Physics Equations, Units, and Constantshomepages.cae.wisc.edu/~callen/appA.pdf · APPENDIX A....

APPENDIX A. PHYSICS EQUATIONS, UNITS, AND CONSTANTS 1

Appendix A

Physics Equations, Units,and Constants

This appendix provides a summary of the fundamental physical laws from otherareas of physics, as they are commonly used in plasma physics. Key equations,units and physical constants are given for mechanics, electrodynamics, statisticalmechanics, kinetic theory of gases, stochastic diffusion processes, fluid mechan-ics and quantum mechanical effects. While the procedures for deriving theseequations are given in outline form, details are omitted. Readers should con-sult the textbook references listed at the end of each section for more detailedexplanations and theoretical developments. In some parts of this appendixextensive use is made of the vector algebra and calculus relations given in Ap-pendix D. The International System of Units (Systeme International d’Unites),often called mks units, are used throughout this appendix, and the book. Phys-ical constants and SI unit interrelationships are given in tables in Section A.8at the end of this Appendix.

A.1 Mechanics

Newton’s second law states that the mass m times the acceleration a of a particleis given by the force F (in units of newtons or kg ·m/s2)

ma = F, Newton’s second law. (A.1)

A conservative force is one that is derivable from the gradient of a potentialthat is independent of time:

F = −∇V (x), conservative force. (A.2)

Since the acceleration in (A.1) is just the time derivative of the particle velocity,a ≡ dv/dt, taking the dot product of the velocity v with Newton’s second law

DRAFT 11:16September 2, 2003 c©J.D Callen, Fundamentals of Plasma Physics


for a conservative force yields:

d

dt

[mv2

2+ V (x)

]= 0 =⇒ ε =

mv2

2+ V (x) = constant,

energy conservation, (A.3)

where mv2/2 ≡ m(v · v)/2 ≡ T is the particle kinetic energy and V (x) is thepotential energy. The SI unit of energy is the joule (J), which is equal to anewton ·meter (N ·m). In plasma physics particle energies are usually quotedin electron volts (eV), which is the energy in joules divided by the elementarycharge [ eV ≡ J/e = J/(1.602× 10−19) ].

The force on a particle of charge q subjected to an electric field E(x, t) anda magnetic induction field B(x, t) is

F = q (E + v×B) , Lorentz force. (A.4)

For electrostatic situations with no magnetic field, the electric field can be writ-ten in terms of the electrostatic potential φ(x) : E = −∇φ. Then, the Lorentzforce becomes conservative [see (A.2)] with V (x) = −qφ(x), and the energyconservation relation (A.3) is applicable.

When only a magnetic field is present, the combination of Newton’s secondlaw and the Lorentz force becomes

mdvdt

= qv×B =⇒ dvdt

= ωc×v, (A.5)

where

ωc ≡ − qB/m, the angular velocity, (A.6)

for gyromotion of the charged particle in the magnetic field. The negative signis needed in this vectorial definition so that charged particles gyrate accordingto the right-hand rule with the thumb pointing in the direction of ωc. Themagnitude of ωc gives the radian frequency (rad/s) for the gyromotion:

ωc = qB/m, gyrofrequency, (A.7)

which is also called the cyclotron (the source of the subscript c) or Larmor1

frequency. This formula is unchanged for relativistic particles except for thefact that then the mass becomes the relativistic mass: m → m/

√1− v2/c2.

Since the dot product of (A.5) with the velocity v vanishes, the particle kineticenergy is constant — a magnetic field does no work on a charged particle in itsgyromotion. In gyromotion a charged particle executes a circular motion aboutthe magnetic field B with a radius of

% ≡ v⊥/ωc, gyroradius, (A.8)

in which v⊥ is the magnitude of the velocity component perpendicular to themagnetic field direction [v⊥ ≡ −B×(B×v)/B2].

1Actually, the Larmor frequency is defined to be half the cyclotron frequency.



For situations where both electric and magnetic fields are present, it is conve-nient to write them in terms of the scalar potential φ(x, t) and vector potentialA(x, t): E = −∇φ−∂A/∂t, B =∇×A — see (A.55). Then, Newton’s secondlaw (A.1) for a nonrelativistic charged particle subjected to the Lorentz force(A.4) can be written as

mdvdt

= − ∂U

∂x+d

dt

(∂U

∂v

), U = qφ− q(v ·A) (A.9)

in which U is a generalized potential energy, ∂U/∂x ≡ ∇U , ∂U/∂v ≡ ∇vU ,where∇v is the gradient in velocity space. The single particle Lagrangian, whichhas units of energy and is given by the kinetic energy minus the generalizedpotential energy, is defined by

L(x,v, t) ≡ T − U =mv2

2− qφ+ q(v ·A), Lagrangian, (A.10)

where again T = mv2/2 is the particle kinetic energy. The vector equation ofmotion for a charged particle (i.e., Newton’s second law with the Lorentz force)can be written in terms of the Lagrangian as

d

dt

(∂L

∂v

∣∣∣∣t,x

)− ∂L

∂x

∣∣∣∣t,v

= 0, Lagrangian equations of motion. (A.11)

For an orthogonal coordinate system with unit base vectors ek, the orthogonalprojections of this vector equation yield

d

dt

(∂L

∂qk

)− ∂L

∂qk= 0, k = 1, 2, 3 Lagrange’s equations. (A.12)

Here, the spatial coordinates are qk ≡ ek · x and the velocity coordinates areqk ≡ ek · v = ek · dx/dt. Note that, like Newton’s second law, Lagrange’sequations are in general second order ordinary differential equations in time.

It is often convenient to change the charged particle equation of motion intotwo coupled first order differential equations. To effect this change one firstdefines

p ≡ ∂L

∂v= mv + qA, canonical momentum, (A.13)

in which v ≡ dq/dt. Next, the single particle Hamiltonian function H, whichalso has units of energy, is defined through the Legendre transformation:

H(x,p, t) ≡ p · dxdt− L =

|p− qA|22m

+ qφ ≡ T + V, Hamiltonian. (A.14)

It is the sum of the kinetic energy (T = |p − qA|2/2m = mv2/2) and thepotential energy (V = qφ), and by construction is independent of velocity:∂H/∂v|x,p,t = 0. The equation of motion for a charged particle can be written



(for both orthogonal and nonorthogonal coordinate systems) in terms of theHamiltonian function H, as two coupled first order vector differential equationsin time:

dpdt

= − ∂H

∂x

∣∣∣∣t,p

,dxdt

=∂H

∂p

∣∣∣∣t,x

, Hamilton’s equations of motion. (A.15)

The total time derivative of the Hamiltonian is given via chain-rule partialdifferentiation by

dH

dt=

∂H

∂t

∣∣∣∣x,p

+dxdt· ∂H∂x

∣∣∣∣t,p

+dpdt· ∂H∂p

∣∣∣∣t,x

.

Using Hamilton’s equations of motion, the sum of all terms except the explicitpartial time derivative vanish — because the Hamiltonian does not vary alongthe charged particle’s motion in the relevant (x,p) six-dimensional phase space:dx/dt · ∂H/∂x + dp/dt · ∂H/∂p = 0. Thus, the total time derivative of theHamiltonian is simply

dH

dt=

∂H

∂t

∣∣∣∣x,p

= − ∂L

∂t

∣∣∣∣x,v

= q

(∂φ

∂t− v · ∂A

∂t

), (A.16)

which indicates the increase in energy due to a temporally increasing potentialφ and due to the work v · qE done by the inductive component −∂A/∂t of theelectric field.

Projecting out the geometrical components of the Hamiltonian form of theequations of motion (A.15) for a charged particle in the orthogonal directionsek yields

dpkdt

= − ∂H

∂qk,

dqkdt

=∂H

∂pk, k = 1, 2, 3, Hamilton’s equations, (A.17)

in which pk ≡ ek · p are the canonical momentum coordinates and qk ≡ ek · xare the conjugate spatial coordinates.

The various equations of motion have been written in forms that are inde-pendent of the coordinate system and they are valid in the initial coordinatesystem as well as transformed ones. Also, the equations are valid for non-relativistic particles (v << c) and are all Galilean invariant. That is, they areunchanged upon transformation to another inertial (non-accelerating) frame ac-cording to v′ = v + Vf and E′ = E + Vf×B, where Vf is the velocity of thesecond inertial frame (subscript f) relative to the first.

When the potential φ and vector potential A do not depend explicitly ontime, (A.16) shows that the Hamiltonian is a constant of the motion:

H =|p− qA|2

2m+ qφ =

mv2

2+ qφ = ε = constant,

for∂φ

∂t= 0,

∂A∂t

= 0, energy conservation. (A.18)



In such conservative systems the particle energy and time are canonical conju-gate Hamiltonian coordinates: p = ε and q = t.

When the gradient of the Hamiltonian vanishes in a particular direction ek,(A.15) shows that the conjugate canonical momentum in the same direction isa constant of the motion:

pk ≡ ek · p = ek · (mv + qA) = constant for ek ·∂H

∂x= 0,

canonical momentum conservation. (A.19)

In an orthogonal coordinate system the base vector ek becomes the unit vectorek; then, the criterion for canonical momentum conservation in the ek directionbecomes simply ∂H/∂qk = 0 (i.e., H independent of the coordinate qk, whichimplies symmetry in the ek direction).

Lagrange’s or Hamilton’s equations of motion can be derived (by consideringvariations with x and v ≡ dx/dt or p as the independent variables, respectively)from Hamilton’s variational principle of least action (time integral of differencebetween kinetic and potential energy): δ

∫Ldt = 0. It can also be shown using

(A.14) that for a conservative system where the Hamiltonian is a constant of themotion [see (A.18)], the action

∫p · dq is a variational quantity along a particle

trajectory.For periodic motion in a given coordinate qi it is convenient to introduce as

a variable the action integral over a cycle:

Ji =1

2π

∮pi · dqi =

12π

∫ 2π

0

dθi pi ·∂qi∂θi

, action variable. (A.20)

The action variable, which is a momentum-like quantity, is the “area” in pi, qiphase space encompassed by the periodic motion. The canonically conjugateaction-angle θi is the angular or cyclic variable corresponding to periodic motionaround the perimeter of this area. Hamilton-Jacobi theory (see references at endof this section) can usually be used to determine the action angle coordinateqi ≡ θi. Writing the Hamiltonian in terms of the action variable Ji, the Hamiltonequation dqi/dt = ∂H/∂pi [see (A.17)] becomes:

dθidt

=∂H

∂Ji≡ ωi, action angle evolution equation. (A.21)

The period of the oscillatory motion can be determined in general from

τi ≡∮dt =

∮dqi

dqi/dt=∮

dqi∂H/∂pi

, oscillation period. (A.22)

The radian frequency for the periodic motion is

ωi(Ji) = 2π/τi, oscillation frequency. (A.23)

The Hamiltonian for a periodic system in action-angle variables is thus simply

Hi = ωiJi, action-angle Hamiltonian. (A.24)



For nearly periodic motion in situations where the generalized potential Uin (A.9) varies slowly and aperiodically in space and time (compared to theoscillations), the action in (A.20) is nearly constant and given by the ratio ofthe oscillation energy to the oscillation frequency:

Ji 'Hi

ωi=εiωi, action for nearly periodic motion. (A.25)

For slow, temporal changes that are characterized by a parameter a(t) and arenot themselves periodic, it can be shown that, while the slow variations cause“linearly small,” oscillatory [∼ (a/ωia) sinωit << 1] changes in J , the averageof dJ/dt over an oscillation period τ is “quadratically small” in the rate oftemporal change:⟨

dJidt

⟩θi

≡ 1τ

∮dθi

θi

dJidt

= 0 + O(

a

ωia

)2

,a

ω2i a

Jiτ, (A.26)

where the dots over quantities indicate their time derivatives. For such situ-ations the action Ji is called an adiabatic invariant; it is often a very usefulapproximate constant of the motion. When the small variations in the poten-tial oscillate at a slow frequency ωi >> ωa ≡

√−(1/a) ∂2a/∂t2, harmonics of

this slower oscillation that are resonant with the fundamental oscillations (i.e.,ωi = nωa, n ≡ an integer) can lead to secular changes in the action Ji that growslowly in time. Hence they can break the constancy of the adiabatic invariantover a long time period. This usually occurs when the slow oscillations exceed asmall critical amplitude (typically ∼ 0.1 of the main oscillations). The relevantmultiple time scale analysis and conditions for such breakdowns of adiabaticityare discussed in E.6.

