Unravelling the Flow of Information in a Nonequilibrium Process in the Presence of Hydrodynamic Interactions

Technical Summary: Unravelling the Flow of Information in a Nonequilibrium Process in the Presence of Hydrodynamic Interactions

Problem Statement
Identifying the origin of nonequilibrium characteristics in interacting systems with multiple degrees of freedom remains a central challenge in nonequilibrium physics. While mutual information (MI) has been successfully used to quantify statistical dependencies in mesoscopic systems, it is inherently symmetric and time-independent for stationary processes. Consequently, standard MI fails to capture directional information flow or causal driving, which are essential for understanding nonequilibrium functionality. A specific conundrum arises in hydrodynamically coupled systems: previous work demonstrated that increasing interaction strength (by reducing particle separation) lowers the total irreversibility of the full system, yet increases irreversibility when observed in coarse-grained subspaces containing only the particles. The fundamental question addressed here is how to systematically disentangle internal dynamics from environmental influences to identify the true source of irreversibility and driving, and how hydrodynamic interactions shape the spatiotemporal structure of information flow to produce these opposing trends.

Methodology
The authors investigate a minimal experimental model consisting of two hydrodynamically coupled colloidal particles confined in separate optical harmonic traps. One trap is driven by an external exponentially correlated noise (Ornstein-Uhlenbeck process), modeling active fluctuations with memory, while the other remains fixed. The system is described by coupled linear Langevin equations where hydrodynamic interactions introduce dissipative, non-conservative couplings between the particles.

To analyze the system, the authors employ a suite of information-theoretic measures applied to both experimental trajectories (recorded at 10 kHz) and numerical simulations of the corresponding Langevin model:

1. Mutual Information (MI): Used to quantify the statistical dependence between variables ($x_1$, $x_2$, and the drive $\lambda$) in the steady state.
2. Time-Delayed Mutual Information (TDMI): Defined as $MI_{x \to y}(\tau)$, this asymmetric measure quantifies how the past of $x$ predicts the future of $y$, allowing for the detection of causal directionality and effective time lags.
3. Transfer Entropy (TE): Defined as $TE_{x \to y}(\tau)$, this measure quantifies the reduction in uncertainty of $y$'s future given the history of $x$, conditioned on $y$'s own history. This isolates the information exchanged specifically between subsystems, excluding self-correlations.

The authors utilize a k-nearest neighbor (kNN) estimator for data-driven computation of MI and TDMI from trajectories, and a Markov-state-model (MSM)-based technique for TE estimation. Analytical solutions derived from the Gaussian nature of the system's steady-state distribution serve as benchmarks for the numerical and experimental results.

Key Results

• Statistical Dependencies: The study confirms that the driven particle ($x_1$) exhibits the strongest dependence on the external drive ($\lambda$). Interestingly, as hydrodynamic interaction strength ($\epsilon$) increases, the mutual information between the drive and the driven particle ($MI_{\lambda x_1}$) decreases, while the mutual information between the two particles ($MI_{x_1 x_2}$) and between the drive and the passive particle ($MI_{\lambda x_2}$) increases. This suggests that strong hydrodynamic coupling effectively screens the driven particle from the direct impact of the non-equilibrium drive.
• Directional Information Flow: Time-delayed measures reveal a hierarchical information flow. For moderate nonequilibrium strengths, the primary flow is $\lambda \to x_1 \to x_2$. However, a non-zero reverse flow ($x_2 \to x_1$) is detected, indicating that information can backflow from the passive particle to the driven one due to hydrodynamic coupling.
• Transition to Unidirectionality: The authors find that the information pathway is bidirectional at moderate driving strengths but becomes fully unidirectional ($\lambda \to x_1 \to x_2$) when the nonequilibrium driving strength is significantly enhanced ($\theta \gg 1$).
• Asymmetry and Coarse-Graining: An asymmetry parameter based on transfer entropy is introduced to quantify causal directionality. The results show that increasing interaction strength ($\epsilon$) enhances the asymmetry (and thus the nonequilibrium character) in the coarse-grained subspaces ($x_1, x_2$) and ($\lambda, x_2$), while reducing it in the ($\lambda, x_1$) subspace. This differential behavior across subspaces provides a mechanistic explanation for the previously observed counterintuitive trend where total irreversibility decreases with interaction strength, while coarse-grained irreversibility increases.

Significance and Claims
The paper claims that information-theoretic tools, specifically time-delayed mutual information and transfer entropy, provide a systematic, data-driven framework for dissecting information flow in complex, hydrodynamically coupled nonequilibrium systems. The primary contribution is the demonstration that these measures can:

1. Identify the driving node and reconstruct the directional dependencies between particles.
2. Explain the apparent contradiction in irreversibility trends observed at different levels of coarse-graining.
3. Uncover the hidden structure of nonequilibrium dynamics, specifically how dissipative hydrodynamic interactions create emergent non-multipartite structures that influence entropy production.

The authors conclude that these directional information measures are essential for understanding how interactions shape dynamics and thermodynamics in mesoscopic systems, offering a pathway to study similar effects in more complex biological or physical systems without inventing new experimental proposals beyond the scope of the current study.