Force Geometry and Irreversibility in Nonequilibrium Dynamics

Abstract: Recent experiments have revealed heterogeneous dissipation in optically trapped systems, often anticorrelated with local positional fluctuations, exposing a structural gap in the scalar stochastic thermodynamic description. While the conventional scalar framework successfully quantifies dissipation through currents and entropy production rates, it does not reveal the underlying vectorial force geometry that shapes spatial dissipation patterns. Here, we bridge this gap by identifying force geometry as an organizing principle for nonequilibrium thermodynamics and introducing force alignment as a geometric determinant of irreversibility. We show that entropy production depends not only on force magnitudes but also on the relative orientation between deterministic driving forces and entropic gradients, vanishing only under exact anti-alignment with matched magnitudes. We formalize this geometric alignment through a time-dependent force-correlation coefficient, quantifying the relative orientation between the forces. This yields an instantaneous geometric lower bound on entropy production that remains valid even when force magnitudes are matched. For overdamped dynamics, perfect anti-alignment defines a thermodynamic stall where net transport vanishes and the lower bound on entropy production is saturated. This force-level perspective provides a structural explanation for the experimentally observed fluctuation-dissipation anticorrelation and nonuniform dissipation. We construct geometric control charts for both constant dragging and sinusoidal driving protocols, explicitly locating experimental operating points within this force-space representation. Together, these results position force geometry as a unifying structural perspective on irreversibility, spanning active biological systems, microrheology, and naturally extending to underdamped dynamics.

• The paper shows that force geometry, by analyzing the alignment between external and entropic forces, offers a new framework for understanding irreversible dissipation.
• It presents a force-level decomposition of entropy production, detailing contributions from force variances and cross-correlations measured via the Pearson coefficient.
• The study connects theory to experiment using harmonic trap examples and geometric design charts, providing actionable insights for controlled nonequilibrium processes.

Force Geometry as a Structural Determinant of Irreversibility

Motivation and Problem Statement

Traditional stochastic thermodynamics predominantly quantifies irreversibility via scalar observables such as entropy production rates and probability currents, offering limited insight into the structural organization of dissipative processes at the force level. Recent spatially resolved experiments on driven biological systems [DiTerlizzi 2024] demonstrate pronounced spatial heterogeneity in entropy production, correlated with anti-aligned dissipation and positional fluctuations. This empirical observation exposes inadequacies in scalar formulations, motivating the need for a geometric perspective on thermodynamic forces. The present paper introduces force geometry—specifically the relative alignment between deterministic driving and entropic (information-theoretic) forces—as a unifying principle for the organization of nonequilibrium thermodynamics.

Force-Level Decomposition and Entropy Production

The paper formalizes entropy production in overdamped stochastic dynamics as a quadratic function of the net thermodynamic force, decomposed into external ($F_{\mathrm{ext}}$) and information-theoretic ($F_{\mathrm{info}}$) components. The entropy production rate is expressed as

$$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$

where $F_{\mathrm{ext}} = -\nabla U$ encodes deterministic driving (e.g., optical trapping) and $F_{\mathrm{info}} = -k_B T \nabla \ln \rho$ represents the entropic gradient from spatial probability inhomogeneity. Expanding yields three distinct contributions: the variance of $F_{\mathrm{ext}}$, the variance of $F_{\mathrm{info}}$, and the force-force cross-correlation term, which exclusively encodes geometric alignment. This decomposition is functionally richer than scalar current-based descriptions, revealing that even with matched force variances, spatial configuration and mutual orientation crucially determine dissipation.

Figure 1: Force correlation diagram depicting instantaneous joint fluctuations of $F_{\mathrm{ext}}$ and $F_{\mathrm{info}}$; green quadrants show anti-alignment, red quadrants show alignment, and dashed diagonal indicates force cancellation.

Geometric Stall, Reversibility, and Correlation Hierarchy

Perfect anti-alignment—where $F_{\mathrm{info}} = -F_{\mathrm{ext}}$ everywhere—defines a reversible regime with vanishing entropy production and transport. In practical nonequilibrium operation, a global anti-alignment (Pearson force-correlation coefficient $F_{\mathrm{info}}$0) leads to zero mean transport but typically nonzero entropy production due to local imbalances. The paper introduces "geometric waste" as entropy production arising from departures either in alignment ($F_{\mathrm{info}}$1 deviating from $F_{\mathrm{info}}$2) or magnitude matching. This hierarchy clarifies the energetic distance between stall regimes (zero mean transport, finite dissipation) and true reversibility.

