along the patch or stay (nearly) unchanged. As a result, for instance, the local energy density in the water column can become an order of magnitude larger in certain areas merely based on where the downstream topography is. This may result in the formation of steep waves in specific areas of the ocean, leading to breaking and enhanced mixing. At a particular distance, the wall or the second patch may also result in a complete disappearance of the trace of the seabed undulations on the upstream and the downstream wave field.

-symmetry breaking in the steady state of microscopic gain–loss systems

The phenomenon of (parity- and time-reversal) symmetry breaking is conventionally associated with a change in the complex mode spectrum of a non-Hermitian system that marks a transition from a purely oscillatory to an exponentially amplified dynamical regime. In this work we describe a new type of -symmetry breaking, which occurs in the steady-state energy distribution of open systems with balanced gain and loss. In particular, we show that the combination of nonlinear saturation effects and the presence of thermal or quantum noise in actual experiments results in unexpected behavior that differs significantly from the usual dynamical picture. We observe additional phases with preserved or ‘weakly’ broken -symmetry, and an unconventional transition from a high-noise thermal state to a low-amplitude lasing state with broken symmetry and strongly reduced fluctuations. We illustrate these effects here for the specific example of coupled mechanical resonators with optically induced loss and gain, but the described mechanisms will be essential for a general understanding of the steady-state properties of actual -symmetric systems operated at low amplitudes or close to the quantum regime.

Equilibrium distribution of the wave energy in a carbyne chain

The steady-state energy distribution of thermal vibrations at a given ambient temperature has been investigated based on a simple mathematical model that takes into account central and noncentral interactions between carbon atoms in a one-dimensional carbyne chain. The investigation has been performed using standard asymptotic methods of nonlinear dynamics in terms of the classical mechanics. In the first-order nonlinear approximation, there have been revealed resonant wave triads that are formed at a typical nonlinearity of the system under phase matching conditions. Each resonant triad consists of one longitudinal and two transverse vibration modes. In the general case, the chain is characterized by a superposition of similar resonant triplets of different spectral scales. It has been found that the energy equipartition of nonlinear stationary waves in the carbyne chain at a given temperature completely obeys the standard Rayleigh–Jeans law due to the proportional amplitude dispersion. The possibility of spontaneous formation of three-frequency envelope solitons in carbyne has been demonstrated. Heat in the form of such solitons can propagate in a chain of carbon atoms without diffusion, like localized waves.

GRAVITATIONAL ENCOUNTERS AND THE EVOLUTION OF GALACTIC NUCLEI. IV. CAPTURES MEDIATED BY GRAVITATIONAL-WAVE ENERGY LOSS

Direct numerical integrations of the two-dimensional Fokker-Planck equation are carried out for compact objects orbiting a supermassive black hole (SBH) at the center of a galaxy. As in Papers I-III, the diffusion coefficients incorporate the effects of the lowest-order post-Newtonian corrections to the equations of motion. In addition, terms describing the loss of orbital energy and angular momentum due to the 5/2-order post-Newtonian terms are included. In the steady state, captures are found to occur in two regimes that are clearly differentiated in terms of energy, or semimajor axis; these two regimes are naturally characterized as "plunges" (low binding energy) and "EMRIs," or extreme-mass-ratio inspirals (high binding energy). The capture rate, and the distribution of orbital elements of the captured objects, are presented for two steady-state models based on the Milky Way: one with a relatively high density of remnants and one with a lower density. In both models, but particularly in the second, the steady-state energy distribution and the distribution of orbital elements of the captured objects are substantially different than if the Bahcall-Wolf energy distribution were assumed. The ability of classical relaxation to soften the blocking effects of the Schwarzschild barrier is quantified.These results, together with those of Papers I-III, suggest that a Fokker-Planck description can adequately represent the dynamics of collisional loss cones in the relativistic regime.

When Rate Constants Are Not Enough.

Real-world chemical systems consisting of multiple isomers and multiple reaction channels often react significantly prior to attaining a steady state energy distribution (SED). Detailed elementary reaction models, which implicitly require SED conditions, may be invalid when non-steady-state energy distributions (NSED) exist. NSED conditions may result in reaction rates and product yields that are different from those expected for SED conditions, although this problem is to some extent reduced by using phenomenological models and rate constants. The present study defines pragmatic diagnostics useful for identifying NSED conditions in stochastic master equation simulations. A representative example is presented for each of four classes of common combustion species: RO2 radicals, aliphatic hydrocarbons, alkyl radicals, and polyaromatic radicals. An example selected from the seminal work of Tsang et al. demonstrates that stochastic simulations and eigenvalue methods for solving the master equation predict the same NSED effects. NSED effects are common under relatively moderate combustion conditions, and accurate simulations may require a master equation analysis.

