Recently there has been growing interest in the use of maximum relative entropy (MaxREnt) as a tool for statistical inference in ecology. In contrast, here we propose MaxREnt as a tool for applying statistical mechanics to ecology. We use MaxREnt to explain and predict species abundance patterns in ecological communities in terms of the most probable behaviour under given environmental constraints, in the same way that statistical mechanics explains and predicts the behaviour of thermodynamic systems. We show that MaxREnt unifies a number of different ecological patterns: (i) at relatively local scales a unimodal biodiversity–productivity relationship is predicted in good agreement with published data on grassland communities, (ii) the predicted relative frequency of rare vs. abundant species is very similar to the empirical lognormal distribution, (iii) both neutral and non-neutral species abundance patterns are explained, (iv) on larger scales a monotonic biodiversity–productivity relationship is predicted in agreement with the species-energy law, (v) energetic equivalence and power law self-thinning behaviour are predicted in resource-rich communities. We identify mathematical similarities between these ecological patterns and the behaviour of thermodynamic systems, and conclude that the explanation of ecological patterns is not unique to ecology but rather reflects the generic statistical behaviour of complex systems with many degrees of freedom under very general types of environmental constraints.