  Particles in turbulence live complicated lives. It is nonetheless sometimes
possible to find order in this complexity. It was proposed in [Falkovich et
al., Phys. Rev. Lett. 110, 214502 (2013)] that pairs of Lagrangian tracers at
small scales, in an incompressible isotropic turbulent flow, have a statistical
conservation law. More specifically, in a d-dimensional flow the distance
$R(t)$ between two neutrally buoyant particles, raised to the power $-d$ and
averaged over velocity realizations, remains at all times equal to the initial,
fixed, separation raised to the same power. In this work we present evidence
from direct numerical simulations of two and three dimensional turbulence for
this conservation. In both cases the conservation is lost when particles exit
the linear flow regime. In 2D we show that, as an extension of the conservation
law, a Evans-Cohen-Morriss/Gallavotti-Cohen type fluctuation relation exists.
We also analyse data from a 3D laboratory experiment [Liberzon et al., Physica
D 241, 208 (2012)], finding that although it probes small scales they are not
in the smooth regime. Thus instead of $\left<R^{-3}\right>$, we look for a
similar, power-law-in-separation conservation law. We show that the existence
of an initially slowly varying function of this form can be predicted but that
it does not turn into a conservation law. We suggest that the conservation of
$\left<R^{-d}\right>$, demonstrated here, can be used as a check of isotropy,
incompressibility and flow dimensionality in numerical and laboratory
experiments that focus on small scales.
