Richard A. Blythe

Nonequilibrium steady states of matrix-product form: a solver's guide

We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven-diffusive systems—which includes exclusion processes—focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix-product form. In addition to a gentle introduction to this matrix-product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free-energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix-product state for a given model and more complicated variants of the matrix-product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.

Stochastic models of evolution in genetics, ecology and linguistics

We give an overview of stochastic models of evolution that have found applications in genetics, ecology and linguistics for an audience of non- specialists, especially statistical physicists. In particular, we focus mostly on neutral models in which no intrinsic advantage is ascribed to a particular type of the variable unit, for example a gene, appearing in the theory. In many cases these models are exactly solvable and furthermore go some way to describing observed features of genetic, ecological and linguistic systems.

Modeling language change: An evaluation of Trudgill's theory of the emergence of New Zealand English

Trudgill (2004) proposed that the emergence of New Zealand English, and of isolated new dialects generally, is purely deterministic. It can be explained solely in terms of the frequency of occurrence of particular variants and the frequency of interactions between different speakers in the society. Trudgills theory is closely related to usage-based models of language, in which frequency plays a role in the representation of linguistic knowledge and in language change. Trudgills theory also corresponds to a neutral evolution model of language change. We use a mathematical model based on Crofts usage-based evolutionary framework for language change (Baxter, Blythe, Croft, & McKane, 2006), and investigate whether Trudgills theory is a plausible model of the emergence of new dialects. The results of our modeling indicate that determinism cannot be a sufficient mechanism for the emergence of a new dialect. Our approach illustrates the utility of mathematical modeling of theories and of empirical data for the study of language change.

Cross-Situational Learning: An Experimental Study of Word-Learning Mechanisms

Cross-situational learning is a mechanism for learning the meaning of words across multiple exposures, despite exposure-by-exposure uncertainty as to the word’s true meaning. We present experimental evidence showing that humans learn words effectively using cross-situational learning, even at high levels of referential uncertainty. Both overall success rates and the time taken to learn words are affected by the degree of referential uncertainty, with greater referential uncertainty leading to less reliable, slower learning. Words are also learned less successfully and more slowly if they are presented interleaved with occurrences of other words, although this effect is relatively weak. We present additional analyses of participants’ trial-by-trial behavior showing that participants make use of various cross-situational learning strategies, depending on the difficulty of the word-learning task. When referential uncertainty is low, participants generally apply a rigorous eliminative approach to cross-situational learning. When referential uncertainty is high, or exposures to different words are interleaved, participants apply a frequentist approximation to this eliminative approach. We further suggest that these two ways of exploiting cross-situational information reside on a continuum of learning strategies, underpinned by a single simple associative learning mechanism.

Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra

We study the partially asymmetric exclusion process with open boundaries. We generalize the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and currents valid for all values of the asymmetry parameter q . Due to the relationship between the matrix algebra and the q -deformed quantum harmonic oscillator algebra we find that q -Hermite polynomials, along with their orthogonality properties and generating functions, are of great utility. We employ two distinct sets of q -Hermite polynomials, one for q 1. It turns out that these correspond to two distinct regimes: the previously studied case of forward bias (q 1) where the boundaries support a current opposite in direction to the bulk bias. For the forward bias case we confirm the previously proposed phase diagram whereas the case of reverse bias produces a new phase in which the current decreases exponentially with system size.

Utterance selection model of language change.

We present a mathematical formulation of a theory of language change. The theory is evolutionary in nature and has close analogies with theories of population genetics. The mathematical structure we construct similarly has correspondences with the Fisher-Wright model of population genetics, but there are significant differences. The continuous time formulation of the model is expressed in terms of a Fokker-Planck equation. This equation is exactly soluble in the case of a single speaker and can be investigated analytically in the case of multiple speakers who communicate equally with all other speakers and give their utterances equal weight. Whilst the stationary properties of this system have much in common with the single-speaker case, time-dependent properties are richer. In the particular case where linguistic forms can become extinct, we find that the presence of many speakers causes a two-stage relaxation, the first being a common marginal distribution that persists for a long time as a consequence of ultimate extinction being due to rare fluctuations.

Cross-situational learning: a mathematical approach

We present a mathematical model of cross-situational learning, in which we quantify the learnability of words and vocabularies. We find that high levels of uncertainty are not an impediment to learning single words or whole vocabulary systems, as long as the level of uncertainty is somewhat lower than the total number of meanings in the system. We further note that even large vocabularies are learnable through cross-situational learning.

Learning Times for Large Lexicons Through Cross-Situational Learning

Cross-situational learning is a mechanism for learning the meaning of words across multiple exposures, despite exposure-by-exposure uncertainty as to a words true meaning. Doubts have been expressed regarding the plausibility of cross-situational learning as a mechanism for learning human-scale lexicons in reasonable timescales under the levels of referential uncertainty likely to confront real word learners. We demonstrate mathematically that cross-situational learning facilitates the acquisition of large vocabularies despite significant levels of referential uncertainty at each exposure, and we provide estimates of lexicon learning times for several cross-situational learning strategies. This model suggests that cross-situational word learning cannot be ruled out on the basis that it predicts unreasonably long lexicon learning times. More generally, these results indicate that there is no necessary link between the ability to learn individual words rapidly and the capacity to acquire a large lexicon.

Building social cognitive models of language change

Studies of language change have begun to contribute to answering several pressing questions in cognitive sciences, including the origins of human language capacity, the social construction of cognition and the mechanisms underlying culture change in general. Here, we describe recent advances within a new emerging framework for the study of language change, one that models such change as an evolutionary process among competing linguistic variants. We argue that a crucial and unifying element of this framework is the use of probabilistic, data-driven models both to infer change and to compare competing claims about social and cognitive influences on language change.

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

We present a pedagogical account of the Lee-Yang theory of equilibrium phase transitions and review recent advances in applying this theory to nonequilibrium systems. Through both general considerations and explicit studies of specific models, we show that the Lee-Yang approach can be used to locate and classify phase transitions in nonequilibrium steady states.