Odd pathways speed up self-assembly

arxiv: 2604.22408 · v1 · submitted 2026-04-24 · ❄️ cond-mat.soft

Odd pathways speed up self-assembly

Dawid Dopiera{\l}a , Luca Cocconi , Robert L. Jack , Anton Souslov This is my paper

Pith reviewed 2026-05-08 09:21 UTC · model grok-4.3

classification ❄️ cond-mat.soft

keywords self-assemblynon-reciprocal interactionsodd forcesactive matterbarrier crossingArrhenius ratesprobability currentssoft matter

0 comments p. Extension

The pith

Non-reciprocal odd interactions accelerate self-assembly by enhancing barrier crossing rates while preserving the equilibrium Boltzmann distribution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how certain non-reciprocal interactions can speed up the formation of ordered structures without the usual drawbacks of active driving. These interactions generate probability currents that help particles cross free-energy barriers more quickly than in passive equilibrium dynamics. The speedup occurs through an effective boost to the mobility of the key reaction coordinate, achieved by coupling different mechanical modes in a non-reciprocal way. If correct, the approach opens a path to self-assembly that finishes faster yet still reaches the exact target state and can then stop consuming energy.

Core claim

Non-reciprocal odd interactions induce probability currents that reshape activated barrier crossing, soft-mode relaxation, and transitions between metastable states. These currents enhance Arrhenius rates by driving particles across otherwise inaccessible free-energy barriers. The acceleration arises from an effective increase in the mobility of the reaction coordinate, mediated by non-reciprocal coupling between mechanical modes. A trade-off exists between this kinetic acceleration and the power dissipation incurred when the active forces are engaged.

What carries the argument

Probability currents from odd non-reciprocal interactions, which increase the effective mobility of the reaction coordinate through non-reciprocal coupling between mechanical modes.

Load-bearing premise

Non-reciprocal odd interactions can be engineered so that the induced currents accelerate barrier crossing while exactly preserving the equilibrium Boltzmann distribution and leaving the target structures undistorted.

What would settle it

An experiment or simulation in which odd interactions are introduced, the final structures remain identical to the equilibrium case, yet the measured assembly or relaxation time shows no reduction compared to the passive system.

Figures

Figures reproduced from arXiv: 2604.22408 by Anton Souslov, Dawid Dopiera{\l}a, Luca Cocconi, Robert L. Jack.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: FIG. 3 view at source ↗

read the original abstract

Active self-assembly can bypass equilibrium bottlenecks through external energy injection. However, generic driving typically distorts target structures and requires sustained energy input even after assembly is complete. Here, we investigate a class of non-reciprocal interactions that accelerates assembly while preserving the equilibrium Boltzmann distribution. The probability currents induced by these odd interactions reshape fundamental processes, including activated barrier crossing, soft-mode relaxation, and transitions between metastable states. In particular, these currents enhance Arrhenius rates by driving particles across otherwise inaccessible free-energy barriers. We show that this acceleration arises from an effective increase in the mobility of the reaction coordinate, mediated by non-reciprocal coupling between mechanical modes. In turn, we discover a trade-off between kinetic acceleration and power dissipation when active forces are engaged. Our results suggest a route to energy-efficient, high-fidelity self-assembly via active catalysts that transiently accelerate relaxation toward equilibrium targets and deactivate upon reaching the desired state.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Odd non-reciprocal interactions can accelerate barrier crossing in self-assembly while keeping the exact Boltzmann distribution, but the many-body integrability condition for the currents is the part that needs the closest look.

read the letter

The core point here is that the authors use odd interactions to generate probability currents that raise the effective mobility along reaction coordinates, speeding up Arrhenius processes and soft-mode relaxation without shifting the equilibrium measure or requiring constant driving after assembly finishes. They also flag the dissipation cost that comes with the speedup, which keeps the claim grounded rather than oversold. This combination of preserved targets plus faster kinetics is the piece that stands out from standard active-assembly work, where driving usually warps the structures or keeps running forever. The mode-coupling argument for how the currents act on the reaction coordinate is a clean way to frame the effect, and it follows directly from the non-reciprocal terms they introduce. The paper does well at showing that the mechanism is distinct from generic energy injection and at sketching the trade-off with power consumption. On the soft spots, the stress-test concern about divergence-free currents at equilibrium is real and worth pressing. For arbitrary particle numbers and interaction graphs, the antisymmetric part of the mobility must be constructed so its contribution to the Fokker-Planck current vanishes exactly when the density is Boltzmann; any slip in that cancellation would move the steady state and change the assembled probabilities. The abstract states the preservation, but the full derivations need to make the integrability condition explicit and show it holds beyond the pairwise cases they simulate. Simulation details on statistics, system sizes, and how they quantify the speedup would also help rule out finite-size artifacts. This is for soft-matter people who already work on non-reciprocal forces or kinetic control of assembly. A reader who wants concrete ways to tune relaxation rates without distorting equilibria will find usable ideas. It deserves a serious referee because the mechanism is testable and the claims rest on explicit dynamics rather than fitting. I would send it to peer review and ask the authors to expand the section on the stationary-distribution condition and add error analysis on the reported rates.

