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Micro/Nano Heat Transfer
Published in Sadık Kakaç, Hongtan Liu, Anchasa Pramuanjaroenkij, Heat Exchangers, 2020
Sadık Kakaç, Hongtan Liu, Anchasa Pramuanjaroenkij
Continuum flow regime description fails when flow enters the transition flow regime. The molecular approach is not enough to express the transition flow regime, and the effect of Knudsen number becomes more important. Because of the failure of continuum description, Boltzmann equation should be used to investigate the atomic-level flows in transition regime. Boltzmann equation can be solved by several methods such as molecular dynamics (MD) and direct simulation Monte Carlo (DSMC). In addition, the simplified Boltzmann equation can be solved by using lattice Boltzmann method (LBM). However, the current computational methods (MD and DSMC) cannot provide an effective solution for transition flow regime.9 Thermal creep is an important phenomenon for transition flow regime. Therefore, it should be considered while studying laminar to turbulent transition regime analytically.10
Spin Polarization by Current
Published in Evgeny Y. Tsymbal, Igor Žutić, Spintronics Handbook: Spin Transport and Magnetism, Second Edition, 2019
Sergey D. Ganichev, Maxim Trushin, John Schliemann
In general, the Boltzmann equation describes the time evolution of the particle distribution function f(t,r,k), in the coordinate r and momentum k space. To describe the electron kinetics in presence of a small homogeneous electric field E in a steady state, one usually follows the standard procedure widely spread in the literature on solid state physics. The distribution function is then represented as a sum of an equilibrium f0(Esk) and nonequilibrium f1(s,k) contributions. The first one is just a Fermi-Dirac distribution, and the second one is a time and coordinate independent nonequilibrium correction linear in E. This latter contribution should be written down as a solution of the kinetic equation; however, it might be also deduced from qualitative arguments in what follows.
Lasers
Published in Abdul Al-Azzawi, Photonics, 2017
However, the Boltzman equation only describes conditions of thermal equilibrium. Lasers are not operated in thermal equilibrium. Instead, the upper state is populated by pumping it via some non-equilibrium process. A pulse of light, an electrical spark, or a chemical reaction can all be used to populate the upper laser state.
An Investigation of Uncertainty Propagation in Non-equilibrium Flows
Published in International Journal of Computational Fluid Dynamics, 2022
The Boltzmann equation depicts the time–space evolution of particle probability distribution function. In the absence of external force, it can be written as where f is the particle distribution function, is the position in physical space, is the particle velocity and is the collision term. Considering the possible uncertainties in intermolecular collisions, initial and boundary conditions, we can extend the Boltzmann equation with stochastic settings and reformulate the gas kinetic system, i.e. where is the random variable, and denotes the boundary operator. For brevity, the following analysis will be conducted on basis of the Bhatnagar–Gross–Krook (BGK) model where is the Maxwellian distribution function, ν is the collision frequency and , where m is the particle mass and k is the Boltzmann constant.
Neutronics Calculation Advances at Los Alamos: Manhattan Project to Monte Carlo
Published in Nuclear Technology, 2021
Avneet Sood, R. Arthur Forster, B. J. Archer, R. C. Little
Estimating the critical masses required solutions to the neutron transport equation. The neutron transport equation is a mathematical description of neutron multiplication and movement through materials and is based on the Boltzmann equation, which was used to study the kinetic theory of gasses. The Boltzmann equation was developed in 1872 by Ludwig Boltzmann to explain the properties of dilute gasses by characterizing the collision processes between pairs of molecules.9 It is a mathematical description of how systems that are not in a state of equilibrium evolve (e.g., heat flowing from hot to cold, fluid movement, etc.) and can be used to describe the macroscopic changes in a system. The Boltzmann equation is a nonlinear integro-differential equation that involves terms that depend on particle position and momentum and includes terms describing collisions and free streaming.10
Asymptotic-stability of the inhomogeneous Boltzmann equation in the Robertson–Walker space–time with Israel particles
Published in Applicable Analysis, 2020
Etienne Takou, Fidèle L. Ciake Ciake
The central assumption of relativistic kinetic theory is that the averaged properties of the gas are described by a one-particle distribution function f associated to the gas. f is a nonnegative function which roughly speaking tells us the average number of particles at time , position () and momentum . The Boltzmann equation describes the time evolution of a system where collisions between the gas particles can no longer be neglected. This occurs when the mean free path is much shorter than the characteristic length scale associated with the system. contains a great deal of information, it can be a practical tool for determining detailed properties of dilute gases and plasmas as well as calculating many important physical quantities. For instance; the temperature (it is a measure of the agitation of the particles, more precisely the computation of kinetic energy) and the the pressure (the pressure of a gas on the walls comes from the shocks between it and particles).