Waclecka Statical Physics

June 27, 2011 17:21 WSPC/Book Trim Size for 9in x 6in SMroot

Appendix A

Non-Equilibrium Statistical

Mechanics

The text has focused on equilibrium statistical mechanics and its wide va-

riety of applications. Little has been said about mechanisms and the ap-

proach to equilibrium. This appendix presents some simple considerations

concerning non-equilbrium statistical mechanics within the framework of

the Boltzmann equation and some of its immediate extensions.1

A.1 Boltzmann Equation

A.1.1 One-Body Dynamics

Consider a collection of identical, non-localized, randomly prepared sys-

tems, which in statistical equilibrium becomes themicrocanonical ensemble.

We now just do classical mechanics and assume to start with the systems

are independent. Place them in their appropriate position in six dimen-

sional phase space p,q.2 The distribution function f(p,q, t) is defined

in the following manner

dN = number of systems in d3p d3q

≡ f(p,q, t)d3p d3q

(2π~)3(A.1)

The quantity d3p d3q/(2π~)3 = d3p d3q/h3 counts the number of cells in

this small volume in phase space. A value f = 1 would then imply that

every cell is occupied by one particle.

The probability of finding a system in this region of phase space is the

probability of picking a member of the ensemble at random, which is dN/N .

1See [Boltzmann (2011)], and a good reference here is [Stocker and Greiner (1986)].2Here q = (x, y, z) and p = (px, py, pz).

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336 Introduction to Statistical Mechanics

This quantity is used to compute expectation values.

The goal now is to follow the time evolution of the distribution function.

As a function of time, a particle at p0,q0 at time t0 moves to the point

p,q at time t (see Fig. A.1). Let dλ0 = d3p d3q be the phase space

p

0

t d

d0

t

q

Fig. A.1 An ensemble of systems with dN = f(p,q, t) d3p d3q/(2π~)3 members in asix-dimensional phase space volume dλ0 = d3p d3q, at a position p0,q0 at a time t0,is followed along a phase trajectory to a time t.

volume at the time t0

dλ0 = d3p d3q ; phase-space volume at t0 (A.2)

Then with hamiltonian dynamics, one has Liouville’s theorem, which states

that the phase space volume is unchanged along a phase trajectory3

dλ = dλ0 ; Liouville’s theorem (A.3)

Since the number of systems is conserved, the number of systems within

this phase-space volume does not change

dN = dN0 ; number conserved (A.4)

One concludes that the distribution function is unchanged along a phase

trajectory

f [p(t),q(t), t] = f(p0,q0, t0) ; unchanged along phase trajectory

(A.5)

Now write out the total differential of Eq. (A.5), and divide by dt

df

dt=∂f

∂t+∇pf · dp

dt+∇qf · dq

dt= 0 (A.6)

3A proof of Liouville’s theorem can be found in [Walecka (2000)], or [Fetter andWalecka (2006)].

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Non-Equilibrium Statistical Mechanics 337

Hamilton’s equations of motion for a particle state that

dq

dt= v = ∇pH ; Hamilton’s equations

dp

dt= F = −∇qH (A.7)

Equation (A.6) can then be re-written as

∂f

∂t= ∇qH ·∇pf −∇pH ·∇qf = H, fP.B. (A.8)

where the last equality identifies the Poisson bracket of classical mechanics.

In equilbrium, the time derivative of the distribution function at a given

point p,q vanishes

∂f

∂t= 0 ; equilbrium (A.9)

A solution to Eqs. (A.8) and (A.9) is then provided by

f = f(H) ; equilbrium (A.10)

where H = E is a constant of the motion for the particle.

A.1.2 Boltzmann Collision Term

Consider an extension to what in equilbrium is the canonical ensem-

ble, and include zero-range two-particle collisions. The goal here is

to project the exact dynamics of the many-body distribution function

f(p1, · · · , p3N , q1, · · · q3N ; t) down to an approximate equation for the one-

body distribution function f(p,q, t). As the ensemble evolves with time,

particles are now scattered in and out of the phase-space volume dλ (see

Fig. A.2). Assume the r.h.s. of Eq. (A.8) is augmented by a collision term,

so that Eq. (A.6) becomes

df

dt=

(

∂f

∂t

)

collisions

(A.11)

Momentum is conserved in the collisions so that

p1 + p2 = p′1 + p′2 (A.12)

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p1

p2

p1

p2

d

Fig. A.2 Collisions take particles in and out of the phase space volume dλ, and mo-mentum is conserved in these collisions.

Detailed balance in a collision states that4

Ratei→f = Ratef→i

or ; σ v12 = σ′ v1′2′ (A.13)

We assume zero-range collisions where σv12, while depending on E1+E2 =

E′1 + E′2, is otherwise independent of the kinematics.5

The number of transitions per unit time in the direction i→ f is given

by(

# of transitions

time

)

i→f

= (incident flux)× σ × (# of target particle)

= (n1v12)σ (n2d3q) (A.14)

4It helps here to think of the quantum mechanical expressions (“Golden Rule”) forthe rate and cross section

Rfi =2π

~δ(Ei − Ef )|〈f |H′|i〉|2

σfi =Rfi

Flux

In these expressions:

• Energy conservation is built in;

• This is for a transition to a given final state. One still needs the number of statesd3p′/(2π~)3 in a large volume V ;

• All factors of V have already been removed from these expressions;

• The hamiltonian is hermitian so that |〈f |H′|i〉|2 = |〈i|H′|f〉|2

However, the calculation is still classical until quantum mechanics later explicitlyappears.

5See Prob. A.3.

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where the particle density n is defined by

n ≡ dN

d3q; particle density (A.15)

Since the interaction is of zero range, everything is evaluated at the same

spatial point q, and in the same spatial volume d3q.

