which the total system including heat baths is treated as a Hamiltonian system and the initial states of the heat baths in the forward and backward processes are the canonical distributions. This proof can confirm the physical validity of the detailed fluctuation theorem even for the non-Markovian dynamics with multi-heat baths, as the stochastic dynamics can be reproduced as that of a partial system of the total Hamiltonian system including the heat baths. We also note that several equalities that are similar but not equivalent to Eqs. (\[fluctuation1\]) and (\[fluctuation2\]) have been derived for different situations. For example, the transient fluctuation theorem has been discussed for dynamical systems in Ref. [@Evans2]. The fluctuation theorem for nonequilibrium steady states has been discussed for stochastic systems [@Kurchan0; @Spohn] and dynamical systems [@Evans; @Gallavotti].

From the detailed fluctuation theorem (\[fluctuation2\]), we can show Crooks’ fluctuation theorem as follows. We denote as $P[\sigma ]$ the probability of finding the entropy production $\sigma$ in the forward processes, satisfying $$P[\sigma ] = \int \delta (\sigma - \sigma [X_N]) P[ X_N] dX_N,$$ where $\delta (\cdot )$ is the delta function. On the other hand, let $P^\dagger[ \sigma] $ be the probability of obtaining $\sigma$ in the backward 