Error Bounds in Equilibrium Statistical Mechanics

VOLUME 9, NUMBER 5

JOURNAL OF MATHEMATICAL PHYSICS

MAY 1968

Error Bounds in Equilibrium Statistical Mechanics Roy G. GORDON· Department of Chemistry, Harvard University, Cambridge, Massachusetts (Received 20 October 1967) A new method is presented for the calculation of thermodynamic properties from equilibrium statistical mechanics. Starting from the high-temperature expansion coefficients for the canonical partition function, error bounds are obtained, which are both rigorous and optimal.

I. INTRODUCTION

The high-temperature expansion method is one of the most widely used techniques in statistical mechanics. It has been used to study the thermodynamic properties of crystalline solids, l binary alloys, 2 magnetic properties,3 pure fluids,4 fluid mixtures, 5 and condensation from gases. 6 The chief advantage of the method is its wide applicability. The main difficulties of the method are twofold: (1) The series of approximations converges rather slowly, and in • Sloan Foundation FeUow. H. Thirring, Physik Z. 14,867 (1913); 15, 127, 180 (1914); E. W. MontroU, J. Chern. Phys. 10, 218 (1942); 11,481 (1943); E. W. MonteoU and D. C. Peaslee, J. Chern. Phys. 12,98 (1944); A. A. Maradudin, P. Mazur, E. E. MontroU, and G. H. Weiss, Rev. Mod. Phys. 30, 175 (1958); C. Dornb, A. A. Maradudin, E. W. MontroU, and G. H. Weiss, Phys. Rev. 115, 18,24 (1959); J. C. Bradley, Ann. Phys. (New York) IS, 411 (1961). • J. G. Kirkwood, J. Chern. Phys. 6, 70 (1938); T. S. Chang, J. Chern. Phys. 9, 169 (1941); G. S. Rushbrooke, Nuovo Cirnento Supp!. 6,251 (1949); T. Oguchi, J. Phys. Soc. (Japan) 6, 499 (1951). ·W. Opechowski, Physica 4,181 (1937); 6,1112 (1938); H. A. Krarners and G. H. Wannier, Phys. Rev. 60, 252,263 (1941); E. W. MontroU, J. Chern. Phys. 10, 61 (1942); L. On sager, Phys. Rev. 65, 117 (1944); D. ter Haar, Phys. Rev. 76,176 (1949); c. Dornb, Proc. Roy. Soc. (London) A199, 199 (1949); G. S. Rushbrooke, Nuovo Cirnento 6, Supp!. 2, 251 (1949); V. Zehler, Z. Naturforsch.A5, 344 (1950); E. Trefftz, Z. Physik 1Z7, 371 (1950); J. E. Brooks and C. Dornb, Proc. Roy. Soc. (London) Al07, 343 (1951); A. J. Wakefield, Proc. Cambridge Phil. Soc. 47, 419 (1951); T. Oguchi, J. Phys. Soc. (Japan) 5, 75 (1950); 6, 31 (1951); M. Kurata, R. Kikuchi, and T. Waturi, J. Chern. Phys. 21, 434 (1953); G. S. Rushbrooke and P. J. Wood, Proc. Phys. Soc. (London) A68, 1161 (1955); A70, 765 (1956); C. Dornb and M. F. Sykes, Proc. Phys. 'Soc. (London) B69, 486 (1956); M. F. Sykes, Phil. Mag. 2, 733 (1957); C. Dornb and M. F. Sykes, Phys. Rev. 108, 1415 (1957); S. Katsura, Progr. Theoret. Phys. 20,192 (1958); G. S. Rushbrooke and P. J. Wood, Mol. Phys. 1,257 (1958); C. Dornb, Advan. Phys. 9, 149 (1960); M. F. Sykes, J. Math. Phys. 2,52 (1961); C. Dornb and M. F. Sykes, J. Math. Phys. 2, 63 (1961); G. S. Rushbrooke and J. Eve, ibid. 3, 185 (1962); C. Dornb and M. F. Sykes, Phys. Rev. 128, 168 (1962); G. S. Rushbrooke and P. J. Wood, Mol. Phys. 6, 409 (1963); C. Dornb and D. W. Wood, Proc. Phys. Soc. 86, 1 (1965); G. A. Baker, H. E. Gilbert, J. Eve, and G. S. Rushbrooke, Phys. Letters 20, 146 (1966); G. A. T. AUan and D. D. Betts, Proc. Phys. Soc. (London) 91,341 (1967). 4 R. W. Zwanzig, J. Chern. Phys. 22,1429 (1954); 23,1915 (1955); J. S. Rowlinson, Mol. Phys. 7, 349 (1964); 8, 107 (1964); D. A. McQuarrie and J. L. Katz, J. Chern. Phys. 44, 2393 (1966). 6 J. C. Wheeler, thesis,CorneU University, 1967. 6 C. N. Yang and T. D. Lee, Phys. Rev. 87, 404, 410 (1952); J. Wang, Proc. Roy. Soc. (London) A161, 127 (1937); J. E. Brooks and C. Dornb, Proc. Phys. Soc. (London) A207, 343 (1951). 1