As an example of the use of mechanics theory, consider the “central-force”problem of determining the scattering angle ϑ and elastic cross-section σ for aCoulomb collision of two non-relativistic, charged particles. Assume a chargedparticle of species s with charge qs, mass ms and initial velocity v experiences aCoulomb collision (i.e., interaction via the Coulomb electric field force) with an-other charged particle of species s′ with parameters qs′ ,ms′ and v′. Multiplyingthe force balance equations obtained from (A.1) and (A.4) for each particle bythe mass of the other particle and subtracting, taking account of the equal andoppositely directed electric field forces on the two particles due to the Coulombpotential [qE = −q∇φ, φ = q/(4πε0|x|) — see (A.33)], yields the equationof motion for the two-particle system in the center-of-momentum coordinatesystem:

mss′d

dt(v − v′) = − qsqs′

4πε0∇ 1|x− x′| (A.27)

in which mss′ = msms′/(ms + ms′) is the “reduced” mass for the two particlesystem.

Initially, when the particles are very far apart, one can define the impactspeed as u = |v−v′| and the collision impact parameter as b (distance of closest



approach if the Coulomb electric field did not deflect the particles). Further,one defines the classical (i.e., not quantum mechanical) minimum distance ofclosest approach as

bclmin ≡

qsqs′

4πε0mss′u2, classical minimum impact parameter (A.28)

at which the center-of-momentum kinetic energy is half the Coulomb potentialenergy [qsqs′/(4πε0bcl

min)] and below which large-angle deflections (> 90) canbe expected to occur.

Since the collision takes place in a plane defined by the vectors x − x′ andv − v′, it is convenient to define instantaneous radial and angular coordinatesin the center-of-momentum frame by the radial separation of the particles r ≡|x − x′| and by the angle θ that the line x − x′ makes with the line |x − x′|when the particles were initially very far apart. In these coordinates the angularmomentum pθ [constant because of symmetry of the Coulomb potential in theθ direction — see (A.19)] and total energy ε [constant because the potential φdoes not depend explicitly on time — see (A.18)] can be written as

pθ = mss′r2θ = mss′bu,

ε = T + V =12mss′

(r2 + r2θ2

)+

qsqs′

4πε0r=

12mss′u

2.

Solving the second (energy conservation) equation for r and dividing by the θobtained from the first equation yields

dr

dθ= ±r

b

√r2 − 2 r bcl

min − b2,

where the sign is negative when the particles are approaching each other andpositive as they recede.

At the minimum or closest approach distance rm, dr/dθ = 0. The angle θmat this point is given by

θm =∫ ∞rm

dr

dr/dθ= − arctan

(bclmin

b

)+π

2.

In the center-of-momentum frame the angular deflection ϑ caused by the colli-sion is given by π − 2θm and hence

tanϑ

2=

bclmin

b=

qsqs′

4πε0mss′u2b, scattering angle. (A.29)

Thus, b > bclmin causes Coulomb scattering by less than 90 (ϑ < π/2), while

b < bclmin induces more than 90 scattering.

The differential cross-section dσ (measured in meters2 or barns ≡ 10−28 m2)by which Coulomb collisions of incoming charged particles of species s withimpact parameter b and azimuthal angle ϕ scatter off of charged particles of



species s′ into spherical angles ϑ, ϕ within the differential solid angle dΩ ≡sinϑ dϑ dϕ is thus given [using (A.29) to write b(ϑ) = bcl

min/ tan(ϑ/2)] by

dσ = b db dϕ =b

sinϑ

∣∣∣∣ dbdϑ∣∣∣∣ dΩ =

(bclmin

2 sin2 ϑ/2

)2

dΩ

or,

dσ

dΩ=(

bclmin

2 sin2 ϑ/2

)2

=(

qsqs′

24πε0mss′u2 sin2 ϑ/2

)2

,

Rutherford differential scattering cross-section. (A.30)

Standard intermediate level mechanics textbooks, which include extensionsto relativistic systems, are:

Symon, Mechanics (1971) [?].

Barger and Olsson, Classical Mechanics: A Modern Perspective (1973) [?].

The standard advanced level mechanics textbook is:

Goldstein, Classical Mechanics (1950, 1980) [?].

A.2 Electrodynamics

An electrostatic theory is appropriate for time-independent charge density distri-butions ρq(x), electric fields E(x), and magnetic induction fields B(x). In elec-trostatics the irrotational (∇×E = 0) electric field E with units of volts/meteris written in terms of the scalar potential φ(x) with units of volts, E ≡ −∇φ,and related to the charge density distribution:

∇·E = ρq/ε0, Gauss’ law, (A.31)

−∇2φ = ρq/ε0, Poisson’s equation. (A.32)

The charge density distribution has units of coulombs/meter3. For localizedcharge density distributions [lim|x|→∞ ρq(x) → 0] the general (Green-function-type) solution of Poisson’s equation in an infinite medium is [see also (??)]

φ(x) =∫d3x′

ρq(x′)4πε0 |x′ − x| . (A.33)

For a point charge at x = x0 the charge distribution is ρq(x) = q δ(x− x0) andthe potential becomes

φ(x) =q

4πε0 |x− x0|, Coulomb potential. (A.34)

Here and throughout this book the mks factor 4πε0 is written in braces;eliminating this factor yields the corresponding cgs forms of these electrostaticresponse formulas.



In a dielectric (ponderable) medium the charge density ρq is composed of apart ρfree due to “free” charges and a part due to a polarization charge density,ρpol = −∇·P where P ≡ ε0χEE is the presumed linear and isotropic polariza-tion (units of coulomb/meter2) of the medium induced by the electric field Eand χE is the dimensionless electric susceptibility of the medium:

ρq = ρfree + ρpol = ρfree −∇·P = ρfree −∇· ε0χEE. (A.35)

Thus, in an isotropic dielectric medium Gauss’ law becomes Coulomb’s law:

∇·D = ρfree, D ≡ ε0E + P = ε0(1 + χE) E = εE (A.36)

in which the medium’s dielectric constant ε ≡ ε0(1 + χE) is the constituitiverelation between the displacement vector D and the electric field E.

A magnetostatic theory is appropriate for time-independent current densitydistributions J(x) and magnetic induction fields B(x). In magnetostatics thesolenoidal (transverse, ∇·B = 0) magnetic induction field B which has unitsof weber/meter2 or tesla can be written in terms of the vector potential A, i.e.,B = ∇×A, and related to the current density distribution J which has unitsof ampere/meter2:

∇×B = µ0J =⇒ −∇2A = µ0J, static Ampere’s law (A.37)

in which∇·A = 0 (the Coulomb gauge) has been assumed in the last form. Forlocalized current density distributions [lim|x|→∞ J(x) = 0], the general (Green-function-type) solution for A in an infinite medium is [see also (??)]

A(x) =µ0

4π

∫d3x′

J(x′)|x′ − x| . (A.38)

The magnetic field around an infinite wire carrying a current I (amperes) alongthe z axis of a cylindrical coordinate system can obtained from this equationusing B =∇×A and a current density J = ez(I/2πr) lima→0 δ(r − a):

B =µ0I

2πreθ, magnetic field around a current-carrying wire. (A.39)

To obtain the corresponding cgs form of this and other magnetic field responseequations, eliminate the µ0/4π factor and replace J by J/c, or replace µ0Jby 4πJ/c.

In a magnetizable medium the current density J is composed of a part Jfree

due to the current induced by “free” charges and a magnetization current den-sity Jmag = ∇×M where M ≡ χMH is the presumed linear and isotropicmagnetization (units of ampere-turns/meter) of the medium induced by themagnetic field H (units of ampere-turns/meter) and χM is the dimensionlessmagnetic susceptibility of the medium:

J = Jfree + Jmag = Jfree +∇×M = Jfree +∇×(χMH). (A.40)



Thus, in an isotropic, magnetizable medium the static Ampere’s law becomes

∇×H = µ0J, H = B/µ0 −M, B = µ0(1 + χM ) H = µH (A.41)

in which µ = µ0(1 + χM ) is the magnetic permeability of the medium andB = µH is the constituitive relation between the magnetic induction B and themagnetic field H.

The microscopic Maxwell or electromagnetic (em) equations for determiningtime-varying electric and magnetic fields caused by charge and current densitydistributions ρq(x, t) and J(x, t), respectively, in free space are

Maxwell Equations

name

differential form integral form

Gauss’ law

∇·E =ρqε0

∫©∫S

dS ·E =∫V

d3xρqε0

(A.41a)

Faraday’s law

∇×E = − ∂B∂t

∮C

d` ·E = − ∂

∂t

∫∫S

dS ·B (A.41b)

no magnetic monopoles

∇·B = 0∫©∫S

dS ·B = 0 (A.41c)

Ampere’s law

∇×B = µ0

(J + ε0

∂E∂t

) ∮C

d` ·B = µ0

∫∫S

dS ·(

J + ε0∂E∂t

)(A.41d)

(A.42)

Here, ε0 is the electric permittivity of free space which has units of farad/meter= coulomb/(volt ·meter) = joule/(volt2·meter), µ0 is the magnetic permeabil-ity of free space which has units of henry/meter = weber/(ampere ·meter) =weber2/(joule ·meter) and µ0ε0 = 1/c2 where c is the speed of light in free space.Taking the divergence of Ampere’s law and making use of Gauss’ law yields

∂ρq∂t

+∇· J = 0, continuity equation for charge and current. (A.43)

In Ampere’s law the displacement current ε0∂E/∂t was introduced by Maxwellto make the electrodynamics equations consistent with the charge and currentcontinuity equation (A.43).



The physical significance of the source terms on the right of the integralforms of the Maxwell equations are:

∫Vd3x ρq ≡ Q, net charge (in coulombs)

within the volume V ;∫∫SdS ·B ≡ ψ, magnetic flux (in webers) penetrating

the surface S;∫∫SdS · J ≡ I, the total electric current (in amperes) flowing

through the surface S. The Maxwell equations are relativistically invariant; inparticular, they are invariant under Lorentz transformations, which preserve theconstancy of the speed of light, independent of the motion of the source, upontransformation to another inertial rest frame.

The corresponding macroscopic Maxwell equations in an isotropic, polariz-able, magnetizable medium are written in terms of D ≡ εE = ε0E + P and themagnetic field H ≡ B/µ0 −M, and free charge, current densities ρfree,Jfree:

∇·D = ρfree, ∇×E = − ∂B/∂t, ∇ ·B = 0, ∇×H = Jfree + ∂D/∂t.(A.44)

The total rate at which the electromagnetic (em) fields do work on a mediumin a finite volume V is

∫Vd3x Jfree·E — the magnetic field does no work since

the magnetic force qv×B on charged particles is perpendicular to the velocity.Using the macroscopic Maxwell equations to calculate the rate of doing workyields the energy conservation law for electromagnetic fields:

∂wem

∂t+∇· Sem = −Jfree·E, em field energy conservation, (A.45)

where

wem ≡ wE + wB ≡12

(E ·D + B ·H), em energy density (J/m3), (A.46)

Sem = E×H, Poynting vector (flux of em energy) (J/m2· s), (A.47)

Jfree·E = joule heating (W/m3 = J/m3· s = V·A/m3), (A.48)

in which the energy densities in the electric and magnetic fields are defined by

wE ≡12

(E ·D), electric field energy density (J/m3), (A.49)

wB ≡12

(B ·H), magnetic field energy density (J/m3). (A.50)

A corresponding momentum conservation equation for electromagnetic fieldscan be deduced, for situations where charge and current densities are present infree space, from the microscopic Maxwell equations:

∂gem

∂t+∇·Tem = ρqE + J×B, em field momentum conservation, (A.51)



where

gem ≡1c2

E×B, momentum density in em fields, (A.52)

Tem = ε0

[ |E|22

I−EE]

+1µ0

[ |B|22

I−BB]

em stress tensor, (A.53)

ρqE + J×B = momentum input to em fields from medium. (A.54)

Here, the electromagnetic stress tensor is defined to be opposite in sign fromthe usual Maxwell stress tensor in electrodynamic theory [see Eq. (6.119) inJackson, Classical Electrodynamics, 3rd Edition (1999)[?]] — so that the elec-tromagnetic stress can be added to the pressure tensor P to obtain the total forcedensity in a plasma in the form F = −∇· (P + Tem). For a dielectric mediumthe conservation of momentum for electromagnetic fields depends somewhat onthe medium considered because of the possible ambiguity as to which parts ofρqE + J×B belong to the dielectric and which parts represent free charge andcurrent densities.

