Limit distribution functions of systems with many-particle interaction in classical statistical physics

Abstract Monte Carlo simulations of a system of 200 water and 24 NaCl molecules at 6 different densities in the range from 0.003 g/cm3 to 0.999 g,/cm3 and T = 125 °C and 225 CC were performed to obtain some insight into cluster formation which should precede and determine the formation of aerosol structures and has possibly played some role in prebiotic atmosphere chemistry. Solute hydration occurs already at very low concentrations mainly in the form of hydrated molecules ("contact ion pairs"). At higher densities larger cluster structures are observed, leading rather continuously to the structure of the supersaturated 7.1 M NaCl solution at the same temperature. Radial distribution functions, coordination numbers and particle interaction energies are discussed with respect to the simulation parameters density and temperature

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Ion-driven Instabilities in the Inner Heliosphere. I. Statistical Trends

The Astrophysical Journal ◽

10.3847/1538-4357/ac3081 ◽

2021 ◽

Vol 923 (1) ◽

pp. 116

Author(s):

Mihailo M. Martinović ◽

Kristopher G. Klein ◽

Tereza Ďurovcová ◽

Benjamin L. Alterman

Keyword(s):

Solar Wind ◽

Particle Interaction ◽

Radial Distance ◽

Solar Wind Plasma ◽

Distribution Functions ◽

Nyquist Criterion ◽

Velocity Distribution Functions ◽

Instability Growth ◽

Statistical Trends ◽

The Impact

Abstract Instabilities described by linear theory characterize an important form of wave–particle interaction in the solar wind. We diagnose unstable behavior of solar wind plasma between 0.3 and 1 au via the Nyquist criterion, applying it to fits of ∼1.5M proton and α particle Velocity Distribution Functions (VDFs) observed by Helios I and II. The variation of the fraction of unstable intervals with radial distance from the Sun is linear, signaling a gradual decline in the activity of unstable modes. When calculated as functions of the solar wind velocity and Coulomb number, we obtain more extreme, exponential trends in the regions where collisions appear to have a notable influence on the VDF. Instability growth rates demonstrate similar behavior, and significantly decrease with Coulomb number. We find that for a nonnegligible fraction of observations, the proton beam or secondary component might not be detected, due to instrument resolution limitations, and demonstrate that the impact of this issue does not affect the main conclusions of this work.

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Closure of multi-fluid and kinetic equations for cyclotron-resonant interactions of solar wind ions with Alfvén waves

Nonlinear Processes in Geophysics ◽

10.5194/npg-5-111-1998 ◽

1998 ◽

Vol 5 (2) ◽

pp. 111-120 ◽

Cited By ~ 27

Author(s):

E. Marsch

Keyword(s):

Velocity Component ◽

Ion Beam ◽

Wave Spectrum ◽

Particle Interaction ◽

Kinetic Equations ◽

Distribution Functions ◽

Average Intensity ◽

Temperature Anisotropy ◽

Quasilinear Theory ◽

Resonant Interactions

Abstract. Based on quasilinear theory, a closure scheme for anisotropic multi-component fluid equations is developed for the wave-particle interactions of ions with electromagnetic Alfvén and ion-cyclotron waves propagating along the mean magnetic field. Acceleration and heating rates are calculated. They may be used in the multi-fluid momentum and energy equations as anomalous transport terms. The corresponding evolution equation for the average wave spectrum is established, and the effective growth/damping rate for the spectrum is calculated. Given a simple power-law spectrum, an anomalous collision frequency can be derived which depends on the slope and average intensity of the spectrum, and on the gyrofrequency and the differential motion (with respect to the wave frame) of the actual ion species considered. The wave-particle interaction terms attain simple forms resembling the ones for collisional friction and temperature anisotropy relaxation (due to pitch angle scattering) with collision rates that are proportional to the gyrofrequency but diminished substantially by the relative wave energy or the fluctuation level with respect the background field. In addition, a set of quasilinear diffusion equations is derived for the reduced (with respect to the perpendicular velocity component) velocity distribution functions (VDFs), as they occur in the wave dispersion equation and the related dielectric function for parallel propagation. These reduced VDFs allow one to describe adequately the most prominent observed features, such as an ion beam and temperature anisotropy, in association with the resonant interactions of the particles with the waves on a kinetic level, yet have the advantage of being only dependent upon the parallel velocity component.