Referee Report

2 major / 2 minor

Summary. The paper proposes a class of non-reciprocal odd interactions in active self-assembly that induce probability currents to accelerate processes such as activated barrier crossing, soft-mode relaxation, and metastable transitions, while exactly preserving the equilibrium Boltzmann distribution. The acceleration is attributed to an effective increase in the mobility of the reaction coordinate through non-reciprocal coupling between mechanical modes, with an accompanying trade-off in power dissipation. The approach is positioned as enabling transient, energy-efficient catalysis of assembly toward equilibrium targets.

Significance. If the mechanism is robust, the result offers a route to high-fidelity active self-assembly that avoids structural distortion and continuous energy input after completion. This could be relevant for colloidal and soft-matter systems where equilibrium bottlenecks limit practical assembly. The work is credited for identifying the mobility-enhancement pathway and the dissipation trade-off as concrete, testable consequences of the odd couplings.

major comments (2)

[Model section / Fokker-Planck derivation] The central claim that odd non-reciprocal terms generate non-zero circulatory currents while leaving the stationary density exactly equal to the Boltzmann measure exp(−βU) is load-bearing. The manuscript must explicitly demonstrate that the antisymmetric contribution to the probability current is divergence-free at ρ_eq for the many-body Langevin/Fokker-Planck dynamics, including the required integrability condition on the mobility matrix for arbitrary particle numbers and interaction topologies (see the skeptic note on this point).
[Results on barrier crossing] The effective-mobility increase for the reaction coordinate (abstract and results) is derived from non-reciprocal mode coupling. The derivation should be shown to remain valid when the odd terms are many-body and not reducible to a single effective coordinate; otherwise the Arrhenius-rate enhancement claim may not generalize beyond the two-particle or low-dimensional cases.

minor comments (2)

[Abstract] The abstract states both the currents and exact Boltzmann preservation but does not indicate where in the text the integrability proof appears; a brief forward reference would improve readability.
[Methods / Results] Simulation details, error bars, and parameter ranges for the reported acceleration factors are not summarized in the provided excerpt; these should be added to the main text or SI for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address the major points raised below and have revised the manuscript to incorporate explicit demonstrations and clarifications where appropriate.

read point-by-point responses

Referee: [Model section / Fokker-Planck derivation] The central claim that odd non-reciprocal terms generate non-zero circulatory currents while leaving the stationary density exactly equal to the Boltzmann measure exp(−βU) is load-bearing. The manuscript must explicitly demonstrate that the antisymmetric contribution to the probability current is divergence-free at ρ_eq for the many-body Langevin/Fokker-Planck dynamics, including the required integrability condition on the mobility matrix for arbitrary particle numbers and interaction topologies (see the skeptic note on this point).

Authors: We agree that an explicit demonstration is necessary to fully support the central claim. In the revised manuscript, we will add a new subsection to the Model section together with an appendix that derives the Fokker-Planck operator for the general many-body Langevin dynamics with a position-dependent, antisymmetric mobility matrix. We explicitly show that the divergence of the odd (antisymmetric) contribution to the probability current vanishes identically when evaluated at ρ_eq = exp(−βU). This follows directly from the antisymmetry M_odd^T = −M_odd together with the equilibrium force balance; the resulting current is purely circulatory and divergence-free. The proof is algebraic and holds for arbitrary particle number N and arbitrary interaction topologies. We will also state the required integrability condition on the mobility matrix (ensuring a consistent stochastic calculus, e.g., Stratonovich interpretation) and verify that it is satisfied by the odd couplings employed in the work. revision: yes
Referee: [Results on barrier crossing] The effective-mobility increase for the reaction coordinate (abstract and results) is derived from non-reciprocal mode coupling. The derivation should be shown to remain valid when the odd terms are many-body and not reducible to a single effective coordinate; otherwise the Arrhenius-rate enhancement claim may not generalize beyond the two-particle or low-dimensional cases.

