Contains a report on a research project.United States Air Force, Office of Scientific Research, Air Research and Development Command (Contract AF49(638)-95

Quay, P. M.

Tisza, Laszlo

Contains reports on two research projects.United States Air Force, Office of Scientific Research, Air Research and Development Command (Contract AF49(638)-95

DSpace@MIT

XIII. STATISTICAL THERMODYNAMICSProf. L. Tisza P. M. Quayt D. SwansontG. SchayA. ON THE NATURE OF THE X-LINE OF LIQUID HELIUMWe have shown in a recent note (1) that the observed singularity of the specific heatC of liquid helium allows one to draw a significant inference on the structure of thepsuperfluid state. However, as we pointed out in footnote 9, the argument still containsa gap. The singularity of Cp was observed at the saturation vapor pressure po, andthese measurements are consistent with two significantly different extrapolations con-cerning the nature of C along the X-line in the p, T plane.(i) C is singular along the X-line.(ii) C has a finite maximum along the X-line, and this maximum tends to infinityas p - P .The results of the previous note rest, in an essential fashion, on assumption (i).Although plausibility arguments support this choice, the definitive exclusion of possi-bility (ii) was left to further experiments.Since such an experimental decision is hampered by technical difficulties, it is inter-esting that experiments performed on the saturated liquid along with the known value ofthe slope E = dTX/dp of the X-line can be analyzed in a fashion that provides strongarguments in favor of assumption (i).The discussion centers around relations recently advanced by Pippard (2). Theserelations are analogous to the well-known relations of Ehrenfest, but they assume sin-gularities of Cp, a (expansion coefficient), and K (isothermal compressibility), ratherthan the discontinuities characteristic of second-order transitions.Apart from minor modifications, we follow Pippard and propose to represent theentropy function of the system in the vicinity of the X-line in terms of the followingexpression:S(p, T) = S (p) + f[T - Tx(p)] (1)Here S (p) is the entropy along the X-line; it is assumed to be a slowly varyingfunction of p, while f(t) is a singular function with f(0) = 0, f'(0) = oo. Equation 1 canbe considered as the analytic expression of assumption (i): We deal all along with thesame singularity contained in the single-variable function f, modified by pressure onlyThis research was supported in part by the U. S. Air Force, Office of ScientificResearch, Air Research and Development Command, under Contract AF49(638)-95.Reproduction in whole or in part is permitted for any purpose of the United StatesGovernment.tNational Science Foundation Predoctoral Fellow.(XIII. STATISTICAL THERMODYNAMICS)through the pressure dependence of the X-temperature.From Eq. 1, we obtain(8 Cp f (2)T- TS) - VaS - f'E (3)p Twhere V is the volume. From Eqs. 2 and 3, we haveCpVa E= - S' (4)T XSimilarly, we obtainK = aE + K (5)where K is the compressibility at a point with a = 0. Equations 4 and 5 are Pippardrelations. As derived by Pippard (2), the relations contain arbitrary integration con-stants. The physical meaning of the latter is made evident in the present derivation.E. Maxwell and C. E. Chase, of Lincoln Laboratory, verified the validity of Eq. 4by using the results (3) of their own measurements of a. The essential feature of thesemeasurements is that a has a singularity similar to that of C .pIt can be shown by a simple geometrical argument concerning the surface S = S(p, T)that such a singularity of a is inconsistent with assumption (ii). The reason that meas-urements at the saturation vapor pressure (in a narrow interval this means a very nearlyconstant pressure po) yield results concerning the behavior of the entropy at higherpressures is not surprising if we keep in mind the well-known thermodynamic relation(8 ~ (8 SOur conclusion is that on the basis of the available facts, assumption (i) is over-whelmingly favored over assumption (ii). Also, Pippard's equation (Eq. 1) seems tobe an adequate analytic expression of this assumption. At the same time, we ought tokeep in mind that this relation is not a necessary consequence of thermodynamics. Itwill have to be refined to account for precise measurements over a wider range of thevariables.L. TiszaReferences1. L. Tisza, Phys. Rev. 110, 587 (1958).2. A. B. Pippard, Phil. Mag. (8) 1, 473 (1956).3. Private communication, June 1958.(XIII. STATISTICAL THERMODYNAMICS)B. DISTRIBUTION FUNCTIONS IN STATISTICAL THERMODYNAMICSThis study was completed and presented by the author as a thesis in partial fulfill-ment of the requirements for the degree of Doctor of Philosophy, Department of Physics,M.I.T., September 1958.P. M. Quay

