The Boltzmann Transport Equation

The lowest form of Eq. (3.24) is obtained by setting s = 1, i.e., we have contracted the Liouviiie equation over all space except one molecule. Thus, 

for so-called dilute gases we need only consider the behavior of any two molecules, i.e., we truncate the BBGKY hierarchy at s = 2 and set /s = 0. Thus, we also have 

The molecular pair function f2 is only weakly dependent on time 

The spatial part of the molecular pair function depends primarily on the separation distance vector ri2 = r2 - ri, i.e., 

Following the methods of Chap. 2, it can be readily shown that the characteristic equations of Eq. (3.29) lead to the following relationships (Prob. 3.8) 

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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. 

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] ... 

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-... 

The motion of particles in a fluid is best approached through the Boltzmann transport equation, provided that the combination of internal and external perturbations does not substantially disturb the equilibrium. In other words, our starting point will be the statistical thermod5mamic treatment above, and we will consider the effect of both the internal and external fields. Let the chemical species in our fluid be distinguished by the Greek subscripts a,p,.. . and let j (r, c, t) d Ld d d be the number of molecules of t5q)e a located in volume dVatr and having velocities between and c + dc etc. Note that we expect c and r are independent. Let the external force on molecules of type a be F. At any space point, r, the rate of increase of dfjd t), will be determined by ... 

In this section we will survey both the informal and formal versions of the kinetic theory of gases, starting with the simpler informal version. Here the basic idea is to combine both probabilistic and mechanical arguments to calculate quantities such as the equilibrium pressure of a gas, the mean free distance between collisions for a typical gas particle, and the transport properties of the gas, such as its viscosity and thermal conductivity. The formal version again uses both probabilistic and mechanical arguments to obtain an equation, the Boltzmann transport equation, that determines the distribution function,/(r, v, t), that describes the number of gas particles in a small spatial region, 5r, about a point r, and in a small region of velocities,... 

Finally, all of the F -terms can be inserted in (A3.1,3 IF and dividing by 5t6r6v gives the Boltzmann transport equation... 

To avoid the account of the edge effects let us consider rather long structures (L > 50 nm), i.e. we will consider the armchair single-wall carbon nanotubes with the length greater than electron mean free path [2-6]. To describe the electron-phonon transport in nanotubes like that the semiclassical approach and the kinetic Boltzmann equation for one-dimensional electron-phonon gas can be used [4,6]. In this connection the purpose of the present study is to develop a model of electron transport based on a numerical solution of the Boltzmann transport equation. 

A Round-off Free Solution of the Boltzmann Transport Equation in Slab Geometry... 

The radiative transfer equation is a simplification of the Boltzmann transport equation (developed by Ludwig Boltzmann in 1872 to describe ideal gas of identical particles) made possible by two characteristics of photons as particles ... 

The Ziman-Faber model for liquid metals (Ziman, 1961 Faber and Ziman, 1965) has been widely used to describe the resistivity behaviour of amorphous metals. It is based on the nearly-free-electron approach and the Boltzmann transport equation. When all multiple two-site scattering corrections are neglected, the resistivity for a pure liquid metal can be represented by means of the equation... 

V11.15. Computational methods that directly solve forms of the Boltzmann transport equation to obtain k j are preferred for use in the criticality safety analysis. The deterministic discrete ordinates technique and the Monte Carlo statistical technique are the typical solution formulations used by most criticality analysis codes. Monte Carlo analyses are prevalent because these codes can better model the geometry detail needed for most criticality safety analyses. Well documented and weU validated computational methods may require less description than a limited-use and/or unique... 

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