Authors: The effective-mobility enhancement is obtained by projecting the full many-body dynamics onto the reaction coordinate. Because the projection integrates out the orthogonal modes, the contribution of the odd couplings to the effective mobility along the reaction coordinate remains valid even when the odd terms are genuinely many-body and cannot be reduced to a single effective coordinate. We will expand the Results section with a general mode-decomposition argument (using the committor or the slowest eigenmode of the linearized dynamics) that demonstrates the Arrhenius-rate increase for arbitrary dimensionality. We will also add numerical illustrations for systems with more than two particles to confirm that the enhancement persists beyond the low-dimensional cases already presented. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation of accelerated self-assembly

full rationale

The paper constructs non-reciprocal odd interactions within a many-body Langevin or Fokker-Planck framework such that the induced probability currents are divergence-free at the equilibrium Boltzmann measure, thereby preserving exp(−βU) exactly while accelerating barrier crossing and relaxation via effective mobility enhancement. This follows directly from the antisymmetric structure of the mobility matrix and the resulting steady-state condition, without reducing any claimed prediction to a fitted parameter, self-citation loop, or ansatz smuggled from prior work. The trade-off with power dissipation is likewise obtained from the same equations of motion. The central result is therefore self-contained against the model assumptions and does not rely on load-bearing external uniqueness theorems or renaming of known patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the domain assumption that non-reciprocal forces can be added without breaking detailed balance in the steady state and on standard overdamped Langevin dynamics for colloidal particles.

axioms (2)

domain assumption Overdamped Langevin dynamics with additive non-reciprocal forces
Standard modeling choice for active colloidal systems invoked to derive probability currents.
ad hoc to paper Non-reciprocal interactions preserve the equilibrium Boltzmann distribution
Central premise stated in abstract; if false the acceleration-without-distortion claim collapses.

pith-pipeline@v0.9.0 · 5458 in / 1296 out tokens · 47366 ms · 2026-05-08T09:21:40.461721+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

65 extracted references · 1 canonical work pages

[1]

Self- assembly: from crystals to cells

Bartosz A Grzybowski, Christopher E Wilmer, Jiwon Kim, Kevin P Browne, and Kyle JM Bishop. Self- assembly: from crystals to cells. Soft Matter, 5(6):1110– 1128, 2009

2009
[2]

Equilibrium mechanisms of self-limiting assembly

Michael F Hagan and Gregory M Grason. Equilibrium mechanisms of self-limiting assembly. Reviews of modern 7 physics, 93(2):025008, 2021

2021
[3]

Reaction-rate theory: fifty years after kramers

Peter H¨ anggi, Peter Talkner, and Michal Borkovec. Reaction-rate theory: fifty years after kramers. Reviews of modern physics, 62(2):251, 1990

1990
[4]

The statistical mechanics of dynamic pathways to self-assembly.Annual review of physical chemistry, 66(1):143–163, 2015

Stephen Whitelam and Robert L Jack. The statistical mechanics of dynamic pathways to self-assembly.Annual review of physical chemistry, 66(1):143–163, 2015

2015
[5]

An active approach to colloidal self-assembly

Stewart A Mallory, Chantal Valeriani, and Angelo Cac- ciuto. An active approach to colloidal self-assembly. Annual review of physical chemistry, 69:59–79, 2018

2018
[6]

Self-assembly and time-dependent control of ac- tive and passive triblock janus colloids

Juri Franz Schubert, Salman Fariz Navas, and Sabine HL Klapp. Self-assembly and time-dependent control of ac- tive and passive triblock janus colloids. The Journal of Chemical Physics, 163(4), 2025

2025
[7]

Activity-controlled annealing of colloidal mono- layers

Sophie Ramananarivo, Etienne Ducrot, and Jeremie Palacci. Activity-controlled annealing of colloidal mono- layers. Nature communications, 10(1):3380, 2019

2019
[8]

Statistical mechanics of ac- tive ornstein-uhlenbeck particles

David Martin, J´ er´ emy O’Byrne, Michael E Cates, ´Etienne Fodor, Cesare Nardini, Julien Tailleur, and Fr´ ed´ eric Van Wijland. Statistical mechanics of ac- tive ornstein-uhlenbeck particles. Physical Review E, 103(3):032607, 2021

2021
[9]

Non-reciprocal phase transitions

Michel Fruchart, Ryo Hanai, Peter B Littlewood, and Vincenzo Vitelli. Non-reciprocal phase transitions. Nature, 592(7854):363–369, 2021

2021
[10]

Self-assembly of catalytically active colloidal molecules: tailoring activ- ity through surface chemistry

Rodrigo Soto and Ramin Golestanian. Self-assembly of catalytically active colloidal molecules: tailoring activ- ity through surface chemistry. Physical review letters, 112(6):068301, 2014

2014
[11]

Accelerating diffusions

Chii-Ruey Hwang, Shu-Yin Hwang-Ma, and Shuenn-Jyi Sheu. Accelerating diffusions. The Annals of Applied Probability, 15(2), 2005

2005
[12]

Acceleration of convergence to equilibrium in markov chains by breaking detailed balance.Journal of statistical physics, 168(2):259–287, 2017

Marcus Kaiser, Robert L Jack, and Johannes Zimmer. Acceleration of convergence to equilibrium in markov chains by breaking detailed balance.Journal of statistical physics, 168(2):259–287, 2017

2017
[13]

Sampling efficiency of trans- verse forces in dense liquids

Federico Ghimenti, Ludovic Berthier, Grzegorz Szamel, and Fr´ ed´ eric van Wijland. Sampling efficiency of trans- verse forces in dense liquids. Physical Review Letters, 131(25):257101, 2023

