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We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter 
$\nu $: the regularized dynamics is globally defined for each 
$\nu> 0$, and the original singular system is recovered in the limit of vanishing 
$\nu $. We prove that this limit yields a unique statistical solution independent of regularization when the deterministic system possesses a chaotic attractor having a physical measure with the convergence to equilibrium property. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.
We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the magnetohydrostatic (steady three-dimensional incompressible Euler) equations with a small force. The solutions are also ‘nearly’ quasisymmetric. The primary idea is, given a desired quasisymmetry direction 
$\xi$, to change the smooth structure on space so that the vector field 
$\xi$ is Killing for the new metric and construct 
$\xi$–symmetric solutions of the magnetohydrostatic equations on that background by solving a generalized Grad–Shafranov equation. If 
$\xi$ is close to a symmetry of Euclidean space, then these are solutions on flat space up to a small forcing.
A Lagrangian fluctuation–dissipation relation has been derived in a previous work to describe the dissipation rate of advected scalars, both passive and active, in wall-bounded flows. We apply this relation here to develop a Lagrangian description of thermal dissipation in turbulent Rayleigh–Bénard convection in a right-cylindrical cell of arbitrary cross-section, with either imposed temperature difference or imposed heat flux at the top and bottom walls. We obtain an exact relation between the steady-state thermal dissipation rate and the time 
$\unicode[STIX]{x1D70F}_{mix}$ for passive tracer particles released at the top or bottom wall to mix to their final uniform value near those walls. We show that an ‘ultimate regime’ with the Nusselt number scaling predicted by Spiegel (Annu. Rev. Astron., vol. 9, 1971, p. 323) or, with a log correction, by Kraichnan (Phys. Fluids, vol. 5 (11), 1962, pp. 1374–1389) will occur at high Rayleigh numbers, unless this near-wall mixing time is asymptotically much longer than the free-fall time 
$\unicode[STIX]{x1D70F}_{free}$. Precisely, we show that 
$\unicode[STIX]{x1D70F}_{mix}/\unicode[STIX]{x1D70F}_{free}=(RaPr)^{1/2}/Nu,$ with 
$Ra$ the Rayleigh number, 
$Pr$ the Prandtl number, and 
$Nu$ the Nusselt number. We suggest a new criterion for an ultimate regime in terms of transition to turbulence of a thermal ‘mixing zone’, which is much wider than the standard thermal boundary layer. Kraichnan–Spiegel scaling may, however, not hold if the intensity and volume of thermal plumes decrease sufficiently rapidly with increasing Rayleigh number. To help resolve this issue, we suggest a program to measure the near-wall mixing time 
$\unicode[STIX]{x1D70F}_{mix}$, which is precisely defined in the paper and which we argue is accessible both by laboratory experiment and by numerical simulation.