Statistical Thermodynamics

XI. STATISTICAL THERMODYNAMICSProf. L. TiszaDr. P. M. QuayA. THE THERMODYNAMIC THEORY OF LINE SINGULARITIES IN ONE-COMPONENTSYSTEMSAccording to the theory of thermodynamic stability, the experimentally observedX-anomalies are manifestations of singularities in the thermodynamic formalism (1).Thus for a one-component system the second derivatives of the Gibbs functionG = G(p, T) tend to infinity, as the X-line is approached in the pressure-temperatureplane.The phenomenological theory does not allow us to calculate the nature of the above-mentioned singularity. However, it is possible to develop a scheme for the analysis andcorrelation of the experimental data.We shall tentatively assume that the Gibbs function can be represented asG(p, T) = f(p) + h(T-T ) (1)where f(p) is a slowly varying function representing the Gibbs function along the X-line,and h(t) is a singular function with the properties thath(O) = 0, h'(0) = finite, h"(0) = -co (2)We confine ourselves to a region in the p,T-plane that is small enough to ensurethat the slope of the X-line is constant. We denote this slope bydp dT (3)dTThe idea of separating the regular and singular parts of the thermodynamic functions,which was first advanced by Pippard (2), has been discussed in an earlier progressreport (3). Pippard's point of departure was the entropy S = S(p,T). However, it isadvantageous to start instead from the Gibbs function, since this constitutes a funda-mental equation and contains implicitly the answer to every thermostatic question.The special form of Eq. 1 is the simplest that is consistent with the broad featuresof the experimental facts. An increased precision and an extended range of the experi-ments will presumably call for a more elaborate functional form.This research was supported in part by the U. S. Air Force (Office of ScientificResearch, Air Research and Development Command) under Contract AF49(638)-95.Reproduction in whole or in part is permitted for any purpose of the United StatesGovernment.135(XI. STATISTICAL THERMODYNAMICS)We note that general thermodynamic principles bring about further restrictions forthe functions f(p) and h(t):f' > 0, h' < 0, f" < 0, h" < 0 (4)where the primes denote differentiation.We shall show that the significant thermostatic functions can be obtained from Eq. 1through direct computation.First we obtain the entropy and the volume:SaT f' h'S - - - - ' - (5)V 8 )T f -h' (6)The specific heat Cp, expansion coefficient a, and the isothermal compressibilityKT are given byCp 2TSV K _ f" -h" (9)8V (9)After eliminating the singular function h" from Eqs. 7, 8, and 9 we obtain the ana-logs of Ehrenfest's relations:CP 2Va t - + 2f" 2 (10)TVK ~ = Va - 2f" (11)Eliminating f", we haveC2 P2 P(12)VTKTFrom Eqs. 7, 8, and 9 we obtain by standard thermodynamic methods the specificheat at constant volume, the ratio of the specific heats, and the adiabatic compressibility:CV 4f" h"1 (13)T 1f" + h" 02136(XI. STATISTICAL THERMODYNAMICS)C p (f" + h" 1/) 2= - - (14)Cv 4f" h"1 4f" h"VK = VK - h" (15)S y T f" + h"It is apparent that y - oo, and CV and KS remain finite as h" -* -oo. This is in accord-ance with the general theory of stability.The foregoing relations can be used to express the derivatives of the functions f andh in terms of measurable quantities. This can be achieved in a number of ways, and theresulting identities serve as checks for the consistency of the theory.We shall now enumerate the formulas corresponding to the choice of V, S, C, Cp,and KS as measurable quantities:1 1f' l V - S (16)2 2h' - V -- S (17)2 22 C p VTK S 2 1/2 -V T K S-f - - C (18)2T pC Ft VTK S,2 1/2(p VTKS (19)h" - I + I -2T pIn the vicinity of the k-line, withVTKS 2C << 1 (20)pEqs. 18 and 19 reduce toK S V4 (21)C 2p VKS2- h" ___ 4 (22)TFinally, we note that, for certain substances, such as liquid helium, a < 0 close tothe X-line. In such cases, there is a line in the p,T-diagram where a = 0, and henceaccording to Eq. 8137(XI. STATISTICAL THERMODYNAMICS)dp =/ (23)dT) a=OEquations 10-12 and 21-23 offer numerous possibilities for experimental verification,although the existing data are usually not accurate enough for an extensive check.Pippard (2) has shown that the linear relation between a and Cp (Eq. 10) is in agree-ment with experiment. C. E. Chase and E. Maxwell, of Lincoln Laboratory, came tothe same conclusion for helium (4).It would be desirable to perform experiments that are designed to check the finerfeatures of the theory.L. TiszaReferences1. L. Tisza, Phys. Rev. 110, 587 (1958).2. A. B. Pippard, Phil. Mag. (8) 1, 473 (1956).3. L. Tisza, On the nature of the k-line of liquid helium, Quarterly ProgressReport No. 51, Oct. 15, 1958, p. 92.4. Private communication, June 1958.B. DISTRIBUTION FUNCTIONS IN STATISTICAL THERMODYNAMICSCertain questions left open in the author's thesis (1) were solved, and the materialof the thesis has been revised for future publication.P. M. QuayReferences1. P. M. Quay, Distribution functions in statistical thermodynamics, Ph. D. Thesis,Department of Physics, M. I. T., September 1958.138

