Entropy-related principles for non-equilibrium systems : theoretical foundations and applications to ecology and fluid dynamics

Abstract

The standard contemporary approaches to modelling non-equilibrium, many-body phenomena are to use detailed computational models or to analyse large datasets. Here we explore an alternative approach: the use of simple idealized models based on the novel application of statistical mechanics in a non-equilibrium, many-body setting. This thesis has two main elements. The first element is the development of idealized statistical mechanics models to address particular topics in physics and biology. The particular topics to be addressed are as follows: (i) we show using a statistical mechanics approach that global patterns of tree/grass co-existence in savanna ecology can be interpreted as primarily the result of the different water use of trees and grasses subject to considerable fluctuation due to other factors such as disturbances; (ii) we discuss the unification of stochastic and mechanistic approaches in community ecology using a statistical mechanics framework, and develop a statistical mechanical model of a plant ecosystem to predict species abundance patterns, stability-diversity relationships and a phase transition between vegetated and non-vegetated community states, aspects of ecosystem behaviour which have so far been studied largely in isolation from one another; (iii) we clarify aspects of Malkus's variational approach to anisotropic, inhomogeneous fluid turbulence and revisit his statistical stability argument for maximising flow quantities, discussing possible connections to statistical mechanical justifications for flow quantity maximisation. The second main element of this thesis is the discussion of some of the broader methodological concerns surrounding the use of statistical mechanics in a generalised non-equilibrium setting. A central theme of this discussion is emphasizing the distinction between statistical mechanics and statistical inference.