Boltzmann transport theory

Kinetic theory formulation assumes local thermodynamic equilibrium in space and time. It cannot be applied when length scale Lc l or and time scale f t or t. The Boltzmann transport theory is more fundamental and can be applied at these situations. The BTE applies to all particles, that is, electrons, ions, photons, phonons, gas molecules, and so on. 

t) is the statistical distribution of ensemble of particles, r is the position vector, p is the momentum vector, v is the average heat carrier velocity, and F is the force applied to the particle. The term scattering is the time rate of change of the scattering term or the so-called collision term. The Boltzmann equation describes the time rate of change of the heat carrier distribution due to diffusion. The Boltzmann equation in one dimension without any external force in reduced form is 

Note that among all particles, only electrons and ions can experience appreciable force due to electric and magnetic fields. As discussed in the earlier chapter, BGK model can be used for simplifying collision integral as 

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The same k p scheme has been extended to the study of transport properties of CNTs. The conductivity calculated in the Boltzmann transport theory has shown a large positive magnetoresistance [18], This positive magnetoresistance has been confirmed by full quantum mechanical calculations in the case that the mean free path is much larger than the circumference length [19]. When the mean free path is short, the transport is reduced to that in a 2D graphite, which has also interesting characteristic features [20]. 

The self-consistent theoretical models based on the Boltzmann transport theory are used to characterize the microscale heat transfer mechanism by explaining mutual interactions among lattice temperature, and number density and temperature of carriers [12]. Especially, a new parameter related with non-equilibrium durability is introduced and its characteristics for various laser pulses and fluences are discussed. This study also investigates the temporal characteristics of carrier temperature distribution, such as the one- and two-peak structures, according to laser pulses and fluences, and establishes a regime criterion between one-peak and two-peak sttuctures for picosecond laser pulses. 

Although the kinetic theory has been successfully applied to predict the thermal conductivity, it cannot be used under nonequilibrium conditions. For such cases, the Boltzmann transport theory is required. 

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-... 

In 1872, Boltzmaim introduced the basic equation of transport theory for dilute gases. His equation detemiines the time-dependent position and velocity distribution fiinction for the molecules in a dilute gas, which we have denoted by /(r,v,0- Here we present his derivation and some of its major consequences, particularly the so-called //-tlieorem, which shows the consistency of the Boltzmann equation with the irreversible fomi of the second law of themiodynamics. We also briefly discuss some of the famous debates surrounding the mechanical foundations of this equation. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

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. 

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

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 ... 

Thus, the particle charge distribution is approximated by the Boltzmann equation. This expression holds best for particles larger than about 1 /.tm. For smaller particles, the flux terms (2,49) based on continuum transport theory must be modified semiempirically. The results of calculations of the fraction of charged particles are given in Table 2.2. The fraction refers to particles of charge of a given sign. 

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. 

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,... 

Perturbation theory in reactor physics is usually formulated and used in the integrodifferential formulation of transport theory (or in the diffusion approximation to this formulation). This formulation is convenient for use because (1) the perturbations in the operators of the Boltzmann equation are linear with the physical perturbations in the system, and (2) most computer codes in use solve the integrodifferential (or the diffusion) equations. 

The assumptions of transport theory. As you well know, the theory which permits one to calculate the neutron densities or fluxes is an essentially statistical theory and is called transport theory. This theory goes back to the last century and Boltzmann s book on the kinetic theory of gases [1] can still be read to advantage. The fimdamental concept is the so-called neutron flux x,EySl t), This quantity gives for the time t the number of neutrons, multiplied with their speed, which satisfy the following conditions ... 

The calculation of transport coefficients and inverse transport coefficients, such as conductivity and viscosity, is an aim of transport theory. Calculations from first principles in transport theory start from non-equilibrium statistical mechanics. Because of the difficulties involved in calculations in non-equilibrium statistical mechanics, transport theory uses approximate methods, including the kinetic theory of gases and kinetic equations, such as the Boltzmann equation. 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

Second, we know that the Boltzmann-Enskog transport theory gives a reasonable description of transport coefficients and the short-time properties of correlation functions, " so the approximation for G should be formulated in such a way that the leading term corresponds to the Boltzmann-Enskog approximation. Third, in deriving the correction to the leading term we must consider the effects of recollision processes. ... 

The approximation wherein one retains the first term in the memory function expression (105) is of special interest because it leads to a kinetic equation that is closely related to the Boltzmann-Enskog equation in transport theory. In this section we will investigate this particular approximation in some detail not only from the standpoint of further analytical analysis but also from the standpoint of practical calculations. We will see that within certain limitations the approximation results in a reasonably realistic description of dense gases and liquids, and in this sense represents the first step in a systematic microscopic calculation. 

More elaborate theories of transport phenomena make use of the Boltzmann transport equation or computer simulations. 

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