The rate of change of the number of systems in the phase space volume

dλ is then given by the difference of the number of systems scattered in and

the number scattered out per unit time. A combination of Eqs. (A.11)–

(A.15) gives(

∂f

∂t

)

coll

d3p d3q

(2π~)3=

∫

· · ·∫

(σv12)

×[

f(p′1,q; t)d3p′1(2π~)3

] [

f(p′2,q; t)d3p′2 d

3q

(2π~)3

]

d3p

(2π~)3

[

δ(3)(∆p)d3p2

]

−[

f(p,q; t)d3p

(2π~)3

] [

f(p2,q; t)d3p2 d

3q

(2π~)3

]

d3p′1(2π~)3

[

δ(3)(∆p)d3p′2

]

(A.16)

The final factor in each line is just unity. A cancellation of common factors

then gives(

∂f

∂t

)

collision

=

∫

· · ·∫

d3p2d3p′1d

3p′2(2π~)6

δ(3)(p+ p2 − p′1 − p′2)(σv12)

× [f(p′1,q; t)f(p′2,q; t)− f(p,q; t)f(p2,q; t)] (A.17)

This can be written in an obvious shorthand notation as(

∂f

∂t

)

collision

=

∫

· · ·∫

d3p2d3p′1d

3p′2(2π~)6

δ(3)(p+ p2 − p′1 − p′2)

× (σv12) [f′1f′2 − ff2] (A.18)

A few comments:

• This is the Boltzmann collision term;

• It is a classical result;

• It is the difference of the number of particles scattered into the phase-

space volume dλ per unit time, minus those scattered out;

• It assumes a zero-range interaction so everything occurs at the same

point q;

• The angular distribution of scattered particles is correspondingly

isotropic.

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• The Boltzmann collision term represents an approximate projection of

the full N -body dynamics onto the space of the one-body distribution

function;

• The result is a coupled, non-linear, integral contribution to the time

development of f(p,q, t).

A.1.3 Vlasov and Boltzmann Equations

Suppose the particles move in a one-body mean-field potential so the start-

ing particle hamiltonian has the form

H =p2

2m+ U(q) ; one-body hamiltonian (A.19)

Equations (A.8) and (A.11) then take the form

∂f

∂t+ v ·∇qf −∇qU ·∇pf =

(

∂f

∂t

)

collision

; Vlasov eqn (A.20)

This is the Vlasov equation. It is a non-linear, integro-differential transport

equation in a mean field.

If one sets U = 0, the result is the Boltzmann equation

∂f

∂t+ v ·∇qf =

(

∂f

∂t

)

collision

; Boltzmann eqn (A.21)

A.1.4 Equilibrium

In equilibrium, as many particles must be scattered into dλ as are scattered

out, and the collision term must vanish(

∂f

∂t

)

collision

= 0 ; equilibrium (A.22)

From Eq. (A.18), this will be true if

f ′1f′2 − ff2 = 0 (A.23)

Insertion of the result in Eq. (A.10), withH = E, then leads to the following

relation

f(E′1)f(E′2) = f(E)f(E2) (A.24)

Energy is conserved in the collision so that

E + E2 = E′1 + E′2 (A.25)

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Since Eq. (A.24) is to hold for all (E′1, E′2), it must be of the form

g(E + E2) = f(E)f(E2) (A.26)

Differentiation of this relation with respect to E, and then E2, leads to

f ′(E)f(E2) = f(E)f ′(E2) (A.27)

Hence

f ′(E)

f(E)= constant (A.28)

Define the constant to be −1/kBT , so that

f ′(E)

f(E)= − 1

kBT(A.29)

Then

f(E) = e−E/kBT (A.30)

With the restoration of E = H , this is

f(H) = e−H/kBT ; Boltzmann distribution (A.31)

We recover the Boltzmann distribution!

In summary:

• This is classical physics;

• If particles move with the one-body hamiltonian in Eq. (A.19), then the

one-body distribution function develops in time according to Eq. (A.8)

∂f

∂t= ∇qU ·∇pf − v ·∇qf

= H, fP.B. (A.32)

The equilbrium solution to this equation, with (∂f/∂t) = 0, leads to

f = f(H) (A.33)

• The Boltzmann collision term in Eq. (A.17), which includes the effect

of particles scattering into and out of dλ through zero-range, two-body

collisions, augments Eq. (A.32) to

∂f

∂t+ v ·∇qf −∇qU ·∇pf =

(

∂f

∂t

)

collision

(A.34)

where (∂f/∂t)collision is given by Eq. (A.18).

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• This is the extension of the Boltzmann equation to the Vlasov equation,

which includes the one-body mean field U ;

• The equlibrium solution to Eq. (A.34), which now includes the ad-

ditional requirement that (∂f/∂t)collision = 0, provides the functional

form of f(H)

f(H) = e−H/kBT (A.35)

• This is quite a remarkable result. One has derived the distribution of

equilibrium statistical mechanics by studying the dynamical evolution

in phase space of the one-body distribution function.6

A.1.5 Molecular Dynamics

With modern computing capabilites, it is now possible to follow the full

distribution function f(p1, · · · , p3N , q1, · · · , q3N ; t) in phase space with a

large, finite number of systems and model interactions. These are referred

to as molecular dynamics simulations.

A.2 Nordheim-Uehling-Uhlenbeck Equation

The extension of Nordheim, Uehling, and Uhlenbeck includes the Pauli

principle for identical fermions. It is here that quantum mechanics first

enters the dynamics. It is assumed that

The scattering process cannot lead to states already filled with one

identical fermion per unit cell in phase space, implied by f = 1 in

Eq. (A.1).

For these particles, one makes the following replacement in the Boltzmann

collision term of Eq. (A.18)

f ′1f′2 − ff2 −→ f ′1f

′2(1− f)(1− f2)− ff2(1 − f ′1)(1− f ′2) (A.36)

As justification, we show that in equilibrium, where the collision term van-

ishes, one recovers the Fermi distribution.

Let

f(E) =1

eβ(µ−E) + 1≡ 1

D(E)(A.37)

6Boltzmann was also interested in why entropy increases and just how equilibiriumis established [Boltzmann (2011)].

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Then the goal is to show that

f(E′1)f(E′2)[1 − f(E)][1− f(E2)] = f(E)f(E2)[1− f(E′1)][1− f(E′2)]

or ; f(E′1)f(E′2)[1− f(E)− f(E2)] = f(E)f(E2)[1− f(E′1)− f(E′2)]

(A.38)

With the substitution of Eq. (A.37), this becomes

1

D(E′1)

1

D(E′2)

1

D(E)

1

D(E2)[D(E)D(E2)−D(E2)−D(E)]

=1

D(E′1)

1

D(E′2)

1

D(E)

1

D(E2)[D(E′1)D(E′2)−D(E′2)−D(E′1)]

(A.39)

A cancellation of common factors leads to

D(E)D(E2)−D(E2)−D(E) = D(E′1)D(E′2)−D(E′2)−D(E′1)

(A.40)

This equality is now established by direct substitution of Eq. (A.37) and

the use of energy conservation. The l.h.s. is

l.h.s. =[

eβ(µ−E) + 1] [

eβ(µ−E2) + 1]

−[

eβ(µ−E2) + 1]

−[

eβ(µ−E) + 1]

= eβ(2µ−E−E2) − 1

= eβ(2µ−E′1−E

′2) − 1 ; E + E2 = E′1 + E′2

= r.h.s. (A.41)

Thus the one-body Fermi distribution in Eq. (A.37) makes the

Nordheim-Uehling-Uhlenbeck expression in Eq. (A.36) vanish, which leads

to a vanishing of the new collision term. Furthermore, with the one-body

hamiltonian of Eq. (A.19), it is then still true that7

∂f

∂t= ∇qU ·∇pf − v ·∇qf = 0 (A.42)

A.3 Example—Heavy-Ion Reactions

As an application of Boltzmann transport theory, consider a reaction be-

tween two heavy nuclei [Stocker and Greiner (1986)]. Here a program

7See Prob. A.2

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applying the Vlasov-Uehling-Uhlenbeck model exists in the literature and

is available to all [Hartnack, Kruse, and Stocker (1993)].

