Stochastic Thermodynamic Interpretation of Information Geometry

In recent years, the unified theory of information and thermodynamics has been intensively discussed in the context of stochastic thermodynamics. The unified theory reveals that information theory would be useful to understand nonstationary dynamics of systems far from equilibrium. In this Letter, we have found a new link between stochastic thermodynamics and information theory well-known as information geometry. By applying this link, an information geometric inequality can be interpreted as a thermodynamic uncertainty relationship between speed and thermodynamic cost. We have numerically applied an information geometric inequality to a thermodynamic model of a biochemical enzyme reaction.

Figure 1

Schematic of information geometry on the manifold $S_2$. The manifold $S_2$ leads to the sphere surface of radius 2 (see also SM [56]). The statistical length $\mathcal{L}$ is bounded by the shortest length $\mathcal{D}=2\theta =2{\cos }^{-1}({\mathbf{r}}_{\mathrm{ini}}·{\mathbf{r}}_{\mathrm{fin}})$.

Figure 2

Numerical calculation of thermodynamic quantities in the three states model of enzyme reaction. We numerically show the non-negativity of $d{s}^2/d{t}^2\ge 0$ and $d{s}^2/d{t}^2=-⟨dF/dt⟩+⟨d\mathrm{\Delta }{\sigma }^{\mathrm{bath}}/dt⟩$ in the graph. We also show the total entropy change rate $⟨F⟩\ge 0$. We note that $d⟨F⟩/dt$ is not equal to $⟨dF/dt⟩$.

Figure 3

Numerical calculation of the thermodynamic uncertainty relationship in the three states model of enzyme reaction. We numerically show the geometric inequality $\mathcal{L}\ge \mathcal{D}\mathbf{(}\mathbf{r}(0);\mathbf{r}(\tau )\mathbf{)}$, the thermodynamic uncertainty relationship $\tau \ge {\mathcal{L}}^2/(2\mathcal{C})\ge [\mathcal{D}\mathbf{(}\mathbf{r}(0);\mathbf{r}(\tau )\mathbf{)}{]}^2/(2\mathcal{C})$, and the efficiency $\eta$ in the graph.