Boltzmann equation transport

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

Shizgal et al. start with the Boltzmann transport equation and after a number of standard approximations write it in the space-independent form as follows ... 

About Rep, it decreases as temperature decreases, due to the fact that the number of phonons decreases. A full treatment of the problem, however, can only be obtained by solving the Boltzmann transport equation, which has only been solved for the case of quasi-free electrons. Further information and approximate solutions can be found in ref. [7,106,107], The general result of these calculations shows that at low temperature T < 0D/1O), the thermal resistance Rep is of the form b- T2. 

Instead of the usual Boltzmann transport equation approach, we will present a more phenomenological method of attaining the desired results (Bube, 1974). The hope here is that the physics will not so easily be obscured in the... 

See also Boltzmann s Distribution Law and Boltzmann Transport Equation. 

BOLTZMANN TRANSPORT EQUATION. The fundamental equa tion describing the conservation of particles which are diffusing in a scattering, absorbing, and multiplying medium. It states that the lime rate of change of particle density is equal to the rate of production, minus the rate of leakage and the rate of absorption, in the form of a partial differential equation sucli as... 

Since Boltzmann transport equation (BTE), which is derived to LBKE, is particle assumption-based theory, an SRS model can be implemented to BTE as follows ... 

The diffusive transport phenomena in nanowires can be described by a semiclassical model based on the Boltzmann transport equation. For carriers in a one-dimensional subband, important transport coefficients, such as the electrical conductivity, a, the Seebeck coefficient, S, and the thermal conductivity, Ke, are derived as (Sun et al., 1999b Ashcroft and Mermin, 1976a)... 

Plasmas typical of C02 laser discharges operate over a pressure range from 1 Torr to several atmospheres with degrees of ionization, that is, nJN (the ratio of electron density to neutral density) in the range from 10-8 to 10-8. Under these conditions the electron energy distribution function is highly non-Maxwellian. As a consequence it is necessary to solve the Boltzmann transport equation based on a detailed knowledge of the electron collisional channels in order to establish the electron distribution function as a function of the ratio of the electric field to the neutral gas density, E/N, and species concentration. Development of the fundamental techniques for solution of the Boltzmann equation are presented in detail by Shkarofsky, Johnston, and Bachynski [44] and Holstein [45]. 

In principle, the Boltzmann transport equation (BTE) can cover the regime where die lengdi and time scales are larger than carrier mean free time rand mean free length A. However, tremendous computational efforts are required in practice when the system length scale L and the process time scale t are getting larger. The BTE is, thus, usually... 

In the relaxation time approximation, the Boltzmann transport equation (BTE) takes the form [22,33] ... 

Narumanchi, S.V.J., J.Y. Murthy, and C.H. Amon. Boltzmann Transport Equation-based Thermal Modeling Approaches for Microelectronics, in 2nd International Thermal Sciences Seminar. 2004. Bled, Slovenia. 

Ladd, A., B. Moran, and W.G. Hoover, Lattice Thermal Conductivity A Comparison of Molecular Dynamics andAnharmonic Lattice Dynamics. Physical Review B, 1986. 34 p. 5058-5064. McGaughey, A.J. and M. Kaviany, Quantitative Validation of the Boltzmann Transport Equation Phonon Thermal Conductivity Model Under the Single-Mode Relaxation Time Approximation. Physical Review B, 2004. 69(9) p. 094303(1)-094303(11). 

As shown by Eq. (4), the rate of reactions involving electrons depends on the EVDF, /(r, V, f). Determination of the distribution function is one of the central problems in understanding plasma chemistry. The EVDF is defined in the phase-space element dydr such that /(r, v, f) dy dx is the number of electrons dn at time t located between r and r + dr which have velocities between v and v -I- d. When normalized by the total number of electrons n, it is a probability density function. The EVDF is obtained by solving the Boltzmann transport equation [42, 43, 48, 49]... 

Local thermodynamic equilibrium in space and time is inherently assumed in the kinetic theory formulation. The length scale that is characteristic of this volume is i whereas the timescale is xr. When either L i, ir or t x, xr or both, the kinetic theory breaks down because local thermodynamic equilibrium cannot be defined within the system. A more fundamental theory is required. The Boltzmann transport equation is a result of such a theory. Its generality is impressive since macroscopic transport behavior such as the Fourier law, Ohm s law, Fick s law, and the hyperbolic heat equation can be derived from this in the macroscale limit. In addition, transport equations such as equation of radiative transfer as well as the set of conservation equations of mass, momentum, and energy can all be derived from the Boltzmann transport equation (BTE). Some of the derivations are shown here. 

If the Boltzmann transport equation is multiplied by the factor v,eD(e)de on both sides and integrated over energy, then the equation transforms into... 

Kinetic theory is introduced and developed as the initial step toward understanding microscopic transport phenomena. It is used to develop relations for the thermal conductivity which are compared to experimental measurements for a variety of solids. Next, it is shown that if the time- or length scale of the phenomena are on the order of those for scattering, kinetic theory cannot be used but instead Boltzmann transport theory should be used. It was shown that the Boltzmann transport equation (BTE) is fundamental since it forms the basis for a vast variety of transport laws such as the Fourier law of heat conduction, Ohm s law of electrical conduction, and hyperbolic heat conduction equation. In addition, for an ensemble of particles for which the particle number is conserved, such as in molecules, electrons, holes, and so forth, the BTE forms the basis for mass, momentum, and energy conservation equa-... 

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