  Fast magnetic reconnection is defined by the topology of the magnetic field
lines changing on a timescale that is approximately an order of magnitude
longer than the topology-conserving ideal-evolution timescale. Fast
reconnection is an intrinsic property of Faraday's law when the evolving
magnetic field depends non-trivially on all three spatial coordinates and is
commonly observed -- even when the effects that allow topology breaking are
arbitrarily small. The associated current density need only be enhanced by a
factor of approximately ten and flows in thin but broad ribbons along the
magnetic field. These results follow from the variation in the separation of
neighboring pairs of magnetic field lines, which in an ideal evolution
typically increases exponentially with time, and the existence of a spatial
scale below which magnetic field lines freely change their identities due to
non-ideal effects such as resistivity. Traditional reconnection theory ignores
exponentially large variations and relies on the current density reaching a
magnitude that is exponentially larger than is actually required. Here, an
analysis of the behavior of magnetic field lines in the neighborhood of an
arbitrarily chosen line is used to obtain more precise and rigorous results on
intrinsic reconnection. The maximum parallel kinetic energy of collisionless
charged particles is shown to have an exponential increase in time during a
generic magnetic evolution.
