Dynamical-system models of transport: chaos characteristics, the macroscopic limit, and irreversibility

Abstract

The escape-rate formalism and the thermostating algorithm describe relaxation towards a decaying state with absorbing boundaries and a steady state of periodic systems, respectively. It has been shown that the key features of the transport properties of both approaches, if modeled by low-dimensional dynamical systems, can conveniently be described in the framework of multibaker maps. In the present paper we discuss in detail the steps required to reach a meaningful macroscopic limit. The limit involves a sequence of coarser and coarser descriptions (projections) until one reaches the level of irreversible macroscopic advection-diffusion equations. The influence of boundary conditions is studied in detail. Only a few of the chaos characteristics possess a meaningful macroscopic limit, but none of these is sufficient to determine the entropy production in a general non-equilibrium state.

Chaotic saddles in a gravitational field: The case of inertial particles in finite domains

“…From the point of view of dynamical systems, our main interest is on how the finite vertical extension of the flow leads to transient chaos [22] in the advection dynamics. The problem of large spatial extension has already been treated in the context of chaotic transport [23][24][25] with a constant prescribed drift. Here, in a hydrodynamical context, we shall also be able to investigate how a constant drift sets in due to gravity and viscous drag, and why the constant settling velocity in the presence of a flow differs from that in a medium at rest.…”