Appendices

Appendix A

Highpoints in the History of Statistical Mechanics

400 BC Leukippos of Miletos and Demokritos of Abdera suggest that matter is composed of tiny indivisible particles- the first atomists (Roller, 1981).

1716 Hermann (1716) notes that heat is due to molecular motion, and suggests that in gases pressure is proportional to nv-which was incorrect!

1727 The beginnings of kinetic theory by Euler (1727). He stated that air consists of molecules, gave a theory of humidity, noted that pressure and temperature are gross manifestations of molecular actions, and was able to derive an equation of state for gases: P = ~nv2-also incorrect.

[738 In his Hydrodynamics Daniel Bernouli (1738) reproduces Euler's results in more detail, and proposes that v 2 be taken as a temperature scale. He provided strong hints for energy conservation, and asserted that heat is nothing but atomic motion.

1782 Euler (1782) now proposes tJ2/2 as a definition of heat, which is the first serious attempt to replace phenomenological temperature with a purely mechanical definition in terms of molecular motion.

1798 The qualitative equivalence of work and heat is suggested by Count Rumford (Thomp­son, 1798), and heat is a manifestation of particles in motion. In retrospect, this marks the beginning of the end for the caloric theory.

1814 Laplace (1825) establishes the utility of probability theory in physics.

1816 Laplace (1816) gives a correct adiabatic treatment of the speed of sound.

1821 Herapath (1821) provides a rough kinetic theory, showing that it explains changes of state, diffusion, and sound propagation.

1824 Sadi Carnot (1824) perceives the second law of thermodynamics: no heat engine can be more efficient than a reversible one.

1825 The relation between heat and particle motion is discuused in a careful way by Seguin (1825).

1842 Robert Mayer (1842) clarifies on theoretical (i.e., philosophical) grounds the importance of energy and its conservation in all forms.

1843 Waterston (1893) provides a complete mathematical kinetic theory: P = ~nv2, noting that temperature is proportional to the average tJ 2 • He develops a simple equipartition law, and notes that the mean-free-path (L) behaves as n -3. The first really viable kinetic theory.

1845 Over the period 1840-49 Joule (1845) demonstrates experimentally the mechanical equivalence of heat.

1847 Helmholtz (1847) brings energy conservation and kinetic theory together into the first law of thermodynamics.

337

338 A: Higbpoints in tbe History of Statistical Mecbanics

1851 Joule (1851) reproduces Waterston's work, though crudely, without ever mentioning 'averages', and rederives the expression for the pressure.

1853 Thomson (1853) improves on Joule's work and, together with waterston's results, firmly establishes that P = inv2.

1856 Kronig reviews and summarizes clearly the state of kinetic theory (1856). Although he adds nothing and even makes some errors, his prestige lends great support for the theory.

1857 Clausius (1857), perhaps as early as 1850, renders the earlier ideas more specific, and distinguishes the three states of matter in terms of molecular properties.

At this point the caloric theory of heat is dying rapidly and the kinetic theory is more-or-Iess well established, if not universally accepted. People spoke of 'averages', but nobody really defined clearly what they meant by them. Statistical mechanics begins with the next step.

1858 Clausius (1858) introduces the first explicit notion of probability into kinetic theory, and defines what is meant by 'average'. He defined formally the mean-free-path (L) and distinguished it from the mean interparticle spacing (l .... n-3 ). He perceives the need for some kind of Stopzahlansatz, but fails to formulate it.

1860 Maxwell (1859, 1860) pulls the kinetic theory together and launches the modern view. He derives the distribution for point particles, finds that L = 2(mx:T)1/2 /31r3 / 2d2 ,

where d is the radius of the 'sphere of influence' of a particle, and predicts that the shear viscosity of a gas is independent of its density. This is the first prediction of new properties using a molecular model.

1865 Clausius (1865) introduces the concept of entropy, providing it with a name as well.

Loschmidt (1865) uses kinetic theory to obtain the estimate d .... 10-8 cm.

1866 Boltzmann (1866) publishes his first paper in statistical mechanics and states that his goal is to derive the first and second laws of thermodynamics as purely mechanical theorems.

1867 Maxwell's second theory is published (Maxwell 1867), in which he straightens out some difficulties with his first approach and lays down a definitive mathematical kinetic theory. In the same year he introduced 'Maxwell's Demon', as well as emphasizing the probabilistic nature of the second law (Maxwell, 1867, 1870).

1868 Boltzmann (1868) extends kinetic theory to encompass the Maxwell-Boltzmann distri­bution, which can include complex molecules and external fields.

1870 Kelvin (1870) provides a discussion of the sizes of atoms, and Clausius (1870) develops the virial theorem.

1872 The Boltzmann equation, the Stopzahlansatz, and the H-theorem are born (Boltzmann, 1872).

1877 Boltzmann (1877a, b) emphasizes the probabilistic nature of the second law, in agree­ment with Maxwell, and introduces the method of most probable values. He expresses the idea that S = In W, but does not write it down explicitly.

1878 Gibbs' monumental work appears: 'On the Equilibrium of Heterogeneous Systems', (Gibbs, 1878).

1896 Boltzmann's book is published: Vorlesungen iiber Gastheorie, (Boltzmann, 1896).

A: Highpoints in the History of Statistical Mechanics 339

1902 Gibbs' book is published: Elementary Principles in Statistical Mechanics, (Gibbs, 1902).

1906 Planck (1906) quantifies Boltzmann's idea by writing it as S = In W.

1908 Perrin (1916) secures the atomistic viewpoint with his experiments on the Brownian motion.

1911 The Ehrenfests' famous and influential critique of statistical mechanics is published (Ehrenfest and Ehrenfest, 1911).

1948 Shannon (1948) generalizes the entropy concept.

1957 Jaynes (1957) re-introduces the PME, thereby extending and modernizing Gibbs' point of view.

Kubo (1957) develops the theory of dynamical response.

REFERENCES

Bernoulli, D.: 1738, Hydrodynamica, Argentorati. Boltzmann, L.: 1866, tUber die mechanischen Bedeutung des zweiten Hauptsatzes der Wiirmetheor­

ie', Wien. Ber. 53, 195. Boltzmann, L.: 1868, 'Studien iiber das Gleichgewicht der lebendigen Kraft zwischen bewegten

materiellen Punkten', Wien. Ber. 58, 517. Boltzmann, L.: 1871, 'Wiirmetheorie aus den Siitzen iiber das Gleichgewicht der lebendigen Kraft',

Wien. Ber. 63, 712. Boltzmann, L.: 1872, 'Weitere studien iiber das Wiirmegleichgewicht unter Gasmolekulen', Wien.

Ber. 66, 275. Boltzmann, L.: 1877a, 'Bemerkungen iiber einige Probleme der mechanischen Wiirmetheorie', Wien.

Ber. 75, 62. Boltzmann, L.: 1877b, tUber die Beziehung zwischen dem zweiten Hauptsatzes der mechanis­

chen Wiirmetheorie und der Wahrscheinlichkeitsrechnung respektive den Siitzen iiber das Wiirmegleichgewicht', Wien. Ber. 76, 373.

Boltzmann, L.: 1887, tUber die mechanischen Analogien des zweiten Hauptsatzes der Thermody­namik', J. r. ang. Math. 100, 201.

Boltzmann, L.: 1895, 'On Certain Questions of the Theory of Gases', Nature 51, 413, 581. Boltzmann, L.: 1896, 'Entgegnung auf der Wiirme theoretischen Betrachtungen des Hrn. E. Zer­

melo', Wied. Ann. 57, 773. Boltzmann, L.: 1896, Vorlesungen iiber Gastheorie, Barth, Leipzig(Part I, 1896; Part II, 1898). Carnot, S.: 1824, ReBexions sur la puissance mortice du feu et sur les machines propres a developper

cette puissance, Bachelier, Paris. Clausius, R.: 1857, tUber die Art der Bewegnung, welche wir wiirme nennen', Ann. d. Phys.{2] 100,

353. Clausius, R.: 1858, tUber die mittlere Liinge der Wege, welche bei der Molekularbewegnung gasfarm­

iger Karper von den einzelnen Moleciilen zuriickgelegt werden, nebst einigen anderen Be­merkungen iiber die mechanischen Wiirmetheorie', Ann. d. Phys.{2] 105, 239.

Clausius, R.: 1865, 'Uber verschiedene fiir die Anwnedung bequeme Formen der Hauptgleichungen der mechanische Wiirmetheorie', Ann. d. Phys.[2] 125, 390.

Clausius, R.: 1870, 'Uber einen auf die Wiirme anwendbaren mechanischen Satz', Ann. d. Phys.{2} 141, 124.

Ehrenfest, P., and T.Ehrenfest: 1911, BegrifHiche Grundlagen der statistischen Auffassung in der Mechanik, in VoLIV, Part 32, Encyklopiidie der Mathematischen Wissenschaften, Teubner, Leipzig.

340 A: Highpoints in the History of Statistical Mechanics

Euler, L.: 1727, 'Tent amen explicationis phaenomenorum aeris', Comm. Acad. Sci. Petrop. 2, 347. Euler, L.: 1782, Acta Acad. Sci. Petrop. 1, 162. Gibbs, J.W.: 1876, 'On the Equilibrium of Hetrogeneous Substances', Trans. Conn. Acad. S, 108,

343. Gibbs, J.W.: 1902, Elementary Principles in Statistical Mechanics, Yale Univ. Press, New Haven. Herapath, J.: 1821, 'A Mathematical Inquiry into the Causes, Laws and Principal Phenomena of

Heat, Gases, Gravitation, etc.', Ann. Phil.{2} 1, 273, 340, 40l. Hermann, J.: 1716, 'Phoronomia sive de viribus et motibus corporum soidorum et fuuidorum libri

duo, Amsterdam. Jaynes, E.T.: 1957, 'Information Theory and Statistical Mechanics', Phys. Rev. 108, 17l. Joule, J.P.: 1845, Phil. Mag. {3}, 27, 205. Joule, J.P.: 1851, 'Some Remarks on Heat and the Constitution of Elastic Fluids', Mem. Manchester

Lit. Phil. Soc. 9, 107. Kronig, A.K.: 1856, 'Grundziige einer Theorie der Gase', Ann. d. Phys.{2} 99, 315. Kubo, R.: 1957, 'Statistical-Mechanical Theory of Irreversible Processes.I. General Theory and

Simple Applications to Magnetic and Conduction Problems', J. Phys. Soc. Japan 12, 570. Laplace, P.S.: 1816, Ann. Phys. Chim. S, 288. Laplace, P.S.: 1825, Essai philosophiqe sur les probabilites, 5th ed., Bachelier, Paris. Maxwell, J.C.: 1859, 'letter to G.G. Stokes on 30 May', in J. Larmor (ed.), Memoir and Scientific

Correspondence of the late George Gabriel Stokes, Vol.2, Cambridge Univ. Press, Cambridge. Maxwell, J.C.: 1860, 'Illustrations of the Dynamical Theory of Gases'-full references on p.30 of

this volume. Maxwell, J.C.: 1867, 'On the Dynamical Theory of Gases', Phil. Trans. Roy. Soc. London 157, 49. Maxwell, J.C.: 1867, 'letter to P.G. Tait on 11 December', in C.G. Knott, Life and Scientific Work

of Peter Gutherie Tait, Cambridge Univ. Press, Cambridge, 1911. Maxwell, J.C.: 1870, 'letter to J.W. Strutt on 6 December', in R.J. Strutt, Life of John William

Strutt, Third Baron Rayleigh, Univ. Wisconsin Press, Madison1968. Mayer, J.R.: 1842, 'Bemerkungen iiber die Krafte der unbelebten Natur', Ann. Chemie und Phar-

macie 42, 233. Perrin, J.: 1916, Atoms, Van Nostrand, Princeton. Planck, M.: 1906, Vorlesungen iiber die Theorie der Warmestrahlung, J.A. Barth, Leipzig. Seguin, M.: 1825, 'Letter to Dr. Brewster on the Effects of Heat and Motion', Edinburg J. Sci. S,

276. Shannon, C.E.: 1948, 'A mathematical Theory of Communication', Bell System Tech. J. 27, 379,

623. Thompson, B. (Count Rumford): 1798, Phil, Trans. Roy. Soc. London 80. Thomson, W. (Lord Kelvin): 1853, 'On the Mechanical Action of Heat, and the Specific Heats of

Air', Cambridge and Dublin Math. J. 96, 270. Thomson, W.: 1870, 'The Size of Atoms', Nature 1, 55l. von Helmholtz, H.: 1847, Uber die Erhaltung der Kraft, G. Reimer, Berlin. Waterston, J.J.: 1893, 'On the Physics of Media that are Composed of Free and Perfectly Elastic

Molecules in a State of Motion', Phil. Trans. Roy. Soc. London 18SA, 79. [Published posthu­mously; first submitted in 1843; abstract published in Proc. Roy. Soc. (London) 5, 604(1846)].