Since the magnetic induction field B is a solenoidal or transverse field (∇·B= 0), it can be represented in terms of a vector potential A, i.e., B = ∇×A.Using this representation, Faraday’s law can be written as∇×(E+∂A/∂t) = 0,which indicates that E+∂A/∂t can be represented in terms of the gradient of ascalar potential φ. Thus, the electromagnetic fields E and B can be representedin terms of the potentials φ (units of volts) and A (units of weber ·meter):

E = −∇φ− ∂A/∂t, B =∇×A, em fields in terms of potentials. (A.55)

In terms of the potentials φ,A the inhomogeneous, microscopic Maxwell equa-tions (Gauss’ and Ampere’s laws) become (µ0ε0 = 1/c2)(

∇2 − 1c2∂2

∂t2

)φ = − ρq

ε0,

(∇2 − 1

c2∂2

∂t2

)A = −µ0J, (A.56)

in which

∇·A +1c

∂φ

∂t= 0, Lorentz gauge condition, (A.57)

which provides a constraint relation between the potentials, has been used. [Ifthe Coulomb gauge (∇·A = 0) is used, the equations (A.56) are different.]

For a dielectric medium [i.e., a medium that is polarizable (ρq = −∇·P)and magnetizable (J = ∇×M) but not significantly conducting which wouldimply J = σE], equations (A.56) become scalar wave equations of the form(∇2 − µε

∂2

∂t2

)u(x, t) = S(x, t), dielectric medium wave equation. (A.58)



Sinusoidal plane wave solutions of this equation are in general of the form

u(x, t) = uk,ω ei(k·x−ωt) ≡ u eiϕ, Fourier plane wave Ansatz, (A.59)

where by convention u is a complex constant for a given k, ω and physical wavesu(x, t) are obtained by taking the real part: u(x, t) ≡ Reu eiϕ. Substitutingthis Ansatz into the sourceless (S = 0) wave equation yields the dispersionrelation (relationship between ω and k) for nontrivial (normal mode) solutions:

ω2 =k2

µε=k2c2

n2=⇒ ω = ±k c

n, light waves in a dielectric, (A.60)

in which

n ≡ c k

ω, index of refraction. (A.61)

The index of refraction is the ratio of the speed of light in vacuum to that inthe medium.

For a given k, ω, a point of constant wave phase in u(x, t), which is definedby 0 = dϕ/dt = k · dx/dt− ω ≡ k ·Vϕ − ω, moves at

Vϕ ≡ω

kek, wave phase velocity, (A.62)

in which ek ≡ k/k is the unit vector along k. The phase velocity for light wavesin a dielectric medium is the speed of light in the medium in the direction ofwave propagation (k): Vϕ = ± (c/n) ek. Since a steady, monochromatic (singlek, ω) “carrier” wave carries no information, the wave phase speed can be greaterthan the speed of light. A wave packet, which results from superposing wavesof different k, ω, carries information at

Vg ≡∂ω

∂k= ∇k ω(k), wave group velocity, (A.63)

whose magnitude must, by causality, be less than or equal to the speed of light.For nondispersive media [∂n/∂k = 0 =⇒ n = n(ω)], the group velocity is thesame as the phase velocity. Thus, the group velocity of light waves in typical(nondispersive) dielectric media (e.g., water for visible light) is the same as theirphase velocity. Since plasmas are typically dispersive media for ranges of k, ω ofinterest, the group velocities of waves in plasmas are often different from theirphase velocities.

The electric field for the most general homogeneous transverse (k×E = 0)plane wave propagating in the direction k can be represented by

E(x, t) = (ε1E1 + ε2E2) ei(k·x−ωt), polarization representation. (A.64)

Here, ε1, ε2 are mutually orthogonal “wave polarization” unit vectors in direc-tions perpendicular to the direction of wave propagation (ε1×ε2 ≡ ek) andE1, E2 are in general complex numbers. If E1 and E2 have the same complexphase, the wave is linearly polarized. If E1 and E2 have the same magnitude, but



differ in phase by 90 degrees the wave is circularly polarized. A representationthat is useful for circularly and elliptically polarized waves is

E(x, t) = (E+ε+ + E−ε−) ei(k·x−ωt), alternative representation, (A.65)

in which E+ and E− are complex amplitudes and

ε± ≡1√2

(ε1 ± iε2), rotating polarization unit vectors. (A.66)

The E+ ≡ |E+|eiϕ+ term represents a positive angular momentum and helicity(“left circularly polarized” in optics2) wave that rotates (for decreasing phaseϕ ≡ k · x − ωt + ϕ+ at a fixed point in space) in the clockwise direction rel-ative to the k direction since ReE(x, t)+ = (|E+|/

√2)(ε1 cosϕ − ε2 sinϕ).

Conversely, the E− term represents a negative angular momentum and helicity(“right circularly polarized”) wave that rotates in the opposite direction. Circu-larly polarized waves are represented by either E+ or E−, depending on whetherthey have positive or negative helicity. A wave is elliptically polarized if it hasboth E+ and E− components and they are dissimilar — when E+/E− = ±1,one reverts to a linearly polarized wave.

Standard intermediate level textbooks for electrodynamics, or electricity andmagnetism as it has been called historically, are:

Reitz, Milford and Christy, Foundations of Electromagnetic Theory (1979) [?]

Lorrain, Corson and Lorrain, Electromagnetic Fields and Waves (??) [?]

Barger and Olsson, Classical Electricity and Magnetism: A Contemporary Per-spective (1987) [?].

The standard advanced level textbooks are:

Jackson, Classical Electrodynamics (1962, 1975) [?]

Panofsky and Phillips, Classical Electricity and Magnetism (1962) [?].

A.3 Statistical Mechanics

A closed system of particles is in equilibrium in a statistical mechanics senseif for subsystems thereof all relevant macroscopic parameters are equal to theirmean values to a high degree of accuracy. The particles in a system are weaklyinteracting and thus statistically independent if the total system Hamiltonianis approximately just the sum of the Hamiltonians for the individual particles.That is, the part of the total system Hamiltonian that represents interactionsbetween particles must be small, or vanishing, except for infrequent collisions.

2In optics the rotation direction is determined by the direction of polarization rotationthat would be seen by an observer facing into the oncoming wave. This direction of rotationis opposite to the modern physics definition which is determined by the direction of rotationrelative to the wavevector k.



Liouville’s theorem, which follows from the incompressibility [see (A.79)] ofthe x,p Hamiltonian phase space for particle trajectories, states that the den-sity of a system of N particles in their 6N -dimensional phase space is constantalong the particle phase space trajectories. A consequence of Liouville’s the-orem is that the probability density in the 6N dimensional phase space mustbe expressible entirely in terms of constants of the motion. In the macroscopicrest frame (where the average momentum and angular momentum vanish) of asystem of weakly interacting particles, the only relevant constant (or additiveintegral) of the motion is the single particle Hamiltonian.

Statistical mechanics predicts that the most probable distribution of a sub-system of a large number of weakly interacting, free (i.e., monoatomic gas orunbound) particles in equilibrium with an even larger system of such particlesat a thermodynamic temperature3 T will have a probability density distributionin the macroscopic rest frame of the system that is given by

ρ (p,q) = ρ0 e−H(p,q)/T , Gibb’s distribution (A.67)

in which ρ0 is a constant and H is the Hamiltonian for a single particle. Theconstant ρ0 is the density of particles in the six-dimensional phase space, which isobtained from the normalization

∫d3p∫d3q ρ (p,q) = 1. Thus, for example, the

most probable distribution function for weakly interacting charged particles inthe presence of a potential φ that is constant in time or slowly varying (comparedto the rate for thermal motion over a relevant scale length — for an adiabaticresponse, subscript A) is

fA(x,v) = n0

( m

2πT

)3/2

e−H/T = n0

( m

2πT

)3/2

e−mv2/2T−qφ/T , (A.68)

in which n0 is the equilibrium density (m−3) of charged particles in the absenceof the potential φ. The normalization here has been chosen such that integratingf over the three-dimensional velocity space yields the density distribution

nA(x) ≡∫d3v f(x,v) = n0 e

−qφ(x)/T , Boltzmann relation. (A.69)

This result is applicable for adiabatic processes, i.e., ones that vary slowly com-pared to the reversible inertial or oscillatory time scales. As an example ofan application of the Boltzmann relation, the gravitational potential near theearth’s surface (qφ → V = mgx) confines neutral molecules in the atmospherenear the earth’s surface according to the law of atmospheres — see (A.137).

In the absence of a potential, (A.68) becomes the Maxwell distribution func-tion:

fM (v) = n0

( m

2πT

)3/2

e−mv2/2T =

n0e−v2/v2

T

π3/2v3T

, Maxwellian distribution,

(A.70)

3Temperatures (and particle energies) in plasma physics are usually quoted in electronvolts, abbreviated eV, and the Boltzmann factor kB that usually multiplies the temperatureT in equations such as (A.67)–(A.75) is usually omitted for simplicity.



Figure A.1: Properies of a Maxwellian distribution function: a) speed depen-dence; b) number of particles per unit speed v.

in which vT ≡√

2T/m. The dependence of the Maxwellian distribution onparticle speed v is shown in Fig. A.1a. In spherical velocity-space coordinatesthe normalized (by density) integral of the Maxwellian distribution over allvelocity space is [see (??)]∫

d3vfM (v)n0

=4√π

∫ ∞0

dv

vT

v2

v2T

e−v2/v2

T =4√π

∫ ∞0

dxx2e−x2

= 1. (A.71)

Some of the characteristic speeds that can be deduced from the Maxwelliandistribution are (see Fig. A.1b):

vT = vmax ≡√

2T/m, thermal, most probable speed,

v =√

8T/πm = (2/√π) vT , average speed,

vrms =√

3T/m =√

3/2 vT , root mean square speed.