In phase space, the initial configuration is as sketched in Fig. A.3.

p

q

Fig. A.3 Sketch of initial phase-space configuration for two colliding heavy nuclei.

The program proceeds through the following series of steps:

(1) Choose random positions and momenta from the initial Fermi gases in

the two nuclei;

(2) Follow the particles with classical dynamics and zero (short)-range two-

body collisions leading to random final states, while conserving energy

and momentum;

(3) Take a statistical average over many “events” run in parallel to deter-

mine the one-body distribution function8

f(p,q; t) ≡ 1

Nevents

∑

events

f event(p,q; t)

(4) Compute the nuclear density n at time t from f ;

(5) Use a density-dependendent mean-field potential U(n) in determining

the one-body motion;

(6) Take an average of “runs” to get the best f ;

(7) Use this f to compute particle distributions, mean values, etc.

The authors include inelastic N -N processes producing (π,∆) in their pro-

gram, where ∆(1236MeV) with (Jπ , T ) = (3/2+, 3/2) is the first nucleon

resonance. Locate the program and learn to run it. It’s fun.

The experimental results are typically used to study the nuclear equa-

tion of state through U(n).9

8As in an actual experiment.9Compare Probs. A.4–A.5.

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Index

absolute activity, 119, 123, 171, 183,202classical statistics, 196quantum statistics, 196

angular momentum, 68, 77, 86eigenvalues, 70, 87magnetic moment, 101principal axes, 86

angular velocity, 86assembly, 12, 56, 96, 99, 112, 114,

127, 137, 181, 182, 184, 207, 249,250, 277, 289, 315

asymmetric top, 89asymptotic freedom, 283, 288, 331atomic structure

Thomas-Fermi theory, 322Avogadro’s number, 29, 47

binomial theorem, 19, 30black-body spectrum, 206Bohr magneton, 101, 319Bohr radius, 322Bohr-Van Leeuwen theorem, 233, 319Boltzmann distribution, 21, 25, 127,

129, 189, 191, 341Boltzmann equation, 335, 340

Boltzmann distribution, 341collision term, 331, 337, 339

detailed balance, 338energy conservation, 338, 340momentum conservation, 337zero range, 331, 339

distribution function, 335, 336equilibrium, 337, 340Liouville’s theorem, 336one-body dynamics, 335phase-space volume, 336

Boltzmann statistics, 21, 49, 195, 199,210, 221classical statistics, 49, 196criterion, 49, 195, 199distribution numbers, 27, 198, 199grand partition function, 196localized systems, 17magnetic susceptibility, 110, 232non-localized systems, 39, 48, 199validity, 48, 49, 199

Boltzmann’s constant, 16, 43, 47Born-Oppenheimer approximation,

66, 304Bose condensation, 207

Bose-Einstein, 217condensates, 218two dimensions, 315

Bose gas, 207Bose condensation, 207chemical potential, 209, 210, 215energy, 208, 213equation of state, 207, 209gluons, 330heat capacity, 213, 315

slope discontinuity, 214, 217liquid 4He, 207, 217non-relativistic, 207, 209

349

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particle number, 209, 213phonons, 323photons, 199pressure, 208, 209, 213spin degeneracy, 208transition temperature, 209, 210,

212two dimensions, 315

Bose-Einstein statistics, 198bosons, 49, 74, 196, 323

Bose gas, 207gluons, 330liquid 4He, 217phonons, 323photons, 201

boundary conditions, 42, 147, 310clamped boundaries, 145periodic, 147, 153, 155, 200, 207,

219, 272, 285, 320, 325, 326Bragg-Williams approximation, 258,

264, 325

canonical ensemble, 127, 189and microcanonical ensemble, 129,

130, 132, 137applications, 141, 158, 249, 263assembly, 128

degeneracies, 128energies, 128, 189Helmholtz free energy, 129mean energy, 129mean entropy, 132

Boltzmann distribution, 129canonical partition function, 127,

129, 141, 159classical limit, 132configuration integral, 159configuration partition

function, 257normal modes, 143solutions, 250

classical limit, 132configuration integral, 159, 160configuration partition function,

257constant-T partition function, 127

energy distribution, 135, 137, 189,308

hamiltonian, 130, 158Helmholtz free energy, 129, 141,

159imperfect gases, 158independent localized systems, 130independent non-localized systems,

132Ising model, 272large N , 132, 134, 137, 163largest term in sum, 130, 137, 250mean energy, 132, 190

mean-square deviation, 190mean entropy, 132normal modes, 142order-disorder transitions, 263perfect gas, 133

effective Ω, 133, 135energy distribution, 137partition function, 134

solids, 141canonical partition function, 127, 129

classical limit, 132, 159configuration integral, 159configuration partition function,

257, 263hamiltonian, 130imperfect gases, 159Ising model, 272lattice gauge theory [U(1)], 288normal modes, 143perfect solution, 250solutions, 249