Instantaneous Force-Correlation as a Geometric Measure

The key quantity organizing force geometry is the instantaneous Pearson correlation coefficient $F_{\mathrm{info}}$3 between $F_{\mathrm{info}}$4 and $F_{\mathrm{info}}$5, capturing their spatial and temporal alignment. $F_{\mathrm{info}}$6 is readily accessible in experiments and provides a direct metric for proximity to stall. The authors derive an instantaneous geometric lower bound on entropy production, saturated when $F_{\mathrm{info}}$7, encoding a physically interpretable regime distinct from scalar bounds like thermodynamic uncertainty relations.

Paradigmatic Example: Harmonic Trap Dynamics

The moving harmonic trap is analytically tractable and bridges theory with experimental microrheology. The system's state is characterized by lag $F_{\mathrm{info}}$8 (mean position minus trap center) and variance $F_{\mathrm{info}}$9, yielding explicit force expressions:

$$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$0

The correlation coefficient is found to depend solely on the dimensionless ratio $$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$1, not on individual fluctuation or lag scales:

$$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$2

Regions with $$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$3 operate near perfect anti-alignment and minimal entropy production, while $$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$4 produces strong dissipation. This geometric dependency explains the fluctuation-dissipation anticorrelation observed in experiments.

Figure 2: Schematic of force configurations in a harmonic trap, illustrating anti-aligned and aligned regimes and dependency on $$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$5.

Geometric Interpretation of Experimental Observations

The theoretical framework maps directly onto spatially resolved experimental observations. It clarifies that energetically efficient flickering in red blood cell membranes is not due to anomalously low noise, but to spatial regimes with strong force anti-alignment buffering dissipation.

Figure 3: Conceptual mapping between spatial heterogeneity in experimentally inferred entropy production and force geometry.

Geometric Control Charts and Operational Trade-offs

The authors construct geometric design charts for constant dragging and sinusoidal driving (the two main experimental protocols) in the $$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$6 and $$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$7 parameter spaces. The charts showcase contours of constant force-correlation $$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$8, with energetic cost (e.g., injected power) orthogonal to geometric alignment, enabling systematic navigation of nonequilibrium trade-offs. These charts locate actual experimental operating points—e.g., the dragged red cell system resides in a regime with $$\dot{S}_i(t) = \frac{D}{k_B T^2} \langle (F_{\mathrm{ext}} + F_{\mathrm{info}})^2 \rangle$$9, demonstrating strong force anti-alignment at finite power input.

Figure 4: Heat map of steady-state force-correlation $F_{\mathrm{ext}} = -\nabla U$0 under constant dragging, showing iso-correlation and iso-power curves and placement of the experimental operating point.

Figure 5: Heat map of time-averaged force-correlation $F_{\mathrm{ext}} = -\nabla U$1 for sinusoidal driving, organized by driving strength $F_{\mathrm{ext}} = -\nabla U$2 and phase lag $F_{\mathrm{ext}} = -\nabla U$3.

Figure 6: Geometric design chart matching experimental dissipation, with contours of $F_{\mathrm{ext}} = -\nabla U$4 and experimental operating point under empirical friction coefficient.

Comparison to Scalar Thermodynamic Bounds

Force geometry complements, but does not duplicate, existing scalar thermodynamic bounds such as TURs or Fisher-information speed limits. At stall, TURs become trivial, yet force geometry clarifies persistent entropy production due to fluctuation. The vectorial decomposition thus enriches the theoretical landscape, organizing dissipative structure beyond efficient or precise current generation.

Extension to Underdamped Dynamics and Multidimensionality

For underdamped regimes, momentum provides additional freedom: perfect spatial anti-alignment does not necessarily suppress transport, allowing "coasting" states where strong anti-alignment coincides with directed motion. This indicates momentum as a thermodynamic buffer, further decoupling force geometry from entropy production. The framework extends directly to higher dimensions via inner products, suggesting broad applicability for optimal control and nonequilibrium regulation across complex driven systems.

Conclusion

The paper establishes force geometry, via the alignment of external and entropic forces, as a principal structural determinant of irreversibility in nonequilibrium stochastic systems (2603.29416). The force-correlation metric $F_{\mathrm{ext}} = -\nabla U$5 provides experimentally accessible diagnostics for mapping operating regimes, tuning dissipation, and interpreting spatial heterogeneity. Geometric control charts enable protocol design beyond traditional scalar bounds, offering a pathway to systematic regulation of energetic cost in driven biological and soft matter systems. The perspective invites further exploration in underdamped dynamics, multidimensional systems, and active matter, where force organization and transport can be decoupled for optimizing stochastic machine performance.