Appendix B

The Law of Succession

Recall Eq.(2-27), which describes Laplace's unique solution to the Bernoulli inversion problem. Suppose an event is related to a mutually exclusive and exhaustive set of possible causes (A1 ,··· , An), and the event E has occurred. Then the posterior probability that its cause was A. is

p(AoIX) p(EIA.X) p(A.IEX) = E; p(EIXA;) p(A;IX) , (B-1)

and the p(A.IX) are a priori probabilities. If an event E occurs, this constitutes new information, and we may wish to use it to readjust our probability p(E'IEX) that an event E' will occur. As in Eq.(2-26), we can expand this as

p(E'IEX) = E p(E'IA.EX) p(A.IEX). (B-2)

One can obtain an alternative expression for this probability by multiplying Eq.(B-l) by p(E'IA.EX) and summing over i:

(E'IEX) = E. p(A.IX) p(EIAiX ) p(E'IA.EX) (B-3) P E; p(EIA;X) p(A;IX) ,

which is one form of the original statement of Bayes' theorem. Note that if A. is given, then E is irrelevant to E' and thus p(E'IA.EX) can be replaced by p(E'IA.X). Consequently, all quantities on the right-hand side of Eq.(B-3) are presumed known or can be calculated.

A continuum form of Eq.(B-3) can be formulated by considering a continuum of proba­ble causes, and in small intervals we can take the priors to be equal. This obviates the need to actually know the priors, for we can now write p(A.IX) = dp and Eq.(B-3) becomes

p(E'IEX) = I: P(~lp) p(E'lp) dp (B-4) Io p(Elp)dp

We emphasize that this result is valid only in situations in which the basic priors are equal. As a specific application of Eq.(B-4) consider a process of the type we have called ex­

changeable sequences, where in N trials n successes have occurred. Given this information, we ask for the probability that m successes will occur in M future trials. The appropriate functins in the numerator of Eq.(B-4) are just binomial distributions:

p(Nnlp) = (~)pn(l_ p)N-n,

P(Mmlp) = (!!")pm(1 _ p)M-m.

The desired probability is then

]',1 pm+n(1 _ p)M+N-m-n dp P(MmINn) = (!!") 0 I01 pn(l _ p)N-n dp

= (n~m) (N+r;:=:;-") (N+::+1)

341

(B-5a)

(B-5b)

(B-6)

342 B: The Law of Succession

where we have employed the ,8-function integrals:

{l r B r!s! 10 x (1 - x) dx = (r + s + 1)1' (B-7)

If one is interested only in the probability that the outcome A occurs in the next trial, we set M = m = 1. Then,

n+1 p(AINn ) = N + 2 '

which is Laplace's law of succession (Laplace, 1774).

(B-8)

It is almost as simple to derive a generalized law ofsuccession, following Jaynes (1958). Suppose that there are k possible outcomes of an experiment, A l , ... , Ak, and that in N = L:i ni trials Al occurs nl times, A2 occurs n2 times, etc. We ask for the probability that in the next M = L:i mi trials Al occurs ml times, etc. Rather than the binomial distributions of Eqs.B-S), we now must empoy the multinomial distributions

(B-9a)

(B-9b)

In place of the simple integrals (B-7) we now must consider

(B-I0)

subject to the conditions

Pi ~ 0, Pl + P2 + ... + Pk = 1. (B-ll)

Evaluation of I is facilitated by the useful device of incorporating the second condition of Eq.(B-ll) diectly into the integral. Define

(B-12)

and eventually we shall be interested in 1(1). First take the Laplace transform:

1000 e-a ... I(z) dz = 1000 dPl" .1000 dPkp~l ". pZ- e-a(Pl +"'+Pk)

nll nk! = a"l+l'" a".+l·

(B-13)

Then calculate the inverse transform from the calculus of residues:

(B-14)

B: The Law of Succession 343

The desired integral is thus

(B-15)

With this result, and the obvious extension of Eq.(B-4) to multiple integrals, the probability of interest is

(B-16)

Suppose now that we are only interested in the probability that Al occurs in the next trial. Then M = ml = 1, with all other mi = 0, and we obtain the generalized law of succession:

(B-17)

When there are only two alernatives, A or a, then k = 2 and this last result reduces to Eq.(B-8).

Alternatively, suppose that there is no evidence available: N = nl = O. Then the probability is k- l and we obtain a novel derivation of the PIR. Note, however, that one must still append to this the presumption that it is at least possible for Al to occur .

• REFERENCES

Jaynes, E.T.: 1958, Probability Theory in Science and Engineering, Socony Mobil Oil Company, Inc., Dallas.

Laplace, P.S.: 1774, 'Memoire sur la probabilite des causes par lee evenements', Mem. Acad. Sci. (Paris) 6, 621.

Appendix C

Method of Jacobians

o ne of the most frustrating aspects of phenomenological thermodynamics has always been the tedious and complicated algebra associated with the manipulation of partial derivatives. For example, the set of Maxwell relations in Eq.(3-16) exhibits only some of the possible ways of combining derivatives for the thermodynamic system defined by E = E(T, S, P, V), and the enormous number of possibilities implies a great need for systematic procedures. The truly complicating feature in thermodynamics is the necessity for maintaining su b­scripts on all partial derivatives so as to indicate which variables are being held constant in the process.

This situation is not unfamiliar in mathematical physics, for the use of vectors and tensors long ago supplanted the cumbersome notation originally employed in mechanics and electrodynamics. Such coordinate-independent schemes would be equally useful in thermodynamics, because one is dealing with an affine space of state variables in which the notion of distance has no physical meaning. Thus, if the theory could be formulated in a way which expresses the relations among physical quantities so that they do not depend on the choice of independent variables, much would be gained in economy of notation, and possibly in insight.

Such a scheme was originally developed by Clausius over one hundred years ago, and used by Gibbs as well, but it never came into widespread use. This method utilizes the concept of Jacobian-determinants to formulate coordinate-independent expressions-it has been redeveloped by Jaynes in unpublished notes, and used extensively by Tribus (1961).

Recall that for functions u(x, V), v(x, V) the elements of area in the xv-plane and uv­plane are related by the Jacobian of the transformation between the two sets of variables:

J (u,v) == 1 ~~ x,V ~:

a"'l all au . all

(C-l)

This function is quite familiar from problems in which one considers a function z(x, V) ranging over an area r in the xv-plane. If we wish to map r into a corresponding area R in the uv-plane, the integrals over z in the two regions are related by

r zdudv = r zJ (u,v) dxdV. JR Jr x,v (C-2)

Suppose now that we consider a small rectangular element of area dx dV and let this el­ement be denoted by [x, V] = al' Presume also the existence of two functions u(x, V), v(x, V) with sufficient continuity properties that they provide a one-to-one mapping of al into the element of area [u, v] = a2 in the uv-plane. We shall prove that the Jacobian of the mapping can be written

(C-3)

344

c: Method of Jacobians 345

where the integrals are taken around the respective boundaries. The essential point for our purposes is the second equality, expressing the Jacobian as the ratio of two well-defined entities. Note that the idea here is a local one.

The proof is actually trivial. One first expands u(x, y) about (xo, Yo),

u=uo+(::) dX+(:u) dy+O(dx2,dy2) , "'0 y 110

and then evaluates the numerator integral in Eq.(C-3) in four pieces. We find that

f (au av au av) 2 2 udv= axay - ayax dxdy+O(dx ,dy).

Now divide by [x, yJ and take the limit as the two areas become vanishingly small:

[u,vJ au av au av [x, yJ = ax ay - ay ax '

(C-4)

which is precisely the definition (C-l). An essential feature of this new notation emerges, for our purposes, if we consider a

mapping from the AB-plane to the C B-plane:

[A,BJ I AdB I BdA [C,BJ = ICdB = I BdC'

But the numerator integral is just

Hence, in the limit of small areas the partial derivative can come outside the integrtal and we find that

(C-5)

As a consequence of this exercise we learn how to represent complicated partial derivatives by bracket symbols.

Note the basic properties of the bracket symbols:

[u,vJ = -[v,uJ, [u,uJ=o, (C-6)

which follow from the definitions. Thus, the algebra of partial derivatives is reduced to the algebra of antisymmetric bracket symbols, reminiscent of the linear-algebraic meth­ods appearing in many other areas of theoretical physics: Lie algebras, Poisson brackets, quantum-mechanical operators, etc.

One readily verifies the properties of linearity

[A± B,CJ = [A,CJ ± [B,CJ, (C-7)

composition [AB,CJ = [A,CJB+A[B,C]' (C-8)

346 G: Method of Jacobians

and the following cyclic identities:

[A,B]dG+ [B,G]dA+ [G,A]dB = 0,

[A,[B,GII + [B,[G,A]] + [G, [A, BII = 0,

[A,Bj[G,D] + [B,Gj[A,D] + [G,Aj[B,D] = O.

Of particular value is the relation stemming from the expression

dA=b(:~)C dB+e(:~)B dG.

Then, for arbitrary X and z, consideration of (8Aj8z)x yields

[A, X] = b[B,X] + e[G, X].

(C-9a)

(C-9b)

(C-9c)

(C-lO)

(C-ll)

The power of the J a.cobian notation can be illustrated immediately by observing that all four Maxwell relations of Eq.(3-16) are contained in the identity

[T,S] = [P,V]. (C-12)

One merely divides this expression successively by all other independent combinations in brackets of the four variables, and then employs Eq.(C-5). Therefore, we see that the Maxwell relations are all expressions in different 'coordinate' systems of the same basic fact (C-12). That is, the mapping from the PV- plane to the TS-plane preserves areas, an observation expressed more physically by integrating Eq.(3-8) over a cyclic path. Similar insight emerges with respect to the set of Maxwell relations involving magnetic variables, (T,S,H,M):

[T,S] = [M,H]. (C-13)

Let us recall the differential forms of Eqs.(3-14), (3-15), (3-8), and (4-75). These can be converted, respectively, into the following forms by utilizing Eqs.(C-10) and (C-ll):

[E,X] = T[S, X] - P[V, X] , [H,X] = T[S,X] + V[P,X] , [F,X] = -S[T,X] - P[V, X] , [G,X] = -S[T,X] + V[P ,X],

(C-14a)

(C-14b)

(C-14c)

(C-14d)

where X is arbitrary. The utility of these expressions as calculational aids is enormous. Although the above results are already impressive, the real power of the method is

found in the ease with which it yields expressions for thermodynamic quantities in terms of measurable parameters. In order to see this, let us first note from Eq.(3-21) that the heat capacities at constant X are all given by

[S,X] Gx=T[T,X]' (C-15)

We can then relate certain ratios of the brackets to these measurable quantities. Also, for all systems in which V is a function of P and T,

(C-16)

c: Method of Jacobians

Hence, it is useful to define

_ 1 (8V) _ 1 [V,P] aT = y- 8T p - V [T, P] ,

_ 1 (8V) _ 1 [V,T] itT=-Y- 8P T --V [P,T] ,

341

(C-17)

(C-IS)

the (isobaric) thermal coefficient of expansion, and the (isothermal) compressibility, respec­tively. It is also convenient to write

(C-19)

(C-20)

which are readily verified. AB a further example, we sometimes find it useful to know the derivative (8E/8P)s in

terms of measurable quantities. This can be found by first writing

( 8E) = [E,S] = T[S,S] - pry,S] 8P s [P,S] [P,S]

= _p [V,S] P,S] ,

utilizing Eqs.(C--6) and (C-14a). From Eq.(C-15),

so that finally

( 8E) = _p(Cv/T)[T,V] 8P s (Cp/T)[T,P] ,

( 8E) _ PVitT 8P s - -"1-.