Carnot cycle, 3, 297catalysis, 123center-of-mass (C-M), 65, 332chemical equilibria, 112

absolute activity, 119, 123chemical potential, 118, 123, 229

components, 119species, 118

chemical reaction, 114, 116, 306,307

complexions, 115, 121components, 8, 119

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Index 351

concentration, 307entropy, 116equilibrium constant, 116, 307Helmholtz free energy, 117, 122law of mass action, 114, 116, 307mean values, 116, 122number constraints, 115, 121open sample, 117, 120partition functions, 116solid-vapor equilibrium, 120

chemical potential, 123component, 120Helmholtz free energy, 122localized systems, 121non-localized systems, 121partition functions, 122species, 120vapor pressure, 122, 308

species, 8, 112, 117surface adsorption, 123

complexions, 124component, 123fractional site occupation, 125Langmuir isotherm, 125, 308localized sites, 124number constraints, 124partition functions, 125perfect gas, 125species, 123

temperature, 117total energy, 115

chemical potential, 8, 112, 118, 183,185, 229, 332Bose gas, 209chemical reaction, 118components, 8, 119, 120Fermi gas, 220–222, 227Gibbs free energy, 9, 171, 311gluon gas, 330Helmholtz free energy, 9Landau diamagnetism, 240open sample, 8Pauli spin paramagnetism, 229perfect gas, 47, 210phase equilibrium, 11, 123, 301phonon gas, 324

photon gas, 202quark-gluon plasma, 330solid-vapor, 123species, 8, 118, 120thermodynamic potential, 184

classical limit, 50, 53entropy, 60, 62number of complexions, 55, 56, 59partition function, 53, 78, 132, 159,

303weighting, 50

classical mechanics, 50canonical momenta, 50, 68generalized coordinates, 50Hamilton’s equations, 50, 236, 337hamiltonian, 50lagrangian, 50Liouville’s theorem, 50Newton’s second law, 152, 154, 234normal coordinates, 142normal modes, 63, 142phase space, 50Poisson bracket, 337rigid-body motion, 85

classical statistics, 49, 53, 176, 196,300Bohr-Van Leeuwen theorem, 319Boltzmann statistics, 49, 196criterion, 49, 195, 300equipartition theorem, 55, 302validity, 48, 199

cluster, 165, 171commutation relations, 236complex variables, 32

analytic function, 32, 299Cauchy-Riemann equations, 33,

299contour integral, 32harmonic functions, 34Laurent series, 33modulus, 34, 299power series, 32residue theorem, 33Taylor series, 35, 316

complexion, 13, 17, 40, 56, 74, 95,115, 124

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component, 8, 119, 249composites, 207configuration partition function, 257,

264Bragg-Williams approximation, 258Ising solution (Z=2), 268largest term in sum, 264, 269lattice gas model, 324mean number of pairs, 258nearest neighbors, 257order-disorder transitions, 263properties, 257

coordinate representation, 237coordination number, 255, 279

effective, 279, 280, 290corresponding states, 175critical opalescence, 193critical point, 174, 178, 180

De Haas-Van Alphen effect, 247Debye T 3-law, 149Debye equation, 109Debye model, 145, 148, 324Debye temperature, 148deformed nuclei, 90degeneracy, 94, 128, 130

spin, 208density functional theory, 323, 333diamagnetism, 111

Bohr-Van Leeuwen theorem, 319Landau diamagnetism, 233

diamond, 149, 309, 310dielectric medium, 104

Clausius-Mosotti relation, 108Debye equation, 109dielectric constant, 106electric susceptibility, 104polarization, 106

Dirichlet integral, 58, 301Dulong and Petit law, 29, 148

Einstein model, 27, 141, 150, 249Einstein’s theory of specific heat, 27,

308energy, 28heat capacity, 28

Helmholtz free energy, 28limiting cases, 29partition function, 28

elasticitykeratin molecules, 314one-dimensional chain, 157, 314

electric dipole moment, 95, 97, 304electromagnetic radiation, 200

absolute activity, 202black-body spectrum, 206boundary conditions, 200chemical potential, 202Coulomb gauge, 200energy in cavity, 203, 204

energy density, 203Rayleigh-Jeans law, 205Stefan-Boltzmann law, 207Wien’s law, 205

equation of state, 204field energy, 201fields, 200heat capacity, 204Maxwell’s equations, 200normal modes, 200, 201, 314photon gas, 201photons, 201Planck distribution, 206pressure, 315spectral weight, 203ultraviolet catastrophe, 205vacuum, 201vector potential, 200

energy, 2, 21, 57, 115, 127, 128, 135,190, 208, 310conservation, 2, 338, 343electromagnetic field, 201external field, 97, 229first law, 2, 7, 298internal, 2, 21, 56, 310normal modes, 142of assembly, 25, 28, 129, 143, 186,

199, 257, 275of mixing, 259

ensemble, 12, 27, 127, 137, 180–182,184, 328

enthalpy, 298, 304, 332

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Index 353

entropy, 5, 6, 24, 39, 64, 184, 297, 311and number of complexions, 15, 16,

55, 60calorimetric, 92, 93canonical ensemble, 132–134, 137chemical equilibria, 116classical limit, 60configuration entropy, 266extensive, 15Fermi gas, 227grand canonical ensemble, 184Helmholtz free energy, 7microcanonical ensemble, 25, 39,

132, 134, 137of mixing, 259perfect gas, 46, 47, 137properties, 15randomness, 15second law, 5, 7, 298spectroscopic, 92, 93spin entropy, 277, 298thermodynamic potential, 184third law, 12, 16variation, 9, 299

equation of state, 175Bose gas, 207, 209Fermi gas, 219imperfect gases, 170, 177, 178, 188isotherms, 173, 175Landau diamagnetism, 241nuclear, 344perfect gas, 47, 122, 159, 186, 297photon gas, 204, 209quark-gluon plasma, 330relativistic Fermi gas, 318Van der Waal’s, 160, 173–175, 295,

311equilibrium, 1, 10, 299

Gibbs criteria, 9, 10equipartition theorem, 46, 55, 149,

205derivation, 302

ergodic hypothesis, 14euclidian metric, 285Euler’s theorem, 311extensive, 178, 192

Fermi energy, 221–223, 227metals, 227, 318nuclear matter, 317white dwarf, 318

Fermi gas, 219chemical potential, 220–222distribution numbers, 221, 343energy, 219, 333equation of state, 219, 220, 318Fermi energy, 221–223, 227, 319heat capacity, 224Landau diamagnetism, 233, 319

Bohr-Van Leeuwen theorem,233, 319

chemical potential, 240, 244counting of states, 238, 239De Haas-Van Alphen effect,

247eigenvalue spectrum, 238grand partition function, 240,

243high-temperature limit, 240low-temperature limit, 242magnetization, 240, 243, 244orbits, 235particle in magnetic field, 234,

235, 238particle number, 240, 243susceptibility, 242

low temperature, 224basic integral, 226chemical potential, 226entropy, 227heat capacity, 224, 227particle number, 226thermodynamic potential, 226

metal, 315, 318non-relativistic, 219, 220nuclear matter, 315–317

symmetry energy, 317particle number, 220Pauli spin paramagnetism, 228, 316

Boltzmann result, 232chemical potential, 229grand partition function, 229hamiltonian, 229

