ANOMALOUS TRANSPORT:_FROM THEORIES ABOUT OSMOTIC PRESSURE TO A NEW APPROACH IN THE FRAMEWORK OF DYNAMICAL SYSTEMS

Abstract

Osmosis is a fundamental physical process involved in many biological phenomena and is exploited in many thecnological applications. In the last decade, the interest for problems has grown with the ever-incresing development of nano-thecnologies and the classical theories, like the Van't Hoff or Morse equations for osmotic pressure, have been observed to fail in many circumstances and this has prompted the development of a quantity of models trying to describe osmotic processes in non-classical regimes. Recently some authors tried to modify the classical laws introducing some parameters of the problem neglected before such as the solute-solvent interaction and the finite volume of solute molecules, or the mechanism of transport of molecules inside the pores of the membrane. This last issue concerning is of particular interest and deserves further investigation. In a current article that focuses on a new kind of phenomenon called transient osmosis we find, first of all, a clarification of the phenomenon of interest, because is an equilibrium phenomenon and does not depend on the process which leads to it. Moreover the proposed model presents the features of a process similar to osmosis, but influenced by the anomalous beahvior of transport inside the pores. In accordance with these ideas, the aim of our work is to build a mathematical model to describe the behavior of molecules inside the pores of membranes at the nanometric scale, where a wide variety of anomalous transport phenomena occurs. We take our inspiration observing the features of a tipical tool used to simulate transport of matter in nanopores: the polygonal billiards. An idealization of them has been given in terms of chains of deterministic maps, and this is the way we undertake in this work. Therefore we introduce a class of chains of deterministic maps, which we call slicer maps, which enjoy some features of the dynamics of polygonal billiards, trying to reproduce the same wide range of anomalous behaviors. We manage to do this by furnishing our map of infinitely many scales and we study the transport behaviors trhough the analysis of the mean square displacement produced by the dynamics of these maps. We also compare the behavior of our maps with a sthocastic model often used to study anomalous transport problems: the Levy walk. We evince that a trivial deterministic map, like our slicer map, seems to behave indistinguishably from a Levy walk. Nevertheless these chain of slicer maps have an infinite range of scales introduced by hand, hence they are rather different fro polygonal billiards, from this point of view. Therefore, we try to overcame this difficulty different kind of modifications, both in the dynamics and in the process of producing infinitely many scales. These variations of the model raise some new questions that remain open to further investigations