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Connection to the Microscopic View

The branch of physics known as statistical mechanics (or kinetic theory of gases) attempts to related the macroscopic properties of an assembly of particles to the microscopic properties of the particles themselves. Statistical mechanics, as its name implies is not concerned with the actual motions or interactions of individual particles, but investigates instead their most probable behavior. The state of a system of particles is completely specified classically at a particular instant if the position r and velocity v of each of its constituent particles are known. The number of particles occupying an infinitesimal cell in the phase space r and v is determined by the distribution function f (r,v,t), where t is the time. The distribution function is normally conserved except from the effect of collisions. Thus, the most general formula for the evolution of the distribution function can be expressed as:

---------- (7)
where the pairing indices (in the subscript and superscript) indicate a sum over i = 1, 2, 3; ai is the acceleration which is equal to the force per unit mass on the particles, and the right-hand side of the equation represents the effect of collisions.

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., v0, v1, and v2, the continuity equation, Navier-Stokes equations, and conservation of energy respectively in fluid dynamics can be derived directly from Eq.(7) by taking the average over the velocity space. Thus, the density is defined by:
(xi, t) = f (xi,vk,t) d3vk
and any average quantity such as the fluid velocity ui is given by:
ui(xi, t) = (f (xi,vk,t) vk d3vk) /

Boltzmann Equation 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)
The collision term on the right hand side of Eq.(7) is substituted by a phenomenological term in Eq.(8), where is the relaxation time - a characteristic decay constant for returning to the equilibrium state, and f0 is the equilibrium distribution. The solution for this equation is:

f = fi e-t/ + f0 ( 1 - e-t/ ) ---------- (9)

where fi is the initial distribution. It shows that f approaches f0, and the collision term vanishes for time t >> .

In thermodynamic equilibrium the distribution function f 0 does not change with time, it can be expressed in the form:

---------- (10)
where the density and temperature can be a function of r in general, and v0 is the velocity of the gas moving as a whole.

In the special case when there are no external forces such as gravity or electrostatic interactions, the density and temperature are constant, with v0 = 0, and by summing over all 3-D space, Eq.(10) becomes:
---------- (11)
which is called Maxwell-Boltzmann distribution, where N is the total number of particles. It is actually a formula about the distribution of kinetic energy E = mv2/2 among the particles.

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., , where a and b denote two different quantum states. They cannot be the same, because the wave function and hence the probability of such occurrence becomes zero. On the other hand the two bosons wave function is symmetric, e.g., ; the wave function does not vanish when a = b. Thus the bosons can occupy the same states. Figure 13 shows the formula and graph for each distribution, where A = e is a normalization constant. The classical and Bose-Einstein distribution are similar except when kT >> E. Near absolute zero
Distribution Functions 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 Ef 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 = Ef for all temperatures. At low temperature most of the low energy states with
E < Ef are filled. At high temperature with kT >> (E - Ef), the distribution function

Figure 13 Distribution Functions
[view large image]

becomes f (E) ~ (1/2) (1 - (E - Ef) / 2kT). Thus in this case, the energy states with
E < Ef are more than half-filler; while for E > Ef they are less than half-filled.

Maxwell Distribution 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 vrms and the temperature T can be derived from such distribution function:

m vrms2 = 3 k T    or   M vrms2 = 3 R T ---------- (12)

where m denotes the mass of the molecule, M = mN0 is the molecular weight/mole, N0 is the Avogadro's number, and k = R / N0 = 1.38x10-16 erg/Ko 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 vrms of the particles and the macroscopic property T.

Classical/Quantum Criterion 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 = vrms), which yields:

= h / (3mkT)1/2 ---------- (13) ,

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)1/3 where V is the volume and N is the number of particles. When the thermal de Broglie wavelength is much smaller than the inter-particle distance, the gas can be considered to be a classical or Maxwell-Boltzmann gas. On the other hand, when the thermal de Broglie wavelength is on the order of, or larger than the inter-particle distance, quantum effects will dominate and the gas must be treated as a Fermi gas or a Bose gas, depending on the nature of the gas particles (Figure 15a, where TF = EF/k, TBEC ~ 3.3x[(2n2/3)/(mk)], n is the number density, m the boson mass and TC = TF or TBEC). It follows from Eq.(13) that massive particles in hot systems should not behave quantum mechanically.

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 l to a representative physical length scale L, i.e., K = l / L . Problems with Knudsen numbers at or above unity, i.e., long mean free path; must be evaluated using statistical mechanics for reliable solutions. Dense system with K<1 can be treated as continuum.
Mean Free Path
The mean free path (Figure 15b) can be expressed mathematically as:

l = 1 / nA = (l1 + l2 + l3 + ... + lN) / N---------- (14)

where n is the number density, A is the collision cross section, li 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

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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