  The event-by-event fluctuations of suitably chosen observables in heavy ion
collisions at SPS, RHIC and LHC can tell us about the thermodynamic properties
of the hadronic system at freeze-out. By studying these fluctuations as a
function of varying control parameters, it is possible to learn much about the
phase diagram of QCD. As a timely example, we stress the methods by which
present experiments at the CERN SPS can locate the second-order critical
endpoint of the first-order transition between quark-gluon plasma and hadron
matter. Those event-by-event signatures which are characteristic of freeze-out
in the vicinity of the critical point will exhibit nonmonotonic dependence on
control parameters. We focus on observables constructed from the multiplicity
and transverse momenta of charged pions. We first consider how the
event-by-event fluctuations of such observables are affected by Bose-Einstein
correlations, by resonances which decay after freeze-out and by fluctuations in
the transverse flow velocity. We compare our thermodynamic predictions for such
noncritical event-by-event fluctuations with NA49 data, finding broad
agreement. We then focus on effects due to thermal contact between the observed
pions and a heat bath with a given (possibly singular) specific heat, and due
to the direct coupling between the critical fluctuations of the sigma field and
the observed pions. We also discuss the effect of the pions produced in the
decay of sigma particles just above threshold after freeze-out on the inclusive
pion spectrum and on multiplicity fluctuations. We estimate the size of these
nonmonotonic effects which appear near the critical point, including
restrictions imposed by finite size and finite time, and conclude that they
should be easily observable.