Contains reports on two research projectsU. S. Air Force (Office of Scientific Research, Air Research and Development Command) under Contract AF49(638)-9

XIII. STATISTICAL THERMODYNAMICS'Prof. L. TiszaP. M. QuayiA. THEORY OF PHASE EQUILIBRIUMIt was pointed out in the Quarterly Progress Report of April 15, 1957, page 167, thatthe scope of Gibbs' phase theory can be significantly extended by including the specifi-cation of symmetry in the definition of a homogeneous phase. However, the new defini-tion cannot be used efficiently in conjunction with the traditional formulation ofthermodynamic postulates (1). Instead, a more appropriate set of definitions and postu-lates has been found, which can be outlined as follows.Isolated thermodynamic systems tend toward states of equilibrium that consist of asmall number of homogeneous regions in space called "phases." The systems aredescribed in terms of the independent variables Q1, . ., which are interpreted asthe energy, the volume, and the mole numbers of the independent components.The precise definition of homogeneous phases in equilibrium is implicit in thefollowing postulates:I. The independent variables Qi are additive:Qi = 7Qi (1)a awhere the index a refers to spatially separated subsystems, such as different phasesor volume elements. The Qi are also called "extensive parameters."II. A homogeneous phase in equilibrium is formally represented by a primitivefundamental equationS =S a  (2a a .... /which is a first-order homogeneous function with continuous derivatives of at least thesecond order. The superscript a denotes the symmetry character of the phase a.III. The entropy S of the system isS(Q .2,...) max LSaa (1 Q (3)The maximum is taken under the constraints provided by Eq. 1 with the Qi as constants.This research was supported in part by the U. S. Air Force, Office of ScientificResearch, Air Research and Development Command, under Contract AF49(638)-95.Reproduction in whole or in part is permitted for any purpose of the United StatesGovernment.National Science Foundation Predoctoral Fellow.117(XIII. STATISTICAL THERMODYNAMICS)DEFINITION:aS 1 (4)aU - Twhere T is the temperature.IV. The temperature varies over the range 0 < T < co.V. For all phases in equilibrium near absolute zero,lim = Sa-oT -0While the traditional formulation deals with quasi-static processes in order to arriveeventually at state functions, the present version considers equilibrium states exclu-sively. We have a postulational foundation of thermostatics, and the discussion of proc-esses is relegated to irreversible thermodynamics. This splitting of classical thermo-dynamics into more sharply delimited special theories can be carried even farther.Thus, the five postulates imply that thermodynamic stability is independent of the sizeof the system. This result is, indeed, a salient feature of phase equilibrium, althoughit fails for very small systems (droplets) and also for very large ones (stars). For suchcases the present postulates have to be modified.Within the clearly understood limits of validity, the postulates form the basis of arigorous phase theory. It is not necessary to introduce ad hoc assumptions which ren-der the existing thermodynamic theories unreliable.L. TiszaReferences1. For a recent formulation, with bibliography, cf. P. T. Landsberg, Revs. ModernPhys. 