Classical Distribution Function and Transport Equation

2017 ◽  
pp. 121-141
Author(s):  
Massimo Rudan
Keyword(s):  
Distribution Function ◽  
Transport Equation ◽  
Classical Distribution

scholarly journals Spherical Harmonic Modeling of a 0.05 μm Base BJT: A Comparison with Monte Carlo and Asymptotic Analysis

VLSI Design ◽  
1998 ◽  
Vol 8 (1-4) ◽  
pp. 147-151 ◽  
Author(s):  
C.-H. Chang ◽  
C.-K. Lin ◽  
N. Goldsman ◽  
I. D. Mayergoyz
Keyword(s):  
Monte Carlo ◽  
Distribution Function ◽  
Band Structure ◽  
Asymptotic Analysis ◽  
Energy Distribution ◽  
Transport Equation ◽  
Spherical Harmonic ◽  
Boltzmann Transport Equation ◽  
System Of Equations ◽  
Boltzmann Transport

We perform a rigorous comparison between the Spherical Harmonic (SH) and Monte Carlo (MC) methods of solving the Boltzmann Transport Equation (BTE), on a 0.05 μm base BJT. We find the SH and the MC methods give very similar results for the energy distribution function, using an analytical band-structure, at all points within the tested devices. However, the SH method can be as much as seven thousand times faster than the MC approach for solving an identical problem. We explain the agreement by asymptotic analysis of the system of equations generated by the SH expansion of the BTE.

scholarly journals A Quantum Mechanical Calculation of the Radial Distrbution Function for a Plasma

1968 ◽  
Vol 21 (2) ◽  
pp. 121 ◽  
Author(s):  
AA Barker
Keyword(s):  
Distribution Function ◽  
Radial Distribution Function ◽  
Radial Distribution ◽  
Bound States ◽  
Quantum Mechanical Calculation ◽  
Quantum Mechanical ◽  
Temperature Dependent ◽  
Mechanical Calculation ◽  
Relative Contribution ◽  
Classical Distribution

A quantum mechanical calculation of the radial distribution function g".m.(r) for unlike particles in a hydrogenous plasma is presented. Results for a neutral plasma over a range of temperatures show that gq.m.(r) differs significantly from the corresponding classical distribution function g.(r) = exp(fJel/r) when r is less than a chosen distance r" the value of which is temperature dependent. The effect of shielding, the relative contribution from scattered and bound states, and the relation to percentage ionization are discussed.

scholarly journals Quarkonium semiclassical transport in quark-gluon plasma: factorization and quantum correction

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Xiaojun Yao ◽  
Thomas Mehen
Keyword(s):  
Distribution Function ◽  
Transverse Momentum ◽  
Transport Equation ◽  
Quark Gluon Plasma ◽  
Parton Distribution ◽  
Gluon Distribution ◽  
Reaction Rate ◽  
Quantum Correction ◽  
Parton Distribution Function ◽  
Gluon Plasma

Abstract We study quarkonium transport in the quark-gluon plasma by using the potential nonrelativistic QCD (pNRQCD) effective field theory and the framework of open quantum systems. We argue that the coupling between quarkonium and the thermal bath is weak using separation of scales, so the initial density matrix of the total system factorizes and the time evolution of the subsystem is Markovian. We derive the semiclassical Boltzmann equation for quarkonium by applying a Wigner transform to the Lindblad equation and carrying out a semiclassical expansion. We resum relevant interactions to all orders in the coupling constant at leading power of the nonrelativistic and multipole expansions. The derivation is valid for both weakly coupled and strongly coupled quark-gluon plasmas. We find reaction rates in the transport equation factorize into a quarkonium dipole transition function and a chromoelectric gluon distribution function. For the differential reaction rate, the definition of the momentum dependent chromoelectric gluon distribution function involves staple-shaped Wilson lines. For the inclusive reaction rate, the Wilson lines collapse into a straight line along the real time axis and the distribution becomes momentum independent. The relation between the two Wilson lines is analogous to the relation between the Wilson lines appearing in the gluon parton distribution function (PDF) and the gluon transverse momentum dependent parton distribution function (TMDPDF). The centrality dependence of the quarkonium nuclear modification factor measured by experiments probes the momentum independent distribution while the transverse momentum dependence and measurements of the azimuthal angular anisotropy may be able to probe the momentum dependent one. We discuss one way to indirectly constrain the quarkonium in-medium real potential by using the factorization formula and lattice calculations. The leading quantum correction to the semiclassical transport equation of quarkonium is also worked out. The study can be easily generalized to quarkonium transport in cold nuclear matter, which is relevant for quarkonium production in eA collisions in the future Electron-Ion Collider.

