Boltzmann transport equation

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For a nonequilibrium system, the mean free path theory is no longer valid, and the Boltzmann equation should be used to describe the molecular velocity distribution in the system. For low-density nonreacting monatomic gas mixtures, the random molecular movement can be described by the molecular velocity distribution function {f_i}({\mathbf{c}},{\mathbf{x}},t), where {\mathbf{c}} is the particle velocity and {\mathbf{x}} is the position vector in the mixture. At time t, the probable number of molecules of the ith species that are located in the volume element dx at position x and have velocity within the range d{\mathbf{c}} about {\mathbf{c}} is {f_i}({\mathbf{c}},{\mathbf{x}},t)d{\mathbf{c}}d{\mathbf{x}}. The evolution of the velocity distribution function with time can be described using the Boltzmann equation

\frac{{D{f_i}}}{{Dt}} = \frac{{\partial {f_i}}}{{\partial t}} + {\mathbf{c}} \cdot {\nabla _{\mathbf{x}}}{f_i} + {\mathbf{a}} \cdot {\nabla _{\mathbf{c}}}{f_f} = {\Omega _i}(f) \qquad \qquad(1)

where {\nabla _{\mathbf{x}}} and {\nabla _{\mathbf{c}}} are the \nabla operator with respect to x and c, respectively (see Appendix G), a is the particle acceleration (m/s2), and Ωi is a five-fold integral term that accounts for the effect of molecular collision on the change of velocity distribution function fi. The Boltzmann equation can also be considered to be a continuity equation in a six-dimensional position-velocity space ({\mathbf{x}}{\rm{ and }}{\mathbf{c}}). The velocity distribution function is related to the number density (number of particles per unit volume) by

Failed to parse (unknown function\mathmathbf): \int {{{\text{f}}_i}({\mathmathbf{c}},{\mathmathbf{x}},t)d{\mathmathbf{c}}} = {\mathfrak{N}_i}({\mathmathbf{x}},t) \qquad \qquad(2)

Note that the density is Failed to parse (unknown function\mathmathbf): \rho ({\mathmathbf{x}},t) = m\mathfrak{N}({\mathmathbf{x}},t)

where m is the mass of the particle. The total number of particles N inside the volume V as a function of time is

Failed to parse (syntax error): N(t) = \int_V {\int_c {{\rm{f}}({\mathbf{c}},{\mathbf{x}},t)d{\mathbf{c}}} } d{\mathbf{x}} \qquad\\qquad(3)

In thermodynamic equilibrium, f is independent of time and space, i.e., f({\mathbf{c}},{\mathbf{x}},t) = f({\mathbf{c}}). It can be demonstrated that the stress tensor, heat flux, and diffusive mass flux can be obtained from the solution of the velocity distribution function fi. More detailed information about the Boltzmann equation and its applications related to transport phenomena in multiphase systems can be found in Chapter 3. Most macroscopic transport equations, such as Fourier’s law of conduction, Navier-Stokes equation for viscous flow, or the equation of radiative transfer for photons and phonons can be developed from the Boltzmann transport equation using local equilibrium assumptions. More detailed information concerning formulation and solution techniques of the Boltzmann equation can be found in Tien and Lienhard (1985) and Ceracignani (1988). In addition, the Boltzmann equation can also be used to describe transport of electrons and electron-lattice interaction and to yield the two-step heat conduction model for electron and lattice temperatures for nanoscale and microscale heat transfer (Qiu and Tien, 1993; Chen, 2004; Zhang and Chen, 2007; Zhang, 2007).