Fast Magnetic Reconnection and Spontaneous Stochasticity
  Magnetic field-lines in astrophysical plasmas are expected to be frozen-in at
scales larger than the ion gyroradius. The rapid reconnection of magnetic flux
structures with dimensions vastly larger than the gyroradius requires a
breakdown in the standard Alfv\'en flux-freezing law. We attribute this
breakdown to ubiquitous MHD plasma turbulence with power-law scaling ranges of
velocity and magnetic energy spectra. Lagrangian particle trajectories in such
environments become "spontaneously stochastic", so that infinitely-many
magnetic field-lines are advected to each point and must be averaged to obtain
the resultant magnetic field. The relative distance between initial magnetic
field lines which arrive to the same final point depends upon the properties of
two-particle turbulent dispersion. We develop predictions based on the
phenomenological Goldreich & Sridhar theory of strong MHD turbulence and on
weak MHD turbulence theory. We recover the predictions of the Lazarian &
Vishniac theory for the reconnection rate of large-scale magnetic structures.
Lazarian & Vishniac also invoked "spontaneous stochasticity", but of the
field-lines rather than of the Lagrangian trajectories. More recent theories of
fast magnetic reconnection appeal to microscopic plasma processes that lead to
additional terms in the generalized Ohm's law, such as the collisionless Hall
term. We estimate quantitatively the effect of such processes on the
inertial-range turbulence dynamics and find them to be negligible in most
astrophysical environments. For example, the predictions of the
Lazarian-Vishniac theory are unchanged in Hall MHD turbulence with an extended
inertial range, whenever the ion skin depth $\delta_i$ is much smaller than the
turbulent integral length or injection-scale $L_i.$
