Exact Identity Linking Entropy Production and Mutual Information

Abstract

Linking entropy production (EP) to information is a key step toward data-driven nonequilibrium thermodynamics. We derive an exact identity for overdamped Langevin dynamics that equates the total EP rate to the mutual-information rate between an infinitesimal displacement and its time-symmetric midpoint, up to a bulk mean-flow contribution. This mapping elevates information theory to a thermodynamic calculus: the chain rule yields a canonical, nonnegative split into self and interaction EP, and leads to a tighter bound on learning rate with interaction EP as the necessary cost. As a proof of concept, applying the estimator to red-blood-cell flickering shows that interaction EP robustly exposes active signatures that conventional summaries can miss.

Figures (4)

• Figure 1: Schematic illustration of displacement $\Delta\bm{x}$ conditioned on $\bm{x}_m$. (Left) In equilibrium, detailed balance (DB) implies that $\bm{x}_m$ contains no information about $\Delta\bm{x}$. (Right) In nonequilibrium, a finite local mean velocity $\bm{v}_t(\bm{x}_m)$ breaks DB and biases $\Delta\bm{x}$ conditioned on $\bm{x}_m$, rendering the midpoint informative. In the infinitesimal-time limit $\mathrm{d}t\to 0$, $\Delta\bm{x}$ corresponds to $\mathrm{d}\bm{x}$ in the main text.
• Figure 2: Physical mechanism and numerical validation of Eq. \ref{['eq:main_identity']}. (a–-e) Linear model defined by $\dot{x}=-kx+(s+a)y+\xi_{x}$ and $\dot{y}=-ky+(s-a)x+\xi_{y}$ with $k=10$ and $\boldsymbol{\mathsf{D}}=\mathbf{I}$. Parameters are $(s,a)=(5,0)$ for equilibrium (a, c), $(0,50)$ for nonequilibrium steady state (NESS) (b, d), and $s=0$ for (e). (a, b) Drift $\bm{F}(\bm{x})$ (black) and local mean velocity $\bm{v}(\bm{x})$ (yellow) overlaid on steady-state density $p(\bm{x})$. Crosses indicate sampling locations for time midpoint $\bm{x}_{m}$. (c, d) Conditional displacements $\Delta\bm{x}|\bm{x}_{m}$ at the marked locations. Circles are centered at sample means with radius $1.96\times\mathrm{SD}$; dashed lines mark the means and black solid lines denote the origin. In equilibrium (c), distributions are indistinguishable. In NESS (d), they separate by the local mean velocity $\bm{v}(\bm{x}_{m})\Delta t$, rendering $\bm{x}_{m}$ informative about $\Delta\bm{x}$. ($\Delta t=10^{-3}$ in simulations) (e, f) Convergence of information rates $4\,I(\Delta\bm{x};\bm{x}_m)/\Delta t$ (dotted) to the total EP rate (solid) as $\Delta t \to 0$. (f) Nonlinear model $\dot{x}=-x+y+\epsilon y^{3}+\xi_{x}$, $\dot{y}=-y+\xi_{y}$. Eq. \ref{['eq:main_identity']} remains exact in both linear and nonlinear models.
• Figure 3: Information-theoretic EP decomposition and learning-rate bound in linear models. (a) Cascade model $\dot{x}=-k_x x + \alpha y + \xi_x$, $\dot{y}=-k_y y + \beta z + \xi_y$, and $\dot{z}=-k_z z + \xi_z$, partitioned into $A=\{x,y\}$ (blue) and $B=\{z\}$ (green). The coupling $\alpha$ controls internal within $A$ (not necessarily symmetric), while $\beta$ drives $A$ from $B$. (b, c) Local EP rate $\sigma_\mathrm{A}$ (solid blue) and its nonnegative components, self $\sigma_A^{\mathrm{self}}$ (dashed blue) and interaction $\sigma_{A|B}^{\mathrm{int}}$ (dashed orange), as functions of coupling strength ($k_{x,y,z}=2$); the black solid line denotes the total EP $\sigma_{\mathrm{AB}}$. (b) Varying $\alpha$ (with $\beta=1$) predominantly increases $\sigma_A^{\mathrm{self}}$, while $\sigma_{A|B}^{\mathrm{int}}$ remains nearly unchanged. (c) Varying $\beta$ (with $\alpha=1$) selectively increases $\sigma_{A|B}^{\mathrm{int}}$ with minimal change in $\sigma_A^{\mathrm{self}}$. (d) Numerical illustration of the system-level learning-rate bound Eq.\ref{['eq:tight_bound_total']} for linear Langevin dynamics $\dot{\bm{x}}=\boldsymbol{\mathsf{A}}\bm{x}+\sqrt{2\boldsymbol{\mathsf{D}}}\,\bm{\xi}$. We randomly sample stable $\boldsymbol{\mathsf{A}}$ and diagonal $\boldsymbol{\mathsf{D}}$, together with random bipartitions $A\cup B=\{1,\dots,d\}$ (see SM), and plot the ratio$|\dot I^{\mathrm A}|^{2}\mathcal{R}_{\mathrm{AB}}/\sigma^{\mathrm{int}}_{AB}$ versus $\sigma^{\mathrm{int}}_{AB}$ (log scale). Colors indicate dimension $d$; the dashed line marks saturation at unity.
• Figure 4: Application to OT-sensing RBC fluctuations terlizzi2024variance. (a) Schematic (not to scale) of the OT-sensing setup and the effective coordinates: measured outer-membrane $x$, hidden membrane--cortex attachment $y$, and a stochastic active force $\eta$ driving the cascade $\eta\!\to\! y\!\leftrightarrow\! x$. (b) Tightness ratios $\rho_{\mathrm{tot}}$ and $\rho_{\mathrm{int}}$ [Eq. \ref{['eq:Q_definitions']}] for passive (blue) and active (red) cells, evaluated using the parameter values reported in Ref. terlizzi2024variance. Each colored marker denotes one cell. Active cells cluster markedly closer to saturation in $\rho_{\mathrm{int}}$, whereas passive cells remain far from it.