  In view of the relation between information and thermodynamics we investigate
how much information about an external protocol can be stored in the memory of
a stochastic measurement device given an energy budget. We consider a layered
device with a memory component storing information about the external
environment by monitoring the history of a sensory part coupled to the
environment. We derive an integral fluctuation theorem for the entropy
production and a measure of the information accumulated in the memory device.
Its most immediate consequence is that the amount of information is bounded by
the average thermodynamic entropy produced by the process. At equilibrium no
entropy is produced and therefore the memory device does not add any
information about the environment to the sensory component. Consequently, if
the system operates at equilibrium the addition of a memory component is
superfluous. Such device can be used to model the sensing process of a cell
measuring the external concentration of a chemical compound and encoding the
measurement in the amount of phosphorylated cytoplasmic proteins.
