Phase transitions and zeros in several physical variables

PHYSICS

Volume 28A. number 2

PHASE

TRANSITIONS

AND

ZEROS

4 November

LETTERS

IN SEVERAL

PHYSICAL

1968

VARIABLES

W. BESTGEN, S. GROSSMANNand W. ROSENHAUER Institut fzir Theoretische

Physik:

Philipps-Universitiit.

Marburg,

Germany

Received 6 September1968 free enthdpy is expressed by entire functions of complex temperature T and pressure p. Using a generalized Weierstrass theorem an analogonof a Lehmannrepresentationin p and T is given appropriate

The

for the description

of phase

transitions.

The theory of phase transitions using distributions of complex zeros in one variable [1,2] can be generalized to several physical variables which have to be continued to complex values. We considered phase transitions with respect to pressure p and temperature /3 = ~/KT studying the pressure ensemble, - exp - (H+pV), and got the following results. 1) The thermodynamic limit N - 00 for the free enthalpy per particle,

g(N, P, P)

(1)

exists under the very general condition of stable and tempered interaction W&-l,. . . ,rN). 2) Conjointly, the proof yields the equivalence of the pressure ensemble with canonical statistics. 3) As the main result, however, it is proved by generalizing a method used in ref. 2, that the partition function can be replaced by a double finite Laplace integral which gives the same free enthalply in the thermodynamic limit: z(N,P,B)

= Jwo

d5 7 du exp{-N(Mu+

-w

PS)]piv(s,u)

.

(2)

0

5, v) is the interaction phase volume; P and @ have to be restricted to finite (but arbitrary) intervals; w, ~0, vo are independent of p ,/3, N. As a finite Laplace integral Z( N,fi f fl) is an entire holomorphic function in zl =pfl and 22 = /3 or in fl and p respectively. In two complex variables, 2 has no point zeros; 2 = 0 defines a zero hyperplane % instaed. Therefore there is no Weierstrass product representation of 2. But physically only ln Zis needed. The sum over zeros could appear as an integral in this case. A Weierstrass representation generalized in this sense has been given by Eneser [3] and Lelong [4]. Let F(z1,. . .zn) be entire in cl,. . .zn ,n 2 2: +N(

lnlF(zI ,...,

+)I

=sJdu(cl,.-. %

&+Z, . . . ,an) = -

,Un)I(UI,. * - ,an*z1,-

(g (up~)(upz~))’ -n+il P&q,.-.

* - J,)

f

,a,,21

,...

.Zn) .

do is the element of area on the zero hyperplane 91; Pv are known polynomials of degree u in Zi,Zi; q characterizes the order of growth of F. With respect to phase transitions the analytic contributions arising from the polynomials P may be neglected. From eq. (3) we obtain (n = 2):

AN, P,P)

1 2+;, dO(p”p’) (Rep’-p)2+(Imp)2+(ReP’-/3)2+(Imp’)2 N

’

N

117

Volume 28A, number

2

PHYSICS

LETTERS

Let p@‘, 0’) be the limiting density of zeros per particle; &‘,P)

= &I

dF(P’,P’)

4 November

1968

then the limit Nd 03ofg(N,P, 8) is

@L(p’,P’) (Rep’-fi)2+(Imp’)2+(Rept-P)2+(Imp7)2

(5)

’

This is an analogon of a Lehmann representation in temperature and pressure instead of energy. Thermodynamics are fixed by a single distribution function 1-1= ~(a). As a physical application one can study the T-dependence for fixed P (cp (T)) at phase transitions [2,5]. For p = const. analogous results can be derived for v(p) or the compressibility KT(fi). In particular, (5) allows to study connections between singularities with respect to /3 at fixed p and p at fixed p, e.g. scaling laws.

References 1. T.D. Lee and C. N.Yang, Phys. Rev. 87 (1952) 410. 2. S.Grossmann and W.Rosenhauer, Z. Physik 207 (1967) 138. 3. H.Kneser. Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl. 1936, 446; H.Kneser, Jahresber.d.Deutsch.Math.VereinigungXLVIII, l.Abt.Heft l/4 (1938) 1. 4. P. Lelong, J.Anal.Math. 12 (1965) 365. 5. W. Bestgen, S.Grossmann and W. Rosenhauer, Proc. Int. Conf. Stat. Phys., Kyoto, 1968, Suppl. Phys. Sot. Japan. *t***

CHANGES

IN LOW-TEMPERATURE SPECIFIC HEATS OF Cu-Pd RESULTING FROM CHANGES IN SHORT-RANGE ORDER

ALLOYS

Y. SATO, J. M. SIVERTSEN and L. E. TOTH School of Mineral and Metallurgical

Engineering,

University

Received

Studies of the effects of short-range Cu-Pd alloys decrease on aging.

order

of Minnesota,

3 September

on low temperature

X-ray diffuse scattering studies have shown that the local atomic order is not random in Pdrich Cu-Pd solid solutions [l]. The degree of order is significantly changed between the quenched and aged states of the alloy. These differences in the short-range order of quenched

Minneapolis,

Minnesota 55455, USA

1968

specific

heats show that y values of quenched

and aged alloys result in small changes in electrical resistivity and magnetic susceptibility [l]. In this note we report on changes in the low temperature specific heats induced by different heat treatments, which according to the previous study [l] should alter the short-range order. Table 1

118

OK2

eD:‘K

Compositions

Heat treatment

y;mJ/mole

58 at.% Pd

Quenched Aged

2.31 * 0.006 2.31 zt 0.006

307.5 * 1.2 308.4 l 1.2

67.5 at.% Pd

Quenched Aged

3.45 f 0.008 3.40 * 0.008

305.0 * 1.2 305.0 l 1.2

75 at.% Pd

Quenched Aged

4.32 f 0.009 4.24 zt 0.009

302.4 l 1.2 304.1 * 1.2