some cases the series does not converge at all at low temperatures. (2) The results are of unknown accuracy, when the series are extrapolated. 7 The purpose of this paper is to develop a new method of high-temperature expansion which helps both of these difficulties. Starting from the coefficients in the usual (truncated) high-temperature series, we derive a new sequence of approximations, which (1) converges much more rapidly than the usual high-temperature expansion, and (2) gives precise upper and lower bounds for the partition function (and some other thermodynamic properties), at each order of approximation. These bounds are optimal in the sense that they are the most precise bounds possible, given the coefficients in the usual high-temperature series. II. STATEMENT OF PROBLEM: ASSUMPTIONS

We consider closed equilibrium classical or quantumstatistical systems which are described by canonical distribution law over energy E which we write as

e-PE dtp(E),

(1)

where fJ is the reciprocal temperature, and dtp(E) is a density of states. We always assume that tp(E) is a nondecreasing function of E. We choose the (arbitrary) zero of energy to be the energy of the ground state of the system. Thus dtp(E) vanishes for E < O. The problem we pose is to find upper and lower bounds for the canonical partition function defined by the Stieltjes integral

1 00

Q(fJ) ==

e-PE dtp(E),

(2)

when we are given values for the first 2M moments 7 D. Park, Physica 12, 932 (1956); G. A. Baker, Phys. Rev. 124, 768 (\961); G. A. Baker and J. L. Gammel, J. Math. Anal. Appl. 2, 21 (1961); G. A. Baker, J. L. Gammel, and J. G. Wills, ibid. 2, 405 (1961); M. E. Fisher and J. W. Essarn, J. Chern. Phys. 38, 802 (1963); G. A. Baker, Phys. Rev. 129, 99 (1963); D. S. Gaunt and M. E. Fisher, J. Chern. Phys. 43, 2840 (1965); G. A. Baker and D. S. Gaunt, Phys. Rev. 155, 545 (1967).

655 Copyright

©

1968 by the American Institute of Physics

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656

ROY G. GORDON

1-' .. of the distribution function, defined by I-'n

==

f'"

En dtp(E)

(3)

for n = 0, 1, 2, ... ,2M. In order for these moments to e.xist and be finite in a classical mechanical system, it is important to separate out the kinetic energy, and treat (2) as the configurational integral, and the moments I-'n as averages of powers of the potential energy only: I-'n

=

1"

vn dtp(V).

(4)

m.

The integrals then are just configuration-space averages of powers of the potential energy. Having a potential-energy function of bounded variation is then sufficient to guarantee their existence. In quantum-mechanical spin systems (such as the Heisenberg model of ferromagnetism) the energy E in (3) may be taken to be the full spin Hamiltonian H. The moments may then be evaluated from the quantum-mechanical trace formula I-'n = Tr [Hn]. (5) The power of this method is that one may calculate the trace in any convenient basis. The usual form of the high-temperature expansion is now readily obtained by expanding the exponential in (2) in its power series and assuming that one can interchange the order of summation and integration, 00

Q(f3)

= 2, (-

n ,....., 10), that one might be lead to the hasty conclusion that a small finite number of terms in the high-temperature expansion tells one little about the partition function. We hope to demonstrate in the following sections of this paper that such a conclusion is unduly pessimistic. We will construct a new set of error bounds, based on precisely that information contained in the coefficients of the usual high-temperature series. Our new error bounds are far more precise, by more than a factor of a million in an example given in Sec. V.

f3t I-'nl n !.