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Dynamical evolution of the seismic coda wave increments during the 2011-2012 Santorini's caldera unrest. A Non-Extensive Statistical Physics approach.

10.5194/egusphere-egu2020-8190 ◽

2020 ◽

Author(s):

Theodoros Aspiotis ◽

Ioannis Koutalonis ◽

Georgios Michas ◽

Filippos Valianatos

Keyword(s):

Statistical Physics ◽

Distribution Functions ◽

Coda Waves ◽

Coda Wave ◽

Tectonic Activity ◽

Volcanic Complex ◽

Dynamical Structure ◽

Gaussian Shape ◽

Caldera Unrest ◽

Seismic Coda

Santorini's caldera being unrest during 2011-2012, led several studies to raise the important question of whether seismicity is associated with an impending and potential volcanic eruption or it solely relieves the accumulated tectonic energy. In the present work we study seismic coda waves generated by local earthquake events prior, during and after the seismic crisis that occurred within the caldera area. Coda waves are interpreted as scattered seismic waves generated by heterogeneities within the Earth, i.e. by faults, fractures, velocity and/or density boundaries and anomalies, etc. In particular, we utilize the three components of the seismograms recorded by three seismological stations on the island of Santorini and estimate the duration of the coda waves by implementing a three step procedure that includes the signal-to-noise ratio, the STA/LTA method and the short time Fourier transform. The final estimation was verified or reestimated manually due to the existent ambient seismic noise. Due to the nature and the path complexity of the coda waves and towards achieving a unified framework for the study of the immerse geo-structural seismotectonic complexity of the Santorini volcanic complex, we use Non-Extensive Statistical Physics (NESP) to study the probability distribution functions (pdfs) of the increments of seismic coda waves. NESP forms a generalization of the Boltzmann-Gibbs statistical mechanics, that has been extensively used for the analysis of semi-chaotic systems that exhibit long-range interactions, memory effects and multi-fractality. The analysis and results demonstrate that the seismic coda waves increments deviate from the Gaussian shape and their respective pdfs could adequately be described and processed by the q-Gaussian distribution. Furthermore and in order to investigate the dynamical structure of the volcanic-tectonic activity, we estimate the q-indices derived from the pdfs of the coda wave time series increments during the period 2009 - 2014 and present their variations as a function of time and as a function of the local magnitude (ML) of the events prior, during and after the caldera unrest. &#160; &#160;Acknowledgments.&#160;We acknowledge support by the project &#8220;HELPOS &#8211; Hellenic System for Lithosphere Monitoring&#8221; (MIS 5002697) which is implemented under the Action &#8220;Reinforcement of the Research and Innovation Infrastructure&#8221;, funded by the Operational Programme "Competitiveness, Entrepreneurship and Innovation" (NSRF 2014-2020) and co-financed by Greece & European Union (ERDF)

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Assessing sediment transport energetics with instrumented particles for above threshold of motion turbulent flows

10.5194/egusphere-egu2020-21727 ◽

2020 ◽

Author(s):

Zaid Al-Husban ◽

Manousos Valyrakis

Keyword(s):