(A.72)

It is customary in plasma physics to use vT as the reference particle speed sincethis is the speed that appears naturally in the exponent of the Maxwellian. Thisis the most probable speed because in spherical velocity space the maximum inthe number of particles with speeds between v and v+dv (∝ 4πv2e−v

2/v2T ) occurs

at this speed (cf., Fig. A.1b). The average speed v (average of v ≡ |v| over theMaxwellian distribution) is relevant in calculations of the random particle fluxto one side of a plane that is introduced into a medium whose particles havea Maxwellian distribution:

∫d3v v 1

2 | cosϑ| fM = π∫∞

0dv v3fM = n0 v/4 [see

(??)] where the z axis of the spherical velocity space coordinate system has beentaken to be perpendicular to the plane being introduced. The root mean squarespeed vrms (square root of average of v2) is relevant in calculations of the averagekinetic energy mv2

x/2 in a given direction x since all directions are equivalent for



an isotropic Maxwellian distribution (v2x = v2

y = v2z = v2

T /2 = v2/3 ≡ v2rms/3):

mv2x

2≡ 1

n0

∫d3v

(mv2

x

2

)fM (v) =

mv2rms

6=

T

2,

one-dimensional particle thermal energy. (A.73)

The total thermal energy of a particle is given by

mv2

2≡ 1

n0

∫d3v

(mv2

2

)fM (v) =

mv2rms

2=

3T2,

three-dimensional particle thermal energy. (A.74)

Finally, the kinetic pressure embodied in the Maxwellian distribution is

p ≡∫d3v (mv2

x)fM (v) =∫d3v

(mv2

3

)fM (v) =

nv2rms

3= nT,

kinetic pressure. (A.75)

Entropy is the state of disorder of a closed system. It never decreaseswith time: it remains constant for reversible (e.g., Hamiltonian dynamics) pro-cesses, but increases for irreversible processes. Irreversible increases in theentropy of a system are caused by dissipative processes such as the cumula-tive effects of a large number of random collisions. For a system of weaklyinteracting, free particles the entropy is given by the logarithm of the aver-age volume [= 1/ρ0 — see (A.67)] of six-dimensional phase space occupied bya single particle, i.e., s = ln(1/ρ0). [For quantum mechanical systems it isthe logarithm of the number of statistically independent states, which is quan-tized to be the number of states that fit in the relevant phase space volume:Nqm =

∫d3Np d3Nq e−H(p,q)/T /(N !h3N ) in which h is Planck’s constant and

N is the number of degrees of freedom for the system being considered.] Thus,neglecting constants and using ρ0 = n0(m/2πT )3/2 ∼ n0/v

3T for an x,v phase

space, for classical systems one has

s = ln (1/ρ0) = ln (T 3/2/n0) + constant, entropy. (A.76)

For a volume V of uniform density (i.e., n0 = 1/V ) monotonic gas, the entropyis given by

s = lnV + (3/2) lnT + constant,

which, when multiplied by the molar gas constant R = kBNA is the conventionalform of the entropy for an ideal gas. Alternatively, writing T = p/n0 in (A.76)so that s = (3/2) ln(pV 5/3) + constant, one obtains the constant entropy (isen-tropic) equation of state pV Γ = constant for an ideal gas in a three-dimensionalsystem where Γ = 5/3.

Standard intermediate level textbooks for statistical mechanics are:

Kittel, Elementary Statistical Physics (1958) [?]



Reif, Fundamentals of Statistical and Thermal Physics (1965) [?]

Callen, Thermodynamics (1960) [?]

Kittel and Kroemer, Thermal Physics (1960). [?]

Some advanced level books on statistical mechanics are:Huang, Statistical Mechanics (1963) [?]

Tolman, The Principles of Statistical Mechanics (1938) [?]

Landau and Lifshitz, Statistical Physics (1959) [?]

Prigogine, Introduction to Thermodynamics of Irreversible Processes (1961). [?]

A.4 Kinetic Theory of Gases

Kinetic theory is a rigorous formalism that is used to provide a description ofthe behavior of a large collection of neutral molecules (or atoms) in a gas, partic-ularly when the assumptions of equilibrium statistical mechanics are not valid.In kinetic theory d3x d3v f(x,v, t) is the (assumed large) number of moleculeslocated in the six-dimensional (x,v) phase space with spatial positions lyingbetween x and x + dx and velocity vectors lying between v and v + dv, attime t. The quantity f(x,v, t), which has units of #/(m3·m3/s3), is called thedistribution function. It is governed by the equation

d f(x,v, t)dt

=∂f

∂ t+ v·∇f +

Fm·∇vf =

δf

δ t

)c

, kinetic equation (A.77)

in which F/m is the acceleration of a molecule due to the force F [e.g., theconservative force in (A.2)], ∇v ≡ ∂/∂v|x,t is the gradient in velocity space,and δf/δ t)c ∼ −νf represents the effects of “abrupt,” binary (microscopic)collisions at rate ν that result from force fields not included in F.

The (mathematical) characteristics of the first order differential operator (inthe 7 variables x,v, t) on the left of (A.77) represent the trajectories of themolecules in the absence of collision effects. The first order differential equa-tions governing the trajectories of the particles can be most generally writtenusing Hamilton’s equations. Thus, the kinetic equation for f(q,p, t), where p isthe canonical momentum defined in (A.13) and q is the canonically conjugateposition vector or for f(z, t) where z ≡ (q,p) = (x,p) is a six-dimensional vari-able that represents all of phase space, can be written most generally in termsof the Hamiltonian variables:

d f(q,p, t)dt

=∂f

∂ t+dqdt· ∂f∂q

+dpdt· ∂f∂p

=∂f

∂ t+dzdt· ∂f∂z

=δf

δ t

)c

,

or, using Hamilton’s equations [see (A.15)], as

d f(q,p, t)dt

=∂f

∂ t+∂H

∂p· ∂f∂q− ∂H

∂q· ∂f∂p

=δf

δ t

)c

, kinetic equation,

(A.78)



Particle motion in the z ≡ (p,q) six-dimensional Hamiltonian phase space isincompressible:

∂

∂z· dzdt

=∂

∂x· dxdt

+∂

∂p· dpdt

=∂

∂x· ∂H∂p− ∂

∂p· ∂H∂x

= 0,

phase space incompressibility. (A.79)

Thus, the kinetic equation can also be written in the “conservative” form

d f(q,p, t)dt

=∂f

∂ t+

∂

∂q·(dqdtf

)+

∂

∂p·(dpdtf

)=∂f

∂ t+

∂

∂z·(dzdtf

)=δf

δ t

)c

.

(A.80)

In the absence of collisions, or for time scales shorter than the collision time,the solution of (A.77) or (A.78) is that f must be a function of the constants ofthe motion — see (A.18), (A.19). For “collisionless” cases where the potentialsφ,A do not change in time and the Hamiltonian is the only constant of motion,the solution is f = f [H(p,q)] = f [H(z)]. Assuming further that there are alarge number of molecules which are interacting weakly (e.g., via collisions) withan even larger number of molecules that have a thermodynamic temperature T ,and hence that the requirements for the validity of statistical mechanics aresatisfied, the distributions given in (A.67) and (A.68) can be derived from thekinetic theory of gases.

The microscopic binary collision effects are most generally represented by

δf

δ t

)c

= CBf(x,v, t)

≡∫d3v′

∫dΩ

dσ

dΩ| v − v′| [ f(v1)f(v′1)− f(v)f(v′) ] ,

Boltzmann collision operator, (A.81)

in which v,v′ and v1,v′1 are the velocities of the colliding particles before andafter the collision and dσ/dΩ is the differential scattering cross-section for thecollisions [cf., (A.30)]. Here, for simplicity f(x,v, t) has been written as f(v)inside the collision operator. In deriving the Boltzmann collision operator it isassumed that the force F on the left of (A.77) is negligible during the collisionprocess, that the gas is sufficiently dilute so that binary or two-body collisionprocesses are predominant (i.e., three-body and many-body collisions or collec-tive particle interactions are negligible), and that the collisions only change thevelocity vectors of the particles (i.e., the collisions abruptly scatter the veloc-ity vectors of the particles at a given “point” x, t along a particle trajectory).The Boltzmann collision operator is a bilinear [because of f(v)f(v′)], integraloperator in velocity space. In the absence of radiative effects, since binary colli-sions conserve particle number, momentum mv and energy mv2/2, so does theBoltzmann collision operator:∫

d3vΨ(v) CB(f) = 0 for Ψ(v) = 1, mv, mv2/2. (A.82)



The functions Ψ(v) are sometimes called summational invariants because linearcombinations of them are also invariants of the collision operator.

For homogeneous (∇f = 0) gases in equilibrium (∂f/∂ t = 0) with noexternal forces on the molecules (F = 0), the kinetic equation (A.77) becomes

0 =δf

δ t

)c

= CB(f0), (A.83)

where the subscript zero on f indicates the equilibrium or lowest order solution.The general solution of this equation is

f0 = fM (v) = n0

(m

2πT0

)3/2

e−m|v−V0|2/2T0 =n0e−|v−V0|2/v2

T

π3/2v3T

,

Maxwellian distribution. (A.84)

This Maxwellian differs from the statistical mechanics result in (A.70) only byits explicit inclusion of the macroscopic flow velocity V0 of the gas (V0 ≡∫d3v vf0/n0), which is not present in (A.70) because that result is obtained

in the rest frame of the gas (i.e., in the V0 rest frame). However, the result isarrived at by different methodologies in statistical mechanics and kinetic theory.Kinetic theory provides the more extendable framework for investigating morecomplicated situations that do not satisfy the assumptions used in deriving(A.70) and (A.84).

The Boltzmann collision operator also has the important property of irre-versibility: entropy increases until the distribution function is given by (A.84).Specifically, taking the entropy functional to be −f ln f and defining HB ≡∫d3v f ln f , it can be shown that

dHB

dt=∫d3v

∂f

∂ t(1 + ln f) =

∫d3v CB(f) ln f ≤ 0,

Boltzmann H-theorem, (A.85)

with the equal sign being applicable only when f becomes equal to the equilib-rium, Maxwellian distribution given in (A.84).

In situations close to thermodynamic equilibrium the lowest order distribu-tion is the Maxwellian given by (A.84) and the distortions of the distributionfunction are higher order and small. In order to understand he nature of thesedistortions and to obtain approximate solutions of the kinetic equation (A.77)for this situation, consider the expansion of the distribution in a combination ofLaguerre and Legendre polynominials (see Appendix B), which are the completeorthogonal basis functions for speed (with the weighting function v2e−v

2/v2T that

comes from the lowest order Maxwellian distribution in spherical velocity space)



and spherical angle dependence::

f = fM

(1 +

[δn

n0L

(1/2)0 +

δT

T0L

(1/2)1 + · · ·

]P0

(vvT

)moments

+2v2T

v ·[δVL(3/2)

0 + V1L(3/2)1 + · · ·

]P1

(vvT

)moments

+vv − (v2/3)I

2mn0v4T

:[πL

(5/2)0 + π1L

(5/2)1 + · · ·

]P2

(vvT

)moments

+ · · ·)

...

=∑lmn

flmn Ylm (ϑ, ϕ)L(l+1/2)n (v2/v2

T ) e−v2/v2

T , moment expansion, (A.86)

in which the Pl(v/vT ) are Legendre polynomial (spherical velocity space an-gular) functionals [≡ 1, v/vT , (vv − (v2/3)I)/(2v2

T /3) for l = 0, 1, 2 ], theL

(l+1/2)n (x) are (energy functional) Laguerre polynomials with arguments x ≡

mv2/2T = v2/v2T , and Ylm(ϑ, ϕ) are the usual spherical harmonics that are

proportional to Pml (cosϑ) eimϕ. Useful properties of these special functions aregiven in B.5 and B.6. The lowest order parameters of this expansion, whichare the

∫d3v Pl(v/v)L(l+1/2)

n moments of the distribution function, correspondphysically to: the density (m−3), flow velocity (m/s) and temperature (eV)distortions δn, δV and δT away from their equilibrium Maxwellian values ofn0,V0 and T0; the heat flow vector q (W/m2), since V1 ≡ − 2q/5nT ; andthe traceless anisotropic part π (N/m2) of the pressure tensor [see (A.95) be-low], which has 5 nonvanishing parameters and is sometimes called a kineticstress tensor. An approximation in which the moments δn, δV, δT,q and π(= 1 + 3 + 1 + 3 + 5 = 13 moments) are used to represent f is usually called aGrad 13 moment approximation.

