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

Quay, P. M.

Tisza, Laszlo

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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

Statistical Thermodynamics

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Statistical Thermodynamics

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