The Boltzmann equation

All the derivations of the Boltzmann equation are based on a number of assumptions and hypotheses, making the analyzes somewhat ad hoc irrespective of the formal mathematical rigor and complexity accomplished. So in this book a heuristic theory, which is physically revealing and equally ad hoc to the more fundamental derivations of the Boltzmann equation, is adopted. It is stressed that the notation used resembles that introduced by Boltzmann [6] and is not strictly in accordance with the formal mathematical methods of classical mechanics. However, some aspects of the formal formulations and vocabulary outlined in sect. 2.2 are incorporated although somewhat based on intuition and empirical reasoning rather than fundamental principles as discussed in sect. 2.3. 

The Boltzmann equation can be derived using a procedure founded on the Liouville theorem . In this case the balance principle is applied to a control volume following a trajectory in phase space, expressed as 

Applying numerous theorems, similar but not identical to those used in chap 1 deriving the governing equations in fluid dynamics, one arrives at the Boltzmann equation on the form 

The simplified interpretation of the LiouviUe theorem discussed in sect. 2.2 is used here deriving the translational terms, intending to make the formulation more easily available. 

The model derivation given above using the Liouville theorem is in many ways equivalent to the Lagrangian balance formulation [83]. Of course, a consistent Eulerian balance formulation would give the same result, but includes some more manipulations of the terms in the number balance. However, the Eulerian formulation is of special interest as we have adopted this framework in the preceding discussion of the governing equations of classical fluid dynamics, chap 1. 

2 are incorporated although somewhat based on intuition and empirical reasoning rather than fundamental principles as discussed in Sect. 2.3. 

This relation is denoting that the rate of change of / for a system of a number of particles moving with the phase space velocity vector (, equals the rate at which / is altered by collisions. 

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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. 

A3.1.3.2 THE CHAPMAN-ENSKOG NORMAL SOLUTIONS OF THE BOLTZMANN EQUATION... 

Fox R F and Uhlenbeck G E 1970 Contributions to non-equilibrium thermodynamics. II. Fluctuation theory for the Boltzmann equation Rhys. Fluids 13 2881... 

McNamara G R and Zanetti G 1998 Use of the Boltzmann equation to simulate lattice-gas automata Phys. Rev. Lett. 61 2332... 

An interesting historical application of the Boltzmann equation involves examination of the number density of very small spherical globules of latex suspended in water. The particles are dishibuted in the potential gradient of the gravitational field. If an arbitrary point in the suspension is selected, the number of particles N at height h pm (1 pm= 10 m) above the reference point can be counted with a magnifying lens. In one series of measurements, the number of particles per unit volume of the suspension as a function of h was as shown in Table 3-3. 

In a plasma, the constituent atoms, ions, and electrons are made to move faster by an electromagnetic field and not by application of heat externally or through combustion processes. Nevertheless, the result is the same as if the plasma had been heated externally the constituent atoms, ions, and electrons are made to move faster and faster, eventually reaching a distribution of kinetic energies that would be characteristic of the Boltzmann equation applied to a gas that had been... 

The Boltzmann equation (Equation 18.2) shows that, under equilibrium conditions, the ratio of the number (n) of ground-state molecules (A ) to those in an excited state (A ) depends on the energy gap E between the states, the Boltzmann constant k (1.38 x 10" J-K" ), and the absolute temperature T(K). 

The thermodynamic probability is converted to an entropy through the Boltzmann equation [Eq. (3.20)] so we can write for the entropy of the mixture (subscript mix)... 

Application of the Boltzmann equation to Eq. (8.33) gives the entropy of the mixture according to this model for concentrated solutions ... 

There is a stack of rotational levels, with term values such as those given by Equation (5.19), associated with not only the zero-point vibrational level but also all the other vibrational levels shown, for example, in Figure 1.13. However, the Boltzmann equation (Equation 2.11), together with the vibrational energy level expression (Equation 1.69), gives the ratio of the population of the wth vibrational level to Nq, that of the zero-point level, as... 

In order to calculate the distribution function must be obtained in terms of local gas properties, electric and magnetic fields, etc, by direct solution of the Boltzmann equation. One such Boltzmann equation exists for each species in the gas, resulting in the need to solve many Boltzmann equations with as many unknowns. This is not possible in practice. Instead, a number of expressions are derived, using different simplifying assumptions and with varying degrees of vaUdity. A more complete discussion can be found in Reference 34. 

The critical size of the stable nucleus at any degree of under cooling can be calculated widr an equation derived similarly to that obtained earlier for the concentration of defects in a solid. The configurational entropy of a mixture of nuclei containing n atoms widr o atoms of the liquid per unit volume, is given by the Boltzmann equation... 

This is die fonn diat chemists and physicists are most accustomed to. The probabilities are calculated from the Boltzmann equation and the energy difference between state t and state it — 1. Because we are using a ratio of probabilities, the normalization factor, i.e., the partition function, drops out of the equation. Another variant when 6 is multidimensional (which it usually is) is to update one component at a time. We define 6, = 6, i,... 

