Entropy production of entirely diffusional Laplacian transfer and the possible role of fragmentation of the boundaries

TL;DR

This paper analyzes entropy production in Laplacian diffusion around fractal boundaries, comparing analytical predictions with numerical simulations, and explores the applicability of the active-zone approximation in the near-field region.

Contribution

It provides the first detailed analytical and numerical study of entropy production near fractal boundaries, especially in the near-field, and reinterprets the active-zone approximation for Laplacian fields.

Findings

Active-zone approximation remains valid in the near-field.

Analytical predictions match numerical results.

Near-field behavior differs from traditional assumptions.

Abstract

The entropy production and the variational functional of a Laplacian diffusional field around the first four fractal iterations of a linear self-similar tree (von Koch curve) is studied analytically and detailed predictions are stated. In a next stage, these predictions are confronted with results from numerical resolution of the Laplace equation by means of Finite Elements computations. After a brief review of the existing results, the range of distances near the geometric irregularity, the so-called "Near Field", a situation never studied in the past, is treated exhaustively. We notice here that in the Near Field, the usual notion of the active zone approximation introduced by Sapoval et al. is strictly inapplicable. The basic new result is that the validity of the active-zone approximation based on irreversible thermodynamics is confirmed in this limit, and this implies a new interpretation of this notion for Laplacian diffusional fields.