Sediment Transport ◽

Turbulent Flows ◽

Particle Transport ◽

Particle Interaction ◽

Turbulent Channel Flow ◽

Optimal Operation ◽

Distribution Functions ◽

Sensor Data ◽

Flow Rates ◽

Test Configuration

Despite the fact sediment transport has been studied for decades, there is still a need to gain a further insight on the nature and driving mechanisms of bed particle motions induced by turbulent flows, for the low transport stages where the particle transport is relatively intermittent. A custom designed and prototyped instrumented particle, embedded with inertial sensors is used herein to study its transport over hydraulically rough bed surfaces. The calibration and error estimation for its sensors is also undertaken before starting the experiments, to ensure optimal operation and estimate any uncertainties.The observations and results of this research are obtained from experiments carried out at the University of Glasgow 12 meters long and 0.9 meters wide, tilting and water recirculating flume. The flume walls comprise of smooth transparent glass that enables observing particle transport from the side (also with underwater video cameras) and the bed surface generally is layered with coarse gravel.The particle is initially located at the upstream end of the test configuration, fully exposed to the uniform and fully developed turbulent channel flow. The top and side cameras are set in their suitable positions to monitor and study the behaviour of particle motion by capturing the dynamical features of sediment motion and to not interfere with flow field that pushes particle downstream.&#160;Using the sensor data to calculate the kinetic energy for a range of sets of sediment transport experiments with varying flow rates and particle densities, the probability distribution functions (PDFs) of particle transport features, such as particle&#8217;s total energy, are generated which give information about particle interaction with the surface bed during its motion. In addition, the effects of different flow rates, particle densities on particle energy are assessed.

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The generation of Ganymede's diffuse aurora through pitch angle scattering

Annales Geophysicae ◽

10.5194/angeo-35-239-2017 ◽

2017 ◽

Vol 35 (2) ◽

pp. 239-252

Author(s):

Arvind K. Tripathi ◽

Rajendra P. Singhal ◽

Onkar N. Singh II

Keyword(s):

Electron Temperature ◽

Pitch Angle ◽

Atomic Oxygen ◽

Particle Interaction ◽

Distribution Functions ◽

Dissociative Excitation ◽

Kappa Distribution ◽

Loss Cone ◽

Whistler Mode ◽

Whistler Mode Waves

Abstract. Diffuse auroral intensities of neutral atomic oxygen OI λ1356 Å emission on Ganymede due to whistler mode waves are estimated. Pitch angle diffusion of magnetospheric electrons into the loss cone due to resonant wave–particle interaction of whistler mode waves is considered, and the resulting electron precipitation flux is calculated. The analytical yield spectrum approach is used for determining the energy deposition of electrons precipitating into the atmosphere of Ganymede. It is found that the intensities (4–30 R) calculated from the precipitation of magnetospheric electrons observed near Ganymede are inadequate to account for the observational intensities (≤ 100 R). This is in agreement with the conclusions reached in previous works. Some acceleration mechanism is required to energize the magnetospheric electrons. In the present work we consider the heating and acceleration of magnetospheric electrons by electrostatic waves. Two particle distribution functions (Maxwellian and kappa distribution) are used to simulate heating and acceleration of electrons. Precipitation of a Maxwellian distribution of electrons can produce about 70 R intensities of OI λ1356 Å emission for electron temperature of 150 eV. A kappa distribution can also yield a diffuse auroral intensity of similar magnitude for a characteristic energy of about 100 eV. The maximum contribution to the estimated intensity results from the dissociative excitation of O2. Contributions from the direct excitation of atomic oxygen and cascading in atomic oxygen are estimated to be only about 1 and 2 % of the total calculated intensity, respectively. The findings of this work are relevant for the present JUNO and future JUICE missions to Jupiter. These missions will provide new data on electron densities, electron temperature and whistler mode wave amplitudes in the magnetosphere of Jupiter near Ganymede.

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THERMODYNAMIC LIMIT IN STATISTICAL PHYSICS

International Journal of Modern Physics B ◽

10.1142/s0217979214300047 ◽

2014 ◽

Vol 28 (09) ◽

pp. 1430004 ◽

Cited By ~ 17

Author(s):

A. L. KUZEMSKY

Keyword(s):

