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Here's a draft of a review article for the proceedings of the workshop I helped run at NIMBioS
Abstract. In this paper we review various information-theoretic characterizations of the approach to equilibrium in biological systems. The replicator equation, evolutionary game theory, Markov processes and chemical reaction networks all describe the dynamics of a population or probability distribution. Under suitable assumptions, the distribution will approach an equilibrium with the passage of time. Relative information - that is, the Kullback-Leibler divergence, or various generalizations thereof - provides a quantitative measure of how far from equilibirum the system is. We explain various theorems that give conditions under which relative information is nonincreasing. In biochemical applications these results can be seen as versions of the Second Law of Thermodynamics, stating that free energy can never increase with the passage of time. In ecological applications, they make precise the notion that a population gains information form its environment as it approaches equilibrium.
Perhaps the title should say 'relative information' since I believe that term makes more sense than the usual 'relative entropy' (as I explain in the paper). Unfortunately more people use the term 'relative entropy'. So, right now the title is a bit at odds with the rest of the paper, which annoys me. But I'm fairly happy with most of the paper. It needs just a bit more work. I'll blog about it in a while.