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magnetic susceptibility, 231,232

magnetization, 230, 319particle number, 230thermodynamics, 232zero temperature, 316

pressure, 219, 220, 317, 333spin degeneracy, 219sum to integral, 222, 320symmetry energy, 317thermionic current, 319two dimensions, 247, 320ultra-relativistic, 317

energy, 317equation of state, 318pressure, 317

white-dwarf star, 318zero temperature, 222

energy, 223Fermi energy, 224particle density, 223, 224pressure, 223, 224ultra-relativistic, 317

Fermi wavenumber, 222Fermi-Dirac statistics, 198fermions, 49, 74, 77, 197, 218, 219,

342electrons, 218, 239, 318, 321, 322Landau diamagnetism, 233low temperature, 224nucleons, 77, 218, 316Pauli spin paramagnetism, 228quarks, 218ultra-relativistic, 317zero temperature, 222

ferromagnetism, 270Heisenberg hamiltonian, 270Ising model, 277, 278, 280, 326linear chain, 277

fluctuations, 189fundamental constants, 43, 47, 48

Gamma function, 58, 135, 211gas constant, 29, 47Gibbs free energy, 7, 10, 183, 311, 332

equilibrium, 10

Euler’s theorem, 311grand partition function, 313one-dimensional chain, 313

gluons, 283, 330Golden Rule, 331, 338grand canonical ensemble, 181, 189,

207and canonical ensemble, 181and microcanonical ensemble, 181applications, 186, 195assembly, 184Boltzmann statistics, 195bosons, 199chemical potential, 207distribution numbers, 198, 199

Boltzmann, 198, 199bosons, 198, 199fermions, 198, 199

energy, 199fermions, 218fluctuations, 189grand partition function, 182, 184Landau diamagnetism, 233largest term in sum, 183mean particle number, 191

mean-square deviation, 191particle distribution, 190Pauli spin paramagnetism, 229quantum statistics, 196

bosons, 196fermions, 197

thermodynamic potential, 184grand partition function, 169, 182

absolute activity, 183, 185, 186Boltzmann statistics, 196Bose gas, 208chemical potential, 183energy, 185, 186entropy, 184Fermi gas, 219Gibbs free energy, 313imperfect gases, 188independent localized systems, 187independent non-localized systems,

186Landau diamagnetism, 240

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Index 355

one-dimensional chain, 313particle number, 184Pauli spin paramagnetism, 229photon gas, 204pressure, 184quantum statistics, 197, 313


thermodynamic potential, 184volume, 312

hadron, 283Halley’s formula, 300Hamilton’s equations, 50, 236, 337hamiltonian, 50

canonical ensemble, 130diatomic molecule, 65Heisenberg, 271Ising model, 271many-body, 158mean field, 340normal modes, 142one-body, 340one-dimensional rotor, 91particle in magnetic field, 236perfect gas, 133rigid rotor in external field, 97scalar field, 285, 329spin-1/2 Fermi gas, 229stretched spring, 310symmetric top, 86

hard-sphere gas, 312second virial coefficient, 312third virial coefficient, 312

harmonic oscillator, 27, 142, 201, 238partition function, 28, 70, 143, 241spectrum, 27, 238wave functions, 238, 320

Hartree approximation, 212, 333heat capacity, 28, 92, 190, 300

Bose gas, 217Debye model, 148, 310diatomic molecule, 72, 302Einstein model, 27, 310energy distribution, 190Fermi gas, 227

Ising model, 275, 278one-dimensional rotor, 92order-disorder transitions, 267perfect gas, 300, 304phase transition

λ-point, 262first order, 262second order, 262

phonon gas, 324photon gas, 207, 315specific heat, 28

heavy-ion reactions, 343Heisenberg hamiltonian, 270, 271Heisenberg uncertainty principle, 51Helmholtz equation, 42Helmholtz free energy, 7, 39, 41, 64,

129, 141, 187Bose gas, 207canonical ensemble, 129, 141, 159chemical equilibria, 117, 122diatomic molecules, 72Einstein model of solid, 28equilibrium, 10extensive, 192imperfect gases, 159in external field, 111independent localized systems, 39,

187independent non-localized systems,

41, 187Ising model, 274microcanonical ensemble, 39, 41,

138order-disorder transitions, 266perfect gas, 46, 62photon gas, 202solutions, 251stretched springs, 311

hindered rotation, 90classical partition function, 91heat capacity, 92one-dimensional rotor, 90vibration, 91

Hugenholtz-Van Hove theorem, 332

ice, 95

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imperfect gases, 158absolute activity, 171and perfect gas, 159, 176classical theory, 158configuration integral, 159, 160,

162, 163, 165, 177equation of state, 177, 178linked-clusters, 165, 168pair-interaction, 165summation of series, 169

critical isotherm, 174critical point, 174, 178, 180density, 159, 170density fluctuations, 193equation of state, 170, 178, 188

general, 177interpretation, 171

grand partition function, 188hard-sphere gas

virial coefficients, 312Helmholtz free energy, 159law of corresponding states, 175,

178critical point, 178derivation, 176equation of state, 177, 178experimental results, 179, 180reduced variables, 178

linked-cluster analysis, 162, 188phase separation, 174pressure, 159, 170reduced quantities, 175two-body potential

Lennard-Jones, 176, 179, 311Van der Waal’s, 176, 311

Van der Waal’s gas, 160, 173, 312virial expansion, 160, 171

second coefficient, 160, 163,164, 172, 179, 311, 312

third coefficient, 312intensive, 178Ising model, 270, 275

ferromagnetism, 277, 278, 280hamiltonian, 271mean field theory, 278, 281

critical temperature, 281

dimension d, 278, 281effective coordination number,

279, 280magnetization m, 279, 280,

282, 326total spin, 279

Metropolis algorithm, 328Monte Carlo calculation, 327nearest neighbors, 271, 277, 278,

280one-dimension, 275, 282

canonical partition function,272, 274

energy, 275hamiltonian, 271heat capacity, 275, 328Helmholtz free energy, 274magnetic field, 326matrix solution, 272–274, 326Metropolis algorithm, 328Monte Carlo calculation, 327no phase transition, 277number of unlike pairs, 326

periodic boundary conditions, 272,278, 279, 326

phase transition, 277, 278, 280two-dimensions, 277

energy, 278heat capacity, 278mean field theory, 281, 326Onsager solution, 277, 281phase transition, 278transition temperature, 278,