REFERENCES

Tribus, M.: 1961, Thermostatics and Thermodynamics, Van Nostrand, Princeton.

(C-21)

Appendix D

Convex Functions and Inequalities

W e gather together here a number of definitions and lemmas on convex functions which form the basis for many variational principles and general inequalities in statistical mechan­ics, and then record numerous theorems often found useful in the subject. Unless otherwise referenced, proofs regarding inequalities and convex functions can be found in Hardy, et al (1952).

A function I(x), which shall always be taken as continuous on the real interval (a, b), is said to be convex on that interval if for any (Xl, X2) E (a, b), and for any real A, 0 :::; A < 1,

(D-l)

Geometrically, the chord joining I(xd and I(X2) always lies above the graph of I(x) itself. If - I is convex, I is said to be concave.

Lemma 1. Let I(x) be continuous and twice differentiable in (a,b). Then a necessary and sufficient condition for I to be convex in (a, b) is that

f"(x) ~ o. (D-2)

Lemma 2. If f" > 0 for x> 0, and 1(0) ~ 0, then l(x)lx increases for X > o. Lemma 3. If I (x) is convex in (a, b) it has at every interior point both right-hand and left-hand derivatives: I:, It, respectively, with I: ~ 1/. Both derivatives increase with x, and the derivative f' = I: = It exists everywhere except at a perhaps a countable set of values x E (a, b).

Lemma 4. If I(x) is convex and differentiable in (a,b), then for (XI,X2) E (a, b),

I(XI) - I(X2) - (Xl - x2)/'(X2) ~ 0,

with equality if and only if Xl = X2.

(D-3)

An immediate and useful application of this lemma is to the convex function - In X for x> 0:

x-I -- < Inx < x-I x - - , (D-4)

with equality if and only if x = 1.

Lemma 5. Let I(x) be convex in (a, b) and let {Pi} be a set of n positive numbers such that

n

(D-5)

Then,

I(LP;x;):::; LP;/(x;) , (D-6) i i

This lemma is often called the convex lunction theorem, and frequently N = 1.

348

D: Convex Functions and Inequalities 349

Lemma 6. Let {ail and {bi} each be a set ofn non-negative numbers. Then,

(2:: aib.) 2 ~ 2:: a~ 2:: b~ , (D-7) i i j"

with equality if and only if ai and bi are proportional for all i.

This lemma is originally due to Cauchy, and integral versions were given by Bunyakovsky and Schwarz. A similar inequality is valid for scalar products of linearly independent vectors in linear vector spaces, an observation which also applies to the following lemma.

Lemma 7. Let k' == k/(k - 1), k i= 0, 1. Then, for the sets discussed in the preceding lemma,

2:: a.b. ~ (2:: a7 f1k (2:: bJ'f1k l

, k > 1, (D-8) i i j

and the inequality is reversed if k < 1. Equality obtains if a7 is proportional to bf' for all i, or if the left-hand side vanishes.

This is Holder's inequality. It is not necessarily true that a convex (or concave) function of a convex function is also

convex (or concave), an elementary example being provided by the logarithm. If log f(x) is convex, so is f(x), but not conversely. A counterexample is provided by f(x) = xlnx on (1,00).

Lemma 8. If f(x) is positive and twice differentiable, then a necessary and sufficient con­dition for log f(x) to be convex is that f f" - /,2 ~ o.

There is one case in which the converse is always true. If f(x) possesses derivatives of all orders on (a, b) such that

dnf dxn ~ 0, n = 0,1, ... , (D-9)

then f(x) is said to be absolutely monotonic on (a, b).

Lemma 9. If f(x) is absolutely monotonic on (-00,0), then log f(x) is convex on (-00,0).

This is a corollary to Bernstein's theorem, Eqs.(3-49) and (3-50). Let x = {Xl, X2, ••• ,xn } be a point in an n-dimensional Euclidean space En. A domain

D in En is convex if xED, y ED implies

>.x + (1 - >.)y ED, o~>.~1. (D-I0)

Geometrically, a convex domain contains all of the straight line segments joining any two of its points. A function f(x) defined on a convex domain D is convex if

>.f(x) + (1 - >.)f(y) ~ ![x + (1 - >.)y]. (D-ll)

Lemma 10. If f(x) is convex and continuous in (Xl, . .. , x n ), and if {pd is a set of positive numbers such that Eq.(D-5} holds, then

f (~Pixi) ~ ~p;/(Xi), (D-12)

where xi == {xL ... , x~}.

This is a generalization of the convex function theorem.

350 D: Convex Functions and Inequalities

Lemma 11. If f(x) possesses all of its second derivatives in a convex open domain D, then a necessary and sufficient condition that f be convex on D is that

n

F == 2: !ijUiUj

iJi=1

be a positive-semidefinite quadratic form at each XED, where

a2 f !ii==-a a .

Xi Xi

GENERAL INEQUALITIES

(D-13)

(D-14)

Many arguments in statistical mechanics are facilitated by the availability of numerous rigorous inequalities, almost all of which follow from the theory of convex functions. The quantities of interest in quantum statistical mechanics are certain Hermitian operators corresponding to observables for a given system. In the following discussion, therefore, we shall always have in mind a real vector space 8 PI) of self-adjoint linear operators on a Hilbert space )/. Owing to the possible noncommutativity of these operators, care must be taken in differentiating them with respect to parameters. This becomes of particular concern in the case of the exponential operators encountered in statistical mechanics, so that the following prescription from Wilcox (1967) is very important. If fI = fIp..), where H is not necessarily a Hamiltonian, then

_e-(3H = _ e-((3-u)H _e-uHdu a - 1(3 . afI -a>.. 0 a>.. . (D-15)

One proves this by showing that both sides of the identity satisfy the differential equation

a F((3) H F((3) = _ a fI -(3 H (D-16) a(3 + a>.. e ,

subject to the initial condition F(O) = O. We shall find this identity of some value in the sequel.

Consider a linear operator A E 8 such that AI>") = a>. I>..) , and let the spectrum of A lie in the domain of a function f(x). If {I>")} and {1m)} are complete orthonormal sets of state vectors in )/, then

Aim) = 2: C>.m a>. I>") , (D-17a) >.

with (D-17b)

>.

From elementary quantum mechanics one thus has the following matrix elements:

(D-18a) >.

(D-18b) >.

Theorem 1. If f(x) is convex in a domain including the spectrum of A, then

f((mIAlm)) ::; (mlf(A)lm) , (D-19)

with equality if and only if 1m) is an eigenfunction of A. The proof follows from Lemma 5. The sense of the equality is that f(a>.) = (>''If(A)I>'') is convex over the spectrum of A.

D: Convex Functions and Inequalities

Theorem 2. For any A E 8 with spectrum in the domain of a convex function /(x),

m.

with equality if and only if 1m} is an eigenfunction of A. One proves this by summing over m in Eq.(D-19).

351

(b-20)

Numerous inequalities can be produced from Theorem 2 by making specific choices for the convex function /(x). For example, if A is a positive operator-meaning all its eigenvalues are non-negative-then the choice /(x) = e-:Z; yields an inequality due to Peierls (1938):

(D-21) m

with equality if and only if 1m} is an eigenfunction of A. For the same operator, the choice /(x) = x In x yields

(D-22) m

with equality if and only if 1m} is an eigenfunction of A. In the same sense that A can be considered a positive operator, one can also study

inequalities of the type B ~ A. This means that Blm} ~ Aim} for every 1m} E )(, and equality implies that B = A. Theorem 3. If A and B are bounded self-adjoint operators with bounded inverses, such that B ~ A > 0, then

(b-23)

and log A ~ log B , (b-24)

with equality if and only if B = A. The proof follows from analyzing the matrix elements in an arbitrary representation.

With these ideas in mind one can consider the sense in which an operator function may be convex over the space of operators 8 itself.

Theorem 4. Let /(x) be convex on a domain containing the spectra of all the operators in 8. Then Tr /(A) is convex on 8.

The meaning of the assertion is that

A Tr /(A) + (1 - A) Tr /(B) ~ Tr /(G) ,

for any A, BE 8 and 0 ~ A ~ 1, and with

G == AA+ (1- A)B.

(D-25)

(D-26)

To prove the theorem, consider a representation in which G is diagonal: Gin) = c ... ln}, Cn == Aan + (1 - A)bn, where we define an == (nIAln), bn == (nIBln). In this representation Theorem 2 yields

A Tr /(A) + (1 - A) Tr /(B) ~ A L /(a n ) + (1 - A) L /(bn )

n n

= L[A/(an) + (1 - A)/(bn)]. n

352 D: Convex Functions and Inequalities

But Eq.(D-1) provides a term-by-term domination:

L[.V(an ) + (1 - A)f(bn )] ~ L ![Aan + (1 - A)bn ]

n n

= L f(c n ) = Tr f(C). n

Theorem 5. Tr(Aln A) is convex on B.

This is proved by taking f (x) = x In x in Theorem 4.

Theorem 6. Let A and B be unbounded self-adjoint operators with spectra in the domain of a convex function f(x). Then,

Tr[t(A) - f(B) - (A - B)!'(B)] ~ 0, (D-27)

with equality if and only if A = B. The proof is constructed by letting Aln) = a"ln), Blm) = bmlm), and defining Cnm == (nlm) such that Em \cnml 2 = 1. Then,

(nlf(A) - f(B) - (A - B)!,(B)ln)

= f(a n ) - L Icnm l2 f(bm) - L \cnmI 2 (an - bm)t(bm) m m.

= L ICn m.12 [f(a n ) - f(bm.) - (an - bm)f'(bm )]

m

~ 0,

because Lemma 4 and Eq.(D-3) ensure that every term in the sum is positive. The theorem now follows by summing over n.

An immediate and very useful example emerges from the choice f (x) = x In x: Tr[Aln A - Aln B - (A - B)] ~ 0, (D-28)

with equality if and only if A = B. According to Lanford and Robinson (1968), Theorem 6 is originally due to O. Klein, a fact transmitted to them by R. Jost.

Theorem 7. If f(x) ~ 0 everywhere and log f(x) is convex, then 10gTr f(A) is convex on B and f(x) is also convex.

The convexity of f(x) follows from Lemma 8 if the function is twice differentiable. With the notation used in the proof of Theorem 4, consider the product

[Tr f(A)]>' [Tr !(BW->' ~ [L f( an)] >. [L f(bm )] 1->' , n m

from Theorem 2. Now write f = (I>'F/\ (11->.)1/(1->.) and employ Holder's inequality, Eq.(D-8):

[Tr f(A)f [Tr f(B)r->. ~ Lf>'(an)f1->'(bn ).

n

But the presumed convexity of log f(x) implies, by the definition (D-1), that

f>'(an)f1->'(bn) ~ ![Aan + (1 - A)bn] = f(cn ) ,

so that

[Tr f(A)f [Tr f(B)r->. ~ L f(c n ) = TrC. n

By taking logarithms we see that logTr f(A) has the form illustrated in Eq.(D-25), which proves the theorem.

D: Convex Functions and Inequalities 353

Theorem 8. The operator function log Tr eA is convex increasing on B.