2023
[14]

Transverse forces and glassy liquids in infinite dimensions

Federico Ghimenti, Ludovic Berthier, Grzegorz Sza- mel, and Fr´ ed´ eric van Wijland. Transverse forces and glassy liquids in infinite dimensions. Physical Review E, 109(6):064133, 2024

2024
[15]

Non-reciprocal multifarious self-organization

Saeed Osat and Ramin Golestanian. Non-reciprocal multifarious self-organization. Nature Nanotechnology, 18(1):79–85, 2023

2023
[16]

Escaping kinetic traps using nonrecipro- cal interactions

Saeed Osat, Jakob Metson, Mehran Kardar, and Ramin Golestanian. Escaping kinetic traps using nonrecipro- cal interactions. Physical Review Letters, 133(2):028301, 2024

2024
[17]

Continuous-time multifarious systems

Jakob Metson, Saeed Osat, and Ramin Golestanian. Continuous-time multifarious systems. ii. non-reciprocal multifarious self-organization. The Journal of Chemical Physics, 163(12), 2025

2025
[18]

Spinning janus doublets driven in uniform ac electric fields

Alicia Boymelgreen, Gilad Yossifon, Sinwook Park, and Touvia Miloh. Spinning janus doublets driven in uniform ac electric fields. Physical Review E, 89(1):011003, 2014

2014
[19]

Kilobot: A low cost robot with scalable operations designed for collective behaviors

Michael Rubenstein, Christian Ahler, Nick Hoff, Adrian Cabrera, and Radhika Nagpal. Kilobot: A low cost robot with scalable operations designed for collective behaviors. Robotics and Autonomous Systems, 62(7):966–975, 2014

2014
[20]

Self-assembly of emul- sion droplets through programmable folding

Angus McMullen, Maitane Mu˜ noz Basagoiti, Zorana Zeravcic, and Jasna Brujic. Self-assembly of emul- sion droplets through programmable folding. Nature, 610(7932):502–506, 2022

2022
[21]

Odd elasticity

Colin Scheibner, Anton Souslov, Debarghya Banerjee, Pi- otr Sur´ owka, William TM Irvine, and Vincenzo Vitelli. Odd elasticity. Nature Physics, 16(4):475–480, 2020

2020
[22]

Odd microswimmer

Kento Yasuda, Yuto Hosaka, Isamu Sou, and Shigeyuki Komura. Odd microswimmer. Journal of the Physical Society of Japan, 90(7):075001, 2021

2021
[23]

Odd viscosity and odd elasticity

Michel Fruchart, Colin Scheibner, and Vincenzo Vitelli. Odd viscosity and odd elasticity. Annual Review of Condensed Matter Physics, 14(1):471–510, 2023

2023
[24]

Direct measurement of the nonconservative force field generated by optical tweezers

Pinyu Wu, Rongxin Huang, Christian Tischer, Alexandr Jonas, and Ernst-Ludwig Florin. Direct measurement of the nonconservative force field generated by optical tweezers. Physical review letters, 103(10):108101, 2009

2009
[25]

Brownian vortexes

Bo Sun, Jiayi Lin, Ellis Darby, Alexander Y Grosberg, and David G Grier. Brownian vortexes. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 80(1):010401, 2009

2009
[26]

Hamiltonian curl forces

MV Berry and Pragya Shukla. Hamiltonian curl forces. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2176), 2015

2015
[27]

Odd dynam- ics of living chiral crystals

Tzer Han Tan, Alexander Mietke, Junang Li, Yuchao Chen, Hugh Higinbotham, Peter J Foster, Shreyas Gokhale, J¨ orn Dunkel, and Nikta Fakhri. Odd dynam- ics of living chiral crystals. Nature, 607(7918):287–293, 2022

2022
[28]

Motile dislocations knead odd crys- tals into whorls

Ephraim S Bililign, Florencio Balboa Usabiaga, Yehuda A Ganan, Alexis Poncet, Vishal Soni, Sofia Magkiriadou, Michael J Shelley, Denis Bartolo, and William TM Irvine. Motile dislocations knead odd crys- tals into whorls. Nature Physics, 18(2):212–218, 2022

2022
[29]

Non-reciprocal robotic metamate- rials

Martin Brandenbourger, Xander Locsin, Edan Lerner, and Corentin Coulais. Non-reciprocal robotic metamate- rials. Nature communications, 10(1):4608, 2019

2019
[30]

More is less in unpercolated active solids

Jack Binysh, Guido Baardink, Jonas Veenstra, Corentin Coulais, and Anton Souslov. More is less in unpercolated active solids. Physical Review X, 16(2):021012, 2026

2026
[31]

Geometric the- ory of (extended) time-reversal symmetries in stochas- tic processes: I

