  We investigate the properties of QCD at finite isospin chemical potential at
zero and non-zero temperatures. This theory is not affected by the sign problem
and can be simulated using Monte-Carlo techniques. With increasing isospin
chemical potential and temperatures below the deconfinement transition the
system changes into a phase where charged pions condense, accompanied by an
accumulation of low modes of the Dirac operator. The simulations are enabled by
the introduction of a pionic source into the action, acting as an infrared
regulator for the theory, and physical results are obtained by removing the
regulator via an extrapolation. We present an update of our study concerning
the associated phase diagram using 2+1 flavours of staggered fermions with
physical quark masses and the comparison to Taylor expansion. We also present
first results for our determination of the equation of state at finite isospin
chemical potential and give an example for a cosmological application. The
results can also be used to gain information about QCD at small baryon chemical
potentials using reweighting with respect to the pionic source parameter and
the chemical potential and we present first steps in this direction.