(6)

n=O

By truncating this sum at successively higher (finite) numbers of terms, one obtains a sequence of approximations to the partition function. The higher the temperature, the smaller f3, and hence the series converges most rapidly at high temperatures. Unfortunately, the convergence is often slow at temperatures of interest. Since the terms in the series (6) alternate in sign, successive partial sums give crude upper and lower bounds to the infinite sum. However, these bounds are not too useful in practice, because for sufficiently low temperatures the lower bounds become negative; hence they furnish no new information, since we knew anyway that the partition function cannot be negative. Similarly, these upper bounds become larger than Q(O), for low temperatures. But we knew already that Q(f3) < Q(O), so these upper bounds also fail to yield any information at low temperatures. Even when the temperature is high enough so that these bounds furnish some information, the magnitudes of the error limits are so large (for reasonable

GENERAL THEORY

We make use of mathematical results from the theories of continued fractions,S quasi-orthogonal polynomials and the moment problem,9 matrix algebra,IO and Gaussian-type integration9 ; reference should be made to these works tor further mathematical background. Where possible, we follow the definitions, terminology and notation of Shohat and Tamarkin9 (to be referred to as ST in the following). Considerlla the function fez) defined by the Stieltjes integral l(z)

== roo d tp(E)

(7)

Jo z + E

over the nondecreasing distribution dtp(E). The integrand may be expanded according to a finite geometric series, with remainder term

lIE

E2

--=---+z + E Z Z2 Z3

-'" +

(-Et- 1 zn

(_E)n

+ zn+1(z + E)

(8)

Inserting this series into the integrand in (7) gives 00

l(z) = -1 Z

50 dtp(E) - "2 1 5orl) E dtp(E) 0

z

0

The coefficients of the inverse powers of z, are recognized to be just the moments fln defined by (3). • H. S. Wall, Analytic Theory of Continued Fractions (D. Van Nostrand, Inc., New York, 1948). • J. A. Shohat and J. D. Tamarkin, "The Problem of Moments," Mathematical Surveys 1 (American Mathematical Society, Providence, R.I., 1950) 2nd ed. 10 J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford University Press, London, 1965). l1a T. J. Stieitjes, Recherches sur les fractions continues, Annales de la Faculte des Sciences de Toulouse 8, 1 (1894); 9, 5 (1895).

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ERROR BOUNDS IN EQUILIBRIUM STATISTICAL MECHANICS

Thus the formal expansion of the integral (7) has the form fez) =

I!:!!. _I!:l z

+ fi2

_ .•.

Z3

Z2

+ (-lr-~ fin-l Zn

+ ... + _1_

roo

Jo

Zn+l

(_E)n dW(E). (to) z +E

As long as the remainder term is kept, this is just an identity. However, if we let n -+ 00, (to) becomes a formal expansion of the integral in inverse powers of z:

_I!:l +

fez) '" fio Z

Z2

fi2 -

Z3

~ + •. . . Z4

(11)

C(z)=~ Z

+OC_2_ _ 1+~

z+

OC~4,--_

1

+ ... .

(12)

The coefficients OC n appearing in this expansion are determined from the moments fin' by the requirement that a formal expansion of C(z) in powers of (lIz) have just the same coefficients as those appearing in (11). Values of the first few OC n may be derived directly from this definition by equating coefficients of (l/z)n in these expansions: (l3a) OC1 = fio, (l3b) OC2 = fil/fio, ~

_

fiD

(fi2fio -

""3 -

(13c)

,

fiofil OC4

= (filfia - fi~)fio , fil(fi2fio -

(l3d)

fii) .a

a 2222 ) 2 OC - ( fiofilfi2fi4 - fiofilfi4 - fiofilfi2 - fiofllfia + fiofilfi2fi3 5 -

fioC filfia -

2

fi'0(fiofi2 -

2

•

fil)

(l3e)

Obviously this direct matching method cannot be practically applied to higher orders. Explicit general expressions for the oc n in terms of the fin can be written down,9 but they involve determinants of high

+