326isothermal compressibility, 191

number distribution, 191

Kronecker delta, 314

Lagrange’s method of undeterminedmultipliers, 23, 48, 115, 121, 124,181, 250

Landau diamagnetism, 233, 319Bohr-Van Leeuwen theorem, 233,

319chemical potential, 240, 244

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Index 357

counting of states, 238, 239De Haas-Van Alphen effect, 247eigenvalue spectrum, 238gauge, 235grand partition function, 240, 243harmonic oscillator, 238high-temperature limit, 240low-temperture limit, 242magnetization, 240, 243, 244orbits, 235, 320particle in magnetic field, 234particle number, 240, 243susceptibility, 242

Langevin function, 99, 103, 111Langevin response, 103Langmuir adsorption isotherm, 125,

308Laplace’s equation, 34lattice gas model, 324, 325

Bragg-Williams approximation, 325configuration partition function,

325filling fraction, 325Helmholtz fee energy, 325periodic boundary conditions, 325phase transition, 325

lattice gauge theory, 283U(1) lattice gauge theory, 285

continuum limit, 288, 329coupling, 288coupling constant, 330dimension, 291, 293effective energy, 293effective temperature, 288, 293euclidian metric, 285four dimensions, 291, 329gauge invariance, 287, 330imaginary time, 285improved approximations, 292mean field theory, 289, 290,

293, 330measure, 287, 330numerical results, 291–293,

330partition function, 287, 288phase transition, 291, 293

six dimensions, 293, 330strong-coupling limit, 293,

329, 330three dimensions, 290two dimensions, 289, 329

U(1) mean field theorycritical coupling, 290dimension, 291effective coordination number,

290effective magnetization, 289gauge invariance, 289numerical results, 291–293,

329, 330phase transition, 291plaquette action, 289unit cell, 290

continuum limit, 288, 329fields, 286gauge invariance, 289improved approximations, 292lattice spacing, 286link, 286mean field theory, 289, 293non-abelian theory SU(n), 295periodic boundary conditions, 286plaquette, 286quantum chromodynamics (QCD),

295quantum electrodynamics (QED),

285site, 286strong-coupling limit, 291, 293, 329

law of corresponding states, 173, 178law of mass action, 116, 119, 307Legendre transformation, 298Lennard-Jones potential, 176, 311Liouville’s theorem, 50, 336liquid 4He, 207, 217, 300

heat capacity, 315mass, 315mass density, 217phase transition, 218

λ-point, 218two fluids, 218

superfluid, 218

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liquid hydrogen, 318liquids, 301

solutions, 249long-range order, 264Lorentz force, 234

magnetic dipole moment, 97, 101,110, 315experimental value, 101, 305in quantum mechanics, 101

Maxwell’s equations, 200Maxwell-Boltzmann distribution, 300mean field theory, 278, 280, 282, 289mean values, 25, 116, 122, 129, 189,

253, 292, 336mean-square deviation, 189metastable equilibrium, 79, 94method of steepest descent, 29, 48,

112, 131, 169contour integral, 32, 33distribution numbers, 300entropy, 39generating function, 30, 169Helmholtz free energy, 39, 171large N , 34, 37, 300linked-cluster analysis, 136, 169number of complexions, 32, 33, 37partition function, 39saddle point, 34, 35, 113, 170several species, 112, 114

Metropolis algorithm, 282, 328microcanonical ensemble, 17, 27

and canonical ensemble, 130applications, 41, 63, 64, 97, 112,

300Boltzmann distribution, 21, 25, 300chemical equilibria, 112classical limit, 50, 53classical statistics, 49distribution numbers, 21, 27, 29,

64, 98, 300entropy, 24, 39, 41, 64Helmholtz free energy, 26, 39, 41,

64, 117, 122, 129high-temperature limit, 53identical systems, 40

independent localized systems, 17,29, 55, 113, 124

independent non-localized systems,39, 41, 48, 55, 113

validity, 48, 49, 199internal partition function, 63mean displacement, 311mean energy, 27, 311mean number, 27, 116, 122method of steepest descent, 29, 112molecular spectroscopy, 64most probable distribution, 19number of complexions, 22, 29, 40paramagnetic and dielectric

assemblies, 97partition function, 26, 39, 41, 98

classical limit, 53degeneracy, 26harmonic oscillator, 28internal, 63perfect gas, 45, 62permanent dipole, 98, 101rigid rotor, 71, 72, 90, 98stretched spring, 310surface site, 125, 308symmetric top, 87

quantum statistics, 49summary, 137, 138temperature, 26, 39two levels, 18

molecular dynamics, 342molecular spectroscopy, 64

and deformed nuclei, 90Born-Oppenheimer approximation,

66, 304C-M system, 65conversion factors, 80diatomic molecules, 64, 89

angular momentum, 68carbon monoxide CO, 95electric dipole moment, 81,

84, 95, 303electronic state, 72, 75hamiltonian, 65, 66, 70heat capacity, 72high-T limit, 78, 303

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Index 359

homonuclear, 76, 77, 84hydrogen H2, 79moment of inertia, 69, 83nuclear spin degeneracy, 77,

84, 303nuclear statistics, 74, 76–78oxygen O2, 78partition function, 70, 77, 307rotation, 70rotation-vibration coupling,

304rotational bands, 82selection rules, 81small oscillations, 67symmetry factor, 73thermal population, 81, 83translation, 70typical energies, 80vibration, 70wave function, 74

ortho- and para-H2, 79heat capacity, 80, 303metastable assembly, 79, 303partition function, 79statistical equilibrium, 303statistical mixture, 79

polyatomic molecules, 85asymmetric top, 89eigenfunctions, 87high-T limit, 87, 89hindered rotation, 90–92internal symmetry, 304partition function, 87spectrum, 87spherical top, 89symmetric top, 85vibrations, 90

spectroscopic and calorimetricentropies, 92–94

sources of kB ln Ω0, 93table of, 94

symmetry factor, 74, 91, 304molecular structure, 75

LCAO method, 75moment of inertia, 69, 83, 85momentum, 13, 50, 51, 235, 337

canonical, 50

conservation, 331, 337Monte Carlo calculation, 282, 327

multinomial theorem, 30, 131, 169

neutron stars, 315

Newton’s second law, 152, 154, 234

non-equilibrium statistical mechanics,335

Boltzmann equation, 335

heavy-ion reactions, 343molecular dynamics, 342

Nordheim-Uehling-Uhlenbeck eqn,342

Vlasov equation, 340

Nordheim-Uehling-Uhlenbeck eqn,331, 342

collision term, 342

energy conservation, 343

equilibrium, 342Fermi distribution, 343

normal modes, 63, 90, 142, 201

sum to integral, 146, 147, 308, 309

nuclear matter, 316density, 317

equation of state, 332, 344

Fermi energy, 317

Fermi wavenumber, 316

symmetry energy, 317nuclear structure

Thomas-Fermi theory, 323

numerical methods, 282

Metropolis algorithm, 282, 328Monte Carlo calculation, 282, 327

order-disorder transitions, 261

λ-point, 261

Bragg-Williams approximation, 264

configuration entropy, 266

configuration partition function,263

Helmholtz free energy, 266Ising solution (Z=2), 268

long-range order, 264

one dimension, 270

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quasi-chemical approximation, 267,269