28, 363 (1956).B. CANONICAL DISTRIBUTION FOR INDEPENDENT RANDOM VARIABLESThe purpose of the present report is to develop a statistical theory of fluctuationswhich, in the limit of vanishing fluctuations, reduces to the phenomenological theory ofSection XIII-A. However, the theory is not confined to the Gaussian approximation,which is valid for small fluctuations only. With the terminology introduced in theQuarterly Progress Report of April 15, 1957, page 166, we shall derive an Ic theorycentered around the canonical distribution of independent random variables.The standard procedure for the derivation of the canonical distribution involves theintroduction of statistical concepts into classical or quantum mechanics. Szilard (1)followed an entirely different course. He developed a unified statistical thermodynamics118(XIII. STATISTICAL THERMODYNAMICS)by formulating a statistical generalization of the classical thermodynamic postulates.His fundamental results remained largely unnoticed in thernmodynamics literature.Recently, the question was reopened and further developed by Mandelbrot (2), whopointed out that Szilard's argument can be considered as an instance of estimationtheory of mathematical statistics.The ideas of Szilard and Mandelbrot tie in particularly well with the new axiomati-zation of thermostatics discussed in Section XIII-A. The objective of our program is todevelop a self-contained theory along these lines. The discussion is divided into a num-ber of successive stages, the first of which can be outlined as follows.Let us consider a thermodynamic system specified by the extensive paranmeters Qintroduced in Section XIII-A. All but one of these parameters are to be held fixed, whilethe coupling with the surroundings produces fluctuations of the energy, which is now arandom variable U that takes on values denoted by u. The case of several randomvariables will be discussed later.As idealized surroundings we consider reservoirs H(t), where t denotes an arbi-trary temperature scale defined by the following stipulation: if, on establishing acoupling between R(t 1 ) and R(tZ), on the average, energy flows from the Ri(tl) to R(tZ)'then tl > t .We formulate the following postulates:I. The values u of the random variable U are distributed according to some proba-bility density function P(u/t) that depends parametrically on the temperature t and thefixed parameters of the system.II. Two closed systems with the same fixed parameters and the same energy u arethermodynamically equivalent regardless of the temperatures of the reservoirs withwhich they had been in contact.III. Composition rule. The energy of a composite system is the sum of the energiesof its parts and the partial energies are statistically independent. Briefly, the energiesare additive random variables.From these assumptions we can derive thatP(u/t) = g(u)e-P(t)U/z(p) (1)where g(u) is a "structure function" that depends only on the properties of the system,p = p(t) is a function of temperature only. The definition T = 1/k3 can be shown to leadto the thermodynamic temperature T, and k is Boltzmann's constant. The partitionsum Z(P) is determined from the normalization of the probability.The structure function g(u) can be identified with the degree of degeneracy of theenergy as obtained from the solution of the Schrodinger equation of the system. It isnoteworthy that the existence of the structure function can be inferred from the generalstatistical postulates I, II, and III without any explicit reference to atomic structure.119(XIII. STATISTICAL THERMODYNAMICS)In addition to the canonical distribution we can also consider the microcanonicaldistribution of systems with fixed values of all the extensive parameters, including theenergy. The foregoing definition of the temperature concept can be extended to micro-canonical distribution by means of a "maximum likelihood estimate" (2).The conditions of the equivalence of the canonical and microcanonical methods arenow being studied.L. Tisza, P. M. QuayReferences1. L. Szilard, Z. Physik 32, 753 (1925).2. B. Mandelbrot, Transactions of the 1956 Symposium on Information Theory, M. I. T.,Cambridge, Massachusetts, Sept. 10-12, 1956; cf. E. T. Jaynes, Phys. Rev. 106,620 (1957).120