Both entropic and coulombic contributions are bounded from below and it can be verified that the second variation of is positive definite so that the above equations correspond to a minimum [27]. Using conditions in the bulk we can eliminate //, from the equations. Then we get the Boltzmann equation in which the electric potential verifies the Poisson equation by construction. Hence is equivalent within MFA to the... 

The relationship between the ground-state and excited-state populations is given by the Boltzmann equation... 

Two-Particle Collisions.—One of the basic assumptions in the derivation of the Boltzmann equation is that the gas being described is sufficiently dilute so that only two-particle collisions are of importance. The mechanics of a two-body encounter will thus be described in order... 

Discussion of the Equation.—The Boltzmann equation describes the manner in which the distribution function for a system of particles, /x = /(r,vx,f), varies in terms of its independent variables r, the position of observation vx, the velocity of the particles considered and the time, t. The variation of the distribution function due to the external forces acting on the particles and the action of collisions are both considered. In the integral expression on the right of Eq. (1-39), the Eqs. (1-21) are used to express the velocities after collision in terms of the velocities before collision the dynamics of the collision process are taken into account in the expression for x(6,e), from Eqs. (1-11) and (1-12), which enters into the k of Eqs. (1-21). Alternatively, as will be shown to be useful later, the velocities before and after collision may be expressed, by Eq. (1-20), in terms of G,g, and g the dynamics of the collision comes into the relation between g and g of Eq. (1-19). 

In the derivation of the Boltzmann equation, it was noted that the distribution function must not change significantly in times of the order of a collision time, nor in distances of the order of the maximum range of the interparticle force. For the usual interatomic force laws (but not the Coulomb force, which is of importance in ionized gases), this distance is less than about 10 T cm the corresponding collision times, which are of the order of the force range divided by a characteristic particle velocity (of the order of 10 cm/sec for hydrogen at 300° C), is about 10 12 seconds. 

The assumption that the probability of simultaneous occurrence of two particles, of velocities vt and v2 in a differential space volume around r, is equal to the product of the probabilities of their occurrence individually in this volume, is known as the assumption of molecular chaos. In a dense gas, there would be collisions in rapid succession among particles in any small region of the gas the velocity of any one particle would be expected to become closely related to the velocity of its neighboring particles. These effects of correlation are assumed to be absent in the derivation of the Boltzmann equation since mean free paths in a rarefied gas are of the order of 10 5 cm, particles that interact in a collision have come from quite different regions of gas, and would be expected not to interact again with each other over a time involving many collisions. 

Moreover, since the mean free path is of the order of 100 times the molecular diameter, i.e., the range of force for a collision, collisions involving three or more particles are sufficiently rare to be neglected. This binary collision assumption (as well as the molecular chaos assumption) becomes better as the number density of the gas is decreased. Since these assumptions are increasingly valid as the particles spend a larger percentage of time out of the influence of another particle, one may expect that ideal gas behavior may be closely related to the consequences of the Boltzmann equation. This will be seen to be correct in the results of the approximation schemes used to solve the equation. 

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. 

Taking the time derivative of H(t), and using the Boltzmann equation, Eq. (1-39) for this case ... 

Hydrodynamic Equations.—Before deriving the hydro-dynamic equations, some integral theorems that are useful in the solution of the Boltzmann equation will be proved. Consider a function of velocity, G(Vx), which may also be a function of position and time let... 

Since mass, momentum, and energy are conserved in a collision, successive multiplication of the Boltzmann equation by m, mVj, and mv, and integration over vl9 may be expected to give rise to equations of importance in the macroscopic domain. Multiplying Eq. (1-39) by m and integrating, we have ... 

If the Boltzmann equation is multiplied by mvlx, and integrated over vx, the first two terms become ... 

Expansion Polynomials.—The techniques to be discussed here for solving the Boltzmann equation involve the use of an expansion of the distribution function in a set of orthogonal polynomials in particle velocity space. The polynomials to be used are products of Sonine polynomials and spherical harmonics some of their properties will be discussed in this section, while the reason for their use will be left to Section 1.13. 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

The expansion coefficients, affl, are functions of space and time, and will be determined from the Boltzmann equation. The product Vi/af1 Yf in the expansion forms a complete orthogonal set in... 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

Bather than carrying out the calculation for the general case, which yields rather unwieldy expressions, only equations sufficient to obtain certain approximations will be developed. If we multiply the Boltzmann equation, Eq. (1-39), by 1 = i%( 2)3r )) (0.9>)> the resulting equation is simply the equation of conservation of mass, since integrating unity over the collision integral gives zero ... 

When we multiply Eq. (1-39) by Sfi)2( 2)iY ) (6,Boltzmann equation yields three terms, as follows ... 

The second term on the left of the Boltzmann equation yields... 

Since /S Tj0) = , its integral over the collision term is zero (conservation of momentum in a collision). Thus the result of multiplying the Boltzmann equation by and integrating is ... 

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