specific heat, 267transition temperature, 266, 267,

269

Pade approximant, 293, 329paramagnetic and dielectric

assemblies, 97diamagnetism, 111

Bohr-Van Leeuwen theorem,319

Landau diamagnetism, 233dielectric medium, 104

capacitor, 104Clausius-Mosotti relation, 108Debye equation, 109dielectric constant, 106, 109effective field, 108electric susceptibility, 104, 107ferroelectrics, 111Lorentz-Lorenz effect, 108polarizability, 305polarization, 104, 105, 109

ferromagnetism, 270Ising model, 270, 277, 278, 280

field analogies, 110, 305paramagnetic medium, 109

Curie temperature, 111Curie’s constant, 110Curie-Weiss law, 111effective field, 110, 306ferromagnet, 111field analogies, 110, 305Langevin function, 111magnetic charge, 306magnetic dipole moment, 110magnetic susceptibility, 109solenoid, 109, 305

Pauli spin paramagnetism, 228, 316permanent dipoles

classical gas, 97distribution numbers, 98electric dipole moment, 97external field, 98induced moment, 99, 102

Langevin function, 99, 103Langevin response, 103

magnetic dipole moment, 97,101

partition function, 98, 101

polarizability, 106, 305thermodynamics, 111, 232

Helmholtz free energy, 111

induced dipole moment, 111partition function, 26, 39, 129

canonical ensemble, 129

canonical partition function, 127,129, 250

classical limit, 159configuration integral, 159

configuration partitionfunction, 257

normal modes, 143chemical equilbria, 307

classical limit, 53, 132, 138, 139

diatomic molecule, 72external field, 98

harmonic oscillator, 54

imperfect gases, 158, 159one-dimensional rotor, 91

perfect gas, 54, 133

stretched spring, 310configuration integral, 159

configuration partition function,257, 264

constant-T , 127

degeneracy, 26, 128, 138, 308diatomic molecule, 70

electronic, 72homonuclear, 77, 303

rotation, 70, 303

translation, 70vibration, 70

Einstein model of solid, 28

field theory, 284U(1) lattice gauge theory, 287

action, 285

imaginary time, 284lattice gauge theory, 285

path integral, 285

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Index 361

periodic boundary conditions,285

scalar field, 284grand partition function, 182, 313

imperfect gases, 188independent localized

systems, 187independent non-localized

systems, 186quantum statistics, 197

harmonic oscillator, 28, 91high-temperature limit, 53, 71, 78,

87, 302, 303hindered rotation, 91imperfect gases, 159, 188internal partition function, 63Ising model, 272microcanonical ensemble, 26normal modes, 143perfect gas, 45, 62, 134permanent dipole, 98, 101quantum statistics, 197solutions, 250stretched spring, 310sum replaced by integral, 43, 302symmetric top, 87system on surface, 125, 308zero of energy, 117

Pauli matrices, 295Pauli principle, 76, 197, 232, 271, 342Pauli spin paramagnetism, 228, 316perfect gas, 3, 41, 47, 159, 176, 190

Carnot cycle, 3chemical potential, 47, 210energy, 46, 56enthalpy, 304entropy, 46, 47, 60–62equation of state, 47, 61, 125, 133,

186, 241, 297heat capacity, 46, 304Helmholtz free energy, 46, 62isothermal compressibility, 192number of complexions, 55, 56, 59particle number, 192partition function, 43, 45, 134, 186phase space volume, 57, 59, 133

solutions, 249phase equilibria, 11, 262, 295, 301,

325phase space, 50

and number of complexions, 60area, 52, 302canonical transformation, 50cells, 51, 52, 56classical limit, 60dimension, 50Dirichlet integral, 59distribution function, 335energy distribution, 59, 133, 137equipartition theorem, 302Hamilton’s equations, 50harmonic oscillator, 301Heisenberg principle, 51Liouville’s theorem, 50, 336of assembly, 56particle in box, 51, 52perfect gas, 60phase orbit, 301phase trajectory, 50, 336Sommerfeld-Wilson, 52volume, 50, 133, 336weighting, 51–53

phase transition, 207, 260, 261λ-point, 261Bose gas, 217, 315critical point, 174first-order, 261Ising model, 277, 278, 280lattice gas model, 325lattice gauge theory, 291, 294, 295linear chain, 277liquid 4He, 217one dimension, 270order-disorder, 261regular solutions, 260

amount formed, 324transition temperature, 260,

261, 324second-order, 261

phonon, 323energy, 323

phonon gas, 323

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chemical potential, 324energy, 324heat capacity, 324

photon, 201energy, 206polarization, 201

photon gas, 199absolute activity, 202chemical potential, 202energy, 204energy density, 203equation of state, 204, 209heat capacity, 204number, 201pressure, 204, 315

Planck distribution, 29, 206Planck’s constant, 43, 48, 206Poisson bracket, 337polar-spherical coordinates, 58, 301potential

diatomic molecule, 67interatomic, 158, 161, 163, 164, 311Lennard-Jones, 176, 179, 311many-body, 158, 160one-body, 333, 340thermodynamic, 184two-body, 158, 161, 163–165

general form, 176Van der Waal’s, 176zero-range, 331

pressure, 7, 10, 46, 122, 125, 159, 170,184, 298, 318Bose gas, 208, 209Fermi gas, 219, 220, 317Hartree approximation, 333Helmholtz free energy, 7imperfect gases, 159, 177, 178partial, 252perfect gas, 46photon gas, 204, 315pressure-volume work, 7, 298Raoult’s law, 252thermodynamic potential, 184Thomas-Fermi theory, 224, 318,