Often one desires a reduced, fluid moment description which integrates thekinetic equation over velocity space to obtain equations for the physical quan-tities of density, flow velocity and temperature:

density (m−3): n(x, t) = n0 + δn ≡∫d3v f, (A.87)

flow velocity (m/s): V(x, t) = V0 + δV ≡∫d3v v f/n, (A.88)

temperature (eV): T (x, t) = T0 + δT ≡∫d3v [m(v −V2)/3] f/n. (A.89)

The relevant fluid moment equations for these quantities are obtained by takingthe relevant velocity-space moments [i.e., the Ψ(v) in (A.82)] of the kinetic



equation in (A.77) using the Boltzmann collision operator in (A.81) and theconservation properties in (A.82), to obtain

density equation:∂n

∂t+∇· nV = 0, (A.90)

momentum equation: mndVdt

= nF−∇p−∇·π, (A.91)

energy equation:32ndT

dt+ p∇·V = −∇· q− π :∇V, (A.92)

where

d

dt≡ ∂

∂t+ V·∇, total time derivative, (A.93)

is the total (partial plus flow-induced advection4) time derivative that is some-times called the “material derivative,” and F is the average of the single particleforce F over a Maxwellian distribution. The higher order moments needed forclosure (complete specification) of these equations are

pressure (N/m2): p ≡∫d3v

(m3|vr|2

)f = nT, (A.94)

conductive heat flux (W/m2): q ≡∫d3v vr

(m

2|vr|2 −

5T2

)f, (A.95)

stress tensor (N/m2): π ≡∫d3v m

(vrvr −

13|vr|2 I

)f, (A.96)

=∫d3vmvrvrf − p I ≡ P− p I

in which vr ≡ v−V(x, t) is the relative velocity in the frame of reference movingat the macroscopic flow velocity V. Note also that q = −T

∫d3v vrL

(3/2)1 f. The

total heat flux Q ≡∫d3v (m|vr|2/2)vrf is the sum of the conductive heat flux

and the convective heat flux: Q ≡ q + (5/2)nTV.The Chapman-Enskog procedure is used to obtain the needed closure rela-

tions for “collision-dominated” situations in which the gas density varies slowlyin space (compared to the collision mean free path λ ∼ v/ν) and time (com-pared to the collision time 1/ν). Then, the lowest order kinetic equation thatdescribes the distribution function is given by (A.83). Its solution is

fC−E0 = fM (x,v, t) ≡ n(x, t)[

m

2π T (x, t)

]3/2

exp[− |v −V(x, t)|2

2T (x, t)

],

“dynamic” Maxwellian, (A.97)

which is the usual Maxwellian, but now parameterized in terms of the (total)spatially and temporally varying density, flow velocity and temperature. The

4In fluid mechanics advection means transport of any quantity by the fluid at its flowvelocity V; convection refers only to the heat flow qconv = (5/2)nTV induced by V.



conductive heat flux q and anisotropic stress π vanish for fC−E0 . Thus, in orderto determine these needed closure relations, it is necessary to determine the firstorder distortion of the distribution function: δf ≡ f − fC−E0 . (Note that, byconstruction, since the total density n flow velocity V and temperature T arebuilt into fC−E0 , the density, momentum and energy moments of δf vanish:∫

d3v δf = 0,∫d3vmv δf = 0,

∫d3v (mv2/2) δf = 0, C-E consraints.

(A.98)

The kinetic equation for δf is obtained by substituting the definition f =fC−E0 + δf into (A.77), making use of the density, momentum and energy con-servation equations to remove the dependences on ∂n/∂ t, ∂V/∂ t and ∂T/∂ t.Neglecting higher order corrections that are inversely proportional to the colli-sion frequency, the result is

CB(δf) '[(

m|vr|22T

− 52

)vr ·

(1T∇T

)+m

TW :

(vrvr −

|vr|23

I

)]fC−E0 ,

(A.99)

in which, as above, vr ≡ v −V(x, t), and

W ≡ 12[∇V + (∇V)T

]− 1

3I (∇·V) , rate of strain tensor, (A.100)

which is caused by gradients in the flow velocity V and has units of per sec-ond. The normalized temperature gradient ∇ lnT and rate-of-strain tensor Ware called thermodynamic forces — because they induce distortions δf of thedistribution function away from a dynamic Maxwellian and hence away fromthermodynamic equilibrium. Note that beacause of the invariants of the Boltz-mann collision operator given in (A.82), a proper solution of (A.99) for δf willsatisfy the Chapman-Enskog constraints in (A.98).

The Boltzmann collision operator needs to be specified in detail in orderto properly solve (A.99). However, the nature of the solution for δf can beexhibited by using an approximate collision model:

CK(δf) = − ν δf ≡ − ν (f − fC−E0 ), Krook-type collision operator(A.101)

in which

ν ≡ nσv, collision frequency, (A.102)

where the overbar indicates the “reaction rate” σv has been averaged over aMaxwellian distribution. Using this collision operator in (A.99), solving forδf and using the definitions in (A.95) and (A.96) yields the needed closure(constituitive) relations for the fluid moment equations (A.90)–(A.92):

q = −κm∇T, κm ≡ nχm, χm =54v2T

ν=

54νλ2, conductive heat flux,

(A.103)



which is in the form of a Fourier law for the heat flux and

π = − 2µmW, µm =12nm

v2T

ν=

12nmνλ2, viscous stress, (A.104)

in which

λ ≡ vT /ν, collision mean free path, (A.105)

has been defined for thermal molecules. In these closure relations, κm is theheat conduction coefficient, χm is the heat diffusivity and µm is the viscositycoefficient. The superscript m on the various coefficients indicate that theyarise from the micoscopic (molecular) processes of discrete collisions in the gas.Equations (A.103) and (A.104) give the thermodynamic fluxes q,π inducedby the thermodynamic forces ∇T,W. If the appropriate Boltzmann collisionoperator is used instead of the approximate Krook-type model of (A.101), thescaling of the κm, χm, and µm “molecular diffusion” coefficients with collisionfrequency and thermal speed remains the same; however the numerical factorsin (A.103) and (A.104) change slightly.

The reference cross section σ for atomic and molecular collisions is σ0 ≡πa2

0 ∼ 10−20 m2 in which a0 is the Bohr (atomic) radius (A.154). For standardtemperature and pressure (STP) air at the earth’s surface, the average crosssection for molecular collisions is σ ∼ 40σ0 ∼ 4× 10−19 m2, the density is nn ∼2.5× 1025 m−3, and the thermal speed is vT ∼ 300 m/s. Thus, for standard airν ∼ nσvT ∼ 3×109 s−1, λ ∼ vT /ν ∼ 10−7 m, and µm/nm ∼ νλ2/2 ∼ 1.5×10−5

m2/s, χm ∼ (5/2)(µm/nm).The Chapman-Enskog analysis is valid as long as the collision mean free

path is short compared to the gradient scale lengths (i.e., λ |∇ lnT | << 1,λ |∇V|/|V| << 1) and temporal variations are slow compared to the collisiontime [e.g., ν−1(∂ lnn/∂t) << 1]. Substituting the closure relations given in(A.103) and (A.104) into the momentum and energy conservation equationsyields (neglecting for simplicity the small effects due to gradients of the transportcoefficients κm and µm):

mndVdt

= nF−∇(p− µm

3∇·V

)+ µm∇2V, (A.106)

32dT

dt+ T (∇·V) = χm∇2T + 2

µm

n|W|2 . (A.107)

The diffusive components of these equations indicate that the “molecular” diffu-sion coefficients for momentum (viscous) and heat diffusion are µm/nm and χm,both of which scale as νλ2 and have units of m2/s. A physical interpretation ofthe processes and parametric scalings that underly these diffusion coefficientsare given in the next section.

An equation can also be developed for the evolution of the collisional entropys which is dimensionless and is defined in kinetic theory for f ' fC−E0 by

s ≡ − 1n

∫d3v f ln f = ln

(T 3/2

n

)+ constant, collisional entropy. (A.108)



Note that this entropy is the negative of the Boltzmann HB function [see (A.85)]and yields the same result as that obtained from equilibrium statistical mechan-ics [see (A.76)]. Taking the total time derivative of this equation yields, uponsubstituting in (A.90) and (A.92),

nTds

dt=

32ndT

dt− T dn

dt= T

[∂(ns)∂t

+∇· (nsV)]

= −∇· q− π :∇V, entropy evolution. (A.109)

Alternatively, since the flow of entropy density (entropy flux) is nsV + q/T ,after using the density conservation relation (A.90) and the rate-of-strain tensordefinition (A.100),

∂(ns)/∂t+∇· (nsV + q/T ) = θ ≡ − (1/T ) [ q ·∇ lnT + π : W ] (A.110)

in which θ represents the rate of entropy production due to dissipative (irre-versible) processes, which is positive definite and caused by the thermodynamicfluxes q,π flowing in response to the thermodynamic forces ∇ lnT , W.

For the closure relations given in (A.103) and (A.104) the entropy produc-tion rate simplifies to (again neglecting gradients in the transport coefficientsκm, µm):

θ = nχm |∇ lnT |2 + 2µm |W|2 , entropy production rate. (A.111)

Thus, entropy is produced by the microcopic collisional processes that diffusivelyrelax the gradients of the temperature and flow velocity in the gas. The entropyproduction rate is small under the Chapman-Enskog approximations (large ν,small λ = vT /ν): ds/dt ∼ νλ2 |∇ lnT |2 << ν if heat conduction effects aredominant, or ds/dt ∼ νλ2 |∇V|2 /v2

T << ν if viscous flow damping is domi-nant. Hence, for processes that are rapid compared to the collisional entropyproduction rate and where the entropy flow induced by the conductive heatflux q is negligible (e.g., in a constant temperature gas), it is sufficient to usethe “adiabatic” or isentropic (i.e., non-dissipative, constant entropy) equationof state for an ideal gas obtained from setting ds/dt = 0:

d

dtln(T 3/2

n

)=

d

dtln(p3/2

n5/2

)= 0 =⇒ p

nΓ= constant,

isentropic (“adiabatic”) equation of state, (A.112)

where Γ is 5/3 for the three-dimensional system being considered, but in generalis given by Γ = (N + 2)/N in which N is the number of degrees of freedom inthe system. [In thermodynamics Γ ≡ cP /cV is the ratio of the heat capacity(≡ ∂u/∂T ) at constant pressure to that at constant volume.] Note that for aconstant density gas of volume V = 1/n, (A.112) becomes the familiar equationof state for an ideal gas: pV Γ = constant. The adiabatic or isentropic equationof state can be used in place of the energy balance equation (A.92) or (A.107)for studies of rapid, isentropic processes because there is no significant entropyproduction or heat flow for such processes.



Most of the previously noted standard textbooks on statistical mechanicsprovide intermediate level descriptions of the kinetic theory of gases. Advancedlevel textbooks and monographs that deal specifically with the kinetic theoryof gases include:

Chapman and Cowling, The Mathematical Theory of Non-Uniform Gases (1952)[?]

H. Grad, “Principles of the Kinetic Theory of Gases,” in Handbuch der Physik ,Volume 12 (Springer-Verlag, Berlin, 1957) [?]

R. Herdan and B. S. Liley, “Dynamical Equations and Transport Relationshipsfor a Thermal Plasma,” Rev. Mod. Phys. 32, 731 (1960). [?]

A.5 Stochastic Processes, Diffusion

The heat and momentum diffusion produced by the collision-induced randomsteps or motions of molecules in a gas can be understood in terms of a stochasticor random walk process. Such processes are often called Brownian motion (aftera botanist Robert Brown who, in 1827, observed irregular motions of smallcolloidal size particles immersed in a fluid), or more formally a Markoff process(no memory of previous history or steps).

For a simple one-dimensional mathematical model of the random walk pro-cess, assume that between collisions (or another random process) a moleculemoves a distance ∆x in a random direction (to the right or left) in a time ∆t.For such a process the position xn of a molecule after the nth step is related tothe position xn−1 after the previous step by

xn = xn−1 +Rn ∆x (A.113)

in which Rn is randomly ±1. Using this mapping equation as a recursion rela-tion, one finds that after N random steps the difference of the final position xNfrom the initial position x0 becomes

xN = x0 + ∆xN∑n=1

Rn. (A.114)

In the limit of a large number N of random steps one obtains:

limN→∞

∣∣∣∣xN − x0

N∆x

∣∣∣∣ = limN→∞

|∑nRn|N

= limN→∞

O(1)N

= 0, (A.115)

because Rn is randomly ±1. Thus, after a large number N of random stepsthe average position of a molecule does not deviate much (<< N∆x) from itsinitial position x0.

However, as illustrated in Fig. A.2, the random steps do have an effect:they cause such molecules to wander randomly in the x direction, to ever largerdistances from x0 as the numberN of random steps increases. Thus, after a largenumber of random steps the position of a molecule is described by a probability



Figure A.2: Illustration of the random walk process of molecules stepping adistance ∆x randomly to larger or smaller x for a total of N = [t/∆t] steps: a)an example of a detailed particle trajectory; b) distribution of particle positionsover the N steps. The smooth dashed curves represent the N → ∞ analyticformulas given in (A.117) and (A.122) with ∆ = 4Dt→ ∆x

√2N .

distribution peaked at the initial position x0 with a spatial spread that increaseswith N and a peak magnitude that decreases with N — see Fig. A.2.

