Fluctuation Theorem with Information Exchange: Role of Correlations in Stochastic Thermodynamics

We establish the fluctuation theorem in the presence of information exchange between a nonequilibrium system and other degrees of freedom such as an observer and a feedback controller, where the amount of information exchange is added to the entropy production. The resulting generalized second law sets the fundamental limit of energy dissipation and energy cost during the information exchange. Our results apply not only to feedback-controlled processes but also to a much broader class of information exchanges, and provide a unified framework of nonequilibrium thermodynamics of measurement and feedback control.

Figure 1

The Szilard engine and memory which are initially correlated with one bit ($=\ln 2$ nat) of the mutual information. We can then use this information to extract ${\beta }^{-1}\ln 2$ of work by means of feedback control. Here the information is used for deciding in which direction we perform an isothermal expansion.

Figure 2

Interaction between $\mathit{X}$ and $\mathit{Y}$. System $\mathit{X}$ evolves from $x$ to $x^{\prime }$ along a path $X_F$ in such a manner that depends on the information about $y$, while $y$ does not evolve in time.

Figure 3

(a) Position measurement on the Szilard engine ($\mathit{Y}$) by a memory system ($\mathit{X}$), where $x$ describes the phase-space point of the memory, and $y$ takes on 0 or 1 corresponding to “left” or “right.” The initial correlation is $⟨I_{xy}^i⟩=0$ and the final correlation is $⟨I_{x^{\prime }y}^f⟩=\ln 2$. (b) Feedback control on the Szilard engine ($\mathit{X}$) by a demon ($\mathit{Y}$), where $x$ describes the position of the particle, and $y$ ($=0$ or 1) is the measurement outcome stored in the memory. The initial correlation is $⟨I_{xy}^i⟩=\ln 2$ and the final correlation is $⟨I_{x^{\prime }y}^f⟩=0$.