321, 323two dimensions, 320

quantizedcirculation, 212flux, 212normal modes, 142oscillators, 142, 201particle in box, 42rotor, 70

quantum chromodynamics (QCD),283, 295asymptotic freedom, 283color, 283confinement, 283gluons, 283, 330lattice gauge theory, 283, 295quarks, 283, 330

quantum electrodynamics (QED), 284action, 284coupling constant, 330gauge invariance, 284lagrangian density, 284

quantum statistics, 49, 76, 196, 210,300Boltzmann statistics, 199Bose gas, 207bosons, 196, 207distribution numbers, 198, 199electromagnetic radiation, 202energy, 199Fermi gas, 218fermions, 197, 219grand partition function, 197, 313


interactions, 195occupation numbers, 196photon gas, 199

quark-gluon plasma, 218, 330quarks, 218, 283, 330quasi-chemical approximation, 260,

267Bethe-Peierls, 260Guggenheim, 260

Raoult’s law, 252, 255Rayleigh-Jeans law, 205reduced mass, 66

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Index 363

Riemann zeta function, 211

Sackur-Tetrode equation, 47Schrodinger equation, 41, 237screening, 321

Thomas-Fermi theory, 321semi-empirical mass formula, 317solids, 141

Debye T 3-law, 149Debye model, 145, 148Debye temperature, 148, 149, 309Dulong and Petit law, 148Einstein model, 141, 150lattice model, 154

frequency cut-off, 156, 157spectral weight, 155, 157, 310

longitudinal waves in rod, 151spectral weight, 154, 157wave equation, 152

Nernst-Lindemann approximation,158

normal modes, 151, 153, 155clamped boundaries, 145Debye model, 145, 148, 150,

154Einstein model, 144frequency cut-off, 144, 149lattice model, 154minimum wavelength, 309number of, 144, 148, 154periodic boundary conditions,

147quantization, 142sound waves, 145, 150, 151spectral weight, 143, 144, 147,

151, 153, 157, 310sum to integral, 146, 147, 153,

308, 309sound waves, 150

longitudinal, 150, 151transverse, 150velocity, 150, 309

thermodynamics, 148solutions, 249

canonical partition function, 254localized sites

Einstein model, 250mixing

energy, 253, 256, 259entropy, 253, 259free energy, 253, 259

mole fractions, 252nearest-neighbor energy, 255number of rearrangements, 255partial pressures, 252perfect gas, 249perfect solution, 249

canonical partition function,250

free energy of mixing, 253Helmholtz free energy, 251,

324mixing energy, 253mixing entropy, 253Raoult’s law, 252, 255, 324

regular solutions, 254, 257Bragg-Williams

approximation, 258configuration energy, 256configuration partition

function, 257coordination number, 255Einstein model, 254energy of mixing, 257, 259entropy of mixing, 259free energy of mixing, 257,

259, 324mean number of pairs, 258number of nearest neighbors,

256phase transition, 260, 324quasi-chemical approximation,

260transition temperature, 260,

261Sommerfeld-Wilson quantization, 52sound waves, 145, 152

in solids, 150phonons, 323velocity, 145, 150, 152wave equation, 152

species, 8, 112, 114, 117, 118

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specific heat, 28, 300heat capacity, 28

spectral weight, 143, 144, 203spherical top, 89spin-spin interaction, 271spontaneous symmetry breaking, 283standard model, 283statistical hypotheses

assumption I, 14assumption II, 16

statistical mechanics, 14, 16, 17, 26assembly, 12, 128basic problem, 2Boltzmann statistics, 195canonical ensemble, 127, 129, 141,

180canonical partition function, 129,

141classical limit, 50, 53, 132classical statistics, 48, 49, 199complexion, 13ensemble, 12, 128, 180equipartition theorem, 46, 302ergodic hypothesis, 14external field, 111fluctuations, 189Gibbs free energy, 7grand canonical ensemble, 180, 181grand partition function, 182Helmholtz free energy, 7, 26, 129high-temperature limit, 53internal partition function, 63mean values, 25, 116, 129, 137,

183, 189mean-square deviation, 189

metastable assembly, 79microcanonical ensemble, 17non-equilibrium, 335partition function, 26, 129quantum statistics, 49, 196second fundamental relation, 129state average, 14statistical equilibrium, 79, 80statistical hypotheses

assumption I, 14, 302assumption II, 16

system, 12, 63, 128thermodynamic potential, 184third fundamental relation, 184time average, 13

Stefan-Boltzmann law, 207Stirling’s formula, 20, 21, 23, 30, 46,

115, 121, 124, 131, 134, 137, 187,250, 264, 308derivation, 137, 308

symmetric top, 85diatomic molecule, 89eigenfunctions, 87high-T limit, 87, 89internal symmetry, 304partition function, 87spectrum, 87

system, 12, 63, 92, 112, 114, 127, 255

thermionic current, 319Richardson-Dushman, 319

thermodynamic potential, 184entropy, 184equilibrium, 185external field, 233Fermi gas, 226grand partition function, 184particle number, 184pressure, 184

thermodynamics, 1, 2Carnot cycle, 3, 297chemical potential, 8, 11, 112, 119,

171, 311closed sample, 6, 8enthalpy, 298entropy, 5, 6, 25, 259, 297equilibrium, 9, 10, 298, 299

Gibbs criteria, 9, 10, 185, 298,299

thermodynamic potential, 185external field, 112, 232first and second laws, 7, 25, 61,

117, 138, 139, 298first law, 2, 297–299free energy

Gibbs, 7, 171, 311Helmholtz, 7, 26, 129

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Index 365

heat, 2, 7, 92, 298heat capacity, 28, 92isolated sample, 6isothermal compressibility, 191magnetic work, 112Nernst heat theorem, 12open sample, 8, 117, 120phase equilibria, 11, 301pressure-volume work, 7, 298quasistatic, 4, 7, 298reversible processes, 4, 7, 298sample, 2, 12second law, 3, 298, 299state function, 3stretched springs, 311thermodynamic potential, 184thermodynamic properties, 1third law, 12, 16, 92two dimensions, 320work, 2, 298, 311

Thomas-Fermi theory, 224atomic structure, 322nuclear structure, 323screening, 321white dwarfs, 318

ultra-relativistic Fermi gas, 317ultracentrifuge, 300units, 80, 101, 200, 284, 329

vacuum, 201Van der Waal’s

attraction, 158equation of state, 160, 173, 295

critical isotherm, 174critical point, 174reduced quantities, 175, 312

potential, 164, 176, 311second virial coefficient, 160, 164third virial coefficient, 312

vapor pressure, 122, 308Raoult’s law, 252

virial expansion, 160virial coefficients, 160, 163, 164,

173, 311, 312Vlasov equation, 340

white-dwarf star, 318work function, 319

Yang-Mills theory, 283, 284

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Waclecka Statical Physics