Boltzmann Transport Equation

Uri Lachish, guma science

Abstract: A powerful tool to study of transport phenomena. Currents and transport coefficients are calculated. Thermally induced diffusion discussed.

Contents
1. Introduction
2. Boltzmann transport equation: Derivation of the equation.
3. The currents: particle energy and heat currents.
4. The transport coefficients: Diffusion, electrical and thermal conductivity, Weidmann-Franz and Einstein relations.
5. Particles flow within a medium with a temperature gradient: Knudsen law
6. Summary and conclusions

1. Introduction
Boltzmann Transport Equation is a powerful tool for analyzing transport phenomena within systems that involve density and temperature gradients. The equation is applied to analysis of the general currents within a system, the transport coefficients and the relationships between them. Thermally induced diffusion is discussed.

2. Boltzmann transport equation
Consider a system with non-uniform particle density and temperature. In each place within the system there is a local range where the thermal velocities are given by an equilibrium distribution function. The distribution is temperature dependent and varies from place to place.

A non-equilibrium distribution function determines the probability of a particle within the system to be at some place and to have some local thermal velocity. Boltzmann transport equation expresses the global non-equilibrium distribution in terms of local equilibrium distributions. The equation enables application of properties of equilibrium systems to the study of a non-equilibrium system.

Consider a system of randomly moving particles within a medium with a temperature gradient along the x-axis. Whenever a particle is scattered, or collides with the medium, its thermal velocity immediately after the collision will be that of the equilibrium distribution f0(x, T(x)) at the collision point.

Figure 1: Particle moves within a medium in a x direction.

A particle moves in a x direction at an angle q with the x-axis with a thermal velocity v (figure-1). The projection of x on the axis is:

Dx = Dx Cosq (1)

The probability, dP, that the particle will collide with the medium along a distance element dx is proportional to the length dx , the density of the scattering centers ns, and their scattering cross section ss :

dP = -P dx S nsss = -P dx / l (2)

where the sum over s accounts for all different types of centers, and l = 1/ (Snsss) is the mean free path.

The probability that the last collision of the particle, before reaching a cross section plane at some x , was along the element dx', at a position x' ( x' < x, figure (1)), is:

P(x' ) dx' / l = e(x' - x )/l dx' / l (3)

l is determined by the properties of the medium and the name "mean free path" is justified, since , that is, l is the probable distance where the particle comes from.

The velocity distribution v of a particle, at the cross section plane x, is obtained by adding the local velocity distributions at all the distance elements along the line up to x. Each distribution is multiplied by the probabilty that the particle comes from that distance, namely, that it has collided for the last time with the medium at that place :

(4)

where v = |v| and f0( x', v) is the local equilibrium distribution at x'. If the equilibrium distribution does not change significantly along the mean free path l, it will be possible to use the first terms of Taylor series:

(5)

The change in velocity is a result of a force acting on the particle. For example, a gravitational field, or an electric field for a charged particle. Assuming that the force acts in the x-direction, the velocity change will be:

v(x')2 - v( x)2 ≈ 2 v (v' - v) = 2 a (x' - x) Cosq (6)

where a is the acceleration, equal to eE/m for an electric field E.

Substituting (6) and (5) into (4), integrating over x', and finally substituting x and q for x according to (1):

(7)

The non-equilibrium distribution (7) is the linear Boltzmann transport equation in a somewhat non-traditional form.
f0(x, v) is the local equilibrium distribution given by Maxwell-Boltzman or Fermi-Dirac distributions. The medium properties enter through the mean free path l, and external forces through the acceleration a. The standard form of the linear Boltzmann transport equation includes a constant relaxation time t, , where t = l / v .

3. The currents
A particle moving at a velocity v will cross a plane section at x during a time Dt if its distance from the plane is less than |v| Cosq Dt.

The particle current through a unit cross section at x is obtained by summing all the velocities of all directions in space (spatial angle of 4p), each velocity weighed by the distribution probability:

Jn = ∫f(x, v, q) |v| Cosq d3v (8)

where d3v = 2 p v2 dv Sinq dq . Substitution of the distribution function (7) into (8) yields the current:

(9)

where the f0 contribution in (7) is zero, and the trigonometric integral is ∫Cos2q Sinq dq = 2 / 3.

Since the energy associated with each particle is (1/2)mv2 the energy current will be:

Ju = (m/2) ∫f(x, v, q) |v|3 Cosq d3v (10)

and by similar calculation:

(11)

The currents will now be calculated by applying Maxwell-Boltzmann distribution:

(12)

where n(x) is the local particle density. The pre-exponential factor in (12) is determined by the normalization condition: .

Since, (by (12)):

(13)

then:

(14)

The integral is equal to , therefore, the particle current is:

(15)

A similar calculation yields the energy current:

(16)

or, by combining (15) and (16):

(17)

This equation expresses the energy current as a sum of two terms. The first is convection, the energy associated with the particle flow; and the second is heat conduction, which is proportional to the temperature gradient and independent of the particle current.

