The paper reports a sharp split in a model of self-propelling colloidal chains: the particles' orientations can have zero steady-state entropy production when the chain geometry is fixed, while the full dimer system still has a finite entropy-production rate when rotational diffusion is positive. In other words, the orientation sector can look equilibrium-like even as the complete modeled system remains dissipative.
The distinction is central to the study. For fixed geometries, the monopolar orientational dynamics map to a scalar potential and obey detailed balance, while the authors interpret translational motion as the part that carries dissipation.
This is a theoretical modeling study of far-from-equilibrium autophoretic colloidal chains. It solves the reduced dynamics exactly for dimers and semi-analytically for general N-mers, alongside numerical simulations.
The default runs began with a straight chain whose monomers were spaced by the bond-rest length. Propulsion initially pointed perpendicular to the shared chain axis, and under this condition the geometry stayed mirror-symmetric about the chain's center.
Stochastic and deterministic calculations used the Ito Euler-Maruyama scheme, with noise removed from the deterministic runs. The trajectory-based entropy-production calculation also included a calculable correction for the difference between Stratonovich and Ito products.
The dimer exposes the split
The clearest deterministic example came from the monopolar dimer. Its center of mass approached a finite displacement rather than propelling indefinitely. At the same time, global polar order started at one and fell toward zero as the two monomers settled into antiparallel orientations.
In the stochastic dimer calculation, the separation remained. For positive rotational diffusion, orientational-sector entropy production was exactly zero, but full-system entropy production was finite and nonnegative. The full rate remained nonzero in the long-time dynamic limit.
Longer chains test the shortcut
A self-consistent chain-length sweep gave a gradual increase in polar order. From N = 2 to N = 13, steady-state polar order rose sub-linearly from zero at N = 2 to about 0.34 at N = 13. The least-stable eigenvalue from the linear-stability calculation moved from -0.0998 to -0.0417 over the same range, with a small non-monotonic change between N = 3, at -0.0850, and N = 4, at -0.0842.
The fixed-geometry shortcut then broke down when bond angles were allowed to evolve. At long lag times, the simulated mean-squared displacement did not saturate and exceeded the reduced-model plateau by an order of magnitude or more within the accessible window.
For the trimer and for N = 5, direct stochastic simulations found genuinely nonzero steady-state values in both the full-system and orientational-sector entropy-production rates. The authors attribute this change to bond-angle coupling that the fixed-geometry reduction leaves out. Their proposed explanation involving free whole-chain rotational diffusion, however, was not explicitly verified.
Dipolar coupling changes the outcome
The dipolar-only dimer was a special case. Its sum and difference orientation modes formed two independent one-dimensional gradient dynamics. One relaxed to a zero sum and the other to a half-turn difference, so the dimer retained quasi-equilibrium through two exactly decoupled modes.
For longer chains under the default starting condition, pure dipolar coupling behaved differently. With monopolar coupling absent and dipolar coupling nonzero, the simulations did not reach a static configuration. The polarization-axis order parameter drifted persistently and non-monotonically even at long times, without approaching a fixed limiting value.
With positive monopolar coupling, both the purely monopolar and combined-coupling cases reached a fully polarized C-shape under the same initial condition, with the polarization-axis order parameter tending to zero. Adding dipolar coupling did not enhance formation of that shape in the reported test.
A broader coupling scan showed the same pattern. On a 20-by-20 grid, the polarization-axis order parameter was indistinguishable from zero across essentially the whole positive monopolar region at the final time, while overall polar order showed that the chain had not depolarized.
A result with clear conditions
The analytical result has clear boundaries. The general-chain quasi-equilibrium claim assumes fixed bond-angle geometry. When those angles fluctuate, positional and orientational sectors become coupled, breaking the fixed-geometry closure for longer chains.
That makes the C-shape conclusion conditional as well: the claim that monopolar coupling is sufficient is restricted to the symmetric initial condition studied. The dipolar longer-chain conclusion comes from the specified model and simulated cases, rather than an exhaustive mathematical classification.
The document is labeled arXiv version 1 and dated 28 Aug 2026. Taken together, the reported cases show a fixed-geometry monopolar orientation sector obeying detailed balance, nonzero orientational entropy production in the longer-chain simulations, and no settled configuration in the purely dipolar longer-chain runs.
Paper data and sources
Original title: Mechanics and statistics of a solvable model of an autophoretic colloidal chain
Authors: Arvin Gopal Subramaniam, Rajesh Singh
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text