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.