Ensemble averaged elastic properties

As revealed through experimental works, however, the flow of lubricants in TFL provides a hint that the macroscopic properties, such as the viscosity and the elastic modulus remain to be a measurement of the fluid characteristics. In addition, the transition from EHL to TFL is inherently progressive, wherein no abrupt transform in lubrication states are found. Thus, the continuum theory is validated to some extent. Furthermore, one can arrives at a continuum viewpoint, but in a different way from conventional fluid mechanics, by considering the material to be a continuum one in an ensemble averaged, rather than a spatial averaged, sense. 

The most useful of such properties would appear to be an ensemble-averaged momentum and density (their ratio being velocity) or elastic modulus, weighting the likelihood... 

For a dense system of hard, smooth, and elastic spherical particles, a transport theorem based on the analogy of the kinetic theory of dense gases [Reif, 1965] may be derived. Define an ensemble average of any property xjr of a particle as... 

The mean-squared, end-to-end distance in Equation [9] is the simplest average property of interest for a polymer chain. Among other physical properties, this quantity appears in the equations of statistical mechanical theories of rubber-like elasticity. In Equation [9], the angle brackets denote the ensemble (or time) average over all possible conformations. The subscript 0 indicates that the average pertains to an unperturbed chain (theta conditions no excluded volume effects are present). (See Figure... 

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