J´ er´ emy O’Byrne and Michael E Cates. Geometric the- ory of (extended) time-reversal symmetries in stochas- tic processes: I. finite dimension. Journal of Statistical Mechanics: Theory and Experiment, 2024(11):113207, 2024

2024
[32]

Cluster phases and bubbly phase separation in active flu- ids: Reversal of the ostwald process

Elsen Tjhung, Cesare Nardini, and Michael E Cates. Cluster phases and bubbly phase separation in active flu- ids: Reversal of the ostwald process. Physical Review X, 8(3):031080, 2018

2018
[33]

Fold- ing mechanisms at finite temperature

D Rocklin, Vincenzo Vitelli, and Xiaoming Mao. Fold- ing mechanisms at finite temperature. arXiv preprint arXiv:1802.02704, 2018

work page arXiv 2018
[34]

Theory of protein folding: the en- ergy landscape perspective

Jos´ e Nelson Onuchic, Zaida Luthey-Schulten, and Pe- ter G Wolynes. Theory of protein folding: the en- ergy landscape perspective. Annual review of physical chemistry, 48(1):545–600, 1997

1997
[35]

The protein- folding problem, 50 years on

Ken A Dill and Justin L MacCallum. The protein- folding problem, 50 years on. science, 338(6110):1042– 1046, 2012

2012
[36]

Dynamic path- ways for viral capsid assembly

Michael F Hagan and David Chandler. Dynamic path- ways for viral capsid assembly. Biophysical journal, 91(1):42–54, 2006

2006
[37]

Mechanisms of virus assembly

Jason D Perlmutter and Michael F Hagan. Mechanisms of virus assembly. Annual review of physical chemistry, 8 66(1):217–239, 2015

2015
[38]

Anisotropy of building blocks and their assembly into complex struc- tures

Sharon C Glotzer and Michael J Solomon. Anisotropy of building blocks and their assembly into complex struc- tures. Nature materials, 6(8):557–562, 2007

2007
[39]

Janus particles

Andreas Walther and Axel HE M¨ uller. Janus particles. Soft matter, 4(4):663–668, 2008

2008
[40]

Lock and key colloids

Stefano Sacanna, William TM Irvine, Paul M Chaikin, and David J Pine. Lock and key colloids. Nature, 464(7288):575–578, 2010

2010
[41]

Folding dna to create nanoscale shapes and patterns

Paul WK Rothemund. Folding dna to create nanoscale shapes and patterns. Nature, 440(7082):297–302, 2006

2006
[42]

Dna origami nano-mechanics

Jiahao Ji, Deepak Karna, and Hanbin Mao. Dna origami nano-mechanics. Chemical Society Reviews, 50(21):11966–11978, 2021

2021
[43]

Bioinspired micro- robots

Stefano Palagi and Peer Fischer. Bioinspired micro- robots. Nature Reviews Materials, 3(6):113–124, 2018

2018
[44]

Micro/nanorobotic swarms: from fundamentals to functionalities

Junhui Law, Jiangfan Yu, Wentian Tang, Zheyuan Gong, Xian Wang, and Yu Sun. Micro/nanorobotic swarms: from fundamentals to functionalities. ACS nano, 17(14):12971–12999, 2023

2023
[45]

Stochastic thermodynamics, fluctuation the- orems and molecular machines

Udo Seifert. Stochastic thermodynamics, fluctuation the- orems and molecular machines. Reports on progress in physics, 75(12):126001, 2012

2012
[46]

The Fokker-PlanckEquation Methods of Solution and Applications

Hannes Risken. The Fokker-PlanckEquation Methods of Solution and Applications. Springer, 1989

1989
[47]

Multiscale methods: averaging and homogenization

Grigoris Pavliotis and Andrew Stuart. Multiscale methods: averaging and homogenization. Springer Sci- ence & Business Media, 2008. 9 Appendix A: Methods

2008
[48]

Langevin dynamics and steady-state probability distribution invariance We consider a minimal stochastic model that isolates non-reciprocal mechanical response. Each particle ex- periences a force composed of three contributions: (i) a reciprocal part derived from a potentialE, (ii) a trans- verse, non-reciprocal odd-elastic component, and (iii) ad- ditive...
[49]

Model definitions We consider several models which are all consistent with Eqs. (A2,A3). The number of particles isM. In 10 some cases it is useful to consider the particles as vertices of a polygon whose signed area is A= MX i=1 xiyi+1 −y ixi+1 2 .(A11) in which indices are added moduloM, so (xM+1 , yM+1) = (x1, y1). a. Two particle model.The model of Fi...
[50]

A 2, the two-particle model is de- scribed in polar co-ordinates (detailed calculations are given in Supplementary Information)

Analysis of two-particle model As explained in Sec. A 2, the two-particle model is de- scribed in polar co-ordinates (detailed calculations are given in Supplementary Information). The dynamics of the particle separation decouples from the centre of mass; converting the associated Langevin dynamics for (r(t), θ(t)) to a Fokker-Planck equation for the prob...
[51]

as ⟨ ˙θ⟩ ≈k a s 2T πkr 2 0 exp − kr2 0 2T .(A17)
[52]

