Adiabatic Invariant Research Articles

Quantifying the Expanding and Cooling Effects into the Double Adiabatic Evolution of the Solar Wind Through the Expanding Box Model

One of the fundamental problems in space physics is the expansion dynamics of the solar wind, strongly correlated with collective plasma reactions, such as wave instabilities that tend to relax kinetic anisotropies. The expansion is in general described through the double adiabatic or Chew–Goldberger–Low (CGL) theory, which sets the main ideas and plasma expansion’s major role in describing plasma cooling/heating dynamics. Here, using the expanding box model (EBM) we revisit the CGL description including plasma expansion. Our primary objective is to isolate the expanding effects into the conservation of the double adiabatic invariants, a key aspect of the CGL theory. Following the same approximations and assumptions as in EBM and CGL theory, we developed a CGL-like description in which the expansion modifies the conservation of the double adiabatic invariants. Our results show that the double adiabatic equations are no longer conserved if plasma cooling is introduced through the EBM, with explicit dependence on expanding parameters, magnetic field profiles, and velocity gradients. Solving the equations for different magnetic field and density profiles (obtained self-consistently through the equations), we compute the evolution of temperature anisotropy and plasma beta, which deviates from CGL predictions and empirical observations. This deviation is attributed to the plasma cooling effect induced by the expansion of the plasma. The results suggest that heating mechanisms even play a major role in counteracting plasma cooling during expansion.

Fluctuations and Power Low Distribution Function in Nonequilibrium Systems

The Fokker–Planck equation is formulated for the distribution functions of macroscopic open systems in the space of slowly changing physical variables (energy, adiabatic invariants, etc.). The stationary solution of such equations determines a quasi-equilibrium distribution function in the relevant space. The proposed approach involves the evolution of systems under the action of dissipation and diffusion in the space of the appropriate variables. It is shown that the well-known power law distribution can be obtained by considering internal and external fluctuations in statistical systems.

Perturbed conformal invariance and Mei adiabatic invariants of the generalized perturbed Hamiltonian systems with additional terms

AbstractThis study investigates the relationship between perturbed conformal invariance and Mei symmetry in the generalized perturbed Hamiltonian systems with additional terms. A necessary and sufficient condition is derived to determine whether perturbed conformal invariance can be considered an approximate Mei symmetry. Furthermore, the Mei adiabatic invariants are also obtained. Lastly, an example is presented to demonstrate the key findings.

A Potential Link between Space Weather and Atmospheric Parameters Variations: A Case Study of November 2021 Geomagnetic Storm

On 4 November 2021, during the rising phase of solar cycle 25, an intense geomagnetic storm (Kp = 8−) occurred. The effects of this storm on the outer magnetospheric region up to the ionospheric heights have already been examined in previous investigations. This work is focused on the analysis of the solar wind conditions before and during the geomagnetic storm, the high-latitude electrodynamics conditions, estimated through empirical models, and the response of the atmosphere in both hemispheres, based on parameters from the ECMWF ERA5 atmospheric reanalysis dataset. Our investigations are also supported by counter-test analysis and Monte Carlo tests. We find, for both hemispheres, a significant correspondence, within 1–2 days, between high-latitude electrodynamics variations and changes in the temperature, specific humidity, and meridional and zonal winds, in both the troposphere and stratosphere. The results indicate that, in the complex solar wind–atmosphere relationship, a significant role might be played by the intensification of the polar cap potential. We also study the reciprocal relation between the ionospheric Joule heating, calculated from a model, and two adiabatic invariants used in the analysis of solar wind turbulence.

Hamiltonian formulation of linear non-Hermitian systems

In the case of a linear non-Hermitian system, I prove that it's possible to construct a Hamiltonian in such a way that the equations governing the non-Hermitian system can be exactly expressed using Hamilton's canonical equations. Initially, I demonstrate this within the discrete representation framework and subsequently extend it to continuous representation. Through this formulation employing the Hamiltonian, I can pinpoint a conserved charge using Noether's theorem and identify adiabatic invariants. When this approach is applied to Hermitian systems, all the obtained results converge to the well-known outcomes associated with the Schrödinger equation.

