  It has experimentally been found by Lampo et al. [Biophys. J. 112, 532
(2017)] that, for two different types of cell, the distribution of the
diffusivities of RNA-protein particles over cytoplasm obeys an exponential law.
Then, an interesting issue has been pointed out: this exponential distribution
is the maximal entropy distribution. Here, time evolution of entropy associated
with local fluctuations of the diffusivity is studied. The entropy rate under
the diffusing diffusivity equation, which admits the exponential fluctuation as
its stationary solution, is shown to be positive. The present result is
expected to be useful for studying the dynamics of diffusivity fluctuations.
Furthermore, the distribution of time being required for characteristic
displacement of the RNA-protein particle is found to decay as a power law. A
comment is also made on a formal analogy with the thermodynamic relation
concerning temperature.