Dynamics of the odd square For the odd square model, we explain in the main text that the low-temperature dynamics of the internal an- gleβis described by Eq. (2). We outline the derivation of this result, with details in Supplementary Information. The central assumption is that the configuration remains always close to a rhombus shape, with small deviati...
[53]

Equations of motion.All stochastic dynamics were integrated using an Euler-Maruyama scheme

Numerical simulations a. Equations of motion.All stochastic dynamics were integrated using an Euler-Maruyama scheme. The particle positions were updated according to: ri(t+ ∆t) =r i(t) +F i(t)∆t+ √ 2T∆tξ i(t),(A26) withF i(t) as in Eq. (A3), the time step is ∆tand the components of the noise vectorsξ i were sampled as inde- pendent standard normal variate...
[54]

Flip ratesRand first-passage timesτestimation Flip rates and first-passage times are defined in terms of basins arounds target states, defined using thresholds on suitable order parameters, as appropriate for each model (see below). For any given initial condition, the mean first passage time is obtained by measuring first time at which the system enters ...
[55]

Supplementary Video Supplementary Movies 1 and 2 provide representa- tive stochastic trajectories illustrating the dynamics dis- cussed in the main text. Movie 1 (based on Fig.2b) shows folding of the four-particle square network from a square to a line state under passive (k a = 0) and non-reciprocal (ka = 1) dynamics, together with the time evolution of...
[56]

= (−0.5,0),x 0 2 = (x 0 2, y0
[57]

= (0, √ 3 2 ),x 0 3 = (x 0 3, y0
[58]

= (0.5,0) andX 0 = (x0 1,x 0 2,x 0 3). Considering small deformations of the (passive) spring network about this configuration, the (normalised) eigenmodes are ξx = 1√ 3 (1,0,1,0,1,0), ξy = 1√ 3 (0,1,0,1,0,1), z1 = 1 2 √ 3 1,− √ 3,−2,0,1, √ 3 , z2 = 1 2 √ 3 √ 3,1,0,−2,− √ 3,1 , z3 = 1 2 √ 3 √ 3,1,− √ 3,1,0,−2 , z4 = 1 2 √ 3 −1, √ 3,−1,− √ 3,2,0 . (A40) wh...
[59]

= cos β 2 ,0 , (x0 2, y0
[60]

= 0,sin β 2 , (x0 3, y0
[61]

= −cos β 2 ,0 , (x0 4, y0
[62]

= 0,−sin β 2 . (A45) and we write s0 = x0 1, y0 1, x0 2, y0 2, x0 3, y0 3, x0 4, y0 4 (A46) Considering small deformations about this state, the eigenmodes of the square spring network are ξx = 1 2 (1,0,1,0,1,0,1,0), ξy = 1 2 (0,1,0,1,0,1,0,1), za = 1√ 2 0,cos β 2 ,−sin β 2 ,0,0,−cos β 2 ,sin β 2 ,0 , zb = 1√ 2 sin β 2 ,0,0,−cos β 2 ,−sin β 2 ,0,0,cos β 2...
[63]

Initialize the assembly in the rectangular configuration (a special case of the ladder state) and measure the time required to reach the chevron configuration, as defined in MethodsA.6
[64]

Repeat forNstatistically independent realizations
[65]

This protocol assumes that once the chevron configuration is reached, the double-square is immediately stabilized by a bound cargo and does not revert to the ladder state

Collect and sort the transition times to construct the density plot in Fig.3e, representing the distribution of first-passage times from ladder to chevron. This protocol assumes that once the chevron configuration is reached, the double-square is immediately stabilized by a bound cargo and does not revert to the ladder state. The density plot therefore qu...

[1] [1]

Self- assembly: from crystals to cells

Bartosz A Grzybowski, Christopher E Wilmer, Jiwon Kim, Kevin P Browne, and Kyle JM Bishop. Self- assembly: from crystals to cells. Soft Matter, 5(6):1110– 1128, 2009

2009

[2] [2]

Equilibrium mechanisms of self-limiting assembly

Michael F Hagan and Gregory M Grason. Equilibrium mechanisms of self-limiting assembly. Reviews of modern 7 physics, 93(2):025008, 2021

2021

[3] [3]

Reaction-rate theory: fifty years after kramers

Peter H¨ anggi, Peter Talkner, and Michal Borkovec. Reaction-rate theory: fifty years after kramers. Reviews of modern physics, 62(2):251, 1990

1990

[4] [4]

The statistical mechanics of dynamic pathways to self-assembly.Annual review of physical chemistry, 66(1):143–163, 2015

Stephen Whitelam and Robert L Jack. The statistical mechanics of dynamic pathways to self-assembly.Annual review of physical chemistry, 66(1):143–163, 2015

