processes, satisfying $$P^\dagger[\sigma ] = \int \delta (\sigma - \sigma [X'_N]) P^\dagger[ X'_N] dX'_N.$$ By using the detailed fluctuation theorem (\[fluctuation2\]) and equality $\sigma [X_N] = - \sigma [X^\dagger_N]$, we obtain Crooks’ fluctuation theorem $$\frac{P^\dagger [-\sigma]}{P[\sigma]} = e^{-\sigma}.
\label{fluctuation_Crooks}$$ The detailed fluctuation theorem (\[fluctuation2\]) or Crooks’ fluctuation theorem (\[fluctuation\_Crooks\]) leads to the integral fluctuation theorem $$\langle e^{- \sigma} \rangle = 1,
\label{integral1}$$ where the ensemble average $\langle \cdots \rangle$ is taken over all trajectories under forward protocol (see Eq. (\[ensemble\_average\])). From the concavity of the exponential function, we obtain $$\langle \sigma \rangle \geq 0,
\label{second1}$$ which is an expression of the second law of thermodynamics: the ensemble-averaged entropy production is non-negative. By taking the ensemble average of the logarithm of both sides of Eq. (\[fluctuation2\]), we have $$\langle \sigma \rangle  = \int dX_N P[X_N] \ln \frac{P[X_N]}{P^\dagger[X_N^\dagger]},
\label{relative1}$$ which we will refer to as the Kawai-Parrondo-Broeck (KPB) equality [@Kawai; @Marin]. The right-hand side of Eq. (\[relative1\]) is the Kullback-Leibler divergence (or the relative entropy) of $P[X_N]$ and $P^\dagger[X_N^\dagger]$, which is always positive. Therefore, Eq. (\[relative1\]) reproduces inequality (\[second1\]).

If the probability distribution of $\sigma$ is Gaussian, the cumulant expansion of Eq. (\[ensemble\_average\]) leads to a variant of fluctuation-dissipation relation $$\langle \sigma 