  We consider the amount of energy dissipated during individual avalanches at
the depinning transition of disordered and athermal elastic systems. Analytical
progress is possible in the case of the Alessandro-Beatrice-Bertotti-Montorsi
(ABBM) model for Barkhausen noise, due to an exact mapping between the energy
released in an avalanche and the area below a Brownian path until its first
zero-crossing. Scaling arguments and examination of an extended mean-field
model with internal structure show that dissipation relates to a critical
exponent recently found in a study of the rounding of the depinning transition
in presence of activated dynamics. A new numerical method to compute the
dynamic exponent at depinning in terms of blocked and marginally stable
configurations is proposed, and a kind of `dissipative anomaly'- with
potentially important consequences for nonequilibrium statistical mechanics- is
discussed. We conclude that for depinning systems the size of an avalanche does
not constitute by itself a univocal measure of the energy dissipated.