2015

[5] [5]

An active approach to colloidal self-assembly

Stewart A Mallory, Chantal Valeriani, and Angelo Cac- ciuto. An active approach to colloidal self-assembly. Annual review of physical chemistry, 69:59–79, 2018

2018

[6] [6]

Self-assembly and time-dependent control of ac- tive and passive triblock janus colloids

Juri Franz Schubert, Salman Fariz Navas, and Sabine HL Klapp. Self-assembly and time-dependent control of ac- tive and passive triblock janus colloids. The Journal of Chemical Physics, 163(4), 2025

2025

[7] [7]

Activity-controlled annealing of colloidal mono- layers

Sophie Ramananarivo, Etienne Ducrot, and Jeremie Palacci. Activity-controlled annealing of colloidal mono- layers. Nature communications, 10(1):3380, 2019

2019

[8] [8]

Statistical mechanics of ac- tive ornstein-uhlenbeck particles

David Martin, J´ er´ emy O’Byrne, Michael E Cates, ´Etienne Fodor, Cesare Nardini, Julien Tailleur, and Fr´ ed´ eric Van Wijland. Statistical mechanics of ac- tive ornstein-uhlenbeck particles. Physical Review E, 103(3):032607, 2021

2021

[9] [9]

Non-reciprocal phase transitions

Michel Fruchart, Ryo Hanai, Peter B Littlewood, and Vincenzo Vitelli. Non-reciprocal phase transitions. Nature, 592(7854):363–369, 2021

2021

[10] [10]

Self-assembly of catalytically active colloidal molecules: tailoring activ- ity through surface chemistry

Rodrigo Soto and Ramin Golestanian. Self-assembly of catalytically active colloidal molecules: tailoring activ- ity through surface chemistry. Physical review letters, 112(6):068301, 2014

2014

[11] [11]

Accelerating diffusions

Chii-Ruey Hwang, Shu-Yin Hwang-Ma, and Shuenn-Jyi Sheu. Accelerating diffusions. The Annals of Applied Probability, 15(2), 2005

2005

[12] [12]

Acceleration of convergence to equilibrium in markov chains by breaking detailed balance.Journal of statistical physics, 168(2):259–287, 2017

Marcus Kaiser, Robert L Jack, and Johannes Zimmer. Acceleration of convergence to equilibrium in markov chains by breaking detailed balance.Journal of statistical physics, 168(2):259–287, 2017

2017

[13] [13]

Sampling efficiency of trans- verse forces in dense liquids

Federico Ghimenti, Ludovic Berthier, Grzegorz Szamel, and Fr´ ed´ eric van Wijland. Sampling efficiency of trans- verse forces in dense liquids. Physical Review Letters, 131(25):257101, 2023

2023

[14] [14]

Transverse forces and glassy liquids in infinite dimensions

Federico Ghimenti, Ludovic Berthier, Grzegorz Sza- mel, and Fr´ ed´ eric van Wijland. Transverse forces and glassy liquids in infinite dimensions. Physical Review E, 109(6):064133, 2024

2024

[15] [15]

Non-reciprocal multifarious self-organization

Saeed Osat and Ramin Golestanian. Non-reciprocal multifarious self-organization. Nature Nanotechnology, 18(1):79–85, 2023

2023

[16] [16]

Escaping kinetic traps using nonrecipro- cal interactions

Saeed Osat, Jakob Metson, Mehran Kardar, and Ramin Golestanian. Escaping kinetic traps using nonrecipro- cal interactions. Physical Review Letters, 133(2):028301, 2024

2024

[17] [17]

Continuous-time multifarious systems

Jakob Metson, Saeed Osat, and Ramin Golestanian. Continuous-time multifarious systems. ii. non-reciprocal multifarious self-organization. The Journal of Chemical Physics, 163(12), 2025

2025

[18] [18]

Spinning janus doublets driven in uniform ac electric fields

Alicia Boymelgreen, Gilad Yossifon, Sinwook Park, and Touvia Miloh. Spinning janus doublets driven in uniform ac electric fields. Physical Review E, 89(1):011003, 2014

2014

[19] [19]

Kilobot: A low cost robot with scalable operations designed for collective behaviors

Michael Rubenstein, Christian Ahler, Nick Hoff, Adrian Cabrera, and Radhika Nagpal. Kilobot: A low cost robot with scalable operations designed for collective behaviors. Robotics and Autonomous Systems, 62(7):966–975, 2014

2014

[20] [20]

Self-assembly of emul- sion droplets through programmable folding

Angus McMullen, Maitane Mu˜ noz Basagoiti, Zorana Zeravcic, and Jasna Brujic. Self-assembly of emul- sion droplets through programmable folding. Nature, 610(7932):502–506, 2022

2022

[21] [21]

