Exact results on diffusion in a piecewise linear potential with a time dependent sink

TL;DR

This paper provides exact solutions to the Smoluchowski equation with a time-dependent sink, covering various sink behaviors, and demonstrates how to derive the full probability distribution from the origin's distribution.

Contribution

It introduces an exact analytical method to solve the Smoluchowski equation with different time-dependent sink terms, advancing understanding of diffusion processes with dynamic sinks.

Findings

Exact solutions for linear, constant, inverse, and exponential sink variations

Method to derive full probability distribution from the origin's distribution

Enhanced analytical tools for diffusion in complex potentials

Abstract

The Smoluchowski equation with a time dependent sink term is solved exactly. In this method by knowing the probability distribution at the origin P(0,s), one may derive the probability distribution at all positions i.e., P(x,s). Further the exact solution for Smoluchowski equation are also provided in different cases where the sink term has linear, constant, inverse and exponential variation in time.