To quantify the spatial spreading effect, and hence the width of the proba-bility distribution, one uses the first form of (A.114) to calculate the square ofthe difference of the final from the initial spatial position:

(xN − x0)2 = (∆x)2

(N∑n=1

Rn

)2

= (∆x)2

N∑n=1

R2n +

N∑i=1

Ri

N∑n 6=i

Rn

. (A.116)

In the limit of a large number N of random steps, the mean spread is given by

limN→∞

(xN − x0)2

N(∆x)2= limN→∞

1N

N∑n=1

R2n +

N∑i=1

Ri

N∑n 6=i

Rn

= limN→∞

N +O(1)N

= 1,

(A.117)

because R2n = (±1)2 = 1. Hence, the average square of the spatial spreading

after a large number N of random steps (or a time t = N∆t) will be given by

(xN − x0)2 ' N(∆x)2 = t(∆x)2

∆t, or

d (xN − x0)2

dt' (∆x)2

∆t. (A.118)

In summary, a random walk process produces a spatial spreading, which is calledstochastic diffusion or simply diffusion, of molecules about their initial position,but no net motion of the average position of the molecules.



However, in an inhomogeneous medium there is, on average, a net motionor flux of particles. The particle transport flux produced by a large numberof molecules undergoing such random walk processes in a neutral gas with aspatially varying density n = n(x) can obtained as follows. In a time ∆t theplane x = x0 will be traversed by the half (on average) of the molecules thatexperience collisions in the layer between x0−∆x and x0, and which are movingto the right (+). Thus, the flux (Γ ≡ nV = ndx/dt) of molecules moving to theright is

Γ+ =12

∫ x0

x0−∆x

n(x)dx

∆t=

∆x2∆t

[n(x0)− ∆x

2dn

dx

∣∣∣∣x0

+ · · ·]

in which the density n(x) has been expanded in a Taylor series about x0. Sim-ilarly, the flux of molecules moving through the x = x0 plane to the left (−)is

Γ− =12

∫ x0+∆x

x0

n(x)dx

∆t=

∆x2∆t

[n(x0) +

∆x2dn

dx

∣∣∣∣x0

+ · · ·].

The net particle flux is the difference between these two fluxes:

Γ = Γ+ − Γ− = −D dn

dx

∣∣∣∣x0

, Fick’s diffusion law. (A.119)

For the simple model being considered D is given by

D =(∆x)2

2 ∆t, diffusion coefficient, (A.120)

which has units of m2/s. Thus, the diffusion coefficient D is half the rate ofspatial spreading for a random walk process — see (A.118).

The natural step size ∆x for the motion of molecules between collisions ina neutral gas is λ, the collision mean free path. The characteristic time ∆tbetween collisions of molecules is 1/ν. Thus, one infers from (A.120) that thescaling of diffusivities induced by molecular collisions should be D ∼ νλ2, whichwas what was obtained in (A.103) and (A.104) in the preceding section. In amonoatomic neutral gas there are heat and momentum diffusivities but there isno particle diffusivity (or particle flux Γ) because, while two colliding moleculesexchange energy and momentum during the molecular collisions, the density ofmolecules is usually unchanged as a result of the collisions.

In more realistic situations different molecules may have different ∆x and∆t values; then one must take an appropriate average and D = 〈(∆x)2〉/(2∆t).Since the parametric scaling of the diffusion coefficient is quite general, but ap-propriate averages are often difficult to formulate or evaluate for various physicalprocesses, the expression for D in (A.120) is mostly used to infer the scalingof the diffusion coefficient with physical parameters. Then, kinetic calculationsare used to obtain the relevant numerical coefficients — the “headache factors.”



In the presence of this random walk process induced by molecular collisions,the equation for the density n(x, t) [see (A.90] becomes ∂n/∂t+∂(nVx)/∂x = 0.Using the Fick’s diffusion law (A.116) for nVx = Γ yields a one-dimensionaldiffusion equation:

∂n

∂t=

∂

∂xD∂n

∂x, diffusion equation. (A.121)

To illustrate the properties of solutions of this equation, imagine that a smallnumber δN of molecules are added to the medium at the position x0: n(x, 0) =δNδ(x−x0). After a short time, the appropriate (Green-function-type) solutionof (A.121) is

n(x, t) = δNe−(x−x0)2/4Dt

√π√

4Dt, short-time diffusive distribution, (A.122)

as can be verified by direct substitution. Note that this distribution of particleshas the desired properties for a random walk process and represents it well inthe N → ∞ limit — see Fig. A.2b. In particular, it is peaked at x = x0 andspreads spatially and decreases in magnitude as time progresses. In a time t(assumed >> ∆t) the average spreading of the molecules in the x direction is

(x− x0)2 ≡∫∞−∞ dx (x− x0)2 n(x, t)∫∞

−∞ dxn(x, t)= 4Dt

∫∞0dy y2e−y

2∫∞0dy e−y2 = 2Dt =

(∆x)2

∆tt

(A.123)

in which y ≡ (x− x0)/√

4Dt and the integrals have been evaluated using (??).Note that this rate of spatial spreading of the density agrees with that inferredabove for the random walk process of a molecule — (A.118).

The Gaussian character of this distribution can be emphasized by writingthe short time diffusive density response in (A.122) in the form

n(x, t) = δNe−(x−x0)2/∆2

√π∆

, ∆ ≡√

4Dt. (A.124)

In this form one readily sees from (??) that in the t → 0 limit the solutionbecomes a delta function (see Section B.2) at x = x0: limt→0 n(x, t)=δN δ(x−x0), which was the initial condition. Also, in terms of ∆ the root mean squarespatial spread becomes simply

δxrms ≡(

(x− x0)2)1/2

=√

2Dt =∆√

2= ∆x

√t

∆t= ∆x

√N,

root mean square spatial spread. (A.125)

The last result shows that, as indicated in Fig. A.2, the spatial spreading isproportional to the square root of the number of random walk steps.



Using these formulas, note also that the average time t required for moleculesto diffuse a distance δx ≡ x− x0 from the initial position x0 is

t ∼ (δx)2

2D∼(δx

∆x

)2

∆t, time to diffuse a distance δx. (A.126)

Hence, the time t required to diffuse a short distance δx is the product of the ba-sic random walk time ∆t times (δx/∆x)2 — the square of the number of randomwalk steps ∆x in the distance δx to be traversed. This quadratic dependenceof the spreading time on the spreading distance is an intrinsic property of dif-fusive processes. As a caveat on this analysis, note that the solution (A.122)is only valid for short times: t < L2

n/D ∼ (Ln/∆x)2∆t — so the backgroundmedium density and diffusion coefficient are reasonably constant over the dis-tance ∆/

√2 =√

2Dt = ∆x√t/∆t that typical particles spread over in the time

t [i.e., (∆/n)(dn/dx) ≡ ∆/Ln << 1 and (∆/D)(dD/dx) << 1].In a finite box, as time progresses molecules eventually diffuse to the bound-

aries of the box where it will be assumed the molecules are absorbed. Thequestion then becomes: what is the average confinement time for molecules inthe box? Assume for simplicity that: the diffusion coefficient is constant inspace; a one-dimensional treatment is sufficient; δN molecules are inserted atthe center (x = 0) of a box of width 2L (assumed >> ∆x = λ) at time t = 0;and the density of molecules vanishes at the box boundaries (x = ±L). Then,the solution of the diffusion equation (A.121) for this boundary value problemcan be shown (by separation of variables, expansion in sinusoidal eigenfunctions)to be

n(x, t) =δN

L

∞∑j=0

e−t/τj cos(λjx

L

), with λj ≡

(2j + 1)π2

, τj ≡L2

λ2jD

. (A.127)

For short times (t << τ0) the box boundaries at x = ±L are unimportantand this solution reduces to (A.124), which is a more convenient form then.For intermediate times (t ∼ τj) the sinusoidal eigenfunctions (up to at least2j) must be summed to obtain the response. In the time asymptotic limit(t > τ0 > τ1 > τ2 · · · ) the lowest order eigenmode solution dominates:

n(x, t)t>τ0' δN

Le−t/τ0 cos

(π2x

L

). (A.128)

Thus, an average ”confinement time” for molecules in the box can be identifiedas

τ0 ≡L2

λ20D

=L2

(π/2)2D' L2

2.5D, confinement time. (A.129)

Note that upon using D = (∆x)2/2∆t one obtains τ0 = (2/λ20)(L/∆x)2∆t,

which quantifies (for this specific case where δx → L, t → τ0) the “headachefactors” in the scaling relation (A.126). For cylindrical, spherical “boxes” the



eigenfunctions are Bessel, spherical Bessel functions and the lowest order eigen-values are λ0 ' 2.405, λ0 = π, respectively. Then, using a as the radius of the“box,” one obtains confinement times of τ0 ' a2/6D, a2/10D for cylindrical,spherical systems, respectively.

Many of the references noted at the end of the two preceding sections havediscussions of random walk (Brownian motion) and stochastic diffusion pro-cesses. The classic and ageless reference for such processes is:

S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy,” Rev. Mod.Phys. 15, 1 (1943). [?]

A.6 Fluid Mechanics

The equations of “hydrodynamics” used to describe the behavior of a fluid arethe fluid moment equations obtained from the kinetic theory of gases — (A.90),(A.91) and (A.112). However, they are usually modified by writing them interms of the mass density ρm ≡ nm, which has units of kg/m3:

mass continuity equation:∂ρm∂t

+∇· ρmV = 0, (A.130)

Navier-Stokes equation: ρmdVdt

= ρmFm−∇p′ + µm∇2V, (A.131)

(momentum balance)

isentropic equation of state:d

dtln(p

ρΓm

)= 0, (A.132)

in which p′ ≡ p − (µm/3)∇·V. (The equation of state is often called the“adiabatic” equation of state in hydrodynamics.) In these equations d/dt is thetotal time derivative taking account both of the direct temporal derivative andthe effects of the advection by the flow velocity V in the fluid:

d

dt≡ ∂

∂t+ V·∇, total time derivative. (A.133)

For gases or liquids in the earth’s atmosphere the relevant force on moleculesis the gravitational force, which is a conservative force:

FG = FG = −m∇VG ' −mg ex ≡ mg, gravitational force, (A.134)

where VG = −MEG/R is the gravitational potential. In the last expression usehas been made of the fact that near the earth’s surface (x ≡ R − RE << RE ,radius of the earth) one has VG ' −(MEG/R

2E)(RE − x + · · · ). Also, here,

g ≡ (MEG/R2E) ' 9.81 m/s2 is the gravitational acceleration at the earth’s

surface.The velocity flow field V can in general be decomposed into parts repre-

sentable in terms of scalar, vector potentials ψ, C (see Section D.5):

V = −∇ψ +∇×C, potential representation of a flow field. (A.135)



The scalar potential part represents the longitudinal, irrotational or compress-ible part of the flow since ∇·V = −∇·∇ψ = −∇2ψ. The vector potentialpart is incompressible since ∇·∇×C = 0. However, this component represents“rotation” or vorticity5 (units of s−1) in the flow:

ω ≡∇×V =∇×(∇×C), vorticity. (A.136)

The properties of sound waves in a fluid can be illustrated by consideringcompressible perturbations of air in the earth’s atmosphere. The equilibriumpressure distribution is determined from the “hydrostatic” force balance equi-librium, which is the equilibrium (∂/∂t = 0) and small viscosity limit of theNavier-Stokes equation that in the absence of equilibrium flows (V0 = 0) be-comes simply:

0 = − ρm0∇VG −∇p0.

