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.

Figures (7)

• Figure 1: Force correlation diagram and force geometry. Joint fluctuations of the external force $F_{\mathrm{ext}}$ and the information-theoretic force $F_{\mathrm{info}}$. Each point represents the instantaneous force pair $(F_{\mathrm{ext}}, F_{\mathrm{info}})$ at time $t$. Green quadrants correspond to negative force correlations (anti-alignment), while red quadrants correspond to positive force correlations. The dashed diagonal marks the force-cancellation condition $F_{\mathrm{info}} = -F_{\mathrm{ext}}$, which restores thermal reversibility.The shaded band illustrates configurations of strong global force anti-alignment ($r \approx -1$), which may still exhibit local force imbalance.
• Figure 2: Conceptual hierarchy of force geometry in overdamped nonequilibrium dynamics. Geometric waste is the total entropy production arising from departures from the reversible force-cancellation condition. While equilibrium trivially satisfies $r(t)=-1$, the converse is not true: global force anti-alignment does not preclude irreversible currents or finite entropy production.
• Figure 3: Force correlation regimes in a moving harmonic trap. Schematic illustrating force configurations in a harmonically trapped particle, shown in terms of the particle position $x$, trap center $\lambda$, and mean position $\mu$. Anti-aligned and aligned configurations of the external and information-theoretic forces correspond to negative and positive force correlations, respectively. The ratio $|\Delta|/\sigma$ characterizes the force-correlation regime.
• Figure 4: Geometric interpretation of experimental observations. Schematic illustrating how spatially heterogeneous entropy production and fluctuation--dissipation trends reported by DiTerlizzi2024 arise from local force geometry. (A) Spatial variation of inferred entropy production across coarse-grained membrane patches (circle size schematically indicates positional variance). (B) Local force configurations in a harmonic trap, highlighting spatial variation in force geometry. (C) Force--correlation coefficient $r(t)$ as a geometric alignment measure linking positional variance and entropy production.
• Figure 5: Geometric organization under constant dragging. Heat map of the steady-state force--correlation coefficient $r_{\mathrm{ss}}$ in the $(k,v)$ parameter space of a harmonically trapped particle, computed at fixed friction $\gamma = 0.01\,\mathrm{pN\cdot s}/\mu\mathrm{m}$. Dashed white curves denote contours of constant force correlation, while dashed red curves indicate contours of constant injected power $P=\gamma v^2$. Several representative correlation contours highlight regimes of strong force anti-alignment at finite power. The rescaled experimental operating point of Ref. DiTerlizzi2024, marked by a yellow star, lies within this strong anti-alignment region at the intersection of $r_{\mathrm{ss}}\approx -0.96$ and $P \approx 2.5\,\mathrm{fW}$. The lack of parallelism between iso-power contours (red) and iso-correlation contours (white) demonstrates that force geometry provides a degree of freedom for controlling dissipation that is independent of the total energetic cost.