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A commonly used test for natural selection has been to compare population differentiation for neutral molecular loci estimated by F ST and for the additive genetic component of quantitative traits estimated by Q ST . Past analytical and empirical studies have led to the conclusion that when averaged over replicate evolutionary histories, Q ST ¼ F ST under neutrality. We used analytical and simulation techniques to study the impact of stochastic fluctuation among replicate outcomes of an evolutionary process, or the evolutionary variance, of Q ST and F ST for a neutral quantitative trait determined by n unlinked diallelic loci with additive gene action. We studied analytical models of two scenarios. In one, a pair of demes has recently been formed through subdivision of a panmictic population; in the other, a pair of demes has been evolving in allopatry for a long time. A rigorous analysis of these two models showed that in general, it is not necessarily true that mean Q ST ¼ F ST (across evolutionary replicates) for a neutral, additive quantitative trait. In addition, we used finite-island model simulations to show there is a strong positive correlation between Q ST and the difference Q ST À F ST because the evolutionary variance of Q ST is much larger than that of F ST . If traits with relatively large Q ST values are preferentially sampled for study, the difference between Q ST and F ST will also be large and positive because of this correlation. Many recent studies have used tests of the null hypothesis Q ST ¼ F ST to identify diversifying or uniform selection among subpopulations for quantitative traits. Our findings suggest that the distributions of Q ST and F ST under the null hypothesis of neutrality will depend on speciesspecific biology such as the number of subpopulations and the history of subpopulation divergence. In addition, the manner in which researchers select quantitative traits for study may introduce bias into the tests. As a result, researchers must be cautious before concluding that selection is occurring when Q ST 6 ¼ F ST .

We show that there are no singular pseudo-self-similar solutions of the Navier-Stokes system with finite energy.

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