  Atom counting theory can be used to study the role of thermal noise in
quantum phase transitions and to monitor the dynamics of a quantum system. We
illustrate this for a strongly correlated fermionic system, which is equivalent
to an anisotropic quantum XY chain in a transverse field, and can be realized
with cold fermionic atoms in an optical lattice. We analyze the counting
statistics across the phase diagram in the presence of thermal fluctuations,
and during its thermalization when the system is coupled to a heat bath. At
zero temperature, the quantum phase transition is reflected in the cumulants of
the counting distribution. We find that the signatures of the crossover remain
visible at low temperature and are obscured with increasing thermal
fluctuations. We find that the same quantities may be used to scan the dynamics
during the thermalization of the system.
