scholarly journals A simple, heuristic derivation of the Balescu-Lenard kinetic equation for stellar systems

2020 ◽  
Author(s):  
Chris Hamilton
Keyword(s):  
Kinetic Equation ◽  
Kinetic Theory ◽  
Superposition Principle ◽  
Collective Effects ◽  
Stellar Systems ◽  
Relaxation Theory ◽  
Stellar Orbits ◽  
Collective Interactions ◽  
Straight Lines ◽  
Action Variables

Abstract The unshielded nature of gravity means that stellar systems are inherently inhomogeneous. As a result, stars do not move in straight lines. This obvious fact severely complicates the kinetic theory of stellar systems because position and velocity turn out to be poor coordinates with which to describe stellar orbits – instead, one must use angle-action variables. Moreover, the slow relaxation of star clusters and galaxies can be enhanced or suppressed by collective interactions (‘polarisation’ effects) involving many stars simultaneously. These collective effects are also present in plasmas; in that case they are accounted for by the Balescu-Lenard (BL) equation, which is a kinetic equation in velocity space. Recently, several authors have shown how to account for both inhomogeneity and collective effects in the kinetic theory of stellar systems by deriving an angle-action generalisation of the BL equation. Unfortunately their derivations are long and complicated, involving multiple coordinate transforms, contour integrals in the complex plane, and so on. On the other hand, Rostoker’s superposition principle allows one to pretend that a long-range interacting N-body system, such as a plasma or star cluster, consists merely of uncorrelated particles that are ‘dressed’ by polarisation clouds. In this paper we use Rostoker’s principle to provide a simple, intuitive derivation of the BL equation for stellar systems which is much shorter than others in the literature. It also allows us to straightforwardly connect the BL picture of self-gravitating kinetics to the classical ‘two-body relaxation’ theory of uncorrelated flybys pioneered by Chandrasekhar.

Long-term evolution of directional spectra of wind waves modelled by DNS and kinetic equations, and comparison with airborne measurements

2021 ◽  
Author(s):  
Sergei Annenkov ◽  
Victor Shrira ◽  
Leonel Romero ◽  
Ken Melville
Keyword(s):  
Angular Distribution ◽  
Kinetic Equation ◽  
Kinetic Theory ◽  
Wind Waves ◽  
Kinetic Equations ◽  
Spectral Peak ◽  
Angular Width ◽  
Angular Structure ◽  
Airborne Measurements ◽  
Steady Growth

<p>We consider the evolution of directional spectra of waves generated by constant and changing wind, modelling it by direct numerical simulation (DNS), based on the Zakharov equation. Results are compared with numerical simulations performed with the Hasselmann kinetic equation and the generalised kinetic equation, and with airborne measurements of waves generated by offshore wind, collected during the GOTEX experiment off the coast of Mexico. Modelling is performed with wind measured during the experiment, and the initial conditions are taken as the observed spectrum at the moment when wind waves prevail over swell after the initial part of the evolution.</p><p>Directional spreading is characterised by the second moment of the normalised angular distribution function, taken at selected wavenumbers relative to the spectral peak. We show that for scales longer than the spectral peak the angular spread predicted by the DNS is close to that predicted by both kinetic equations, but it underestimates the corresponding measured value, apparently due to the presence of swell. For the spectral peak and shorter waves, the DNS shows good agreement with the data. A notable feature is the steady growth of angular width at the spectral peak with time/fetch, in contrast to nearly constant width in the kinetic equations modelling. Dependence of angular width on wavenumber is shown to be much weaker than predicted by the kinetic equations. A more detailed consideration of the angular structure at the spectral peak at large fetches shows that the kinetic equations predict an angular distribution with a well-defined peak at the central angle, while the DNS reproduces the observed angular structure, with a flat peak over a range of angles.</p><p>In order to study in detail the differences between the predictions of the DNS and the kinetic equations modelling under idealised conditions, we also perform numerical simulations for the case of constant wind forcing. As in the previous case of forcing by real wind, the most striking difference between the kinetic equations and the DNS is the steady growth with time of angular width at the spectral peak, which is demonstrated by the DNS, but is not present in the modelling with the kinetic equations. We show that while the kinetic theory, both in the case of the Hasselmann equation and the generalised kinetic equation, predicts a relatively simple shape of the spectral peak, the DNS shows a more complicated structure, with a flat top and dependence of the peak position on angle. We discuss the approximations employed in the derivation of the kinetic theory and the possible causes of the found differences of directional structure.</p>

scholarly journals A many-particle approach to the gyro-kinetic theory

2007 ◽  
Vol 73 (5) ◽  
pp. 757-772 ◽  
Author(s):  
ALEXEY MISHCHENKO ◽  
AXEL KÖNIES
Keyword(s):  
Kinetic Equation ◽  
Kinetic Theory ◽  
First Principles ◽  
Conventional Approach ◽  
The Self ◽  
Collision Term ◽  
Lie Transform ◽  
Self Consistent ◽  
The Many ◽  
Particle Approach