Assuming for simplicity that the temperature T is constant, taking x to bethe vertical distance above the earth’s surface, and using p = nT = ρm0T/m(Boyle’s law for this situation), the hydrostatic equilibrium becomes

0 = − ρm0g −T

m

dρm0

dx=⇒ ρm0(x) = ρm0(0) e−mgx/T ,

law of atmospheres. (A.137)

Thus, in equilibrium the density of air decreases with distance above the surfaceof the earth on a scale length of T/mg = v2

T /2g ∼ 104 m.To exhibit the properties of sound waves consider perturbations of the com-

pressible air in this equilibrium:

ρm = ρm0 + ρm, V = V, p = p0 + p, perturbed equilibrium, (A.138)

in which the tilde over a quantity indicates the perturbation in that quantity.Substituting these forms into the fluid equations (A.130)–(A.132) yields, uponneglecting the effect of gravity for simplicity and linearizing the equations (i.e.,neglecting all quantities that are quadratic or higher order in the perturbations):

∂ρm∂t

+ ρm0∇·V + V·∇ρm0 = 0, ρm0∂V∂t

= −∇p+ µm∇2V, p = cHS2ρm

where

cHS ≡√

Γp0/ρm0 =√

ΓT/m, hydrodynamic sound speed, (A.139)

which is typically about 340 m/s at the earth’s surface. Note that for a neutralgas cHS =

√Γ/2 vT . For an equilibrium that is approximately homogeneous

over the collision mean free path λ (λ |∇ ln ρm0| << 1), a perturbed density

5A physical example of vorticity is the circular flow of water around a drain in a bathtubas it is being emptied.



response ρm exists only for compressible perturbations (∇·V 6= 0). For suchperturbations these equations can be combined to yield

∂2ρm∂t2

−∇·(cHS

2∇ρm − µm∇2V)

= 0.

Considering perturbations that are localized relative to the scale length of theequilibrium density gradient so that the∇cHS

2 and∇µm terms can be neglected,but longer scale than the collision mean free path (typically ∼ 10−7 m for air atthe earth’s surface) so the viscosity can be neglected (i.e., cHS

2/g ∼ 104 m >>

perturbation scale length >> λ ∼ 10−7 m), this equation becomes simply

∂2ρm∂t2

− cHS2∇2ρm = 0, sound wave equation. (A.140)

Thus, density perturbations compress (∇·V < 0) and rarefy (∇·V > 0) thefluid as they propagate through it adiabatically (with negligible entropy pro-duction) at the sound speed cHS defined in (A.139).

To exhibit the properties of the most fundamental type of fluid instabilities,the Rayleigh-Taylor (R-T) instabilities, consider perturbations of nearly incom-pressible liquids, in a case where a heavy liquid is above a lighter liquid andthe two fluids are immiscible. For incompressible (∇·V→ 0) perturbations thelinearized continuity equation becomes

∂ρm∂t

= − V·∇ρm0, advective response, (A.141)

which indicates the change in local mass density caused by a perturbed flowin the direction of the gradient in the equilibrium mass density. Combiningthis advective response with the partial time derivative of the linearized Navier-Stokes equation yields

ρm0∂2V∂t2

=(V·∇ρm0

)∇VG −∇

∂p

∂t.

Taking the curl of this equation to eliminate the perturbed pressure gradient andhence the coupling to sound waves, and neglecting gradients in the equilibriumcompared to those in the perturbations (ρm0∇×V >>∇ρm0×V) and viscosityeffects (valid for perturbation scale lengths long compared to the collision meanfree path λ), yields for the perturbed flow vorticity ω ≡∇×V:

ρm0∂2ω

∂t2= ∇

(V·∇ρm0

)×∇VG.

Considering a coordinate system where x is directed vertically upward and y, zare in a plane parallel to the earth’s surface, and assuming wavelike pertur-bations of the type V = ∇×C = − ez×∇Cz, in which C = Czez, Cz ∼exp(ik·x − iωt) with kx << ky is a stream function [see (??,??)], so thatω = − ez∇2

⊥Cz ' ezk2yCz, yields an equation for the perturbation frequency:

ω2 ' −∇VG ·∇ ln ρm0 = g ·∇ρm. (A.142)



When ∇VG ·∇ρm0 < 0 (light liquid above heavy liquid since ∇VG ≡ −g isupward), ω2 is positive and two benign, oscillating waves occur. (Adding vis-cosity effects causes the waves to be damped.) However, when a heavy liquid isplaced over a light liquid (∇VG ·∇ρm0 = −g ·∇ρm0 > 0) the ω2 < 0 indicatescomplex conjugate roots, one of which will be growing exponentially in time atrate:

γ ≡ Imω ' (−g ·∇ ln ρm0)1/2, R-T instability growth rate (A.143)

This is the Rayleigh-Taylor (or interchange) instability by which the interfaceregion between the upper heavy fluid and the lower lighter fluid develops growingundulations that lead ultimately to interchange of the positions of the heavy andlight fluids.

The overall process of the interchange of the two fluids can be thought ofas consisting of the following steps. First, thermal fluctuations excite a modestundulation of the boundary between the two liquids. If the heavy fluid is on top,this spontaneous perturbation grows exponentially in time at the rate indicatedby (A.143). The undulations grow to a large amplitude where the linearizationprocedure used to derive (A.142) becomes invalid. Lagrangian coordinates (i.e.,coordinates that follow particular fluid elements as they move rather than theusual fixed position Eulerian ones) can be used to explore the growth of thestructures into the slightly nonlinear regime. However, ultimately the vortex-like collective motions of the fluids become highly nonlinear, very contorted andlarge enough to encounter adjacent vortices and/or the boundaries of the regionsoccupied by the fluids. Then, turbulence in the fluid develops and it cascadesthe large vortices into smaller ones, turbulently mixing the two fluids until theheavier one is on the bottom.

In order to describe the behavior of the vortices as they evolve nonlinearlytoward the turbulent state, consider the total time derivative of the circulationCK ≡

∮Cd` ·V in the rotational part of the flow V, which is responsible for the

vortex, over the closed curve C within the fluid:

dCKdt

≡ d

dt

∮C

d` ·V =d

dt

∫∫S

dS · ∇×V

=∫∫S

dS · [(∂/∂t+ V·∇) (∇×V)] +∫∫S

(dS/dt) ·∇×V

=∫∫S

dS · ∇×(∂V/∂t)−∇× [V× (∇×V)]

=∫∫S

dS · [∂ω/∂t−∇×(V×ω)] (A.144)

in which use has been made of Stokes’ theorem (??), S is the open surfacebounded by the closed curve C that moves with the encompassed fluid, dS/dt =(∇·V)dS −∇V· dS [see (??)], and the vector identity in (??) has been used.Dividing the Navier-Stokes equation (A.131) with a conservative force F by ρm



and taking its curl to obtain an equation for ∇×∂V/∂t yields, after makinguse of (??):

dCKdt

=∫∫S

dS ·(

1ρ2m

∇ρm×∇[p− µm

3(∇·V)

]+∇×µ

m

ρm∇2V

). (A.145)

For an adiabatic equation of state (A.132), p ∝ ρmΓ and hence ∇ρm×∇p = 0.Thus, the circulation CK is constant in time, except for the dissipative effectsdue to viscosity that are small for all but very short scale lengths of the order ofthe collision mean free path λ because µm/ρm ∼ νλ2. Thus, on most relevantscale lengths

dCKdt

= 0 for µm → 0, Kelvin’s circulation theorem, (A.146)

for inviscid (zero viscosity) fluids.What this theorem shows is that a vortex tube moves with (or is “frozen

into”) the fluid as it evolves, and that the amount of circulation CK in the flowfield V remains constant — except for the effects of viscosity, which becomesimportant in boundary layers near the edge of the fluid or at the edge of vorticesthat come close to other vortices. However, the derivation relied on the use ofStokes’s theorem, which required that the topology of the closed curve C becontinuous and that it remain so. Thus, the invariance of CK could be brokenby nonlinear interactions between vortex structures that break or reconnect thetopology by causing the bounding curve C, which is expected to always movewith the fluid and encompass the same vorticity flux

∫∫SdS ·∇×V, to become

discontinuous. To the extent that the topology of the surfaces of vorticity fluxremains intact there is no motion (or transport) of fluid relative to these surfaces.However, the flux surfaces of the vorticity can distort in shape as they movearound in the fluid. Thus, vortex tubes or eddies are relatively robust objectsin low viscosity fluids.

The nonlinear evolution and interactions of vortices in a fluid are governedby the vorticity evolution equation

∂ω

∂t=∇×(V×ω)− ω (∇·V) +

µm

ρm∇2ω (A.147)

or,

dω

dt≡(∂

∂t+ V·∇

)ω = ω ·∇V +

µm

ρm∇2ω. (A.148)

These equations are obtained by taking the curl of (A.131), which eliminates thecoupling to sound waves, and assuming for simplicity that the mass density isconstant. The ∇×(V×ω) term in (A.147) represents the advection of the vor-ticity vector ω by the flow velocity V — as indicated by the last line of (A.144).The ω ·∇V term on the right of (A.148) represents vortex tube stretching bygradients in the velocity flow; it vanishes for two-dimensional flows. In three-dimensional flows the vortex tube stretching term reduces the area of a vortex



but also increases its vorticity — to keep the vorticity flux CK constant asrequired by Kelvin’s circulation theorem.

The ratio of the nonlinear advection of vorticity [∇×(V×ω), a nonlinear“inertia” term] to the viscous dissipation of vorticity (µm∇2ω) is

Re =∇×(V×ω)

(µm/ρm)∇2ω∼ ρm0V0L0

µm, Reynolds number, (A.149)

in which V0, L0 are typical flow speeds and gradient scale lengths in the fluid.When the vorticity evolution equation is written in terms of dimensionless vari-ables, the reciprocal of the Reynolds number is the only dimensionless parameterin the equation — as the coefficient of the viscous dissipation term. For example,for incompressible flows (i.e., ones that do not excite sound waves and are dom-inated by vorticies), Eq. (A.147) can be written in terms of the dimensionlessvariables t ≡ (V0/L0)t, V ≡ V/V0, and ∇ ≡ L0∇ as

∂ω

∂t= ∇×(V×ω) +

1Re∇2ω. (A.150)

Thus, all incompressible flows with the same Reynolds number and the same flowgeometry will have the same flow properties. At low Reynolds numbers (Re <∼ 1)the flow is laminar. For not too large Reynolds numbers vortex structuresinduced by the particular geometrical situation (e.g., flow past a fixed body)tend to dominate the flow pattern. For high Reynolds numbers (Re >∼ 103) thenonlinear vorticity advection overwhelms the viscous dissipation and the flowbecomes turbulent.

In fully developed turbulence (Re >> 103) there is a cascade of energyfrom macroscopically-induced large-scale vortices through nonlinear interactionsof turbulent eddies of successively smaller dimensions until the scale lengthsbecome so small that the energy in the eddies is viscously dissipated. (Theeffective Reynolds number is close to unity for the dissipative scale eddies.)Since the dominant eddy interaction term is the vortex stretching term ω ·∇V in(A.148), successive “generations” of the turbulent eddies become longer, thinnerand have larger vorticities. Thus, the mean square vorticity, which is known asthe enstrophy (Ω ≡ |ω|2), increases during the cascade.

For sufficiently large Reynolds numbers there is a large “inertial” range ofspatial scale lengths for which the vortex interactions are predominantly non-linear (i.e., where the large-scale “stirring” and small-scale viscous dissipationeffects are negligible). In the inertial range the turbulent eddies are self-similar(i.e., of the same structure, independent of scale size, from one generation to thenext one). The energy flow per unit mass ε = (V·∇)(V 2/2) from one wavenum-ber range k to the next smaller one can be estimated by εk ∼ kV 3

k ∼ V 2k /τk

in which τk ∼ (kVk)−1 is the turbulent decorrelation or eddy turnover timeat a given k. Since energy is input via “stirring” at large scales and dissi-pated at small scales, in steady-state the energy transfer rate from one scaleto the next smaller one in the inertial range must be nearly constant. Thus,εk ' ε ∼ V 3

0 /L0, a constant for a given externally driven situation, and hence



Vk ∼ (ε/k)1/3. The energy E(k) in the turbulent fluctuations between k and2k is given approximately by

∫dk E(k) ∼ k E(k) ∼ V 2

k or E(k) ∼ ε2/3k−5/3,

which is the Kolomogorov spectrum for turbulence within a large inertial range.The successively smaller scale eddies have smaller velocities and energies, butlarger vorticity and faster turnover rates [τk ∼ (kVk)−1 ∼ (εk2)−1/3] to keepthe energy flow rate in k-space constant.