Robust autothermal microchannel reactors

Autothermal microchannel reactors are intensified process units that bring significant energy efficiency benefits over their conventional counterparts. Efficiency gains are obtained, however, at the cost of operational challenges. These stem from the loss of control handles that is inherent to combining several unit operations in a single physical device. In this paper, we investigate the impact of two recently proposed reactor design concepts (a segmented catalyst macromorphology and an embedded layer of phase change material) aimed at improving the steady state energy distribution and, respectively, preventing the advent of hotspots during transient operation, on reactor dynamics and control. Using an autothermal microchannel reactor coupling steam methane reforming with methane catalytic combustion as a prototype system, we demonstrate through rigorous simulations that these design innovations have a synergistic effect, resulting in superior steady-state performance and excellent disturbance rejection ability.

Kinetics of a single trapped ion in an ultracold buffer gas

The immersion of a single ion confined by a radiofrequency (RF) trap in an ultracold atomic gas extends the concept of buffer gas cooling to a new temperature regime. The steady-state energy distribution of the ion is determined by its kinetics in the RF field rather than the temperature of the buffer gas. Moreover, the finite size of the ultracold gas facilitates the observation of back-action of the ion onto the buffer gas. We numerically investigate the system's properties depending on atom–ion mass ratio, trap geometry, differential cross-section and non-uniform neutral atom density distribution. Experimental results are well reproduced by our model considering only elastic collisions. We identify excess micromotion to set the typical scale for the ion energy statistics and explore the applicability of the mobility collision cross-section to the ultracold regime.

Impact of tunneling on hydrogen-migration of the n-propylperoxy radical

The kinetics of three unimolecular reactions of the n-propylperoxy radical were studied by canonical variational transition state theory and multidimensional small curvature tunneling (SCT). The reactions studied were 1,5 and 1,4 H-migration, and HO(2) elimination. Benchmark calculations were carried out at the CCSD(T) level in order to determine which density functional to use for SCT calculations for each reaction. For 1,5 and 1,4 H-migration, and HO(2) elimination, the M05-2X, B3LYP and B1B95 functionals, respectively, performed closest to the benchmark when coupled to the 6-311+G(2df,2p) basis set. The SCT tunneling corrections, κ(T), computed here were much larger than those calculated from the Wigner or zero-curvature tunneling treatments at low temperatures, but the asymmetric Eckart method works surprisingly well in these three reactions. Comparison of energy-dependent transmission coefficients, Γ(E), indicates that not only the magnitude, but also the sign, of the error in the Eckart approximation is a function of energy; therefore, the error introduced by using the Eckart approach depends strongly on the steady state energy distribution. These results may provide guidance for future studies of tunneling effects in reactions of other peroxy radicals.

Steady state energy distribution function of adatoms in reactive adsorption

The interplay between the energy distribution function of adatoms and the rate of diatom formation in catalytic reaction is studied by means of mean field rate equations. The recombination of adatoms is described as a multi-channel process where adatom desorption arises from several energy levels. It is shown that the distribution function can be computed, analytically, as a function of the ratio between recombination probability and rate constant for energy disposal to the solid. This parameter is the key quantity of the kinetics since it governs both reaction rate and the shape of the energy distribution function. It is found that, in order to obtain steady state conditions, the control parameter is constrained within a well defined interval of values which result lower than unity. It is shown that a kinetic transition takes place for the highest value of the parameter, which entails hyperthermal reaction rate.

Non-thermal emission from secondary pairs in close TeV binary systems

Context. Massive hot stars produce dense ultraviolet (UV) photon fields in their surroundings. If a very high-energy (VHE) gamma ray emitter is located close to the star, then gamma rays are absorbed in the stellar photon field, creating secondary (electron-positron) pairs. Aims. We study the broadband emission of these secondary pairs in the stellar photon and magnetic fields. Methods. Under certain assumptions regarding the stellar wind and the magnetic field in the surroundings of a massive hot star, we calculate the steady state energy distribution of secondary pairs created in the system and its radiation from radio to gamma rays. Results. Under the ambient magnetic field, possibly high enough to suppress electromagnetic (EM) cascading, the energy of secondary pairs is radiated via synchrotron and single IC scattering producing radio-to-gamma ray radiation. The synchrotron spectral energy distribution (SED) is hard, peaks around X-ray energies, and becomes softer. The IC SED is hard as well and peaks around 10 GeV, also becoming softer at higher energies due to synchrotron loss dominance. Conclusions. The radio emission from secondary pairs is moderate and detectable as a point-like and/or extended source. In X-rays, the secondary pair synchrotron component may be dominant. At energies <10 GeV, the secondary pair IC radiation may be dominant over the primary gamma ray emission, and possibly detectable by the next generation of instruments.