AbstractA systematic first-principles approach to the many-particle formulation of the gyro-kinetic theory is suggested. The gyro-kinetic many-particle Hamiltonian is derived using the Lie transform technique. The generalized gyro-kinetic equation is obtained following the Born–Bogoliubov–Green–Kirkwood–Yvon approach. The microscopic expression for the self-consistent potential and the polarization density is obtained. It is shown that new terms appear in the gyro-kinetic polarization that can not be derived in the conventional approach. An expression for the collision term is obtained in the Landau approximation.

scholarly journals Analysis and Forecast Based on the Kinetic Equation for Changing the Numerical Composition of Living Systems

2020 ◽  
Vol 2 (2) ◽  
pp. 8-18
Author(s):  
Alexander Alexandrovich Victorov ◽  
Viacheslav Alexandrovich Kholodnov
Keyword(s):  
Kinetic Equation ◽  
Kinetic Theory ◽  
Population Growth ◽  
Biological Object ◽  
Human Life ◽  
Developed Countries ◽  
Living Systems ◽  
Subsequent Decrease ◽  
Final Population ◽  
Theory Of Aging

The possibility of applying the kinetic theory of aging of biological species published earlier by the authors of this work to assess and predict changes in the number of specific populations is evaluated. The populations of the USA, China and Russia, as well as the population of mice observed in the experiment "mouse paradise" of the American scientist John Calhoun are considered. To this end, a historically consistent analysis of the main previously proposed multi-scenario mathematical models describing demographic data and predicting the dynamics of the population was performed. The results of these models show a decrease in the population growth rate, a tendency toward a limit with an increase in historical time, the achievement of such a limit in some developed countries with a relatively high level of social security, a subsequent decrease in the number and further uncertainty of the final population outlook in the distant future. In addition, these models made it possible to establish that the observed population growth in developed countries is unambiguously accompanied by its aging - a relative predominant increase in the number of elderly people compared to the number of the younger generation (people are aging, the population of countries is aging). In this work, the assumption was made and confirmed that the dynamics of the aging of the population of the countries of the World corresponds to the dynamics of aging of a person of one generation and is mathematically described by the differential equation of the kinetic theory of aging of living systems of the same type with close values of the parameters. The biophysical meaning of the parameters of the kinetic equation reflects G. Selye's concept of the determining role of stress in human life and populations. An analysis of the changes in the numbers of the considered populations of humans and mice at various stages of their development is qualitatively commented on from the standpoint of comparative tension according to G. Selye. To assess the degree of aging of a biological object of one population in kinetic theory, the probability of death during life is selected as an indicator of aging. In this work, the probability of reaching the maximum population size was chosen as an indicator of the aging of a biological object of various populations. The published literature predicts various options for changing the population after reaching a maximum - maintaining the reached maximum level and decreasing to a certain limit, less than the maximum achieved. In this paper, based on an analysis of its results and an analogy with the complete degeneration of mice in the “mouse paradise” experiment, a conclusion is drawn about a hypothetically possible third variant of the limiting decrease in the population - its complete degeneration.

A kinetic theory of dense fluids. Test‐particle kinetic equation

1981 ◽  
Vol 74 (4) ◽  
pp. 2477-2493 ◽  
Author(s):  
Myung S. Jhon ◽  
John S. Dahler
Keyword(s):  
Kinetic Equation ◽  
Kinetic Theory ◽  
Test Particle ◽  
Dense Fluids

MODEL KINETIC EQUATION FOR THE DISTRIBUTION OF DISCRETIZED INTERNAL ENERGY

1994 ◽  
Vol 04 (05) ◽  
pp. 669-675 ◽  
Author(s):  
K. NANBU
Keyword(s):  
Kinetic Equation ◽  
Kinetic Theory ◽  
Internal Energy ◽  
Boltzmann Distribution ◽  
Model Kinetic Equation ◽  
Discrete Velocity ◽  
Second Moment ◽  
Exponential Relaxation ◽  
H Theorem

Kinetic equation for discretized internal energy is obtained by using the idea underlying the discrete-velocity kinetic theory. The equation satisfies the Boltzmann H-theorem. The solution of this equation in equilibrium is the Boltzmann distribution. The second moment of distribution shows an exponential relaxation.

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