The function is obviously increasing, and the proof of the theorem follows from taking I(x) = e'" in Theorem 7.

Of equal interest in studying operator inequalities is the possible convexity of operator functions with respect to parameters appearing in the operators.

Theorem 9. For convex I(x) and real x, Tr I(A + xB) is also convex in x.

This is proved by employing Eq.(D-25) for A + xB:

A Tr I(A + xB) + (1 - A) Tr I(A + yB) ~ Tr I[A(A + xB) + (1 - A)(A + yB)] = Tr I[A + {Ax + (1 - A)y}B] ,

which is just the definition of convexity.

Theorem 10. If I(x) ~ 0 for all real x and iflog I(x) is convex, then logTr I(A + xB) is convex in x.

The proof follows from Theorem 7 and the definition of convexity:

A log Tr I(A + xB) + (1 - A) log Tr I(A + yB) ~ log Tr I[A(A + xB) + (1 - A)(A + yB)] = log Tr t[A + {Ax + (1 - A)y}B].

Again we note that this result does not follow from Theorem 9. Rather, one must specify I (x) to be logarithmically convex.

This last result can be used to obtain some useful results when I is a function of a linear combination of a number of operators.

Theorem 11. Let I(x) be lOlIarithmically convex on a domain containing the real numbers Xj, j = 1,2, ... , N, and let Aj , j = 1,2, ... , N be arbitrary Hermitian operators. Then,

a2 (A ) E •. = --log Tr I Lx' A-3 aXiaXj . 3 3

3 •

(D-29)

is a non-negative matrix.

We follow Okubo and Isihara (1971) and show that Li,j tiEijtj ~ 0 for arbitrary ti, i = 1, ... ,N. Let

so that

A= LXjAj, j

B = LtjAj, j

A + xB = L(Xj + xtj)Aj. j

But from Theorem 10 we know that log Tr(A + xB) is convex in x. Hence, calculation of second derivatives with respect to x and with respect to (Xi, Xj), and reference to Lemma 2, yields

'" a2 A A d2

A A

~ titj ax.ax' 10gTr I(A + xB) = dx 2 log Tr I(A + xB) ~ O. . . ,. 3 -,3

354 D: Convex Functions and Inequalities

By setting x = 0 we obtain the theorem. An immediate application of this theorem is that, because €ii is positive-semidefinite,

(D-30)

These inequalities will arise again presently in a more directly physical context. In the application of these theorems to statistical mechanics one is interested primarily

in the Hamiltonian H and the statistical operator p, such that Tr p = 1. Perhaps the result of most general utility is the following.

Theorem 12. If P and p' are any two statistical operators sucb tbat Tr p = Tr p' = 1, tben

Tr(p In p) ~ Tr(p In p') , (D-31)

witb equality if and only if p' = p. The proof follows from Eq.(D-28). The classical version of this theorem, Eq.(2-85), was used by Gibbs to prove the variational theorem of Eq.(2-86).

Theorem 13. Tr(p In p) is convex on tbe convex set of statistical operators.

Take A = P in Theorem 5 for the proof. The canonical ensemble describing a system in thermal equilibrium is characterized by

Eqs.(4-42) and (4-43). In order to study the specific effects of interactions in the system we write the Hamiltonian explicitly as

H>. == Ho +>'V, O:S: >. :s: 1, (D-32)

where Ho refers to free particles and the coupling constant>. measures the strength of the particle interaction potential V. A more explicit description is then given by

(D-33)

(A)>. = Tr(h A) , F>.«(3) = -II:TlnZ>.«(3). (D-34)

Subscripts zero therefore refer to a free-particle system, whereas >. = 1 refers to a fully­interacting system at the same temperature and density. When >. = 1 the subscript is generally omitted.

Theorem 14. In tbe canonical ensemble

(v)o ~ (F - Fa) ~ (V) ,

witb equality if and only ifH = Ho.

(D-35)

In order to prove this we substitute p and Po into Eq.(D-31), both ways, and employ the definition of entropy in the form

S>.(p>') == -II: Tr(hlnh),

= II:lnZ>. + 1I:(3(H>.h, (D-36)

the second form referring to the maximum entropy. The inequalities (D-35) are known as the Gibbs-Bogoliubov inequality and its inverse. Their historiccl aspects are discussed by Girardeau and Mazo (1973), and by Huber (1970).

D: Convex Functions and Inequalities 355

Theorem 15. The canonical partition function is convex with respect to both (:J and ~.

For the proof, one takes /(x) = e-1lJ in Theorem 9 and makes the appropriate choices for ..4, E, and x.

Theorem 16. log Z>.((:J) is convex with respect to both (:J and ~.

Take /(x) = e- 1lJ in Theorem 10 and make the appropriate identifications in order to obtain a proof. These last two theorems can be proved in several other ways, of course.

There are available a number of theorems regarding the entropy itself, as expressed in Eq.(D-36), two of which were considered in Chapter 4. Let p be defined on the Hilbert space )/ = )/1 ® )/2, and define PI as the projection of ponto )/1, and P2 as the projection onto )/2. One can also define the direct product h ® P2 =1= P on )/1 ® )/2. In Eq.(4-35) it was shown that S(h ® P2) = S(h) + S(P2), and a certain inequality was asserted. That was a special case of a property known as the 8ubadditivity of entropy (e.g., Lieb, 1975), encompassed in the following theorem.

Theorem 17.

with equality if and only if p = h ® P2.

A proof is constructed using Theorem 12:

Tr(plnp) - Tr(plnh ® P2) = Tr(plnp) - Tr(pln[(h ® 11)(12 ® P2)j)

= Tr(plnp) - Tr(plnh ® 11) - Tr(pln 12 ® P2) = Tr(p In p) - Tr(p1ln PI) - Tr(p21n P2) ~ o.

(b-37)

The concavity of S(p) can be shown to be a consequence of this last theorem (e.g., Wehrl, 1978). In a slightly different manner we can prove Eq.(4-31).

Theorem 18. Let p and pi be defined on )/, where P~n == Pnnomn in a particular represen­tation. Then,

S(p') ~ S(p) , (b-38)

with equality if and only if pi = p.

For the proof we take /(x) = x In x, ..4 = p, E = pi in Theorem 6. In the above representa­tion,

Tr(p In p) - Tr(p' In pi) ~ Tr[(p - p')(l + In pi] = Tr[(p - pi) In pi]

= ~)ml(p - p')lm)(mllnp/lm) m

=0,

because the indicated matrix elements of p and pi are equal term-by-term in this represen­tation.

Define the difference between the entropies in the noninteracting and interacting sys­tems as

I:l.S == So - S. (D-39)

356 D: Convex Functions and Inequalities

Theorem 19. (H)o - (H) ? Tt:..S ? (Ho)o - (Ho),

with equality if and only ifH = Ho.

(D-40)

The proof follows from substituting p and Po into Eq.(D-31)' both ways, and employing the definition (D-36). This result was obtained by Leff (1969) in demonstrating that the entropy of a classical interacting gas is less than that of the similar ideal gas, because classically the right-hand side of Eq.(D-40) vanishes, a point noted earlier by Jaynes (1965). These and other inequalities can be treated more generally in quantum statistical mechanics, as we now demonstrate.

A special case of Eq.(D-15) is the identity

(D-41)

Now consider a Hamiltonian depending on a number of parameters in the form of coupling constants:

m

H(Al, ... , Am.) = Ho + 2.:>kh , (D-42) k=l

where {Ai} is a set of real variables such that 0 ~ Ai ~ 1. The latter restriction is not necessary, but merely convenient for the physical situations we wish to emphasize. Also, H need not be linear in the Ai and the modifications of the following equations in that case are straightforward to derive-we treat only the linear case here.

Let us write the expectation values of the operators Pk as

(D-43)

and then

B A -1 B2 BAm (Fk) = -/3 BAmBAk In Z

= (PmFk ) - (Fm) (h) (D-44)

As suggested by Eq.(D-15), we have defined the Kubo transform of Pm as

(D-45)

The functions KFG are called covariance functions, because they are just the quantum­mechanical variances and covariances of the theory.

We note several general properties of covariance functions, the first of which is the evident reciprocity in Eq.(D-44). For Hermitian operators P and a, K FG is real and

(D-46)

D: Convex Functions and Inequalities 357

with equality when and only when F = (F) 1. This inequality follows immediately from Eq.(D-44) with Flo = Fm = F. On the linear vector space B of Hermitian operators, KFG

satisfies all the properties for a scalar product. Hence, the Schwarz inequality of Lemma 6 yields

(D-47)

with equality if and only if F is a real scalar multiple of G. The covariance functions define a covariance matrix which, from the proof of Theorem 11, is positive semidefinite. We therefore obtain additional proofs of Theorems 15 and 16.

Let us evaluate K FG in a representation in which H is diagonal, and then perform the x-integration in the Kubo transform of Eq.(D-45) to obtain

1 ~ • • KFG = Z L.J(mI8Gln)(nI8Flm} ~(-,BEn' -,BEm). (D-48)

m,n

We have here introduced the deviation, 8F == F - (F), denoted the energy eigenvalues of H by En, and followed Okubo (1971) in defining a function

e'" - ell ~(x,y)==-­

x-y

From Lemmas 3 and 4, with f(x) = e"', we find

e'" ~ ~(x,y) ~ ell, x> y,

and conclude that He'" + ell) ~ ~(x, y) ~ 0,

where the first equality is achieved only for x = y, and the second when x = y = O. Now note that the correlation function without the Kubo transform is

(oG of) = (GF) - (G}(F)

We have therefore proved

Theorem 20.

= ~ Le-PEm (mI8Gln}(nI8Flm}. n,m

(oF of} ~ KFF ~ 0,

with equality on the upper bound if and only if[F,H] = o.

Theorem 21.

The proof follows from considering KF-G,F-G in Theorem 20.

(D-49)

(D-50)

(D-51)

(D-52)

(D-53)

(D-54)

Let us now return to a consideration of the entropy difference of Eq.(D-39), using the notation of Eqs.(D-33) and (D-34). From Eq.(D-36) we can write this difference explicitly as

(D-55)

358 D: Convex Functions and Inequalities

Theorem 22. With only one interaction term in Eq.(D-42),

In order to prove the theorem we define

in the same notation. Then,

(Ho) - (Ho)o = /(1) - /(O) = 11 a~~>.) d>..

But a straightforward calculation yields

so that

In a similar manner define

such that F(O} = 0, /(1) = K,P(V) + K,ln(Z/Zo}. Then,

F(l} = 11 a~i>'} d>', a~f} = -K,p2>.Kvv(>')'

which completes the proof.

Theorem 23.

p11 KHoV(>'} d>' + p11 Kvv(>'} d>' ;=: TAS ;=: p11 KHoV(>'} d>',

which is just Eq.(D-40).

(D-56)

(D-57)

The lower bound follows from Eq.(D-56}, because Kvv is non-negative. The upper bound arises from the observation that, if /(x) is a non-negative function, then

(D-58)

Note that classically AS ;=: 0, always, because KHov == O. That is, there are no correlations between kinetic and potential energy in a classical system. It should be emphasized that this result has little to do with the classical noncommutativity of Ho and V: although KAB = 0 implies that ([A, E]) = 0, the converse is not true.

D: Convex Functions and Inequalities 359

With the same notation, it is useful to note that the calculations of the last two theorems yield the following identities:

Theorem 24.

8 2 KHH(.\) = 8fJ2 InZ,

82 Kyy(.\) = fJ- 2 8.\2InZ,

8 2

KHY(.\) = 8.\8fJ[P-2InZI.

82 82 (82 )2 8fJ2 In Z 8.\2 In Z ~ fJ 8.\8fJfJ-1ln Z

The proof follows by substitution of Eqs.(D-59) into Eq.(D-47).

