Towards a covariant formulation of the rheol- ogy of dense colloidal suspensions: Stochastic processes on time-dependent manifolds

Kurzfassung

The continuum mechanics description of the deformation of matter rests on the concept of time-dependent manifolds. Material specifics enter through constitutive laws that need to obey symmetry principles. It is one of the main goals of statistical physics to derive these constitutive equations from knowledge about the microscopic dynamics of a given many- particle system. The description of dense colloidal suspensions on microscopic grounds naturally leads to the theory of stochastic differential equations and the associated Fokker-Planck (Smoluchowski) equation. Recent development is based on the integration-through transients (ITT) formalism and mode-coupling theory of the glass transition (MCT). The ITT-MCT constitutive equation for simple shear flows has been used to predict many different phenomena that have been confirmed experimentally. However, the most recent ansatz to extend the theory to (homogeneous) arbitrary flows leads to an upper convected version, while simulation results hint towards a lower convected version of the theory. Thus a detailed analysis of the formal details of ITT-MCT may be beneficial. This also challenges the dogma, that the upper convected time derivative should be preferred in constitutive modeling. We propose that such an analysis should rest on a proper differential geometric formulation of stochastic processes on time-dependent manifolds. This allows to formulate covariance requirements and a careful analysis of how quantities have to be convected is possible. In this vein we will give the general Fokker-Planck equation for stochastical processes on a time-dependent embedded Riemannian manifold and discuss a first naive formula defining the corresponding frame-invariant evolution operator.