Many species of plants are found in regions to which they are alien. Their global distributions are characterised by a family of exponential functions of the kind that arise in elementary statistical mechanics (an example in ecology is MacArthur's broken stick). We show here that all these functions are quantitatively reproduced by a model containing a single parameter – some global resource partitioned at random on the two axes of species number and site number. A dynamical model generating this equilibrium is a two fold stochastic process and suggests a curious and interesting biological interpretation in terms of niche structures fluctuating with time and productivity; with sites and species highly idiosyncratic. Idiosyncrasy implies that attempts to identify a priori those species likely to become naturalized are unlikely to be successful. Although this paper is primarily concerned with a particular problem in population biology, the two fold stochastic process may be of more general interest.