Diffusers, potential slides, and surmountability

A diffuser traversing an infinitely steep 'potential slide' might never overcome it. In such cases, the notion of First Passage Time - 'how long would it take till the diffuser surmounts the slide?' - is inadequate. Rather, the notion of relevance is that of Peak Height - 'how high-up the potential slide would the diffuser reach?'. Placing the 'potential slide' on the non-negative half-line, the Global Minimum attained by the diffuser's trajectory becomes a universal measure for the Peak Height. In this manuscript, the Global Minima of general diffuser systems on the non-negative half-line which are 'pushed away' from the origin are explored. We (i) compute the statistical distributions of the Global Minima; (ii) establish a reverse-engineering scheme which tells us how to design diffuser systems to obtain a pre-desired Global Minimum distribution; (iii) study the probabilistic limit-laws of the Global Minimum distributions; and (iv) characterize the class of diffuser systems with scale-free Global Minimum distributions. Lastly, we investigate the Global Minimum attained by an entire 'flock' of independent diffusers, whose initial locations are randomly scattered along the positive half-line.