  We present some aspects of the fidelity approach to phase transitions based
on lower and upper bounds on the fidelity susceptibility that are expressed in
terms of thermodynamic quantities. Both commutative and non commutative cases
are considered. In the commutative case, in addition, a relation between the
fidelity and the nonequilibrium work done on the system in a process from an
equilibrium initial state to an equilibrium final state has been obtained by
using the Jarzynski equality.
