Symmetry of physical laws. Part I: Symmetry in space-time and balance theorems Symmetry in space-time and balance theorems.
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Authors
Watanabe, Michael Satosi
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Date of Issue
1953
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Monterey, CA; Naval Postgraduate School
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Abstract
In default of the theorem of "detailed balance": Pᵢⱼ = Pⱼᵢ, with regard to elementary transition probabilities, several "balance" theorems are introduced and proved on the basis of symmetry of physical laws in space-time. (1) First theorem of "averaged balance" (#5): We can establish Pᵢⱼ = Pⱼᵢ. by averaging over quantities of "minus class." Table V (#3) gives a list of "minus" quantities. (2) The so-called "detailed balance of collisions" in classical physics is a special case of Theorem (1). (3) Heitler-Coester's theorem of "semi-detailed balance" is also a special case of Theorem (l). (4) Second theorem of "averaged balance" (#5): We can establish Pᵢⱼ = Pⱼᵢ by averaging over quantities with PR =-1. The quantities with PR =-1 are listed on Table II (#2). (5) Theorem of "cyclic balance" (#7): In classical physics, a chain of transitions i→j→k→. . .→i repeats itself cyclically. (6) Theorem of "long-range balance" (#7): The time average of transition probability from i to j is equal to the time average of transition probability from j to i. Theorems (1), (2) and (3) are direct consequences of inversibility (covariance for space-and-time inversion). Theorem (4) is a consequence of reversibility (covariance for time reversal). Theorems (5) and (6) are connected with ergodicity of Markoff's chains. This ergodicity is proved by the condition of bilateral normalization of transition probabilities: Σⱼ Pᵢⱼ = 1, Σᵢ Pᵢⱼ = 1. This bilateral normalization in turn can be derived from either reversibility or inversibility. The limits of validity of all these balance theorems in actual applications are carefully examined in the text.
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Technical Report
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Research Paper No. 2
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52 leaves ; 28 cm.
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This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.