Stokes-Einstein relation formula

Here we show how the modified Kubo formula (187) for p(co) leads to a relation between the (Laplace transformed) mean-square displacement and the z-dependent mobility (z denotes the Laplace variable). This out-of-equilibrium generalized Stokes-Einstein relation makes explicit use of the function (go) involved in the modified Kubo formula (187), a quantity which is not identical to the effective temperature 7,eff(co) however re T (co) can be deduced from this using the identity (189). Interestingly, this way of obtaining the effective temperature is completely general (i.e., it is not restricted to large times and small frequencies). It is therefore well adapted to the analysis of the experimental results [12]. 

Formula (206), together with the out-of-equilibrium generalized Stokes-Einstein relation (203), is the central result of the present section. 

The main results of this sectionjire the out-of-equilibrium generalized Stokes-Einstein relation (203) between Ax2(z) and p(z), together with the formula (206) linking Teff(ffi) and the quantity, denoted as ( ), involved in the Stokes-Einstein relation. One thus has at hand an efficient way of deducing the effective temperature from the experimental results [12]. Indeed, the present method, which avoids completely the use of correlation functions and makes use only of one-time quantities (via their Laplace transforms), is particularly well-suited to the interpretation of numerical data. 

This expression is known as the Stokes-Einstein equation. This formula correctly relates diffusivity to molecular dimensions and viscosity for cases in which Stokes law is applicable. 

This formula follows from the Stokes-Einstein equation, the relation (r ) = 6Dt, and the definition = R/k. 

This constitutes the generalization of the Stokes-Einstein formulas (316). It applies for any choice of origin. The positive-definite character of the resistance matrix assures the existence of its inverse, as required by the preceding relation. 

The diffusion coefficient of natural organics is related to the size of the molecules. Worch (1993) published a formula to relate MW with size for organic molecules (see Eqn. (2.1)). The size can then be related to diffusion coefficient by using the Stokes Einstein equation (see Appendix 5). While this method assumes spherical shapes, it allows the estimation as equivalent spheres. 

The distribution formula (124b) has been used in deriving the above formula. The validity of (150) was shown by Perrin who showed that the Boltzmann constant k could be obtained by relating (150) with the Stokes-Einstein equation... 

For non-sphericd but symmetrical particles, there are two translational diffusion coefficients one parallel and one perpendicular to the symmetric axis However, only the average can be retrieved in most situations. This average translational diffusion coefficient Dt is related to both dimensions of the particle. It can generally he written in the same formula as the Stokes-Einstein equation ... 

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