IMF Bx, By, and Bz dependence of energetic particle precipitation in the magnetosphere

The orientation of Interplanetary Magnetic Field (IMF B) components plays a major role in magnetospheric Energetic Particles Precipitation (EPP). This is because the IMF B components in the Geocentric Solar Magnetospheric (GSM) coordinate system control magnetic reconnection at the subsolar magnetopause. This study intends to unravel the dependence of the preceding conditions of the orientations of IMF B components that drove geomagnetic storms on the EPPs produced in the Van Allen radiation belt. We classified the geomagnetic storms based on their drivers and IMF Bz orientations prior to the storms’ onset. Energetic particle fluxes, IMF B, and solar wind data were used. The study showed that electrons are more reactive to solar wind speed than protons. Additionally, storms with southward IMF Bz orientation caused more EPP enhancement than with northward orientation. Our results also showed that protons are more influenced by magnetospheric compression than electrons at the northward orientation of IMF Bz. In fact, at this point, the proton’s motion violates adiabatic invariants. As particles interact with electric fields, a significant population of EPPs is produced by pitch angle diffusion. Generally, IMF Bx and IMF By are often anticorrelated. The preceding conditions of orientations of the IMF Bx indicated weaker dependence on auroral indices, and invariably on energetic particles precipitation than the IMF By.

Numerical Calculations of Adiabatic Invariants From MHD‐Driven Magnetic Fields

AbstractThe adiabatic invariants (M, J, Φ) and the related invariants (M, K, L*) have been established as effective coordinate systems for describing radiation belt dynamics at a theoretical level, and through numerical techniques, can be paired with in situ observations to order phase‐space density. To date, methods for numerical techniques to calculate adiabatic invariants have focused on empirical models such the Tsyganenko models TS05, T96, and T89. In this work, we develop methods based on numerical integration and variable step size iteration for the calculation of adiabatic invariants, applying the method to the Lyon‐Fedder‐Mobarry (LFM) global magnetohydrodynamics (MHD) simulation code, with optional coupling to the Rice Convection Model (RCM). By opening the door to adiabatic invariant modeling with MHD magnetic fields, the opportunity for exploratory modeling work of radiation belt dynamics is enabled. Calculations performed using LFM are cross‐referenced with the same code applied to the T96 and TS05 Tsyganenko models evaluated on the LFM grid. Important aspects of the adiabatic invariant calculation are reviewed and discussed, including (a) sensitivity to magnetic field model used, (b) differences in the problem between quiet and disturbed geomagnetic states, and (c) the selection of key parameters, such as the magnetic local time step size for drift shell determination. The rigorous development and documentation of this algorithm additionally acts as preliminary step for future thorough reassessment of in situ phase‐space density results using alternative magnetic field models.

Perturbation to Symmetries and Adiabatic Invariants for Generalized Birkhoffian Systems on Time Scales

Time scale is a new and powerful tool for dealing with complex dynamics problems. The main result of this study is the exact invariants and adiabatic invariants of the generalized Birkhoffian system and the constrained Birkhoffian system on time scales. Firstly, we establish the differential equations of motion for the above two systems and give the corresponding Noether symmetries and exact invariants. Then, the perturbation to the Noether symmetries and the adiabatic invariants for the systems mentioned above under the action of slight disturbance are investigated, respectively. Finally, two examples are provided to show the practicality of the findings.