(D-59a)

(D-59b)

(D-59c)

(D-60)

Finally, we point out that it is also possible to obtain lower bounds on ~(x, y), and therefore on KFF (e.g., Okubo, 1971). Likewise, various bounds on (V), such as the Golden­Thompson inequality (Golden, 1965; Thompson, 1965), and the Falk inequality (Falk, 1966), have been calcula.ted. The direct usefulness of these inequa.lities has yet to be exhibited.

REFERENCES

Falk, H.: 1966, 'Upper and Lower Bounds for Canonical Ensemble Averages', J. Math. Phys. '1, 977.

Girardeau, M.D., and R.M. Mazo: 1973, 'Variational Methods in Statistical Mechanics', Adv. Chem. Phys. 24, 187.

Golden, S.: 1965, 'Lower Bounds for the Helmholtz Function', Phys. Rev. B lS'1, 1127. Hardy, G.H., J.E. Littlewood, and G. P6lya: 1952, Inequalities, Cambridge Univ. Press, Cambridge. Huber, A.: 1970, in J.E. Bowcock (ed.), Methods and Problems of Theoretical Physics, North-

Holland, Amsterdam, p.37. Jaynes, E.T.: 1965, 'Gibbs vs. Boltzmann Entropies', Am. J. Phys. SS, 391. Lanford, O.E.,III, and D.W. Robinson: 1969, 'Mean Entropy of States in Quantum-Statistical

Mechanics', J. Math. Phys. 9, 1120. Leff, H.: 1969, 'Entropy Differences between Ideal and Nonideal Systems', Am. J. Phys. S7, 548. Lieb, E.H.: 1975, 'Some Convexity and Subadditivity Properties of Entropy', Bull. Am. Math. Soc.

81, 1. Okubo, S.: 1971, 'Some General Inequalities in Quantum Statistical Mechanics', J. Math. Phys.

12,1123. Okubo, S., and A. Isihara: 1971, 'Some Considerations of Entropy Change', J. Math. Phys. 12,

2498. Peierls, R.E.: 1938, 'On a Minimum Property of the Free Energy', Phys. Rev. 54, 918. Thompson, C.J.: 1965, 'Inequality with Applications in Statistical Mechanics', J. Math. Phys. 6,

1812. Wehrl, A.: 1978, 'General Properties of Entropy', Rev. Mod. Phys. 50, 221. Wilcox, R.M.: 1967, 'Exponential Operators and Parameter Differentiation in Quantum Physics',

J. Math. Phys. 8, 962.

Appendix E

Euler Maclaurin Summation Formula

We derive the formula first for a special case. Let I(x) possess a continuous derivative in the interval a ::; x ::; n, and let 'Y be any integer from a through n - 1. Then one can surely write

1"1+1 +1 1"1+1

(x - 'Y - ~)I'(x) dx = [(x - 'Y - ~)/(x)]~ - I(x) dx. "1 "1

If [x] denotes the greatest integer not exceeding x, then the observation that the value of an integral can in no way be altered by changing the value of the integrand at a point allows us to write

1"1+1 1"1+1

H/"1 + 1"1+d = "1 I(x) dx + "1 (x - [x]- ~)I'(x) dx.

Now let 'Y run through all its possible values and add all these results together:

'" r'" ['" LIp = In I(x) dx + H/a + I",) + In (x - [x]- ~)/'(x) dx, p=a G a

(E-l)

and this is the simplest form of the sum formula (Euler, 1738). In order to generalize the formula so as to make it more useful, we follow the method

of Wirtinger (1902). Let us define

(E-2)

This function is periodic, with period unity, and for every nonintegral value of x one can make an elementary Fourier expansion to obtain

Now set

such that

etc. In general,

Pl(x) = - f sin(2mrx) . mr

",=1

p. ( ) = ~ 2cos(2n1rX) 2 x - L-, (2mr)2 '

",=1

p. () = (_ )"1-1 ~ 2cos(2mrx) 2"1 x_I L-, (2mr)2"1 '

",=1

_ "1-1 ~ 2sin(2mrx) P2"1+1(X) = (-1) ~ (2mr)2"1+1 .

360

(E-3)

(E-4)

(E-S)

E: Euler Maclaurin Summation Formula 361

Clearly, then, we can generalize Eq.(E-l) by repeated integration by parts. For example,

j n PI (x)f'(x) dx = B~ (f~ - f!) + jn P3(x)flll(X) dx, .. 2. ..

and so on, by induction. The coefficients Bn are the Bernoulli numbers, and one can even understand them to be defined in this way. The first few are

and B2n+1 == 0, for n ~ 1. The generalization of Eq.(E-l) now follows, and is

tfp= jn f(x)dx+i(fn+f .. )+~~(f~-f!)+'" p=a G

+ B2k (f(2k-l) _ f(2k-I») (2k)! n ..

+ in P2k+1 (x)J(2k+1) (x) dx. (E-6)

One can let n ---+ 00 under the proviso that the sum on the left-hand side converges, and that all derivatives of fez) are continuous in the appropriate interval.

REFERENCES

Euler, L.: 1738, Comm. Acta Petrop. 6. Wirtinger, W.: 1902, 'Einige Anwendungen der Euler-Maclaurinischen Summen Formel insbeson­

dere auf eine Aufgabe von Abel', Acta. Matb. 26, 255.

Appendix F

The First Four U rsell Functions and Their Inverses

T he first four Ursell functions are

WIG') = UIG,) , W 2 G,~,) = UI G,) UI (:,) + U2 G,:,) ,

W3 (:,:,!,) = UI (:,) UI (:,) UI G,) + UI G,) U2 (:,:,)

+ UI (:,) U2 G,:,) + UI G,) U2 (:,:,)

( 123) + U3 1'2'3' ,

W 4 C~;,:,!,) = UI (:,) UI (:,) UI G,) UI (:,) + UI G,) UI (:,) U2 (:,:,)

+ UI (:,) UI G,) U2 (:,:,) + UI (:,) UI (:,) U2 G,:,) + UI G,) UI G,) U2 (:,:,) + UI (:,) UI (:,) U2 (:,:,)

+ UI G,) UI (:,) U2 (:,:,) + UI G,) U3 (;':':')

( 2) (134) U (3) (124) + UI 2' U3 1'3'4' + I 3' U3 1'2'4'

( 4) (123) U (12)U (34) + UI 4' U3 1'2'3' + 2 1'2' 2 3'4'

( 13) (24) U (14)U (23) U (1234) + U2 1'3' U2 2'4' + 2 1'4' 2 2'3' + 4 1'2'3'4' .

362

F: The First Four Ursell Functions and Their Inverses

The first four inverted equations are

U (123) -W (123) -W (l)W (23) -W (2)W (13) 3 1'2'3' - 3 1'2'3' 1 l' 2 2'3' 1 2' 2 1'3'

U4 C~:,:,:,) = W4C~:,:,:,) - 6W1 G,)W1 G,)W1 G,)Wl (:,) -W2 G,~,) W2 (:':') -W2 G,:,) W2 G,:,) -W2 G,:,)W2 (:':') + 2W1 G,)Wl (:,)W2 (:,:,) + 2W1 G,) W1 (:,) W2 G,~,) + 2W1 G,) W1 (:,) W2 G,:,) + 2W1 G,) W1 G,) W2 G,:,) + 2W1 G,) W1 G,) W2 (:,:,) + 2W1 G,)Wl (:,)W2 G,:,) -W1 G,)W3 (;,:'!,) W (2)W (134) W (3)W (124)

- 1 2' 3 1'3'4' - 1 3' 3 1'2'4'

( 4) (1,2,3) - W1 4' Ws 1'2'4' .

363

Appendix G

Thermodynamic Form of Wick's Theorem

During the course of developing a perturbative calculational scheme for the grand potential function in Chapter 8 it was found necessary to evaluate expectation values of the form

(G-1)

where b>.. (Ti) is either a creation or annihilation operator, .Ai refers to the apprppriate single­particle state labeling the operator, and T is the ordering operator for these operators in the modified interaction picture. As usual, Po is the canonical statistical operator describing a free-particle system in thermal equilibrium:

(G-2)

where the 'grand' Hamiltonian and it eigenvalues are given by

(G-3)

The Wi are single-particle energies, and fl is the chemical potential. Note that the explicit parameter dependence of the operators is given by

(G-4)

Expectation values of the form (G-1) can always be evaluated by repeated application of the basic commutation relations, of course: [A, BJ = AB - eBA, for either bosons or fermions. But for more than a few operators this process becomes extremely tedious. In fact, a perturbation expansion requires consideration of an arbitrary number of operators in Eq.(G-1), so that a systematic procedure is a real necessity.

The original theorem developed by Wick (1950) is an identity in pure operator algebra, and was derived for the purpose of simplifying the evaluation of vacuum expectation values in quantum electrodynamics. Subsequently, Matsubara (1955) generalized the procedure in order to facilitate evaluation of expectation values in quantum statistical mechanics, and Gaudin (1960) discovered a particularly clear derivation of the prescription. We shall carry out a straighforward extension of Gaudin's procedure here and, although not at all necessary, it will be convenient to work in the momentum representation where w? = h2 kll2m.

Define numbers Z" such that Zi = +1 if h. is a creation operator, and Zi = -1 if it is an annihilation operator. Then immediately Eq.(G-4) reduces to

(G-5)

Recall that the effect of the operator T is to reorder the operators h. so that Tl > T2··· > Tr ,

and in such a way that a factor of f appears after each transposition. Thus, the essential

364

G: Thermodynamic Form of Wick's Theorem 365

effect of immediate interest here is the introduction of an overall factor eP , where P is the number of transpositions comprising the particular permutation of T-variables under scrutiny. We can now rewrite Eq.(G-l) as

(G-6)

The case of equal 'time' variables will be discussed later, when we shall also consider the explicit reordering of the variables T .. in completely arbitrary products. At the moment we are considering in Eq.(G-6) just one particular ordering of these variables.

Next, employ the commutation relations to expand the trace in Eq.(G-6) as follows:

Tr [,;ob1b2 ... br ] = Tr [';0 [b1 ,b21bsb4 ... br ]

+ e Tr [,;o[b1, bslb2b4 ... br ]

+ ... + e Tr [';ob2bs··· br b1 ]. (G-7)

From Eq.(G-5), with T. = p, it follows that

(G-8)

because Ko = E .. a! a.f? This result, coupled with invariance of the trace under cyclic permutations, leads to the identity

e Tr [,;ob2bs ... brb1 ] = ee1S1{3E~ Tr [,;ob 1b2 ••• br ] ,

so that we can rewrite Eq.(G-7) as

Tr [,;ob1b2 ... br ] = bi b2 Tr [';obsb4··· br ]

+ ebi b3 Tr [,;ob 2b4 ... br ]

+ ebi b4, Tr [,;ob2bs ... br ]

+ ....

Here we have introduced the symbol for, and defined a simple contraction:

b··b·· = [b· b·l(l-ee1S;{3e?)-1 , 3 - 1, 3 ,

(G-9)

(G-lO)

(G-l1)

which occurs within the trace as a c-number times the unit operator and can therefore be factored out. The important aspect of the lemma (G-lO) is that it reduces a trace over a product of r operators to a !Wm of traces over (r - 2) operators. Consequently, repeated application of the lemma will result in a sum of c-numbers. Note that r must be an even integer; if it is odd, one will eventually obtain factors of the form Tr[';ob .. l, which are seen to vanish when evaluated in a representation in which Ko is diagonal.

A more explicit and general form of the expression (G-lO) is

Tr [,;ob1b2bsb4 ... b4] = bi b2b3"b4" ••• b~·· + ebi b3b2 b4" ••• b~··

+ bi b4,b2 b3" ••• b~·· + ... , (G-12)

366 G: Thermodynamic Form of Wick's Theorem

where operators with the same number of dots form the pair in the simple contraction. If we adopt the rule that

... b~bkbi'" = .,. eb~bibk ... , then Eq.(G-12) can be written compactly as

Tr[pOb1 .. • br ] = L)all fully contracted products).

