×

zbMATH — the first resource for mathematics

Information theory and statistical mechanics. I, II. (English) Zbl 0084.43701
The author attempts to show that a large part of the statistical structure of thermodynamics can be deduced from the fundamental requirements imposed upon a process of statistical inference. The point of departure of that process is provided by the very scant knowledge contained in the specification of the energy and the number of particles of a system. The statement of the author’s aims is very lucid and vigorous. However, his implementation of his aims is unconvincing: basically, he writes as if the uncertainty which is involved in statistical inference were entirely described by Shannon’s information. That is, he does not use at all the modern theory of statistical inference, the need for which seems to be his papers. Actually, he does use some statistical terminology, obvious all through but in a misleading or incorrect way. For example, the concept of “unbiased”, which he refers to, is not that of statistic, and seems very inadequate; the formula for Fisher’s information is incorrect. The Reviewer objects most, however, to the misuse of the concept of “sufficiency”, which is mentioned several times, and never defined. Obviously, the author has not realized the fundamental importance of this concept. lt seems that the author’s aims had been implemented, even before his papers had appeared, in the reviewer’s paper published in “An outline of a purely phenomenological theory of statistical thermodynamics. I: Canonical ensembles”. IRE Trans. Inf. Theory 2, No. 3, 190–203 (1956; doi:10.1109/TIT.1956.1056804)] (See also C. R. Acad. Sci., Paris 243, 1835–1838 (1956; Zbl 0072.21001)).
Reviewer: B. Mandelbrot

MSC:
82B03 Foundations of equilibrium statistical mechanics
82B30 Statistical thermodynamics
94A15 Information theory (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] D. ter Haar, Revs. Modern Phys. 27 pp 289– (1955) · Zbl 0067.18902 · doi:10.1103/RevModPhys.27.289
[2] J. W. Gibbs, in: Elementary Principles in Statistical Mechanics (1928)
[3] C. E. Shannon, Bell System Tech. J. 27 pp 379– (1948) · Zbl 1154.94303 · doi:10.1002/j.1538-7305.1948.tb01338.x
[4] C. E. Shannon, in: The Mathematical Theory of Communication (1949)
[5] E. P. Northrop, in: Riddles in Mathematics (1944)
[6] H. Cramer, in: Mathematical Methods of Statistics (1946)
[7] W. Feller, in: An Introduction to Probability Theory and its Applications (1950) · Zbl 0039.13201
[8] J. M. Keynes, in: A Treatise on Probability (1921)
[9] H. Jeffreys, in: Theory of Probability (1939)
[10] R. A. Fisher, Proc. Cambridge Phil. Soc. 22 pp 700– (1925) · JFM 51.0385.01 · doi:10.1017/S0305004100009580
[11] J. L. Doob, Trans. Am. Math. Soc. 39 pp 410– (1936) · JFM 62.0611.01 · doi:10.1090/S0002-9947-1936-1501855-5
[12] E. Schrödinger, in: Statistical Thermodynamics (1948)
[13] D. ter Haar, in: Elements of Statistical Mechanics (1954) · Zbl 0056.17002
[14] D. Bohm, Nuovo cimento, Suppl. II pp 1004– (1955) · doi:10.1007/BF02744278
[15] M. B. Lewis, Phys. Rev. 101 pp 1227– (1956) · Zbl 0073.45506 · doi:10.1103/PhysRev.101.1227
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.