Adiabatic Invariants during Channeling in a Bent Crystal

Adiabatic Invariants during Channeling in a Bent Crystal

Secular evolution of resonant planets in the coplanar case

Aims. We study the secular evolution of two planets in mutual deep mean-motion resonance (MMR) in the planar elliptic three-body problem framework for different mass ratios. We do not consider any restriction in the eccentricity of the inner planet e1 or in the eccentricity of the outer planet e2. Methods. The method we used is based on a semi-analytical model that consists of calculating the averaged resonant disturbing function numerically. It is assumed for this that all the orbital elements (except for the mean longitudes) of both planets are constant on the resonant timescale. In order to obtain the secular evolution inside the MMR, we make use of the adiabatic invariance principle, assuming a zero-amplitude resonant libration. We constructed two phase portraits, called the ℋ1 and ℋ2 surfaces, in the three-dimensional spaces (e1, Δϖ, σ) and (e2, Δϖ, σ), where Δϖ is the difference between the planetary longitude of perihelia and σ is the critical angle. These surfaces, which are related through the angular moment conservation, allow us to find the apsidal corotation resonances (ACRs) and to predict the secular evolution of e1, e2, Δϖ, and σ (libration center). Results. While studying the 1:1, 2:1, 3:1, and 3:2 MMR, we found that large excursions in eccentricity can exist in some particular cases. We compared the secular variations of e1, e2, Δϖ, and σ predicted by the model with a numerical integration of the exact equations of motion for different mass ratios. We obtained good matches. Finally, the model was applied to study the secular evolution of the resonant exoplanet systems HD 73526 and HD 31527. They both have a pair of planets and are very close to the deep MMR condition. In the first system, we found that the pair of planets that constitutes the system evolves in a symmetrical ACR, whereas in the second system, we found that planets c and d, which are in an unusual 16:3 MMR, are close to an ACR, but outside its dynamical region, where Δϖ circulates.

Adiabatic invariance and its application to Wien's complete displacement law of blackbody radiation

We derive the “complete” or “strong” version of Wien's displacement law from two adiabatic invariants: one of a thermodynamic system composed of a finite-sized segment of frequencies taken from the spectrum of blackbody radiation and one of the individual electromagnetic waves that compose this system. By exploiting the algebra of these invariants, we shift the calculational burden of deriving Wien's displacement law toward the methods of classical thermodynamics. These methods also produce a class of displacement laws that constrain both the particles of a classical ideal gas and the acoustic waves of the Debye model of a solid.

On the thermodynamic entropy in the microcanonical ensemble of classical systems

We demonstrate that the surface entropy given by the volume of an energy shell in the phase space can be the thermodynamically consistent entropy in a classical microcanonical ensemble if the thickness of the energy shell is not an arbitrary constant but a non-extensive function satisfying a specific differential equation. A particular form of the energy shell thickness as a possible solution to the differential equation converts the surface entropy into the volume entropy given by the phase-space volume bounded by a constant energy surface. However, such a form bears a problem: The temperature derived accordingly becomes extensive when the density of states is a non-monotonic function of energy. Based on the adiabatic invariance of the degeneracy of a quantum system and the Weyl correspondence, we propose an alternative solution: the energy shell thickness given by the energy level spacing in the quantum counterpart of the classical ensemble considered, which is illustrated by a few simple examples.

Perturbation to conformal invariance and adiabatic invariants of generalized perturbed Hamiltonian system with additional terms

In this article, by considering the generalized perturbed Hamiltonian systems with additional terms, approximate conformal invariance and adiabatic invariants are studied. These invariants are obtained using approximate Lie point symmetries and without solving the structural equation. The necessary and sufficient condition for the relation between approximate conformal invariance and approximate Lie symmetry is obtained. The fundamental relationship between integrals and variables is presented herein, which is applicable to perturbed dynamic systems of both linear and non-linear nature. The approach is implemented on perturbed Hamiltonian systems of a generalized nature, featuring supplementary components of even dimensions. Additionally, the study derives the Hojman adiabatic invariants for generalized perturbed Hamiltonian systems with additional terms. Finally, an illustrative example is given to demonstrate the practical implementation of the findings.

Evaluating the Adiabatic Invariants in Magnetized Plasmas Using a Classical Ehrenfest Theorem.