(G-13)

(G-14)

It follows directly from the basic commutation relations that the only nonvanishing simple contractions are

Alternatively,

(G-15a)

(G-15b)

(G-16)

which gives the same results. We note that Eq.(G-14) gives a zero result unless the number of creation operators is equal to the number of annhilation operators.

We now return to the problem of evaluating the expression in Eq.(G-6), first noting that exponentials and simple contractions always occur in the combination

But the only nonvanishing simple contractions are given by Eqs.(G-15), so it follows that zi = -Zi. Hence, we can now define a contraction as

b>..(Ti)b>.j{Tj) == Tr[poTh,(Ti)b>.j(Tj)]

In a manner similar to the above we see that the only nonzero contractions are

a>.,(Ti)at(Tj) = e-(T,-Tj)E? [O(Ti - Ti) + e(aLa1,)o]8>.,>.j'

aL(Ti)a>./Ti) = ee(T,-Tj)E? [O(Tj - Ti) + e(aLa>..)o]8>',>'j,

where we have noted that O(Ti - Ti) + O(Ti - Ti) = 1.

(G-17)

(G-18a)

(G-18b)

The last three equations demonstrate that the actual re-ordering induced by T, in­cluding factors of f, is completely accounted for in the definition of a contraction. Thus, the factor of eP in Eq.(G-6) is accounted for in Eq.(G-13) and we now have the desired generalization of Wick's theorem:

(Th (T1) ... b>. (r.))o =" [ all possible fUllY]. 1 T L...J contracted products

(G-19)

Several comments are in order to complete the understanding of this last expression. Inspection shows that the case of equal 'time' variables, is completely accounted for in the final result, because in this case the value 0(0) = 0 yields the correct contraction. For multicomponent systems Eq.(G-19) remains valid as well, for commutators and contractions between operators representing different types of particles always vanish.

Often one is interested in momentum space as the appropriate single-particle represen­tation. In that case we have

e-J3EO(k.)

(at,ak,)o = v(kd = 1 _ eeJ3EO(k.) ,

which is the free-particle momentum distribution.

(G-20)

G: Thermodynamic Form of Wick's Theorem 367

REFERENCES

Gaudin, M.: 1960, 'Une demonstration simplifiee du theoreme de Wick en mechanique statistique', Nucl. Phys 15, 89.

Matsubara, T.: 1955, 'A New Approach to Quantum-Statistical Mechanics', Prog. Theor. Phys. (Kyoto) 14, 351.

Wick, G.C.: 1950, 'The Evaluation of the Collision Matrix', Phys. Rev. 80, 268.

Index

T his is both a subject and name index. The author has attempted to make it as comprehensive as possible, but only with respect to significant items. Matter that is mentioned merely in passing, or which is so broad in meaning that indexing it is pointless, has been studiously omitted. Unless clearly called for, authors names are not indexed to specific pages in the main text, but only to those pages on which a full reference is provided. In this latter case the page number is italicized.

A

Abel, N.H., 61 Abraham, D.B., 990

Abramowitz, M., 158, 195, 249 Aczel, J., 61 Adams, A.N., 244 Adhikari, S.K., 249, 288 Ailawadi, N.K., 990

Alder, B.J., 990

Alers, G.A., 288 Amado, R.D., 249, 288 Amdur, I., 249 Amus'ya, M.Ya., 288 analytic propagator, 274 Andersen, H.C., 990

Anderson, P.W., 28 Andrews, T., 990

annihilation operator defined, 251

approach to equilibrium, 16 Arf, C., 249 Aristotle, 61 Aspnes, D., 991 average value

defined,46

B

Bagrov, V.G., 196 Baierlein, R., 29, 82, 195 Baker, G.A., Jr., 990

Balescu, R., 121 Band, W., 122 Bansal, M., 61 Bardeen, J., 195, 990

Barker, J.A., 249, 990

Baumgartl, B.J., 249 Bayes, T., 61 Bayes' theorem, 37-39, 58, 60 BBGKY hierarchy, 21, 27

Boltzmann distribution and, 22 BCS theory, 182, 328-329

energy gap in, 329 gap equation in, 329 ground state of, 329

Beckenstein, J.D., 61 Beckmann, R., 158 Bernoulli, J., 61, 999 Bernoulli's theorem, 31-32, 45 Bernstein, H.J., 121, 244 Bernstein, S., 82 Bernstein's theorem, 74 Berry, M.V., 28 Beth, E., 249, 248 Bird, R.B., 245 Bjorken, J.D., 195 blackbody radiation, 145-147

energy density of, 146 pressure of, 146 spectral distribution of, 145

Blackett, P.M.S., 28 black hole

spontaneous emission from, 82 Blatt, J.M., 990

Blech, L, 992 Bloch, F., 990

Bloch wavefunctions, 309 Bogoliubov, N.N., 288 Bogoliubov approximation, 257-258 Bogoliubov transformation, 258, 287 Bohr, N., 121, 195

369

370

Boltzmann, L., ~8, 61, 158, ~49, 999 barometric formula of, 73, 124, 165 most probable values and, 57

Boltzmann H-theorem, 16 violation of, 17-19

Boltzmann equation, 14-15 Boltzmann statistics, 130-135 Boole, G., 61 Bordovitzin, V.A., 196 Born, M., ~88 Bose, S.N., 158 Bose-Ebstein condensation, 139-142

in a gravitationa.l field, 167-168 relativistic, 154-156

Bose-Einstein transition, 155 Bose ga.s

charged, 287 free-particle, 138-144 ground-state depletion in, 261 hard-sphere, 256-261 relativistic, 154-156

Bowers, R.G., 8~ Bowers, R.L., 1~1 Boyd, M., ~49 Bricogne, G., 61 Brillouin function, 171 broken symmetry

Bose-Einstein condensation and, 142 Brooks, D.R., 1~1 Bruch, L.W., ~49 Brueckner, K.A., ~88 Brush, S.G., ~8 Bryan, R.K., 61 Brydges, D.C., ~49, ~44 Buckingham, M.J., 159 Buff, F.P., 196 Burgess, R.E., 1~1

Ca.hn, J.W., 99~ Callen, H.B., 1~1 Camky, P., ~48

C

canonical ensemble, See ensemble Canuto, V., 195 Carnot, S., 1~1, 999 Carnot's principle, 108-109 Carr, W.J., Jr., ~88 Chandler, D., 990 Chandra.sekhar, S., 158, ~88 Chandra.sekhar limit, 286

charge neutra.lity, 269 chemica.l potential

defined, 97 Chiu, H.Y., 195 Chiuderi, C., 195 Chudnovsky, E.M., 195 cla.ssical mechanics, 2-9 cla.ssica.l viria.l coefficients, 213-225

experimental survey of, 224-229 hard core plus square well, 221-222 hard spheres, 215-216, 218 Lennard-Jones potentia.l, 223-224 potential models for, 218-224 quantum corrections to, 225-235 repulsive exponential, 220-221 soft spheres, 218-220 Sutherland potential, 222 triangle well, 222 trapezoidal well, 223

Clausius, R.J.E., 1~1, ~44, 999 Clausius-Clapeyron equation, 313 Clayton, D.D., 158 cluster coefficient, 204 cluster functions, 199 cluster integra.ls, 199-201 clustering

physical, 199 Cohen, E.G.D., ~8 Coldwell-Horsfall, R.A., ~88 Collins, D.M., 61 Compaan, K., ~44 compressibility, 68

forward scattering and, 299 g(r) and, 297

compressibility factor, 316 Cooper, L.N., 195, 990 Cooper pairs, 326-328 correlation functions, 292-302

direct, 301 pair·, 295 space-time, 296 spatia.l, 296 total, 301

correlation length, 302, 323 correlations, 293-303 corresponding states

law of, 316 covariance

defined, 41

Index

Index

for general functions, 53 illustration of, 43 matrix, 53, 61, 100

covariance functions, 70, 356-359 Cover, T.M., 69 Cox, R.T., 61

probability axioms of, 35 Cozzolino, J., 61 creation operator

defined, 252 critical density

in a degenerate Bose gas, 140-141 in a relativistic Bose gas, 155

critical field, 325 critical opalescence, 299, 313 critical phenomena, See phase transitions critical point, 311

exponents for, 321-322 critical temperature

in a degenerate Bose gas, 140 in a gravitational field, 167

cross entropy, 56 minimum principle for, 56, 61

crystalline solids, 306-311 Hamiltonians for, 306-307

Curie's law, 81 for free electrons, 172 for spins on a lattice, 173

Currie, D.G., 181 Currie, R.G., 61 Curtiss, C.F., 845 cyclotron frequency

defined, 174

D

D'Arruda, J.J., 844 Darwin, C.G., 195 Darwin-Fowler method, 58 Dashen, R., 181, 844 David, F.N., 61 Davison, S.G., 845 de Boer, J., 158, 159, 844 Debye, P., 888 Debye function, 266 Debye length, 282, 283 Debye temperature, 266 decomposable Hilbert space, 76 de Finetti, B., 61 degenerate Bose gas, 138-143

as a model for He II, 142-144

in a magnetic field, 178-182 fluctuations in, 142 macroscopic occupation in, 139-142 relativistic, 154-156 thermodynamic functions for, 141

degenerate Fermi gas, 135-138 in a magnetic field, 182-185 relativistic, 151-154 thermodynamic functions for, 137-138 zero-point pressure in, 138

de Graff, W., 847 de Haas, W.J., 195 de Haas-van Alphen effect, 184-185 de Llano, M., 990 Delsante, A.E., 195 Dennison, D.M., 159 density ma.trix, See statistical operator density of states, 74

for photons, 145 for spins on a lattice, 172-173 relativistic, 147 See a.lso, structure function

De Rocco, A.G., 844,848 Deutsch, D., 68 DeWitt, H.E., 844 diamagnetism, 173-175

Landau, Boltzmann limit, 174-175 Landau, quantum, 184

diatomic molecules, 132, 157-158 Dirac, P.A.M., 159 distribution

Bernoulli, 32, 47, 59 binomial, See Bernoulli canonical, 50, 51, 54 coarse-grained, 21, 27 Gaussian, 45, 60-61 Maxwell-Boltzmann, 14, 17 over occupation numbers, 127-128 Planck,145 Poisson, 45

Dorfman, J.R., 88 Dorofeev, O.F., 196 Douslin, D.R., 144 Drell, S.D., 195 Drude, P., 990 DuBois, D.F., 888 Dunning-Davies, J., 88, 159 Dymond, J.H., 844 Dyson F.J., 888, 889

371

372

E

Eddington, A.S., 113, 1f1 Edwards, J.C., f.U effective interaction

Coulomb, 280-283 electron-phonon, 328

effective mass, 184, 310 Efimov, V.N., f88 Ehrenfest, P., £8, 999 Ehrenfest, T., f8, 999 Einstein, A., £8, 8£, 159, f88 electromagnetic interactions, 287 electron gas, 138, 267-272

in metals, 272, 307-310 electron-phonon interaction, 307, 328 Ellis, R.L., 6£ energy gap

in liquid helium, 259-260 superconducting, 328-329

ensemble canonical, 20, 64-68 density function for, 20 Gibbs and theory of, 19-24 grand canonical, 95-100 Lorentz, 119 micro canonical, 20 rotational, 93, 117 stationary, 20 translational, 118

entropy absolute maximum of, 54 anthropomorphic nature of, 112 Boltzmann's definition, 16 Clausius' definition, 110 concentration theorem, 57 continuous form of, 55 for free particles, 23, 81, 127-128 inequalities for, 355-358 maximum, See principle of maximum of mixing, 82 of a probability distribution, 49 quantum mechanical definition, 88-89 theoretical (SI), 51

equation of state ideal Bose gas, 139 ideal Fermi gas, 138 ideal gas, 23, 81, 134 relativistic Fermi gas, 153 van der Waals, 314-318

virial expansion of, 204-213 Erdelyi, A., 159, 195 ergodic theory, 24-26

Hamiltonian systems and, 25 Ernst, M.H., £8 Esterman, I., f8 Euler, L., 940, 961

Index

Euler-Maclaurin sum formula, 158, 360-361 Evans, R., 195 exchangeable sequences, 46-47 expectation value

defined, 41 deviation from, 41 quantum mechanical, 85, 88

experimentally reproducible phenomena, 75-77

Falk, H., 959 Fallieros, S., 195 Federbush, P., f44 Fein, A.E., £88 Feinberg, M.J., 1144 Feinstein, A., 8£ Federhof, B.U., 195 Feller, W., 61l Fermi, E., 159 Fermi energy, 135, 137 Fermi gas

free-particle, 135-138 interacting, 267-272 relativistic, 151-154

F

Fermi momentum, 135, 137 Fetter, A.L., 1l88, 990 field operators, 255 Fieschi, F., £47 Fine, P.C., 1188 first law of thermodynamics, 65 Fisher, M.E., 195, 1l88, 990, 991l Fisher, R.A., 61l Huctuations

correlation of, 42 for free particles, 294 in ideal quantum gases, 158 in Lagrange multipliers, 53 in particle number, 293 physical, 71 statistical, 42 thermodynamic, 68-72, 99-100

