  In Echo experiments, imperfect time-reversal operations are performed on a
subset of the total number of degrees of freedom. To capture the physics of
these experiments, we introduce a partial fidelity, the Boltzmann echo, where
only part of the system's degrees of freedom can be time-reversed. We present a
semiclassical calculation of the Boltzmann echo. We show that, as the
time-reversal operation is performed more and more accurately, the decay rate
of the Boltzmann echo saturates at a value given by the decoherence rate of the
controlled degrees of freedom due to their coupling to uncontrolled ones. We
connect these results with NMR spin echo experiments.
