Mean-square displacement Stokes-Einstein relation

In Section VI, we consider a classical particle diffusing in an out-of-equilibrium environment. In this case, all the dynamical variables attached to the particle, even its velocity, are aging variables. We analyze how the drift and diffusion properties of the particle can be interpreted in terms of an effective temperature of the medium. From an experimental point of view, independent measurements of the mean-square displacement and of the mobility of a particle immersed in an aging medium such as a colloidal glass give access to an out-of-equilibrium generalized Stokes-Einstein relation, from which the effective temperature of the medium can eventually be deduced. 

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]. 

As displayed by the out-of-equilibrium generalized Stokes-Einstein relation (203), independent measurements of the particle mean-square displacement and frequency-dependent mobility in an aging medium give access, once Ax2(z) and p(z) = p(co = iz) are determined, to T (z) and to T (co) = T (z = — iffi). Then, the identity (189) yields the effective temperature ... 

When the environment of the particle is itself out-of-equilibrium, as is the case for a particle evolving in an aging medium, we showed how the study of both the mobility and the diffusion of the particle allows one to obtain the effective temperature of the medium. We derived an out-of-equilibrium generalized Stokes-Einstein relation finking the Laplace transform of the mean-square displacement and the z-dependent mobility. This relation provides an efficient way of deducing the effective temperature from the experimental results. 

For a binary mixture, if experimental diffusivities do not exist over the whole range of concentration, an interpolation of the diffusivities at infinite dilution D k] J is used. In calculating the diffusivities at infinite dilution by the Stokes-Einstein relation, we consider small isolated hard spheres, submerged in a liquid, that are subjected to Brownian motion The friction of the spheres in the liquid is given by the Stokes law Einstein used the Stokes law to calculate the mean-square displacement of a particle. The displacement increases linearly with time, and the proportionality constant is the Stokes-Einstein diffusivity... 

Translational and rotahonal diffusion coefficient of a molecule in a liquid provides a quantitahve measure of the dynamic timescales in the liquid. These coefficients are related to viscosity by the Stokes-Einstein [2] and the Debye-Stokes-Einstein relation [3], respectively. Using the definihon of diffusion coefficient in terms of mean-square displacement [2] and the Stokes-Einstein relahon, we can estimate the time needed by a water molecule to translate a distance equal to its molecular diameter a... 

This technique is pretty new and has already been successfully applied to measure the viscoelastic properties of gels based on the gelators. The principle of this technique is based on the tracking of submicrometer tracers embedded within a solution. The mean-squared displacement of the tracers is given by (Ar (r)) a t , where a = 1 for liquids and o < 1 for viscoelastic fluids. It appears that (Ar (r)) is also related to the viscoelastic modules through the generalized Stokes-Einstein relation ... 

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