XII. STATISTICAL THERMODYNAMICS'Prof. L. TiszaP. M. QuayTIn our previous report (Quarterly Progress Report, July 15, 1957, p. 118) we out-lined the statistical phenomenological postulates that are sufficient for a rigorousderivation of the canonical distribution law of a thermodynamic system in which theenergy is the only extensive variable that is subject to random fluctuations.The canonical probability density function of the distribution of the energy u isP(u/p) = g(u) e-pu Z(p)- 1 (1)where p = 1/kT, k is Boltzmann's constant, T is the absolute temperature, g(u) is astructure function that depends only on the properties of the system, and Z(P) is thepartition sum determined from the normalization condition of the probability densityfunction asccZ(p) = g(u) e-u du (2)It is assumed that the energy spectrum has no upper bound, and its lower bound ischosen as the zero of the energy.By introducing the concept of entropy, it is, on the foregoing basis, possible todevelop a theory which is phenomenological in the sense that the probabilistic elementsare all implicit in the entropy definition. Otherwise, the theory is more general thantraditional thermostatics, since fluctuations are taken into account. In the terminologyof the Quarterly Progress Report, April 15, 1957, p. 166, we have an Ic theory insteadof the classical Il theory.As a first step toward the definition of the entropy, we introduce the index ofprobability,g(u)s(u, p) = k In - = k [pu + (p)] (3)P(u/p)Herec( ) = In Z(P) = -A(T)/kT (4)This research was supported in part by the U. S. Air Force, Office of ScientificResearch, Air Research and Development Command, under Contract AF49(638)-95.Reproduction in whole or in part is permitted for any purpose of the United StatesGovernment.-National Science Foundation Predoctoral Fellow.(XII. STATISTICAL THERMODYNAMICS)is the Mathieu function, and A(T) is the Helmholtz free energy. Since the energyu is a random variable, the same is true of s(u, P).In order to define the thermodynamic phenomenological entropy, we have to distin-guish two cases:(i) the system is in contact with a reservoir of temperature T, hence p is constant.(ii) the system is isolated and has a fixed energy u.These two cases correspond to the canonical and microcanonical ensembles ofstatistical mechanics. The discussion of the first case closely parallels the traditionalmethod, and we shall confine ourselves to a short summary of the relevant formulas.The discussion of case ii has entirely novel features.In case i the entropy is defined as the average of the random variable (see Eq. 3),S(p) = s(u, P) = kpu + k O(P)(5)f" g(u)P(u/P) In - duP(u/p)The thermodynamic properties of the system are contained in the fundamental equa-tion(P) = S(p) - pU = -A(T)/kT (6)whereU = u = - (7)apThe energy fluctuation isAu2 > 0 (8)ap2We turn now to case ii. The procedure most frequently applied is to define theentropy for an isolated system asS = k In g(u) (9)The proper definition of the temperature is somewhat problematic in this case (1).Following a suggestion of Mandelbrot (2), we have assigned a temperature to isolatedsystems by using the maximum likelihood method of mathematical statistics (3). Wefound that in the present context the use of this method can be made plausible on thegrounds of simple physical considerations.Let us assume that before its isolation the system reached its