Statistical Mechanics ◽

Thermodynamic Limit ◽

Statistical Physics ◽

Formal Series ◽

Distribution Functions ◽

Particle Systems ◽

Boltzmann Equations ◽

Qualitative Understanding ◽

Statistical Ensembles ◽

Equipartition Of Energy

The thermodynamic limit in statistical thermodynamics of many-particle systems is an important but often overlooked issue in the various applied studies of condensed matter physics. To settle this issue, we review tersely the past and present disposition of thermodynamic limiting procedure in the structure of the contemporary statistical mechanics and our current understanding of this problem. We pick out the ingenious approach by Bogoliubov, who developed a general formalism for establishing the limiting distribution functions in the form of formal series in powers of the density. In that study, he outlined the method of justification of the thermodynamic limit when he derived the generalized Boltzmann equations. To enrich and to weave our discussion, we take this opportunity to give a brief survey of the closely related problems, such as the equipartition of energy and the equivalence and nonequivalence of statistical ensembles. The validity of the equipartition of energy permits one to decide what are the boundaries of applicability of statistical mechanics. The major aim of this work is to provide a better qualitative understanding of the physical significance of the thermodynamic limit in modern statistical physics of the infinite and "small" many-particle systems.

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Revisiting the theory of the evolution of pick-up ion distributions: magnetic or adiabatic cooling?

Annales Geophysicae ◽

10.5194/angeo-25-2649-2007 ◽

2007 ◽

Vol 25 (12) ◽

pp. 2649-2659 ◽

Cited By ~ 23

Author(s):

H. J. Fahr

Keyword(s):

Solar Wind ◽

Particle Interaction ◽

Distribution Functions ◽

Particle Interactions ◽

Ion Distribution ◽

Ion Distributions ◽

Magnetic Cooling ◽

Adiabatic Cooling ◽

Diffusive Acceleration ◽

Nonlinear Energy

Abstract. We study the phasespace behaviour of heliospheric pick-up ions after the time of their injection as newly created ions into the solar wind bulk flow from either charge exchange or photoionization of interplanetary neutral atoms. As interaction with the ambient MHD wave fields we allow for rapid pitch angle diffusion, but for the beginning of this paper we shall neglect the effect of quasilinear or nonlinear energy diffusion (Fermi-2 acceleration) induced by counterflowing ambient waves. In the up-to-now literature connected with the convection of pick-up ions by the solar wind only adiabatic cooling of these ions is considered which in the solar wind frame takes care of filling the gap between the injection energy and energies of the thermal bulk of solar wind ions. Here we reinvestigate the basics of the theory behind this assumption of adiabatic pick-up ion reactions and correlated predictions derived from it. We then compare it with the new assumption of a pure magnetic cooling of pick-up ions simply resulting from their being convected in an interplanetary magnetic field which decreases in magnitude with increase of solar distance. We compare the results for pick-up ion distribution functions derived along both ways and can point out essential differences of observational and diagnostic relevance. Furthermore we then include stochastic acceleration processes by wave-particle interactions. As we can show, magnetic cooling in conjunction with diffusive acceleration by wave-particle interaction allows for an unbroken power law with the unique power index γ=−5 beginning from lowest velocities up to highest energy particles of about 100 KeV which just marginally can be in resonance with magnetoacoustic turbulences. Consequences for the resulting pick-up ion pressures are also analysed.

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Modeling the Dirichlet distribution using multiplicative functions

Nonlinear Analysis Modelling and Control ◽

10.15388/namc.2020.25.16518 ◽

2020 ◽

Vol 25 (2) ◽

Author(s):

Gintautas Bareikis ◽

Algirdas Mačiulis

Keyword(s):

Distribution Function ◽

Limit Distribution ◽

Multiplicative Function ◽

Remainder Term ◽

Dirichlet Distribution ◽

Distribution Functions ◽

Multiplicative Functions ◽

Cesàro Mean ◽

Cesaro Mean ◽

The One

For q,m,n,d ∈ N and some multiplicative function f > 0, we denote by T3(n) the sum of f(d) over the ordered triples (q,m,d) with qmd = n. We prove that Cesaro mean of distribution functions defined by means of T3 uniformly converges to the one-parameter Dirichlet distribution function. The parameter of the limit distribution depends on the values of f on primes. The remainder term is estimated as well.

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