Authors:
Yuri G. Rudoy,Olga I. Chekmareva,DOI NO:
https://doi.org/10.26782/jmcms.2019.03.00054Keywords:
Gibbs theory,Fisher−Rao−Cramer theorem ,Information theory ,Abstract
The geometrical approach due to Gibbs in the equilibrium phenomenological, or Clausius, thermodynamics (CTD) is generalized for the statistical case (STD), which is naturally stipulated by the stochastic nature of the thermal contact between the TD-object and the external surrounding (thermostat). To this end the probabilistic measure p is introduced into the affine space of finite-dimensional space of basic TD-variables, which is parametrized by means of the intensive variables of the thermostat. For the case of strongly additive extensive TD-variables the measure p possess the exponential, or canonical Gibbs form. The ordinary Clausius TD-variables are then Gibbs TD-variables averaged with the measure p and of primary interest are the relevant spontaneous fluctuations; in particular, they determine the accuracy of Zeroth Law of TD fulfillment.Refference:
I. Amari S.-I. (1968). R.A.A.G. Memoirs. 4: 373-418.
II. Amari S.-I. (1982). Ann. Stat.,10(2): 357-385.
III. Amari S.-I., Differential Geometrical Method in Statistics. Lecture Notes Statistics, 28, Springer, Berlin, 1985.
IV. Bogoliubov N.N., Tyablikov S.V. (1959).Sov. Phys.Doklady,126:53-56
V. Chentsov N.N. (1966). Theory of Probabilityand Appl.,11: 425-435.
VI. Dynkin E.B. (1961). Selected Transl. Math. Stat. Prob. (I.M.S.), 23-40. VII. Falk H., Bruch L.W. (1969). Phys. Rev.,180: 442-444.
VIII. Gibbs J.W. (1902). Elementary Principles in Statistical Mechanics.Scribner&Sons, N.Y.Gilmore R.(1985). Phys. Rev.,A 31: 3237-323
IX. Greene R.F., CallenH.B. (1951).Phys. Reiew,83:1231-1235.
X. Hasegawa H., Petz D. (1995, 1997). Non-Commutative Extension of Information Geometry, I, II; InQuantum Communications, Computingand Measurement,ed. ByH. Hirota etal., Plenum Press,N.Y.
XI. Helstrom C.W. (1976). Quantum Detection andEstimation Theory. Acad. Press, N.Y.
XII. Holevo A.S. (1982). Probabilistic and Statistical Aspects of QuantumTheory. North-Holland, Amsterdam
XIII. Ingarden R., Sato Y., Sugawa S.,Kawaguchi M. (1979). Tensor, N.S. 33:348-353.
XIV. Jeffreys H. (1962). Theory of Probability. 3rded., Clarendon Press, Oxford.
XV. Koopman B.O. (1936). Trans. Amer. Math. Soc.,39: 399-409.XVI Lavenda B. (1991). Statistical Physics: Probabilistic Approach, John Wiley, N.Y.
XVII. Luttinger J.M. (1966). Suppl. of the Prog. of Theor. Phys.(in honour of Prof. S. Tomonaga 60thbirthday), 37-38: 35-45.
XVIII. Mandelbrot B. (1956). IRE Trans. Information Theory. IT-2: 190-203.
XIX. Mandelbrot B. (1962).Ann. Stat.,33:1021-1038. Mandelbrot B.(1964).J. Math. Phys. 5: 164-171.
XX. Morozova E.A., Chentsov N.N. (1990).Markov Invariant Geometry on State Manifold(in Russian), Itogi Nauki i Tekhniki, 36: 69-102, VINITY, MoscowMünster A. (1990). Statistische Thermodynamik, Springer, Berlin.
XXI. Petrov N., Brankov J. (1986). The Modern Problems of Thermodynamics(in Russian), Mir, Moscow.
XXII. Petz D., Toth G. (1993). Lett. in Math. Phys.,27: 205-216.
XXIII. Petz D. (1996). Lin.Algebra and Appl.,244:81-96. Pittman E.J.G. (1936). Proc. Cambridge Philos. Soc.,32: 567-579.XXIV. Rudoy Yu. G., Sukhanov A.D. (2000). Physics–Uspekhi,43: 1169-1199.
XXV. Rudoy Yu. G., Sukhanov A.D.(2005).Cond. Matter Physics,8: 507-535.
XXVI. Schlögl F. (1985).Z. Phys.,B.59: 449-456.XVIII. Streater R.F. (2000). In Proc. of the Steklov Institute of Math.,Moscow. 228: 205-223.
XXIX. Sukhanov A.D. (2006). Theor. Math. Phys.,139: 129-137, 2004; 148: 295-308.
XXX. Sukhanov A.D. (2005). Elementary Particles and Atomic Nuclei, 36: 183-195.
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