About diffusion equations in bounded systems

TL;DR

This paper examines the relationship between diffusion equations and boundary conditions, emphasizing that in realistic systems, the boundary often acts as a discontinuity, challenging the assumption of their independence.

Contribution

It demonstrates that the common assumption of boundary conditions being independent of diffusion equations is only valid in peculiar cases, highlighting the importance of microscopic stochastic models.

Findings

Diffusion equations are derived as long-wavelength limits of microscopic stochastic models.

In realistic systems, boundary layers are effectively discontinuous, affecting the correlation between DE and BC.

The assumption of a single type of boundary condition may not hold in practical scenarios.

Abstract

Differential equations need boundary conditions (BC's) for their solution. It is commonly acknowledged that differential equations and BC's are representative of independent physical processes, and no correlations between them is required. Two recent papers [D. Hilhorst, et al, Nonlinear Analysis, 245, 113561 (2024); J-W.Chung, et al, Jour. Math. Phys. 65, 071501 (2024)] focus on diffusion equations (DE's) in a case with continuity of the physics at the boundary, where transport coefficients go smoothly to zero in a very small layer about it. They argue that, once the analytical expression of the DE is chosen, only one kind of BC's may emerge (e.g., Neumann rather than Dirichlet). In this paper, we show that this case is very peculiar. Indeed, DE's generally arise as long-wavelength limit out of a stochastic picture of microscopic dynamics, in the form of an integro-differential Master Equation (ME). Accordingly, they are justified only on a statistical basis, provide accurate pictures of the system's evolution only over large enough length and time scales. In realistic cases, the width of the interface between the interior and exterior of the system is much smaller than transport scales, providing effectively a discontinuity and therefore a decorrelation between DE and BC.