Language change is driven by a constellation of acquisition and usage factors operating at two ontological levels: the level of the individual and the level of the population. This thesis proposes that an improved understanding of processes of language change can be obtained through the use of mathematical models that incorporate detailed mechanisms of language acquisition and use and derive diachronic predictions as mathematical theorems of those mechanisms. Four aspects of linguistic diachrony are singled out for detailed study in the four original publications constituting the core of this journal-format dissertation: the Constant Rate Effect, stable variation in multidimensional grammatical competition, effects of social network topology and rewiring, and the relationship between processes of change and synchronic frequency and spatial distributions of linguistic traits. Through mathematical analysis and computer simulations, the dissertation suggests (i) that diachronic patterns such as the Constant Rate Effect arise through an interaction of acquisition and usage effects, the latter modelled as probabilistic post-acquisition production biases that filter the underlying grammatical state; (ii) that the fundamental result on competition between two grammatical options that outlaws diachronically stable variation does not hold of multidimensional competition; (iii) that finite-size effects such as network topology and dynamic network rewiring may give rise to orderly phenomena such as S-curves even in the absence of traditional biasing factors; and (iv) that typological distributions of linguistic features arise through a dynamic interplay of faithful transmission and mutation, subject to both local and areal effects. As a general conclusion, I propose that language change can be understood as the product of neither acquisition nor usage factors alone, but that both types of factor need to be incorporated in mechanistic models which make the interrelationships between these factors explicit. Acquisition and use constitute a diachronic feedback loop leading to nonlinear equations of linguistic change; it is in many cases a consequence of these nonlinearities that bifurcations arise which may throw change trajectories onto qualitatively unexpected courses simply as a response to minute variation in relevant control parameters. This sheds new light on explanation in diachronic linguistics: to explain a process of change is to model it with mechanisms which derive facets of that process as empirical predictions. Mathematical models are particularly well suited to this task, as they force the modeller to be explicit and exact, both qualitatively and quantitatively, about the mechanisms involved; explanations of this kind, moreover, can be either traditionally deductive-nomological, or probabilistic.