  Strong-coupling statistical thermodynamics is formulated as Hamiltonian
dynamics of an observed system interacting with another unobserved system (a
bath). It is shown that the entropy production functional of stochastic
thermodynamics, defined as the log-ratio of forward and backward system path
probabilities, is in one-to-one relation with the log-ratios of joint initial
conditions of the system and the bath. A version of strong-coupling statistical
thermodynamics where the system-bath interaction vanishes at the beginning and
the end of a process is, as is also weak-coupling stochastic thermodynamics,
related the bath initially in equilibrium by itself. The heat is then the
change of bath energy over the process. It is discussed when this heat is a
functional of system history alone. The version of strong-coupling statistical
thermodynamics introduced by Seifert and Jarzynski is related to the bath
initially in conditional equilibrium with respect to the system. This leads to
heat as another functional of system history which needs to be determined by
thermodynamic integration. The log-ratio of forward and backward system path
probabilities in a stochastic process is finally related to log-ratios of
initial conditions of a combined system and bath. It is shown that the entropy
production formulas of stochastic processes under general class of time
reversals are given by change of a bath energy in a larger underlying
Hamiltonian system. The paper highlights the centrality of time reversal in
stochastic thermodynamics, also in the case of strong coupling.