Fock, V., 1188 Fock space, 250-256

Index

Fock vector, 251 Folkhard, W., 61 Ford, D.!., £46 Fosdick, L.D., £44 fourth virial coefficient

nonadditivity correction, 215 pairwise additive approximation, 215

Fowler, R.H., 159 Frankel, N.E., 195 free energy, 65-66, 107, 121

infinite-volume limit of, 80 free-particle models, 126-130

electrons in metals, 307-310 Friar, J.L., 195 Friedman, K., 6£

Fre, P., £44 Frisch, H.L., £44 fugacity, 104 fugacity expansions

for a degenerate Bose gas, 139-141 free-particle, 133-134 inversion of, 135, 141, 150-151, 205-107 magnetic, 177-178 of equation of state, 204-205

G

Gajzag6, E., 1££ Galilean transformations, 115-116 r-space,3 Gardner, M., 6£ Gaudin, M., 967 Gell-Mann, M., £88 generalized inverse, 78 Gibbs, J.W., £8, 1£1, 940

Gibbs function, 97 Gibbs' paradox, 82, 103

Gibson, W.G., £44 Ginibre, J., £88 Ginzburg, V.L., 990

Girardeau, M.D., 959 Glaser, W., 159 Gnadig, P., 1££ Goldberger, M.L., £44 Golden, S., 959 Goldstein, L., 195 Goldstone, J., £88 Goodman, B.B., 991 Gotze, W., 991 Graben, H.W., £44 grand canonical ensemble, See ensemble

grand potential function, 96 as a Mellin transform, 131 cluster-integral expansion of, 201 for degenerate fermions, 136 perturbation expansion of, 272-278 thermodynamic definition, 97-98

373

Grandy, W.T., Jr., 69, 1£1, 159, 195, 196, £47, £88, £89

Gratias, D., 99£ gravitational field, 164-164

as a long-range interaction, 283-286 Bose-Einstein condensation in, 167-168

Gray, P., £89 Green function, thermodynamic

for a rotating bucket, 163-164 free-particle, 103-104, 133, 162 in a gravitational field, 165 partial-wave expansion of, 231 single-particle, 132

Griffiths, R.B., 991 Gropper, L., £45 Gubbins, K.E., £47 Guggenheim, E.A., £45, 991 Gunton, J.D., 159 Gutierrez, G., 990

Haber, H.E., 159 Hahn, E.L., 1£1 Haines, L.K., £8 Hamblein, D., 991 Hammel, E.F., £46 Handelsman, R.A., £45 Happel, H., £45 Hardy, G.H., 959

H

harmonic confinement, 165-166 Harrison, R.H., £44-Hartree, D.R., £89 Havas, P., 1£1 Hawking, S.W., 8£ heat bath, 67 heat capacity, 68

electronic contribution, 272 in crystals, 264-266 in metals, 309 negative, 283 See also, specific heat

heat engine, 108 efficiency of, 108-109

Hecht, R., 991

374

Heims, S.P., 1££ Heisenberg, W., 991

Helfand, E., £-4-4 helium-4

Landau theory of, 259-261 phase diagram for, 143

Hemmer, P.C., 2-45, 991 Henderson, D., 2-45, 990 Henshaw, D.G., £89 Herapath, J., 9-40

Hermann, J., 9-40 Herring, C., 195 Hertel, P., £89 high-probability manifold, 76, 111-112 Hikita, T., £-46

Hill, R.N., 2-45 Hirschfelder, J.O., £-45, 2-47 Hobson, A., 62 Hohenberg, P.C., 159

Holborn, L., 2-45 Hoover, W.G., £-47

Hopf, E., £8 Huang, K., 1££ Huber, A., 959 Hiickel, E., 288 Hwang, I.K., 159, 289 hypernetted chains equation, 304

identical particles, 101-103, 125-127 Imre, K., 2-49 Imry, Y., 159 infinite-volume limit, 5, 79-81, 107

for inhomogeneous systems, 162-166 in a degenerate Bose gas, 139-141

I-graphs definition of, 276 for ring diagrams, 280 linked, 276-278 rules for, 276

inhomogeneous systems, 161-168 number density in, 162, 194

intensive quantities, 79-80 inverse problems, 1, 77-79 irreversibility, 113-114 Isihara, A., 959

Ising model, 319 isotope effect, 326, 329 Iwata, G., 159

jacobians, 67, 344-347 Jancovici, B., 195, 2-45 Jansen, L., 2-46

J

Index

Jaynes, E.T., 28, 62, 82, 1£2, 9-40, 9-49, 959

classical mechanics notes, 9-13 principle of maximum entropy, 49

Jeans, J., 2-45 Jeffreys, H., 62 Johnson, R.W., 69 Johnston, H.L., 2-48 Johnston, J.R., 159 Jordan, H.F., 2-4-4, 2-46 Jordan, P., 289 Jordan, T.F., 121, 122 Joule, J.P., 9-40 Jiittner, F., 159

Kac, M., 159, 991 Kadanoff, L.P., 991

Kahn, B., 2-45

K

Kamerlingh Onnes, H., 2-45, 991 Kane, J., 991 Karsch, F., 158 Katz, M.J., 992 Kawasaki, K., 289

Keesom, W.H., 2-45

Keller, J.B., £-45 Keller, W.E., £-46, 289

Kelvin, Lord, See Thomson, W. Kestner, N.R., 2-46 Keynes, J.M., 62

probability and, 33 Khinchin, A.!., 28

Kihara, T., 2-45, 2-46 Kilpatrick, J.E., 2-49, 2-46 kinetic theory, 13-19 Kirkwood, J.G., 2-46 Kislinger, M.B., 122

Kittel, C., 1£2

Klein, 0., 122 Koenig, S.H., 9-45 Kofsky, I.L., 28 Kogut, J., 992 Kosevich, A.M., 195 Kozak, J.J., 991 Kroemer, H., 122 Kronig, A.K., 9-40

Index

Kubo, R., 940 Kullback, S., 6£ Kunz, H., 990

Lamb, W.E., Jr., 195 A-point in He\ 143

L

Landau, L.D., 1££, 195, £89, 990, 991 Landau, L.J., 159 Landau levels, 174

degeneracy in, 174 Landsberg, P.T., 6£, 1££, 159 Lanford, O.E., III, £9, 959 Langevin function, 172 Langevin theory, 172 Laplace, P.S., 6£, 8£, 940, 949

probability and, 32-33 Larsen S., £49, £46 latent heat, 313 law of large numbers, 44 law of succession, 40, 341-343 Lebowitz, J.L., £9, 195, £46, 991 LeChatelier's principle, 72 Lee, H.J., 195 Lee, T.D., 1£9, £46, 991, 999 Leff, H., 959 Legendre transformation, 2, 52, 66 Lehmann, M.S., 69 Leighton, R.B., £89 Lenard, A., £88, £89 Lennard-Jones, J.E., £46 Lenz's law, 169 Leonard, A., 159 Leonard, P.J, £49 Levelt, J.M., £47 Levine, D., 991 Levine, R.D., 69 Levinstein, H., £8 Levy-Leblond, J.-M., £89 Lewis, E.A.S., 991 Lewis, M.B., 1££ Lieb, E.H., 1££, £46, 991, 959 Lifshitz, I.M., 195 Liouville, J., £9 Liouville's theorem, 3, 4, 13 liquids, theory of, 303-305 Littlewood, J.E., 959 Livesey, A.K, 6£ London, F., 159, £46, 991 London, H., 991

long-range forces, 278-286 Coulomb, 279-283 gravitational, 283-286

Lorentz, H.A., 991 Lorentz transformations, 118-119 Loschmidt, J., £9 Luks, KD., 991 Lunbeck, R.J., £47 Luszczynski, K, £47 Lynden-Bell, D., £89 Lynton, E.A., 991

Ma, S.-K, £9, 1£1, £44 Macke, W., £89

M

macroscopic uniformity, 11, 75-77 magnetic-moment operator, 170

for spins on a lattice, 172 magnetic susceptibility

defined, 169, 170 gauge dependence of, 175

magnetism classical theory, 169-175 quantum theory, 176-185 for bosons, 178-182

Majumdar, R., £46 many-body quantum mechanics, 101-105 Maradudin, A.A., £88 Margenau, H., £46 Marvin, D.A., 61 Marx, G., 1££ Mascheroni, P.L., £46 Mason, E.A., £49, £46, £48 master equation, 14 Matsubara, T., 967 Maxwell, E., 991 Maxwell, J.C., £9, 1£2, 940 Maxwell relations, 66 May, R.M., 195 Mayer, J.E., 246, 991 Mayer, J.R., 940 Mayer, M.G., £46, 991 Mayr, E., £9 Mazo, R.M., 959 McCullough, J.P., £44 McGinnies, R.T., £46 McKerrell, A., 82 Mead, L.R., 62 Meissner, W., 991 Meissner-Ochsenfeld effect, 325

375

376

in a free Bose gas, 182 mean-field theory, 318-319, 324-325 Merkuriev, S.P., 1145 Mermin, N.D., 195 method of Ursell, 197-203 Metropolis, N., 1146 Michels, A., 1144, 1147 Midzuno, Y., 1146 Mie, G., 1147 Miller, D.E., 158 mixing, 6, 25 Mohling, F., 119, 1147, 1188, 1189 molecular dynamics, 305 momentum distribution, 293

for fermions, 135-136, 300 Monaghan, J.J., 1149 Monte-Carlo method, 305 Moore, R.T., 1144 Mork, K.J., 1145 Morley, P.D., 11111 Muir, T., 1147 multiplicity far.tor, 78 IL-space, 15

defined, 13

Nagamiya, T., 1147 Nashed, M.Z., 811 Nave, C., 61 Neighbours, J.R., 1188 Nelson, R., 811 Nernst, W., 811 Nernst's theorem, 74

N

in a degenerate Fermi gas, 138 Nesbitt, L.B., 9911 Newton, R.G., 195 Nieto, M.M., 159 Nijboer, B.R.A., 1147 Nilsen, T.S., 1147 nonlocal interactions, 253 Norberg, R.E., 1147 Nordsieck, A., 195 Nozieres, P., 9911 number operator, 252 number representation, 250

o occupation number 13, 102

Bose, in a gravitational field, 167 for bosons, 125-127, 138-139

for fermions, 125-127, 135 for photons, 145

Ochsenfeld, R., 991 Okubo, S., 959 Onsager, L., 195 Opat, G.I., 195 Opfer, J.E., 1147 Oppenheim, I., 1189 optical wavelength, 146 order parameter, 312, 318, 321-322 ordering operator, 273 Ore, 0.,611 organizing principles, 1, 13, 306 Ornstein, L.S., 119, 9911 Ornstein-Zernike theory, 301-302 orthohydrogen, 158 Osborn, T.A., 1147 Otto, J., 1145 Ozizmir, E., 1149