In the inertial range the turbulent eddies lose their momentum on a mixinglength scale ∆k ∼ Vkτk ∼ 1/k. This leads to a Prandtl mixing length esti-mate for the effective diffusion coefficient [cf., (A.120)] for turbulent viscosityin the fluid of Deff ∼ µeff/ρm ∼ ∆2

k/τk ∼ ε1/4/k4/3. However, this turbu-

lent mixing is actually dissipationless; all it does is transfer the momentumand energy to shorter scale lengths. Eventually, the eddies reach the (Kolo-mogorov) dissipation scale k−1

d at which 1/τkd ∼ (µm/ρm) k2d, which yields

kd ∼ ε1/4/(µm/ρm)3/4 ∼ (Re)3/4/L0.

Because the effects of viscosity are negligible in the inertial range and becausethe viscous dissipation scale length is so short [k−1

d ∼ L0/(Re)3/4 << L0], itis tempting to neglect it entirely. However, while its effects can be neglectedfor inertial range scale lengths (1/L0 << k << kd), it must be retained ingeneral because it: 1) increases the order of the differential equation governingvorticity; 2) is important in boundary layers near material objects and othernearby vortices; and, 3) most importantly for computer simulation, providesthe only energy sink (at high k) for turbulent fluctuations in a neutral fluid.

Most of the previously noted standard textbooks on mechanics, statisticalmechanics and kinetic theory of gases contain introductory or intermediate leveldescriptions of fluid mechanics. Advanced level monographs and textbooks thatspecifically deal with fluid mechanics include:

Batchelor, Introduction to Fluid Dynamics (1967) [?]

Tennekus and Lumley, A First Course in Turbulence (1972) [?].

A.7 Quantum Mechanical Effects

The fundamental concept in quantum mechanics is that, owing to the wavelikenature of particles on small scale lengths, a particle’s position q and canonicallyconjugate momentum p cannot simultaneously be known to arbitrarily high ac-curacy. Rather, the product of the uncertainties in the position and momentum,δq and δp, respectively, must be Planck’s constant or greater:

δp · δq ≥ h, Heisenberg uncertainty principle. (A.151)

This relation shows the limit of applicability of mechanical causality. The un-certainty principle holds for any pair of canonically conjugate variables. Thus,it applies for energy and time, which for conservative systems are canonicallyconjugate variables (p = H = ε and q = t), as well:

δε δt ≥ h. (A.152)



By Heisenberg’s uncertainty principle, the position of a nonrelativistic particlemoving with velocity v in a force-free region (so that its canonical momentump is simply mv) cannot be known to within

λh = h/mv, de Broglie wavelength. (A.153)

In the Bohr model of the hydrogen atom an electron gyrates at constantradius a0 around the proton nucleus of the atom. Since the rotational angle ϕ isa symmetry coordinate and hence totally uncertain, the Heisenberg uncertaintyprinciple requires that the canonically conjugate action J be quantized to integermultiples (n) of Planck’s constant:

J =∮

p · dq =∮pϕ dϕ = 2πme a

20 ω0 = nh

in which the angular momentum pϕ ≡ me a20 ω0 where ω0 ≡ dϕ/dt is the con-

stant rotation frequency. The equilibrium radial force balance between theelectric field force e2/(4πε0a2

0) and the centripital acceleration force me a0 ω20

on the electron yields the equation

e2

4πε0a20

= me a0 ω20 .

Solving these two simultaneous equations for a0 in the ground state (n = 1)case yields the characteristic radius of the hydrogen atom:

a0 = 4πε0(h/2π)2

me e2' 0.529×10−10 m, Bohr radius. (A.154)

This is the characteristic scale length for the “size” of all atoms — the rangeover which their electrostatic force field extends. The corresponding range overwhich nuclear forces extend is

re =e2

4πε0mec2' 2.82×10−15 m, classical electron radius, (A.155)

which is inferred from equating the electric potential energy ∼ e2/(4πε0re)from a distributed electron charge to the electron rest mass energy mec

2.The binding energy of an electron in a Bohr atom in its ground (lowest

energy) state is given by the (negative of the) potential energy of the electronwhen it is located at the Bohr radius from the proton plus the kinetic energy ofthe electron:

EH∞ =e2

4πε0a0− me

2a2

0ω20 =

14πε02

mee4

2(h/2π)2' 13.6 eV,

Bohr atom binding energy, (A.156)

which is also called the Rydberg energy. For electrons in the nth quantumstate the orbit radius increases by a factor of n2 and the rotation frequency



ω0 decreases by a factor of 1/n3; consequently, the binding energy of the statedecreases by a factor of 1/n2. For electrons gyrating around an ion of charge Zi,the potential and consequently the electric field force increases by a factor of Zi.This causes the Bohr radius to decrease by a factor of 1/Zi and the ionizationenergy to increase by a factor of Z2

i . Thus, neglecting fine-structure effects, thebinding energy of an outer electron in a level labeled by the quantum numbern (≥ 1) which is gyrating around an ion of charge Zi is given by

EZ∞(n) ' Z2i E

H∞/n

2, outer electron binding energy. (A.157)

Note that for a nucleus with a high atomic number Z the binding energy ofthe most tightly bound (n = 1, ground state) electron, which is the last oneto be removed as an atom is ionized, can be very large. For example, for iron(Z = 26) the binding energy of the last electron is ∼ 9 keV while for tungsten(Z = 74) it is ∼ 75 keV.

The degree of ionization in a plasma can be estimated from the Saha equationwhich gives the population density of a particular ionization and quantum stateof an atom in a gas in thermodynamic equilibrium. It can be obtained by equat-ing the rates of ionization [∝ nn exp(−Ui/Te)] and recombination [∝ ni(neλ3

h)]for ions in a partially ionized gas:

ninn' 2

ne

(2πmeTeh2

)3/2

e−Ui/Te = 25/2 neλ3De

(nea30)1/2

e−Ui/Te

' 6× 1027

ne(m−3)[Te(eV)]3/2 e−Ui/Te , Saha equation, (A.158)

in which ne, ni and nn are the electron, ion and neutral density, respectively,Ui is the ionization potential and Te is the temperature in electron volts of theassumed Maxwellian distribution of electrons. The ionization potential Ui forionization of an atom from its ground (neutral) state to the first ionized stateis given by the electron binding energy in the atom [cf., Eq. (A.157)]. It rangesfrom 3.9 eV for Cesium atoms to 24.6 eV for Helium.

The fractional ionization [≡ ni/(nn +ni)] is exponentially small for electrontemperatures Te much lower than the ionization potential Ui. The electrontemperature required to attain a small degree of ionization (∼ ni/nn << 1) canbe estimated by solving the Saha equation iteratively for Te:

Te|ion 'Ui

ln(

6× 1027 [Te(eV)]3/2

(ni/nn)ne(m−3)

) ∼ (0.02–1) Ui, (A.159)

where in the last form the smallest number correponds to interplanetary densi-ties (∼ 106 m−3) and the largest to solid densities (∼ 1029 m−3). The Te requiredto produce a fully ionized state (ni/nn >> 1) is not much larger. Thus, for ex-ample, a nitrogen gas (Ui = 14.5 eV) at a density of 2.5×1025 m−3 (the densityof room temperature air) becomes 1% ionized at Te ' 1.4 eV, and fully ionizedfor Te >∼ 2.2 eV. At lower densities the electron temperature range over which



the transition from a partially to fully ionized gas takes place is even narrower.For some examples of the variation with electron density of the Te required forcomplete ionization, see Fig. ?? at the end of Chapter 1.

Note however that the ions might not be fully stripped of their electrons. Inparticular, for Te ∼ 0.1–10 keV, high Z ions might not be fully stripped becauseof the very large binding energy of their most tightly bound electrons. Suchions would have an ion charge Zi < Z.

Some standard introductory level quantum mechanics textbooks are:

Krane, Modern Physics ( ) [?]

Sproul and Phillips, Modern Physics ( ) [?]

Tipler, Modern Physics ( ) [?]

Gasiorowicz, Quantum Physics ( ) [?]

Powell and Crasemann, Quantum Mechanics ( ) [?].



A.8 Physical Constants

Fundamental Physical ConstantsRelativeUncertainty

Quantity Symbol Best Value6 (×10−6)

electron mass me 9.109 389 7× 10−31 kg 0.59proton mass mp 1.672 623 1× 10−27 kg 0.59elementary charge e 1.602 177 33× 10−19 C 0.30speed of light in vacuum c 299 792 458 m/s exact7

permeability of vacuum µ0 4π × 10−7 H/m exactpermittivity of vacuum ε0 1/µ0c

2 F/m exactgravitational constant G 6.672 59× 10−11 N ·m2/kg2 128Planck constant h 6.626 075 5× 10−34 J · s 0.60Boltzmann constant kB 1.380 658 ×10−23 J/K 8.5

SI Units And Their Abbreviations, Interrelationships

Quantity Name Symbol In Terms Of Other Units

length meter m 102 cm = 1010 Amass kilogram kg 103 gtime second selectric current ampere A C/stemperature kelvin K ' 1/11 604.4 eVamount of substance mole molatomic unit of energy electron volt eV ' 1.602 177 33× 10−19 Jatomic unit of mass amu u ' 1.660 540 2× 10−27 kgfrequency hertz Hz s−1 (cycles per second)force newton N m · kg · s−2

pressure, stress pascal Pa N/m2 = m−1· kg · s−2

energy, work joule J N ·m = m2· kg · s−2

power watt W J/s = m2· kg · s−3

electric charge coulomb C s ·Aelectric potential volt V W/A = m2· kg · s−3·A−1

capacitance farad F C/V = m−2· kg−1· s4·A2

electrical resistance ohm Ω V/A = m2· kg · s−3·A−2

magnetic flux weber Wb V · s = m2· kg · s−2·A−1

magnetic flux density8 tesla T Wb/m2 = kg · s−2·A−1

inductance henry H Wb/A = m2· kg · s−2·A−2

6E.R. Cohen and B.N. Taylor, Physics Today, August 1998, BG7 [?].7The speed of light fixes the length of the meter in terms of the second.8In plasma physics magnetic field strengths are often quoted in Gauss: 1 Tesla ≡ 10 kGauss.



Other Physical Constants

Quantity Symbol Value

Avogadro constant NA 6.022× 1023 #/molMolar gas constant R 8.31 J ·mol−1K−1

Air (20oC and 1 atmosphere)density n 2.49× 1025 molecules/m3

sound speed cS 343 m/satmospheric pressure p 760 Torr = 1.01× 105 Pamolecular weight 28.9 g/molviscous diffusivity µm/ρm 1.5× 10−5 m2/s

Waterdensity n 3.33× 1028 molecules/m3

sound speed cS 1460 m/sviscous diffusivity µm/ρm 10−6 m2/s

Earthmass ME 5.98× 1024 kgmean radius RE 6.37× 106 mgravitational acceleration g 9.81 m/s2

magnetic dipole moment Md 8.0× 1022 A ·m2

Particle Masses

Particle Atomic Best Atomic Energy Unitsor Atom Symbol Number Z Mass9Value (mc2/e, MeV)

electron me 0.000 548 579 903 0.511muon mµ 0.113 428 913 105.658proton mp 1 1.007 276 470 938.272neutron mn 1.008 664 904 939.566deuteron mD 1 2.013 553 214 1 875.613triton mT 1 3.016 050 2 809.853helium mHe 2 4.002 603 3 728.402carbon mC 6 12.011 15nitrogen mN 7 14.006 7oxygen mO 8 15.999 4argon mAr 18 39.948iron mFe 26 55.845molybdenum mMo 42 95.94tungsten mW 74 183.84

9The unified atomic mass unit = 1.660 540 2 × 1027 kg (0.59 × 10−6 relative error) =931.494 32 MeV (0.30× 10−7 relative error). Note also that Avogadro’s constant NA ≡ 1/u.


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