Space-time averages in macroscopic gravity and volume-preserving coordinates

Journal of Mathematical Physics

Volumn 38, Issue 9, 1997, Pages 4741-4757

  Mars, Marc ^a   Zalaletdinov, Roustam M ^a,b,c  

^a QUEEN MARY UNIVERSITY OF LONDON   (United Kingdom)

^b NICOLAUS COPERNICUS ASTRONOMICAL CENTER   (Poland)

^c S YU YUNUSOV INSTITUTE OF THE CHEMISTRY OF PLANT SUBSTANCES   (Uzbekistan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031509691     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532119     Document Type: Article

References (40)

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5. Alternative approaches for deriving macroscopic electrodynamics which apply other averaging procedures (for example, space averaging and statistical ensemble averaging) are not considered here. For such approaches and a discussion about their interrelations, physical significance, etc. see Refs. 34-37 and 9.
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20. ∝-diffeomorphisms.
21. It should be noted here that in classical hydrodynamics, as distinct from macroscopic electrodynamics, discussion on the definition and properties of the averages continues for more than one hundred years. A definition of an average (either over space, time, ensemble, or a combination of such) in hydrodynamics and its properties are vital elements of the theory itself for it is clearly understood that the form of the equations depends on the definition and properties of the average. The definition (i) under conditions (i) and (ii) with the properties (2), (3) and (4), which are part of the Reynolds conditions in hydrodynamics, is known to result in the fundamental equations of hydrodynamics describing the dynamics of turbulence. If one of the Reynolds conditions is absent one must get different equations. For a discussion on averages and their properties in hydrodynamics, see, for example, Ref. 24 and references therein.
22. R. M. Zalaletdinov (unpublished).
23. The case of two scales is discussed here for the sake of simplicity. Of course, very often there is a hierarchy of scales, in which case the arguments are applied for each couple of scales satisfying (14) to be micro-and macroscopic ones, respectively.
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26. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 1 (Nauka, Moscow, 1965) (in Russian) [English translation of Vol. 1 revised by the authors, edited by J. L. Lumley (MIT, Cambridge, Mass., 1971)].
27. It should be pointed out here that within the ensemble average procedure the idempotency follows without problem. However, such averages have their own problems. In particular, the whole body of problems related with idempotency is replaced by the necessity to prove the ergodicity hypothesis which states that ensemble and time (or space) averages are equivalent. Both ensemble and volume averagings have their own advantages and areas of applicability in describing physical phenomena. It is important to realize in this connection that in all macroscopic settings a volume averaging (over space, time, or space-time) is an unavoidable element (Refs. 34-37).
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33. lk,i = 0. This condition is fulfilled also globally on parallelizable manifolds (both orientable and non-orientable) (Ref. 38). This shows that Theorem 4 is valid globally for such manifolds (a manifold is called parallelizable if its tangent bundle is trivial).
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