In electrical conductors (but not in semiconductors) the charge density does not depend on the temperature. The electrical current Jq will then be (by (15):

(18)

4. The transport coefficients
The transport coefficients are determined directly from the currents in equations (15)-(17). The diffusivity D, defined by , is:

(19)

The electrical conductivity s, defined by , is:

(20)

The thermal conductivity, defined by Ju(Jn = 0) = -K (dT/dx), is:

(21)

By comparing these coefficients, the Wiedemann-Franz law:

(22)

and Einstein relation:

(23)

follow directly.

The transport coefficients, and the relationships between them, are determined from the currents (equations (15)-(17)) calculated from Boltzmann transport equation (7).

5. Particles flow within a medium with a temperature gradient
The particle flow may be that of a gas within a narrow pipe, a porous medium, or that of atoms diffusing within a solid. The current of uncharged particles is, (by (15)):

(24)

If there is no time dependent accumulation or depletion of particles the current will not depend on x and then the equation can be integrated to yield:

(25)

where L is the length of the medium, and the subscripts C and H denote the cold and hot ends. If the ideal gas law, p = n kT, is obeyed, then n (kT)1/2 can be replaced in equations (24)-(25) by p / (kT)1/2. Another important result is that the current is determined by the thermodynamic parameters at the ends of the medium and not by their variations along it. In the special case J = 0 the following expression becomes a system invariant:

n (kT)1/2 = p / (kT)1/2 = Const (26)

In particular nC (kTC)1/2 = nH (kTH )1/2, and there is no dependence on the medium properties. This is Knudsen law (Knudsen (1952)).

6. Summary and conclusions
Boltzmann transport equation relates the properties of a non-equilibrium system, expressed by a non-equilibrium distribution, in terms of local equilibrium distributions. The equation is derived by assuming random movement of particles within a medium.

Boltzmann transport equation is applied to calculation of the general currents in a medium with particle and temperature gradients. The currents determine the transport coefficients of the medium.

The calculated diffusivity, electrical and thermal conductivities verify Wiedemann-Franz and Einstein Laws.

The calculated current of uncharged particles verifies Knudsen law.

See:
Expansion of an Ideal Gas
Osmosis and Thermodynamics
van't Hoff's Evidence

Appendix: integrals

0∫ ∞xn exp(-a2x2) dx = G((n+1)/2) / 2an+1

G is the gamma function:

G(n) = (n – 1)!

G(x + 1) = x G(x)

G(x) G(x + 1/2) = G(2 x) p1/2 / 22 x - 1

G(1/2) = p1/2

G(3/2) = (1/2) p1/2

G(5/2) = (3/4) p1/2

References
C. Kittel, Elementary Statistical Physics (Wiley, New york, 1958) pp. 159-165, 192-201.
M. Knudsen, Kinetic Theory of Gases, (Methuen, London, 1952) pp.33-34.
F.W. Sears, G.L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics (Addison-Wesley, 3rd ed 1975) pp 331-336, 355-361.
J.M. Ziman, Principles of the Theory of Solids (Cambridge University Press, 1969) pp 116-123, 179-200.

On the net: May, 2002.

Download as pdf: http://urila.tripod.com/Boltzmann.pdf

By the author:

  1. "Osmosis and thermodynamics", American Journal of Physics, Vol 75 (11), pp. 997-998, November (2007).
  2. "van't Hoff's Evidence", http://urila.tripod.com/evidence.htm, October (2007).
  3. "Osmosis and Thermodynamics", http://urila.tripod.com/osmotic.htm, January (2007).
  4. "Expansion of an ideal gas", http://urila.tripod.com/expand.htm, December (2002).
  5. "Optimizing the Efficiency of Reverse Osmosis Seawater Desalination", http://urila.tripod.com/Seawater.htm, May (2002).
  6. "Boltzmann Transport Equation", http://urila.tripod.com/Boltzmann.htm, May (2002).
  7. "Energy of Seawater Desalination", http://urila.tripod.com/desalination.htm, April (2000).
  8. "Avogadro's number atomic and molecular weight", http://urila.tripod.com/mole.htm, April (2000).
  9. "Vapor Pressure, Boiling and Freezing Temperatures of a Solution", http://urila.tripod.com/colligative.htm, December (1998).
  10. "Osmosis Reverse Osmosis and Osmotic Pressure what they are", http://urila.tripod.com/index.htm, February (1998).
  11. "Calculation of linear coefficients in irreversible processes by kinetic arguments", American Journal of Physics, Vol 46 (11), pp. 1163-1164, November (1978).
  12. "Derivation of some basic properties of ideal gases and solutions from processes of elastic collisions", Journal of Chemical Education, Vol 55 (6), pp. 369-371, June (1978).

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