Reversible escape from a well across a barrier by diffusion in one dimension

Understanding how particles move across potential barriers is a central challenge in statistical physics. Prior research has shown that diffusion processes are influenced by the shape and height of energy landscapes. However, the specific behavior of escape from a parabolic well over a parabolic barrier remains underexplored. This uncertainty drives the need for more detailed studies. The long-time behavior of occupation probabilities is not yet fully understood. No prior work had resolved the role of barrier height in this context. The decay of occupation probability is a key area of interest. The influence of return dynamics on escape rates is still debated. This gap motivated the current investigation into one-dimensional diffusion.

The goal of this study is to explore how particles escape from a parabolic well over a parabolic barrier in one dimension. The focus is on the time-dependent behavior of occupation probabilities. The study addresses the role of barrier height in this process. The researchers aim to understand the long-time decay of the well's occupation probability. They also investigate how repeated returns affect the escape dynamics. The motivation is to clarify the deviation from Kramers' theory for moderately high barriers. The study seeks to model the escape process using a rate equation with memory. The researchers propose to test the validity of a phenomenological approach.

The researchers used a one-dimensional model with a parabolic well and a parabolic barrier. They varied the barrier height to observe its effect on escape dynamics. The study tracked the occupation probability of the well over time. A phenomenological rate equation was applied to describe the time dependence. The equation included a memory term to account for past states. A source term was added to model the return rate from outer space. The researchers compared the results to the Kramers expression. They analyzed how the amplitude of the long-time tail changed with barrier height.

The occupation probability of the well decays inversely with the square root of time at long times. This decay is due to repeated returns after excursions into outer space. The amplitude of the long-time tail increases as the barrier height decreases. The time dependence is well described by a rate equation with memory. The source term accounts for the return rate from the outer region. For moderately high barriers, the rate coefficient deviates from the Kramers expression. The deviation becomes more pronounced as the barrier height increases. The model successfully captures the non-exponential decay behavior.

The study shows that escape from a parabolic well over a parabolic barrier follows a specific time dependence. The long-time decay is inversely proportional to the square root of time. The amplitude of the decay increases with lower barrier heights. The researchers propose that a rate equation with memory accurately describes the process. The source term is essential for modeling return dynamics. The deviation from Kramers' expression is observed for moderately high barriers. The results suggest that the escape mechanism is sensitive to barrier height. The model provides a useful framework for understanding one-dimensional diffusion.