A mathematical model suggests that a hinged swimmer can move in a directed way when thermal fluctuations are combined with two cyclic controls. It asks whether that combination can overcome the deterministic single-degree-of-freedom scallop limitation and produce geometric propulsion. The controls are the swimmer's equilibrium opening angle and the stiffness of a spring that favors that angle.
The study analyzes an idealized single-hinge scallop model. It reports no empirical sample or sampled population, so the result is a prediction from the model rather than a measurement from a real swimmer.
Motion from a loop in control space
The model couples the changing opening angle to translation through a position-dependent effective mobility and a hydrodynamic connection, which describes how a change in shape is tied to movement. The shape dynamics are written as a Smoluchowski equation, a probability equation for how the model's shapes are distributed over time.
The two controls are varied as separate sine waves, each with its own baseline, amplitude and frequency, with a phase shift between them. The first control is the equilibrium opening angle; the second is spring stiffness.
For a closed cycle of these controls, the predicted displacement is geometric: it is set by the path traced through the two-control plane. The paper describes the displacement as a line integral around that loop, or equivalently as the flux of a quantity called BSN curvature through the area inside it. Average swimming velocity is the predicted displacement divided by the cycle period.
The calculation uses a slow-driving expansion around the system's stationary state. Because the probability-evolution operator has a stationary zero mode and is therefore singular, the analysis uses a generalized inverse to calculate the first response to changing controls.
A path can produce motion, but it also has a cost
The same framework assigns the extra, non-adiabatic dissipation to a thermodynamic metric. In ordinary language, this metric measures how costly it is to move through different directions in control space. The leading cost is quadratic in the speeds of the controls, and the paper states that the first-order work correction is nonnegative.
The model also sets a lower limit on that excess cost: excess entropy production multiplied by the cycle period cannot fall below the square of the thermodynamic length of the chosen path. Thermodynamic length is a distance measured using the metric. The bound is reached only when the path is traversed at constant thermodynamic speed, so the timing of a protocol matters as well as its route.
A version of the calculation in which the swimmer's mobility is clamped provides an upper comparison for the freely moving system. In the paper's model, effective free-swimming mobility is no smaller than clamped mobility, while the free-swimmer thermodynamic metric and excess entropy production are no greater than the corresponding clamped quantities.
The shape of the cycle matters
For equal-frequency, 1:1 driving, the two controls trace a simple ellipse when their phase difference is nontrivial. For fixed amplitudes, the enclosed area is largest when the controls are a quarter-cycle apart. But the paper cautions that higher-order Lissajous loops can partly cancel their own geometric displacement because the BSN curvature varies across control space.
Numerical displacement maps show that the result is not determined by enclosed area alone. The predicted displacement changes strongly with the operating point and varies non-monotonically with the ratio of the two amplitudes, even along contours with the same area. A reported map uses a loop radius of 0.2 centered at the point (0.35, 0.55), illustrating how the local region of control space can alter the outcome.
The numerical test of the dissipation bound used a phase-shift family with equal angular frequencies, amplitudes of 0.11 and 0.22, and phase varied across a full cycle. For that tested family, dimensionless entropy production stayed above the squared dimensionless thermodynamic length, with the reported ratio remaining above one.
A result confined to an idealized regime
The paper's formal zero-noise limit removes the stochastic pumping contribution: the predicted displacement and BSN curvature tend to zero, recovering deterministic single-degree-of-freedom scallop behavior. The authors present this as a formal limit of the model.
The calculation assumes slow driving, so higher-frequency corrections are outside the reported analysis. The hydrodynamic connection and effective mobility remain part of an idealized construction, and exact free-swimmer mobility requires full resistance data.
The paper does not establish the 1:1 protocol as universally optimal, because geometric cancellation depends on how the BSN curvature varies across the control plane. It is an arXiv preprint, version 2, dated 28 August 2026. The author states that the work was partially supported by JSPS KAKENHI Grant No. JP26K06960.
Taken together, the paper presents a conditional theoretical result. Within this idealized model and its slow-driving calculation, thermal fluctuations and two coordinated controls offer a geometric route around the deterministic scallop constraint, while the thermodynamic metric tracks the extra dissipation. The framework therefore links how much motion a control loop can generate with how costly that loop is to traverse.
Paper data and sources
Original title: Geometric Thermodynamics of Scallop Motion with Two Control Parameters
Authors: Hisao Hayakawa
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text