In this article, we address the reliance on probability density functions to obtain macroscopic properties in systems with multiple degrees of freedom as plasmas, and the limitations of expensive techniques for solving Equations such as Vlasov's. We introduce the Ehrenfest procedure as an alternative tool that promises to address these challenges more efficiently. Based on the conjugate variable theorem and the well-known fluctuation-dissipation theorem, this procedure offers a less expensive way of deriving time evolution Equations for macroscopic properties in systems far from equilibrium. We investigate the application of the Ehrenfest procedure for the study of adiabatic invariants in magnetized plasmas. We consider charged particles trapped in a dipole magnetic field and apply the procedure to the study of adiabatic invariants in magnetized plasmas and derive Equations for the magnetic moment, longitudinal invariant, and magnetic flux. We validate our theoretical predictions using a test particle simulation, showing good agreement between theory and numerical results for these observables. Although we observed small differences due to time scales and simulation limitations, our research supports the utility of the Ehrenfest procedure for understanding and modeling the behavior of particles in magnetized plasmas. We conclude that this procedure provides a powerful tool for the study of dynamical systems and statistical mechanics out of equilibrium, and opens perspectives for applications in other systems with probabilistic continuity.

Symmetries and perturbations of time-scale nonshifted singular systems

In this work, the symmetries and perturbations of time-scale nonshifted singular Lagrangian and singular nonconservative Lagrangian systems are studied. The differential equations of motion are given. The definitions and criteria of the Noether, Lie, and Mei symmetries of the two systems are presented, along with the corresponding conserved quantities deduced from these symmetries. In addition, the perturbations to each symmetry and the related adiabatic invariants are studied. Finally, examples are used to illustrate the applications of these results.

The Variable Mass- Spring System and the Bessel's Function

Damped harmonic oscillator and Bessel’s function both exhibit similar behaviours. In order to justify this similarity between the two functions, the mass-spring system with variable mass is used. It is shown how the damped harmonic oscillator can be used as an approximated function for the Bessel function. The adiabatic invariance is also employed to make the mathematical and physical comparison between the two functions. It is also shown how to use the zeros of the Bessel's function to obtain the same approximation by solving the boundary-value problems. The zeros of the Bessel's function are used to calculate the time period of the variable mass-spring system.

New adiabatic invariants for disturbed non-material volumes

New adiabatic invariants for disturbed non-material volumes

Unraveling Soft Squeezing Transformations in Time-Variant Elastic Fields

Quantum squeezing, an intriguing phenomenon that amplifies the uncertainty of one variable while diminishing that of its conjugate, may be studied as a time-dependent process, with exact solutions frequently derived from frameworks grounded in adiabatic invariants. Remarkably, we reveal that exact solutions can be ascertained in the presence of time-variant elastic forces, eschewing dependence on invariants or frozen eigenstate formalism. Delving into these solutions as an inverse problem unveils their direct connection to the design of elastic fields, responsible for inducing squeezing transformations onto canonical variables. Of particular note is that the dynamic transformations under investigation belong to a class of gentle quantum operations, distinguished by their delicate manipulation of particles, thereby circumventing the abrupt energy surges commonplace in conventional control protocols.

Further Research for Lagrangian Mechanics within Generalized Fractional Operators

In this article, the problems of the fractional calculus of variations are discussed based on generalized fractional operators, and the corresponding Lagrange equations are established. Then, the Noether symmetry method and the perturbation to Noether symmetry are analyzed in order to find the integrals of the equations. As a result, the conserved quantities and the adiabatic invariants are obtained. Due to the universality of the generalized fractional operators, the results achieved here can be used to solve other specific problems. Several examples are given to illustrate the universality of the methods and results.

Research on the Symmetry of the Hamiltonian System under Generalized Operators

Generalized operators have recently been proposed with great potential applications. Here, we present research carried out on Noether figury and perturbation to Noether symmetry for Hamiltonian systems within generalized operators. There are four parts, and each part contains two kinds of generalized operator. Firstly, Hamilton equations are established. Secondly, the Noether symmetry method is used for finding the solutions to the differential equations of motion, and conserved quantities are obtained. Thirdly, perturbation to Noether symmetry and adiabatic invariants are further explored. In the end, two examples are given to illustrate the methods and results.