Richard Blythe · Research Topics

Nonequilibrium steady states

Statistical mechanics, a theory due to Boltzmann and Gibbs taught to undergraduate physicists, tells us the probability that a physical system adopts a given configuration when it as equilibrium with its environment. When a system is instead driven by its environment, all bets are off. To try and get a handle on what nonequilibrium systems do, I focus on the steady states which are eventually reached when there is a constant driving force. Despite widespread ignorance of many of the properties of these states, we have nevertheless been able to show that heat production can be related to the statistics of microscopic trajectories. Through exact solutions of certain simple model systems, we have also demonstrated the utility of an object similar to an equilibrium partition function in predicting nonequilibrium phase transitions.

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•R. A. Blythe (2008) “Reversibility, heat dissipation and the importance of the thermal environment in stochastic models of nonequilibrium steady states”. Phys. Rev. Lett. 100 010601. Preprint➚ Published article➚
•R. A. Blythe and M. R. Evans (2007) “Nonequilibrium steady states of matrix product form: A solver's guide”. Review article, J Phys A 40 R333. Preprint➚ Published article➚
•R. A. Blythe and M. R. Evans (2002) “Lee-Yang zeros and phase transitions in nonequilibrium steady states”. Phys. Rev. Lett. 89 080601. Preprint➚ Published article➚

Particle reactions and population dynamics

How on earth are the dynamics of chemical reactions and biological populations related? The answer comes if you consider the ancestry of a population: two individuals (or some aspect of those individuals) share a common ancestor. So if you trace back the lines of descent you end up with a set of lineages (particles) that merge (coalesce) on contact. By studying this process, one can predict how a present-day population came to be. The results are relevant to cultural change (see below). We have also examined – and in some cases solved – various other reaction systems of interest to chemists and physicists, for example, those in which one species traps another.

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•G. J. Baxter, R. A. Blythe and A. J. McKane (2008) “Fixation and consensus times on a network: A unified approach”. Phys. Rev. Lett. 101 258701. Preprint➚ Published article➚
•R. A. Blythe and A. J. McKane (2007) “Stochastic models of evolution in genetics, ecology and linguistics”. Review for J. Stat. Mech.: Theor. Exp, P07018. Preprint➚ Published article➚
•A. J. Bray and R. A. Blythe (2002) “Exact asymptotics for one-dimensional diffusion with mobile traps”. Phys. Rev. Lett. 89 150601. Preprint➚ Published article➚

Modelling cultural change

Cultural traits (languages, fashions, technologies) change over time through processes of innovation, replication and selection – by evolution, in other words. Is it possible to predict what kinds of innovations will propagate, how far, and how long it will take them to do so? We’re currently using the formation of the New Zealand English language dialect as a testing ground for new ideas, and in particular to see how well modelling paradigms from physics and genetics (see above) are suited to the task. Meanwhile, learning plays a central role in cultural replication, so we're also trying to come up with some theories for how that works both for individuals and in spatially-extended populations too.