\rangle = \frac{1}{2}(\langle \sigma^2 \rangle - \langle \sigma \rangle^2),
\label{FDT1}$$ which indicates that $\langle \sigma \rangle$ is determined by the fluctuation of $\sigma$. Equality (\[FDT1\]) is an expression of the fluctuation-dissipation theorem of the first kind, which gives a special case of the Green-Kubo formula [@Evans2].

In the case of an isothermal process with a single heat bath, the entropy production reduces to $$\sigma [X_N] = \beta (W[X_N] - \Delta F),$$ where $W[X_N]$ is the work performed on the system during the process, and $\Delta F$ is the difference of the free energies for the initial and final Hamiltonians (see also Appendix A for details). Under this situation, Eq. (\[integral1\]) leads to $$\langle e^{- \beta W} \rangle = e^{-\beta \Delta F}, 
\label{Jarzynski1}$$ which is the Jarzynski equality [@Jarzynski1]. The second law of thermodynamics then reduces to $$\langle W \rangle \geq \Delta F.
\label{second_work}$$

Measurement
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In this section, we formulate and investigate the effect of measurements on nonequilibrium dynamics.

Classical Measurement and Mutual Information
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In this subsection, we review the general framework of a measurement on a probabilistic variable, which can be applied to a broad class of measurements on classical systems. Let $x$ be an arbitrary probability variable 