Path-Integral Quantum Transport Equation for the Study of Homogeneous Semiconductors

1998 ◽  
Vol 12 (06) ◽  
pp. 673-687 ◽  
Author(s):  
C. Schulz-Mirbach
Keyword(s):  
Distribution Function ◽  
Transport Equation ◽  
Quantum Transport ◽  
Homogeneous Problem ◽  
Path Integrals ◽  
Statistical Procedure ◽  
Matrix Elements ◽  
Feynman Path ◽  
Feynman Path Integrals ◽  
Integral Quantum

Based on perturbation theory for real-time Feynman path-integrals, we derive a quantum transport equation for an electron in a semiconductor that uses a time-discretized formulation of the Feynman kernel in momentum space for the inclusion of scattering processes of any kind on the level of matrix-elements of the perturbing potential. The movement of the electron in the electric field is accounted for by the analytical solution of the homogeneous problem while the scattering processes are accounted for in a statistical procedure. We apply the method to calculate the momentum and energy distribution function for homogeneous GaAs.

scholarly journals Determination of partition surface of grained material by means of non-classical approximation methods of distributions functions of particle size and density

2016 ◽  
Vol 32 (1) ◽  
pp. 137-154
Author(s):  
Tomasz Niedoba
Keyword(s):  
Particle Size ◽  
Distribution Function ◽  
Statistical Methods ◽  
Particle Density ◽  
Distribution Functions ◽  
Alternative Methods ◽  
Hard Coal ◽  
Classical Distribution ◽  
Partition Surface ◽  
Non Parametric

Abstract In this paper, the grained material analyzed was hard coal collected from one of the mines located in Upper Silesia. Material was collected from a dust jig where it was separated in industrial conditions by concentrate and waste. It was then screened in sieves and it was separated in dense media into density fractions. Both particle size distribution and particle density distribution for feed and concentrate were approximated by several classical distribution functions. The best results were obtained by means of the Weibull (RRB) distribution function. However, because of the unsatisfying quality of approximations it was decided to apply non-parametric statistical methods, which became more and more popular alternative methods in conducting statistical investigations. In the paper, the kernel methods were applied to this purpose and the Gauss kernel was accepted as the kernel function. Kernel method, which is relatively new, gave much better results than classical distribution functions by means of the least squared method. Both classical and non-parametric obtained distribution functions were evaluated by means of mean standard error, the values of which proved that they sufficiently well approximate the empirical data. Such function forms were then applied to determine the theoretical distribution function for vector (D, P), where D is the random variable describing particle size and P – its density. This approximation was sufficiently acceptable. That is why it served to determine the equation of partition surface dependent on particle size and particle density describing researched material. The obtained surface proves that it is possible to evaluate material separation which occurs during mineral processing operations, such as jigging, by means of more than one feature of researched material. Furthermore, its quality confirms that it is justified to apply non-parametric statistical methods instead of commonly used classical ones.

scholarly journals Cosmic Baryon Kinetics Expressed by Its Velocity Moments over Times after the Matter Recombination

2021 ◽  
Vol 4 (2) ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Distribution Function ◽  
Transport Equation ◽  
Cosmic Time ◽  
Gas Distribution ◽  
Volume Changes ◽  
Expansion Of The Universe ◽  
Velocity Moments ◽  
The Universe

As we are going to show here it is not easily understandable how cosmic gases like H-atoms, after the recombination of cosmic matter, do thermodynamically behave under the ongoing Hubble-like expansion of the universe. The question namely is not easy to answer; how cosmic gas atoms do in fact recognize the expansion of cosmic 3- space. Contemporary mainstream cosmology takes for granted that gas atoms do react polytropically or even adiabatically to cosmic volume changes and thus do get more and more tenuous and colder in accordance with gas- and thermo- dynamics. However, one has to face the fact that cosmic gases at the recombination era are already nearly collisionless over scales of 10 AU, and how gases react to cosmic volume changes under such conditions is not a trivial problem. We derive in this article a kinetic transport equation which describes the evolution of the gas distribution function f(t, v) in cosmic time t and velocity space of v. This partial differential equation does not allow for a solution in form a separation of the two variables t and v, but instead we can find solutions for two moments of f(v, t), i.e. the density n(t) and the pressure P(t). Then we show that using kappa-like functions for the cosmic gas we can derive such functions as function of their velocity moments, i.e. as functions of cosmic time. It means we understand the kinetic evolution of the cosmic gas by understanding the evolution in cosmic time of their moments.

Die Methode der Korrelationsfunktion in der Theorie der Supraleitung

1967 ◽  
Vol 22 (6) ◽  
pp. 845-856 ◽  
Author(s):  
Gerhart Lüders
Keyword(s):  
Distribution Function ◽  
Phase Space ◽  
Electric Current ◽  
Current Distribution ◽  
Current Density ◽  
Diffusion Approximation ◽  
Pair Potential ◽  
Critical Fields ◽  
Ginzburg Landau ◽  
Classical Distribution

The electric current density in a superconductor is calculated up to second order in the pair potential. The current density can be formulated in terms of the classical distribution function in phase space which was introduced in previous papers. The resulting expression is specialized both to the Ginzburg-Landau approximation and to the diffusion approximation. Finally, we discuss the current distribution at the critical fields Hc2 and Hc3

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