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---------- (7) |

This is known as the Boltzmann equation. It is very useful as mathematic tool in treating the process of fluid flow. By multiplying the distribution function with the power of the velocity, e.g., v

(x

and any average quantity such as the fluid velocity u

u

Analytical solutions of the Boltzmann equation are possible only under very restrictive assumptions. Direct numerical methods for computer simulation have been limited by the complexity of the equation, which in the complete 3-D time-dependent form requires seven independent variables for time, space and velocity. A 2-dimensional animation of a flow process is presented by clicking Figure 12. It shows the development of a clump of gas molecules initially released from the left. The particles flow to the right, reflected by the wall at the other end, then established an equilibrium configuration after some 4000 collisions between the particles. | |

## Figure 12 Boltzmann Equation Simulation [view animation] |

Considering the simplest case when the force on the particles is switched off instantaneously. If the distribution is space-independent, then Eq.(7) is reduced to:

---------- (8) |

f = f

where f

In thermodynamic equilibrium the distribution function f

---------- (10) |

In the special case when there are no external forces such as gravity or electrostatic interactions, the density and temperature are constant, with v

---------- (11) |

There are three kinds of energy distribution function depending on whether the particles are treated as classical or quantum. In quantum theory, the wave packets overlapped when the particles come together, it is impossible to distinguish their identities. Thus, it results in different behaviour in quantum statistics. Further modification is caused by the exclusion principle, which allows only one fermion in a given state. This is related to the fact that the two-particle wave function is anti-symmetric for fermion, e.g.,

temperature, most of the bosons occupy the same state with E ~ 0. This is the Bose-Einstein condensate first discovered in 1995. Another example of Bose-Einstein distribution is the black-body radiation. In Fermi-Dirac distribution, the normalization constant A can be re-defined as A = e^{-Ef}, where E_{f} is known as the Fermi energy, which has a value of a few ev (~ 10000 K) for the electron gas in many metals. Note that f (E) = 1/2 at E = E_{f} for all temperatures. At low temperature most of the low energy states with E < E _{f} are filled. At high temperature with kT >> (E - E_{f}), the distribution function
| |

## Figure 13 Distribution Functions |
becomes f (E) ~ (1/2) (1 - (E - E_{f}) / 2kT). Thus in this case, the energy states with E < E _{f} are more than half-filler; while for E > E_{f} they are less than half-filled. |

In classical statistic, the velocity distribution of the ideal gas is given by the Maxwell distribution as shown in Figure 14. A relationship between the root-mean-square velocity v_{rms} and the temperature T can be derived from such distribution function: m v _{rms}^{2} = 3 k T or M v_{rms}^{2} = 3 R T ---------- (12)where m denotes the mass of the molecule, M = mN _{0} is the molecular weight/mole, N_{0} is the Avogadro's number, and k = R / N_{0} = 1.38x10^{-16} erg/K^{o} is the Boltzmann constant .
| |

## Figure 14 Maxwell Distribution [view large image] |
The formula in Eq.(12) provides a link between the microscopic root-mean-square velocity v_{rms} of the particles and the macroscopic property T. |

The criterion to adopt quantum or classical statistic for a system depends on the value of the "Thermal de Broglie wavelength". Originally, the de Broglie wavelength = h/p is defined for a single particle with momentum p = mv (h is the Planck constant). It has been generalized to an aggregate of gas particles in an ideal gas at specified temperature T. The "Thermal de Broglie wavelength" is derived by substituting Eq.(12) to the de Broglie wavelength (with v = v_{rms}), which yields: = h / (3mkT) ^{1/2} ---------- (13) ,
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## Figure 15a Classical/Quantum Criterion [view large image] |

Now we can take the average inter-particle spacing in the gas to be approximately (V/N)

Another criterion is for determining whether to use thermodynamics (a macroscopic description) or statistical mechanics (with microscopic consideration). The Knudsen number K is used to make the selection. It is the ratio of the molecular mean free path length

The mean free path (Figure 15b) can be expressed mathematically as: l = 1 / nA = (l_{1} + l_{2} + l_{3} + ... + l_{N}) / N---------- (14)where n is the number density, A is the collision cross section, l_{i} is the path length between collisions, i.e., length of the free path, and N is the total number of collisions. The concept of mean free path may be visualized by thinking of a man shooting a rifle aimlessly into a forest. Most of the bullets will hit trees, but some bullets will travel much farther than others. The
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## Figure 15b Mean Free Path [view large image] |
average distance traveled by the bullets will depend inversely on both the denseness of the woods and the size of the trees. |

See "Statistical Mechanics" for detail.

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