Hidden Entropy Production at Mechanical Stall: Exact Reconstruction in a Reciprocal Brownian Motor

Technical Summary: Hidden Entropy Production at Mechanical Stall in a Reciprocal Brownian Motor

Problem Statement
In nonequilibrium thermodynamics, a central challenge is quantifying microscopic entropy production when only a subset of system degrees of freedom is observable. Experimentally, one typically measures a single coordinate (e.g., the displacement of a colloid or a molecular motor step) while internal chemical or conformational degrees of freedom remain hidden. A critical qualitative issue arises at "mechanical stall," where the mean displacement of the observed coordinate vanishes ($J_x = 0$). In such states, the system may still be dissipative due to hidden currents in unmeasured variables, yet standard inference methods often fail to quantify this hidden dissipation. Previous work has established that fluctuation-response violations can signal the presence of hidden dissipation, particularly in time-scale-separated models, but a quantitative method to exactly reconstruct the full entropy production from a single stalled coordinate without time-scale separation has remained an open problem.

Methodology
The authors introduce a minimal, analytically solvable model of a reciprocal Brownian motor to address this gap. The system consists of:

1. Observed Coordinate ($x$): A translational variable with period $L$.
2. Hidden Coordinate ($\theta$): An internal angular variable (representing a rotor or chemical cycle) with period $2\pi$.
3. Reciprocal Coupling: The two coordinates interact via a single periodic potential $V(q)$, where $q = x - \ell\theta$ is the relative mechanical phase (slip). The gearing length is $\ell = L/2\pi$.

This specific coupling enforces force–torque reciprocity: the force on $x$ and the reaction torque on $\theta$ are derived from the same potential, satisfying $\tau_{int} = -\ell F_x$ identically. The dynamics are governed by overdamped Langevin equations driven by an active torque $\tau_a$ on the hidden rotor and an external load $f$ on the observed coordinate.

The analysis proceeds by:

• Exploiting translational symmetry to reduce the two-variable Fokker–Planck equation to a one-dimensional equation for the slip coordinate $q$.
• Deriving exact stationary solutions for the probability density and currents.
• Defining mechanical stall as the condition where the mean translational current $J_x$ vanishes, while the hidden current $J_\theta$ remains non-zero.
• Utilizing the Harada–Sasa relation to connect the fluctuation–response violation of the observed velocity to the heat dissipated in that channel.

Key Contributions and Results

1. Exact Reconstruction of Hidden Entropy Production:
   The paper proves that at mechanical stall, the total entropy production rate $\dot{s}^s_{p}$ can be reconstructed exactly from measurements of the observed coordinate alone, provided a "reciprocal mobility factor" $K = 1 + \ell^2\mu_\theta/\mu_x$ is calibrated. The reconstruction formula is:
   $$\dot{s}^s_{p} = K\Phi^s_x + \frac{T K^2 (\Phi^s_x)^2}{\mu_x (K-1) f^2_{st}}$$
   where $\Phi^s_x$ is the Harada–Sasa violation measured at stall, $f_{st}$ is the stall load, and $T$ is temperature. This result relies on a local current identity derived from force–torque reciprocity, which ensures that the information-flow correction term (normally present in coupled subsystems) vanishes identically.

2. Optimized Model-Class Bound:
   In scenarios where the hidden mobility $\mu_\theta$ (and thus $K$) is unknown, the authors derive a strictly stronger lower bound on the total entropy production than the direct visible-dissipation bound ($\dot{s}^s_{p} \ge \Phi^s_x$). The optimized bound is:
   $$\dot{s}^s_{p} \ge \Phi^s_x \left(\sqrt{1+a} + \sqrt{a}\right)^2, \quad \text{with } a = \frac{T\Phi^s_x}{\mu_x f^2_{st}}$$
   The authors demonstrate that this bound is attainable by a specific member of the reciprocal motor class, proving it is not merely a mathematical artifact but a physical limit for this system class.

3. Distinction Between Stall and Reversibility:
   The work clarifies that mechanical stall ($J_x=0$) is distinct from thermodynamic reversibility (detailed balance). Reversibility requires the local probability currents to vanish everywhere ($j_x = j_\theta = 0$), which in this model only occurs when both the load and active torque are zero. At mechanical stall with non-zero driving, the system maintains a non-zero "slip" velocity in the relative coordinate $q$, sustaining hidden dissipation despite the observed coordinate being stationary on average.

4. Analytical Verification:
   All theoretical results are verified through two independent methods:

   • Direct quadrature of the stationary Fokker–Planck solution.
   • Langevin simulations of the original two-variable dynamics.
     The numerical results confirm that the correlation-flow correction vanishes and that the reconstructed entropy production matches the direct calculation across various coupling strengths and mobility ratios.

Significance and Scope
The paper claims to provide an analytically solvable benchmark for entropy-production inference that does not rely on the separation of time scales, a common assumption in previous recovery methods. The central contribution is an inference theorem showing that under the structural condition of reciprocal coupling (force–torque reciprocity), the "hidden" thermodynamic bookkeeping is closed by the "visible" fluctuations.

The authors emphasize that this result is specific to systems where the hidden coordinate is mechanically tied to the observed one via a shared potential. It does not apply to generic hidden-variable models where the hidden coordinate acts as an external clock without back-reaction. However, within the reciprocal class, the work establishes that a mechanically silent motor can be quantitatively characterized as thermodynamically active, allowing for the exact reconstruction of fuel consumption or active power dissipation from a single observable coordinate.