Time-dependent Langevin modeling and Monte-Carlo simulations of diffusion in one-dimensional ion channels

Charge transport in narrow channels exists abundantly in nature and electronic devices. In this article, we adopt the notion of the Fokker-Planck equation, fixed point theorem and spectral analysis to derive a semi-analytical solution for the simple ion transport in one-dimensional channels, driven solely by an external electric field. Various diffusion limits are scrutinized. We find that when an initial state is sufficiently close to the final state, the solution converges for no and low diffusion systems. However, such systems might easily blowup for other cases when the external fields are not properly chosen. In particular, oscillating fields will always ease the blowup. We also find that the closeness of initial charge density distributions to the final one and the strength of external fields can affect the time of convergence under a normal diffusion limit. Intriguingly, systems will converge rapidly under a large diffusion limit almost regardless of the strength of fields. The paper demonstrates the importance of diffusion, initial charge configurations and the nature of external fields on determining the time of convergence or even maintaining the stability of Langevin systems, especially when diffusion is low. Most numerical results are supported by the relevant mathematical analysis and the existence of such Fokker-Planck equation with constant boundary conditions are discussed. Last but not least, a modified Monte-Carlo simulation is used to support the idea of viscosity solutions, proposed by our theoretical outcomes.