Ergodic Hypothesis and Equilibrium Statistical Mechanics in the Quantum Mechanical World View

Shiro Ishikawa

Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan.

DOI: 10.4236/wjm.2012.22014                

Abstract

In this paper, we study and answer the following fundamental problems concerning classical equilibrium statistical mechanics: 1): Is the principle of equal a priori probabilities indispensable for equilibrium statistical mechanics? 2): Is the ergodic hypothesis related to equilibrium statistical mechanics? Note that these problems are not yet answered, since there are several opinions for the formulation of equilibrium statistical mechanics. In order to answer the above questions, we first introduce measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. And we propose the measurement theoretical foundation of equili-brium statistical mechanics, and further, answer the above 1) and 2), that is, 1) is “No”, but, 2) is “Yes”.

Keywords

The Copenhagen Interpretation; Probability; Operator Algebra; Ergodic Theorem; Quantum and Classical Measurement Theory; Liouville’s Theorem; The Law of Increasing Entropy

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. Ishikawa, “A Quantum Mechanical Approach to a Fuzzy Theory,” Fuzzy Sets and Systems, Vol. 90, No. 3, 1997, pp. 277-306. doi:10.1016/S0165-0114(96)00114-5
[2]	S. Ishikawa, “Mathematical Foundations of Measurement Theory,” Keio University Press Inc., Tokyo, 2006.
[3]	S. Ishikawa, “Ergodic Problem in Quantitative Language,” Far East Journal of Dynamical Systems, Vol. 11, No. 1, 2009, pp. 33-48.
[4]	S. Ishikawa, “A New Interpretation of Quantum Mecha- nics,” Journal of Quantum Information Science, Vol. 1, No. 2, 2011, pp. 35-42.
[5]	S. Ishikawa, “Quantum Machanics and the Philosophy of Language: Reconsideration of Traditional Philosophies,” Journal of Quantum Information Science, Vol. 2, No. 1, 2012, pp. 2-9.
[6]	S. Ishikawa, “A Measurement Theoretical Foundation of Statistics,” Journal of Applied Mathematics, Vol. 3, No. 3, 2012, pp. 283-292.
[7]	D. Ruelle, “Statistical Mechanics, Rigorous Results,” World Scientific, Singapore, 1969.
[8]	G. Gallavotti, “Statistical Mechanics: A Short Treatise,” Springer Verlag, Berlin, 1999.
[9]	M. Toda, R. Kubo and N. Saito, “Statistical physics, Springer Series in Solid-State Sciences,” Springer Verlag, Berlin, 1983.
[10]	G. J. Murphy, “C*-Algebras and Operator Theory,” Academic Press, Waltham, 1990.
[11]	J. von Neumann, “Mathematical Foundations of Quantum Mechanics,” Springer Verlag, Berlin, 1932.
[12]	A. N. Kolmogorov, “Foundations of the Theory of Probability (Translation),” 2nd Edition, Chelsea Pub Co, New York, 1960.
[13]	E. B. Davies, “Quantum Theory of Open Systems,” Academic Press, Waltham, 1976.
[14]	U. Krengel, “Ergodic Theorems,” Walter de Gruyter, Berlin, 1985. doi:10.1515/9783110844641
[15]	S. Ishikawa, “The Linguistic Interpretation of Quantum Mechanics,” 2012. http://arxiv.org/abs/1204.3892