Energy distribution and effective temperatures in a driven dissipative model

We investigate nonequilibrium behavior of driven dissipative systems, using a model we recently presented [Phys. Rev. Lett., 93, 240601 (2004)]. We solve the non-Boltzmann steady state energy distribution and the temporal evolution to it, and find its high energy tail to behave exponentially. We demonstrate that various measures of effective temperatures generally differ. We discuss infinite hierarchies of effective temperatures defined from moments of the nonexponential energy distribution, and relate them to the "configurational temperature," measured directly from instantaneous particle locations without any kinetic information. We calculate the "granular temperature," characterizing the average energy in the system, two different "fluctuation temperatures," scaling fluctuation-dissipation relations, and the "entropic temperature," defined from differentiating the entropy with respect to energy.

Suprathermal nitrogen atoms and molecules in Titan's corona

Dissociation and dissociative ionization of molecular nitrogen by solar UV radiation and by photoelectrons and sputtering by magnetospheric ions are the main sources of translationally excited (hot) nitrogen atoms and molecules in the upper atmosphere of Titan. The Boltzmann equations describing the production, thermalization, and transport of suprathermal nitrogen atoms and molecules in Titan's upper atmosphere were solved using the Direct Simulation Monte Carlo method. Steady-state energy distribution functions for the hot nitrogen atoms and molecules were calculated vs. altitude in the thermosphere-exosphere transition region. The contribution of dissociation and atmospheric sputtering to the formation of Titan's hot nitrogen corona and to the escape flux into the Saturnian magnetosphere are described using the calculated energy distribution functions. It is shown that photo-dissociation dominates and, therefore, the principal ejecta are N atoms.

Mode composition of radiation in graded-index waveguides with random microbending of the axis

It is shown that the problem of investigating the mode composition of a partly coherent radiation beam in a randomly inhomogeneous medium can be reduced to a study of evolution of the energy of individual modes and of the coefficients of correlations between the modes. General expressions are obtained for the coupling coefficients of modes in a parabolic waveguide with a random microbending of the axis and an analysis is made of their evolution as a function of the excitation conditions. An estimate is obtained of the distance in which a steady-state energy distribution between the modes is established. Explicit expressions are obtained for the correlation function in the case when a waveguide is excited by off-axial Gaussian beams or Gauss–Hermite modes.

Properties of hot photoexcited holes in Cu-doped germanium at low temperature

A theoretical study is made of the properties of hot photoexcited holes in Cu-doped Germanium as a function of the lattice temperature and compensation density ( N d ). Numerical solutions of the rate equation, obtained by an iterative technique, give the steady-state energy distribution of hot photoexcited holes due to aproximately blackbody room temperature radiation. Considering only a spherical heavy-hole band, simple acoustic deformation scattering and instantaneous optical phonon emission it is found that for the lowest values of N d the energy distribution function is approximately Maxwellian and gives a good fit with experimental photohall mobilities. For higher values of N d , a spherical light hole band is included in the calculations and this involves the use of a two-parameter deformation potential for inter- and intra-valley transverse and longitudinal scattering, and also instantaneous inter- and intra-valley optical scattering. The experimental photohall mobilities can then be used to give information on the deformation potential parameters for the valence band of Ge which cannot be derived from equilibrium transport measurements.

An approximate theory of steady-state and dynamic characteristics of alpha-particle-heated DT-fusion reactors

The steady-state energy distribution function of alpha (α)-particles is derived in an analytical form by directly solving the Fokker-Planck (FP) equation in an approximate manner under the assumption that the functional dependence of the α-particle confinement time τα(E) on its energy is given by τα(E) ∝ E-ℓ (ℓ is a constant). In addition, a new retardation theory is proposed on the basis of the FP equation under the same assumption. Finally, this proposed retardation theory is applied to dynamic problems of an α-particle-heated DT-reactor through the introduction of what is called a “dynamic response (matrix) function”, the result of which enables us to discuss quantitatively problems such as thermal instability and control without numerical analysis of the FP equation.

Some time dependent aspects of fast neutron induced atomic cascades

Abstract Analytical results are obtained for the time-energy distribution of neutrons and the associated displaced atoms slowing down in an amorphous medium according to a general force law. Explicit results are given for the inverse power law, and applications to hard-sphere and Coulomb scattering are discussed. Complete results are obtained for the steady state energy distribution of particles arising from a primary knock-on, and from a neutron initiated cascade. The speed of the slowing down process is assessed by calculating the slowing down time of particles. Two different concepts of slowing down time are discussed, one based upon a density average and the other on a slowing down density average. It is shown that the latter definition is physically more realistic and mathematically simpler.