A Lagrangian formalism for treating the stretching of material line and surface elements convected with a turbulent fluid is developed, and consequences of statistical isotropy for the distribution function of the components of a tensor are derived. Incompressibility and isotropy are then used to prove rigorously that the expectation value of the logarithmic change of the length of a line element (and area of a surface element) is greater than zero. A connection between the distribution function of the velocity shear tensor and that of the symmetric time-development tensor used in the Lagrangian formalism is shown.
The hypothesis of complete time symmetry in oscillating cosmologies in which there exist locally irreversible processes is examined. A time-symmetric formalism for purely statistical processes is developed, and all aspects of quantum and statistical mechanics are shown to be time-symmetric within this framework. We conclude from analysis of a simple example that in a completely time-symmetric oscillating cosmology, statistical processes which produce entropy in the expanding phase will reverse themselves in the contracting phase, although microscopic reversal of motion need not occur. The analysis provides a self-consistent formalism for general problems involving the coexistence of different entropic directions of time in the observable universe. Boltzmann's H theorem is discussed in this framework.
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