state of energy uothrough contact with a reservoir of temperature T = i/kp. In classical thermostatics,(XII. STATISTICAL THERMODYNAMICS)an Ii theory, there is a unique functional relation between uo and P. This is not truein the Ic theory, in which the energy is allowed to fluctuate within a canonical distri-bution. However, it is still possible to estimate the temperature.Following the method of maximum likelihood estimation, we require that the proba-bility density function (1), or more conveniently its logarithm, should be maximum inits dependence on P. Hencea In P(u IP) 8-- u = 0 (10)P a3<0 (11)ap2if Po is the solution of Eq. 10.If we compare Eq. 7 with Eq. 10, we see that 1/k is that temperature for whichthe canonical average is i = u . This is consistent with the I1 theory, in which fluctu-ations are altogether neglected. However, in the present case the determination of thetemperature from Eq. 10 is not sharp. Expanding In P(uI P) as a function of p aroundPo, and breaking off at the quadratic terms, we find that the estimated reciprocaltemperatures have a standard deviation Ap that obeys the relationAu ap = k (12)where Au is the standard deviation (rms fluctuation) of the energy in the canonical dis-tribution of the temperature 1/kp oThis relation has an interesting physical meaning. The transition from the presentIc theory to the Il theory corresponds to a limiting process:k-0, Au -0, A - 0 (13)In the Ii theory there is no difference between an isolated system of energy uo and asystem that, is in contact with a reservoir, provided that u0 = i. Moreover, both uand p = 1/kT have sharp values, i. e., the energy of a system uniquely determines thetemperature of the surroundings with which it might be in equilibrium, and, conversly,the temperature of a reservoir uniquely determines the energy of the system with whichit is in contact.The approximate validity of Eq. 13 assures us that the Il theory is approximatelycorrect. However, large values of Au or Ap are indications that in singular situationsthe phenomenological theory is close to the limits of its usefulness.At a critical point, the energy fluctuations are very great (Nu - oo), hence theenergy is not precisely determined by the reservoir temperature. The singularity atabsolute zero exhibits the opposite features: Au -0, A3 -oo. Physically, this means(XII. STATISTICAL THERMODYNAMICS)that with such low temperatures that the specifice heat very nearly vanishes, the know-ledge of the energy of the system does not enable us to draw any reliable inferencesabout the temperature of the surroundings. This means, in simple terms, that a sub-stance with nearly vanishing specific heat is not an appropriate thermometric substance,which is well known to low-temperature physicists.While the Ic theory is a great improvement over the Il theory, as far as describingthe aforementioned singular situations is concerned, the difficulties connected withthese situations have not all been removed and call for further analysis.L. Tisza, P. M. QuayReferences1. A. I. Khinchin, Statistical Mechanics (Dover Publications, Inc., New York, 1949),p. 142.2. B. Mandelbrot, 1956 Symposium on Information Theory, held at M.I. T., Cambridge,Massachusetts, Sept. 10-12, 1956; Trans. IRE, vol. IT-Z, no. 3, pp. 190-203(Sept. 1956).3. Cf. B. L. Van der Waerden, Mathematische Statistik (Springer Verlag, Berlin, 1957),p. 148.

