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https://stockholmuniversity.zoom.us/j/530682073
Mean-field theories facilitate practical modeling of secular, large-scale properties of astrophysical or laboratory systems with fluctuations. Practitioners commonly assume wide scale separation between mean and fluctuating quantities to justify equality of ensemble and spatial or temporal averages. Often however, real systems do not exhibit such scale separation. This raises two questions: (I) what are the appropriate generalized equations of mean-field theories in the presence of mesoscale fluctuations? (II) how precise are theoretical predictions from them? We discuss both by considering examples in galactic dynamo and accretion disk theories. We first derive mean-field equations for different types of averaging, along with mesoscale correction terms that depend on the ratio of averaging scale to variation scale of the mean. We then show that even if these terms are small, predictions of mean-field theories can still have a significant precision error. This error has an intrinsic contribution from the theory input parameters and a filtering contribution from differences in the way observations and theory are projected through the measurement kernel. Minimizing the sum of these contributions can produce an optimal scale of averaging that makes the theory maximally precise. The precision error is important to quantify when comparing to observations because it quantifies the resolution of predictive power.