Friction in liquids arises from conservative forces between molecules and atoms
and, even in simple liquids, must not be thought of as a single constant;
rather it is a function of the temporal resolution of the experiment. As a
consequence, the effective dynamics of single molecules and solutes is
non-Markovian, being one essential ingredient to vitrification mechanisms
and viscoelasticity of soft materials. Based on high-precision simulations of
three prototypical liquids, including water, we have obtained
frequency-resolved friction data from atomistic to hydrodynamic scales [1].
Combining with theory, we show that friction in liquids emerges abruptly at a
characteristic frequency, beyond which viscous liquids appear as
non-dissipative, elastic solids. Concomitantly, the molecules experience
Brownian forces that display persistent correlations and long-lasting memory. A
critical test of the generalised Stokes-Einstein relation (GSER), mapping the
friction of single molecules to the viscoelastic response of the macroscopic
sample, disproves the relation for Newtonian fluids, but substantiates it
exemplarily for water and a moderately supercooled liquid.
In the second part, I will introduce a generalised master equation (GME) that
results from coarse-graining of diffusion processes in partitioned spaces [2].
The spatial domains can differ with respect to their diffusivity, geometry, and
dimensionality, but can also refer to transport modes alternating between
diffusive, driven, or anomalous motion. This is motivated by a range of
biological applications related to the transport of molecules in cells, but
finds also use in other fields. Mapping random trajectories in a continuous
space to a small set of spatial domains or states yields a non-Markovian jump
process. The GME governing the time evolution of this process is then obtained
under a renewal assumption of the original dynamics. The approach is
exemplified and validated for a target search problem in a two-domain model.
First-passage time (FPT) densities are solved analytically in Laplace domain
and, after a numerical backtransform, FPT densities are obtained that cover
many decades in time. Our results confirm that the geometry and heterogeneity
of the space can introduce additional characteristic time scales.
[1] A. V. Straube, B. G. Kowalik, R. R. Netz, and F. Höfling, Commun. Phys. 3, 126 (2020).
[2] D. Frömberg and F. Höfling, J. Phys. A: Math. Theor. 54, 215601 (2021).