  We compute statistical properties of the stochastic entropy production
associated with the nonstationary transport of heat through a system coupled to
a time dependent nonisothermal heat bath. We study the 1-d stochastic evolution
of a bound particle in such an environment by solving the appropriate Langevin
equation numerically, and by using an approximate analytic solution to the
Kramers equation to determine the behaviour of an ensemble of systems. We
express the total stochastic entropy production in terms of a relaxational or
nonadiabatic part together with two components of housekeeping entropy
production and determine the distributions for each, demonstrating the
importance of all three contributions for this system. We compare the results
with an approximate analytic model of the mean behaviour and we further
demonstrate that the total entropy production and the relaxational component
approximately satisfy detailed fluctuation relations for certain time
intervals. Finally, we comment on the resemblance between the procedure for
solving the Kramers equation and a constrained extremisation, with respect to
the probability density function, of the spatial density of the mean rate of
production of stochastic entropy.