Contains research objectives and reports on one research project.U.S. Air Force (Office of Scientific Research, Air Research and Development Command) under Contract AF49(638)-9

XV. STATISTICAL THERMODYNAMICSProf. L. TiszaP. M. QuayRESEARCH OBJECTIVESIn recent years various authors have noted that statistical thermodynamics can bederived from general statistical assumptions without the specification of a statisticalmechanical model. Our research project is devoted to the systematic development ofthis idea. A thermodynamic system is a region of space specified by its volume, energy,and mass content (including its chemical composition). If an isolated thermodynamicsystem is subdivided into parts that are allowed to exchange energy and/or mass, thesequantities become random variables. They exhibit rapid fluctuations around valueswhich drift slowly toward asymptotic equilibrium levels. The study of the asymptoticvalues is the objective of classical thermostatics. The systematic approach hasyielded new results even in this case, as described in Section XV-A. The fluctuationaround equilibrium may be described by assuming that the values of the fluctuatingquantities at discrete intervals of time are independent random variables. This assump-tion leads to a theory that is similar to that of the canonical and microcanonical ensem-bles in statistical mechanics, but contains less mechanical and more statistical detail.Finally, the drift toward equilibrium and the time-dependent aspects of fluctuationsare discussed by means of a theory in which the random variables form Markov chains.Thus far, we have considered Markov processes almost exclusively in the Gaussianapproximation. Only a simple combinatorial model was in the non-Gaussian category.This line of thought is to be pursued in the near future.A. SECOND PHASE RULE AND SUPERFLUIDITYA theory of phase equilibrium has been recently reported (1, 2). Its essentialfeature is a new definition of the phase concept which requires specification of itssymmetry. This theory leads to a relation (3) which might be called the "secondphase rule." It plays a role for X-anomalies that is similar to the role of the firstphase rule for first-order transitions.The application of the second phase rule to the helium X-line yields the followingtheorem.THEOREM: If the X-line of helium represents a locus of singularities of the spe-cific heat C in the p-T plane, the second phase rule requires that the super-Pfluid state exist in two modifications that are related to each other by a binarysymmetry operation.The experiments of Buckingham, Fairbank, and Kellers (4) provide convincing evi-dence for a singularity of the specific heat at the saturated vapor pressure. Thereare reasons to believe that the X-point has a similar character even under higherpressure, although a direct experimental check would be desirable.This research was supported in part by the U.S. Air Force, Office of ScientificResearch, Air Research and Development Command under Contract AF49(638)-95.Reproduction in whole or in part is permitted for any purpose of the United StatesGovernment.109(XV. STATISTICAL THERMODYNAMICS)If we assume tentatively the existence of a line of singularities, our theoremrequires that the ground-state wave function of helium be doubly degenerate.This conclusion contradicts the existing quantum-mechanical theories for helium,which assume that the ground-state wave function is nondegenerate. While this con-clusion might be invalidated by a negative result of the aforementioned experiment, itis still pertinent to ask, How reliable is our theorem ? This question has many sides,since there is no general agreement on the basic assumptions of phase theory and theadmissible rules of inference. A partial list of the problems that we have investigatedfollows.The most important basic assumptions of the theory were listed previously (2).These include, in addition to the laws of thermodynamics, the assumptions that thesystem consists of homogeneous parts and that the energy of spatially disjoint sub-systems is additive. These assumptions are of limited validity, but they are quitejustified in the case of helium. However, in the case of superconductivity, the magneticinteraction of current elements is in contradiction to strict additivity, and hence theapplication of the theory calls for special considerations. The same is true of othertransitions involving magnetization or electric polarization.In the existing thermodynamic theories of phase transitions there are two proceduresthat are objectionable and render the conclusions unreliable. The first is the method ofpower-series expansion. This method is indispensable in the study of stability withrespect to small displacements, but becomes questionable at the critical point. Theexpansion of the Gibbs function around a critical point is demonstrably wrong. This isnot so obvious in the expansion of the internal energy.We have made a detailed study of this matter, but shall not report it, since wefound that expansion techniques are always unsatisfactory in the critical region. Itis therefore important to note that the derivation of the second phase rule does notrequire such expansions and depends only on the topological properties of thermo-dynamic phase space.The second objectionable procedure is the well-known Maxwell rule of equal areas,which is generally used in connection with the van der Waals equation. This procedureis often believed to be an integral part of thermodynamic phase theory; at the same time,it is justly criticized for involving integration over a region of essential instability (5).Actually, this method can be avoided by using Gibbs' construction: the instable partsof the fundamental surface in the thermodynamic U-S-V space are replaced by a ruledsurface representing the two-phase region.It is interesting to point out the rationale for the use of the Maxwell rule. The appli-cation of Gibbs' construction requires a knowledge of the fundamental equation, whichis equivalent to the equation of state, and also of the specific-heat function C = C (T)at a fixed pressure. If these functions are known, the coexisting states of the liquid110(XV. STATISTICAL THERMODYNAMICS)and vapor may be connected by integration around the critical point within the stablerange of the fluid. In the conventional treatment of the van der Waals condensation theC (T) function is unspecified; instead, information is supplied concerning the essen-tially instable range of the system. This information is spurious; it has no theoreticalbasis and could not be supported by any conceivable experiment.Our conclusion is that the second phase rule rests on solid ground and may beapplied to helium, provided that direct experiments confirm the existence of a lineof singularities.L. TiszaReferences1. L. Tisza, Generalized phase rule and superfluidity, Quarterly Progress Report,Research Laboratory of Electronics, M.I.T., April 15, 1957, p. 167.2. L. Tisza, Theory of phase equilibrium, Quarterly Progress Report, ResearchLaboratory of Electronics, M.I.T., July 15, 1957, p. 117.3. Quarterly Progress Report, Research Laboratory of Electronics, M.I.T., April 15,1957, p. 168; see Eq. 2.4. M. J. Buckingham, W. M. Fairbank, and C. F. Kellers, International Conferenceon Low Temperature Physics and Chemistry, University of Wisconsin, Madison,Wisconsin, August 1957.5. L. Van Hove, Revs. Modern Phys. 29, 200 (1957).111

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Statistical Thermodynamics

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