P

pair-correlation function, 295 pair-distribution function, 293 Pais, A., 159, 1147 Palciauskas, V.V., 991 Papanicolaou, N., 611 parahydrogen, 158 paramagnetism, 169 Park, J.L, 11111 partition function, 4

classical free-particle, 8 diamagnetic, 7, 174 for a spherical pendulum, 6 in probability theory, 50 magnetic, 175 paraelectric, 194 paramagnetic, 171 rotational, 163 single-particle, 132

Partovi, M.H., 611 Pauli, W., 196 Pauli paramagnetism, 184

high-field, 189-194 Pauli principle, 102, 126, 130, 279, 300 Peierls, R., 196, 959 Penrose, 0., 119, 1146, 1147, 991 Percus, J.K., 195, 9911 Percus-Yevick equation, 303

hard-sphere solutions to, 304 Perrin, J., 340

Index

Index

perturbation expansion of the grand potential, 272-278

phase space, 3 phase transitions, 311-325

cluster expansions and, 320 first order, 312 scaling laws and, 322 second order, 319 universality and, 323

phase volume defined, 3

phonons, 261-266 density of states for, 265 gas of, 263-266

photon gas, 144-147 thermodynamic functions for, 146

Pines, D., 992 Pippard, A.B., 992 Plancheral, M., 29 Planck, M., 29, 122, 159, 940 Poincare, H., 29 Poincare recurrence, 6, 16 P6lya, G., 959 Pompe, A., 249 Poston, T., 992 Prausnitz, J.M., 248 Present, R.D., 244 pressure ensemble, 105 pressure operator, 97, 99-100 principle of indifference, See principle of

insufficient reason principle of insufficient reason, 31-32, 39-40 principle of maximum entropy, 49-53

absolute maximum, 54 applications of, 59 breakdown of, 55 for microscopic systems, 93-95 Gibbs and, 49 historical features of, 128-130 quantum statistical, 88-95 objections to, ziii, 56

probable inference, 34-61 algebra of, 34-48 calculus of, 48-61

probability, 31-61 a priori, See prior axiomatic formulation, 35-37 continuous, 44-45 definition of, 35

377

density, 44-45 distribution, See distribution frequencies and, 45-48, 56-58 inverse, 32, 39 notation for, 35 prior, 37-39 uncertainty in, 48-49

propagator, See analytic propagator propositions

algebra of, 34-35 exhaustive set of, 38 mutually exclusive, 38 mutually irrelevant, 38

PVT-surface, 290, 291 PT-projection, 311 PV-projection,312

quasicrystals, 306

Q

R

radial distribution function, 297-304 experimental behavior of, 298, 303 for ideal quantum fluids, 297, 300-301 thermodynamic functions and, 297, 330 virial expansion of, 297-298

radiation pressure, 146 Rainville, E.D., 196 Ramsey, N.F., 122 Raval, S.P., 195 Rayl, M., 991 reciprocity laws, 52, 53 rectilinear diameter

law of, 312 recurrence paradox, See Wiederkehreinwand reduced statistical operators, 294 reductionism, 1 Ree, F.H., 247 Reed, T.M., 247 Rehr, J.J., 196 Reif, F., 992 Reiner, A.S., P,47 Reynolds, C.A., 992 relativistic statistics, 119-120, 147-157

thermodynamic functions for, 150, 153 paramagnetism and, 189-194

removing coherences, 91 removing correlations, 91 renormalization group, 324-325 reversibility paradox, See Umkehreinwand

378

Rietsch, E., 6£ Rice, S.A., f89 Rice, W.E., f47 Ricka.yzen, G., 99f ring dia.gra.ms

defined, 279 sum over, 271, 279-283

Robinson, D.W., 959 Robinson, J.E., 159, 196 Rosa., S.G., 159,196 Rosen, J.M., 6£ Rosen, P., f47 Rosenbluth, A.W., 99f Rosenbluth, M.N., 99f Rosentha.l, A., f9 rota.ting bucket, 163-164 rota.tions, 115, 117 Rothstein, J., 69 Rowlinson J.S., f47, f48 Rubin, T., f48 Ruelle, D., 8f, 1ff, f48, f88 Runnels, L.K., 99f Rushbrooke, G.S., 99f

S

Sa.ckur-Tetrode equa.tion, 81 Sa.lzberg, A.M., f89 Sca.la.pino, D.J., 8f, 1ff, 196 sca.ling la.ws, 322, 323 Schick, M., f89 Schiff, L.l., 159 Schneider, W.G., f48 Schrieffer, J.R., 195, 990 Schrodinger, E., f9, 69 8f screening, 279-283 second la.w of thermodyna.mics, 12, 17, 108-114

Cla.usius' sta.tement of, 110 wea.k a.nd strong forms, 110

second qua.ntiza.tion, 255 second viria.l coefficient

cla.ssica.l, 213-214 classica.l ha.rd-sphere, 216, 218 experimenta.l results for, 226-228 for idea.l qua.ntum ga.ses, 207-208 forma.l expression for, 211-213 He4 da.ta. for, 230 in terms of Green functions, 213 qua.lita.tive beha.vior of, 216 rela.ted to ra.dia.l distribution function, 298 tra.ce structure of, 211, 243

Seguin, M., 940 Serin, B., 99£ Sha.nnon, C.E., 69, 8f, 940

a.nd informa.tion measure, 49 Shechtma.n, D.S., 99f Sherwood, A.E., £48 Sheynin, O.B., 69

Shimony, A., 6f Shizume, T., £46 Shore, J.E., 69 Siegert, A.J.F., 1ff, £48

Simpson, O.C., f8 Skilling, J., 6f Sina.i, Ya.. G., f9 Sma.le, S., f9 Smith, C.R., 69 Smith, C.W., 1f£

Smith, E.B., f44 Smith, R.A., £48 Sommerfeld, A., 99f Sondheimer, E.H., f9, 159, 196 spa.ce-time correla.tion function, 296 spa.ce-time tra.nsforma.tions, 114-119 spa.tia.l correla.tion function, 296

in idea.l qua.ntum fluids, 300-301 specific hea.t, 68

in the Debye model, 266 in the Einstein model, 264 See also, hea.t ca.pa.city

Spitzer, J.J., 195 Spruch, L., f89 Spurling, T.H., f46 sta.bility

of equilibrium sta.tes, 72-73 thermodyna.mic, 69, 80, 100, 107

sta.nda.rd devia.tion, See va.ria.nce Sta.nley, H.E., 99f sta.tistica.l opera.tor

defined, 85, 90 equa.tion of motion for, 87 for the gra.nd ca.nonica.l ensemble, 99 for inhomogeneous systems, 161 inequa.lities for, 354-355 reduced, 294 single-pa.rticle, 132

Stea.rns, M.B., 99f Stefa.n-Boltzma.nn la.w, 146 Stegun, l.A., 158, 195, f49 Steinha.rdt, P.J., 991

Index

Index

Stern, 0., B8 Stewart, I., 99B

Stillinger, F.H. Jr., 196 Stone's theorem, 116 Stoner, E.C., 196 Sto/3zahlansatz, 14, 17 structure factor, 299 structure function, 4-12

classical fre~particle, 8 defined,4 diamagnetic, 7 for an harmonic oscillator, 7 for a spherical pendulum, 6 See also, density of states

Sudarshan, E.C.G., lBl

superconductivity, 325-330 isotope effect and, 326, 329 Meissner-Ochsenfeld effect and, 325

Sutherland, W., B48

Swenson, R.J., lBl Swift, J., 991 symmetrized state vectors, 101

Taylor, J.R., 196, B48 tempered potentials, 80 ter Haar, D., lBB Ternov, I.M., 196

T

thermal coefficient of expansion, 69 thermal equilibrium, 11, 64-65, 67

complete, 110 thermal radiation, See blackbody radiation thermal wavelength, 104 thermodynamic reversibility, 113 thermometer, 67-68 Theumann, A., B48 Thiele, E., 99B third virial coefficient

experimental results for, 227 nonadditivity correction, 214 pairwise-additive approximation, 214 qualitative behavior of, 216 trace structure of, 211

Thirring, W., B89 Thompson, B., 940

Thompson, C.J., 959 Thomson, W., lBB, 940 't Hooft, A.H., 159 Thouless, D.J., B89 Tikoshinsky, Y., 69

Tinkham, M., 99B Tishby, N.Z., 69 Titchmarsh, E.C., 196 Titus, W.J., lBB

Todhunter, I., 69 Tranah, D., 6B transformations, See specific type Tribus, M., 947 Trugman, S.A., 196 Tsang, T.Y., B47

U

Uhlenbeck, G.E., B9, 159, B49, B45, B47, B48 Umkehreinwand, 15 universality hypothesis, 323, 324 Ursen, H.D., B48

Ursen functions, 199, 362-363 Ursen method, See method of Ursen

V

van Alphen, P.M., 195 Van Campenhout, J.M., 69 van der Waals, J.D., B48, 99B

van der Waals equation, 314-317 'derivation' of, 317

van der Waals limit, 317 Van Hove, L., 8B, lBB, B47, 99B Van Hove's theorem, 108 van Kampen, N.G., 99B

van Kranendonk, J., B44

van Laar, J.J., B48

van Leeuwen, H.-J., 196 van Leeuwen's theorem, 176 Varghese, J.N., 69 variance

defined,41 illustration of, 42

Velo, G., B88

Venn, J., 69 virial coefficients

and pairwise additivity, 209-210 classical, 213-225 for ideal quantum gases, 208-209 general analysis of, 209-213 in terms of cluster coefficients, 207 potential models for, 218-224 quantum, 235-239

virial expansion convergence of, 207 of the equation of state, 204-213

379

380

of paramagnetic susceptibility, 239-242 of the radial distribution function, 297-298,

304 Visser, A., !41 von Hehnholtz, H., 940 von Karman, Th., f88 von Mises, R., 69 von Neumann, J., 1££ von Wijk, W.R., £9

W

Wainwright, T.E., 990 Walecka, J.D., £88, 990 Waterston, J.J., 940 Watson, G.N., 196 Watson transformation, 178 weak degeneracy, 133-135

relativistic, 150-151 weakly-interacting systems, 10-11

composition law for, 10 Weast, R.C., 99£ Weaver, W., 89 Weeks, J.D., 990 Wehrl, A., 1£9, 959 Weiss, P., 99£ Weldon, H.A., 159 Wenzel, R.G., 99£ Wergeland, H., 1££ Wertheim, M.S., 99£ White, D., £48 Wick, G.C., £89, 961 Wick's theorem, 274, 364-367 Widom, A., 196 Widom, B., £48, 99£

Wiederkehreinwand, 16 Wightman, A.S., £9

on ergodic theory, 25-26 Wigner, E.P., 8£, £48, £89, 99£ Wijker, H.K., £41 Wijker, Hub., £41 Wilcox, R.M., 1£9, 959 Wilde, I.F., 159 Wiley, E.O., 1£1 Wilkins, S.W., 69 Wilks, J., £89 Wilks, S.S., 69 Wilson, A.H., £9, 159, 196 Wilson, K.G., 99£, 999 Wirtinger, W., 961 Wolkers, G.J., £41 Wood, R., £89 Woods, A.D.B., £89 Wu, T.M., £89 Wu, T.T., £89

Y

Yang, C.N., 1£9, £46, 991, 999 Yarnell, J .L., 99£ Yeh, H.-C., 1£9 Yevick, G.J., 99£ Yntema, J.L., £48

Zahner, M., 61 Zellner, A., 69 Zermelo, E., 90 Zernike, F., 99£ Ziff, R.M., 159 Zimmerman, R.L., 1£1

Z

Index