  In macroscopic systems behavior is usually reproducible and fluctuations,
which are deviations from the typically observed mean values, are small. But
almost all inverse problems in the physical and biological sciences are
ill-posed and these fluctuations are highly 'amplified'. Using stochastic
thermodynamics we describe a system in equilibrium kicked to a state far from
equilibrium and the following dissipative process back to equilibrium. From the
observed value at a certain time after the kick the magnitude of the kick
should be estimated, which is such an ill-posed inverse problem and
fluctuations get relevant. For the model system of a kicked Brownian particle
the time-dependent probability distribution, the information loss about the
magnitude of the kick described by the Kullback-Leibler divergence, and the
entropy production derived from the observed mean values are given. The
equality of information loss caused by fluctuations and mean entropy production
is shown for general kicked dissipative processes from stochastic
thermodynamics following the derivation of the Jarzynski and Crooks equalities.
The information-theoretical interpretation of the Kullback-Leibler divergence
(Chernoff-Stein Lemma) allows us to describe the influence of the fluctuations
without knowing their distributions just from the mean value equations and thus
to derive very applicable results, e.g., by giving thermodynamic limits of
spatial resolution for imaging.
