It is widely believed that statistical closure theories for dynamical systems provide statistics equivalent to those of the governing dynamical equations from which the former are derived. Here, we demonstrate counterexamples in the context of the widely used mean-field quasi-linear approximation applied to both deterministic and stochastic two-dimensional fluid dynamical systems. We compare statistics of numerical simulations of a quasi-linear model (QL) with statistics obtained by direct statistical simulation via a cumulant expansion closed at second order (CE2). We observe that although CE2 is an exact statistical closure for QL dynamics, its predictions can disagree with the statistics of the QL solution for identical parameter values. These disagreements are attributed to instabilities, which we term rank instabilities, of the second cumulant dynamics within CE2 that are unavailable in the QL equations.

Electrophoretic characterization of nano- and micro-metre scaled bubbles and drops is increasingly important in environmental and health sciences. Despite more than a hundred years of study, the interpretation of bubble electrophoresis data remains an unresolved fundamental problem that bridges fluid mechanics and interfacial science. This paper examines, from a theoretical perspective, how the electrophoretic mobility of small drops and bubbles responds to the interfacial kinetic-exchange rate and interfacial-charge mobility: factors that have been largely overlooked, but which provide new insights on the interpretation of $\zeta$-potentials, which are routinely used to assess surface charge density. A variety of outcomes are demonstrated, each reflecting subtle balances of hydrodynamic and electrical forces, modulated by interfacial thermodynamics and transport. Among the findings is that irreversibly bound charge with low interfacial mobility furnishes rigid-sphere behaviour; whereas interfacial charge with high mobility produces the characteristically high electrophoretic mobilities of non-conducting, uniformly charged fluid spheres. Outcomes are more complex when drops and bubbles have interfacial charge that seeks local equilibrium with the immediately adjacent electrolyte. For example, the present model shows that interfacial-charge mobility regularizes the singular behaviour predicted by theories for fluid spheres bearing high, perfectly uniform surface charge.