Odd elasticity

Colin Scheibner, Anton Souslov, Debarghya Banerjee, Pi- otr Sur´ owka, William TM Irvine, and Vincenzo Vitelli. Odd elasticity. Nature Physics, 16(4):475–480, 2020

2020

[22] [22]

Odd microswimmer

Kento Yasuda, Yuto Hosaka, Isamu Sou, and Shigeyuki Komura. Odd microswimmer. Journal of the Physical Society of Japan, 90(7):075001, 2021

2021

[23] [23]

Odd viscosity and odd elasticity

Michel Fruchart, Colin Scheibner, and Vincenzo Vitelli. Odd viscosity and odd elasticity. Annual Review of Condensed Matter Physics, 14(1):471–510, 2023

2023

[24] [24]

Direct measurement of the nonconservative force field generated by optical tweezers

Pinyu Wu, Rongxin Huang, Christian Tischer, Alexandr Jonas, and Ernst-Ludwig Florin. Direct measurement of the nonconservative force field generated by optical tweezers. Physical review letters, 103(10):108101, 2009

2009

[25] [25]

Brownian vortexes

Bo Sun, Jiayi Lin, Ellis Darby, Alexander Y Grosberg, and David G Grier. Brownian vortexes. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 80(1):010401, 2009

2009

[26] [26]

Hamiltonian curl forces

MV Berry and Pragya Shukla. Hamiltonian curl forces. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2176), 2015

2015

[27] [27]

Odd dynam- ics of living chiral crystals

Tzer Han Tan, Alexander Mietke, Junang Li, Yuchao Chen, Hugh Higinbotham, Peter J Foster, Shreyas Gokhale, J¨ orn Dunkel, and Nikta Fakhri. Odd dynam- ics of living chiral crystals. Nature, 607(7918):287–293, 2022

2022

[28] [28]

Motile dislocations knead odd crys- tals into whorls

Ephraim S Bililign, Florencio Balboa Usabiaga, Yehuda A Ganan, Alexis Poncet, Vishal Soni, Sofia Magkiriadou, Michael J Shelley, Denis Bartolo, and William TM Irvine. Motile dislocations knead odd crys- tals into whorls. Nature Physics, 18(2):212–218, 2022

2022

[29] [29]

Non-reciprocal robotic metamate- rials

Martin Brandenbourger, Xander Locsin, Edan Lerner, and Corentin Coulais. Non-reciprocal robotic metamate- rials. Nature communications, 10(1):4608, 2019

2019

[30] [30]

More is less in unpercolated active solids

Jack Binysh, Guido Baardink, Jonas Veenstra, Corentin Coulais, and Anton Souslov. More is less in unpercolated active solids. Physical Review X, 16(2):021012, 2026

2026

[31] [31]

Geometric the- ory of (extended) time-reversal symmetries in stochas- tic processes: I

J´ er´ emy O’Byrne and Michael E Cates. Geometric the- ory of (extended) time-reversal symmetries in stochas- tic processes: I. finite dimension. Journal of Statistical Mechanics: Theory and Experiment, 2024(11):113207, 2024

2024

[32] [32]

Cluster phases and bubbly phase separation in active flu- ids: Reversal of the ostwald process

Elsen Tjhung, Cesare Nardini, and Michael E Cates. Cluster phases and bubbly phase separation in active flu- ids: Reversal of the ostwald process. Physical Review X, 8(3):031080, 2018

2018

[33] [33]

Fold- ing mechanisms at finite temperature

D Rocklin, Vincenzo Vitelli, and Xiaoming Mao. Fold- ing mechanisms at finite temperature. arXiv preprint arXiv:1802.02704, 2018

work page arXiv 2018

[34] [34]

Theory of protein folding: the en- ergy landscape perspective

Jos´ e Nelson Onuchic, Zaida Luthey-Schulten, and Pe- ter G Wolynes. Theory of protein folding: the en- ergy landscape perspective. Annual review of physical chemistry, 48(1):545–600, 1997

1997

[35] [35]

The protein- folding problem, 50 years on

Ken A Dill and Justin L MacCallum. The protein- folding problem, 50 years on. science, 338(6110):1042– 1046, 2012

2012

[36] [36]

Dynamic path- ways for viral capsid assembly

Michael F Hagan and David Chandler. Dynamic path- ways for viral capsid assembly. Biophysical journal, 91(1):42–54, 2006

2006

[37] [37]

Mechanisms of virus assembly

Jason D Perlmutter and Michael F Hagan. Mechanisms of virus assembly. Annual review of physical chemistry, 8 66(1):217–239, 2015

2015

[38] [38]

Anisotropy of building blocks and their assembly into complex struc- tures

Sharon C Glotzer and Michael J Solomon. Anisotropy of building blocks and their assembly into complex struc- tures. Nature materials, 6(8):557–562, 2007

2007

[39] [39]

Janus particles

Andreas Walther and Axel HE M¨ uller. Janus particles. Soft matter, 4(4):663–668, 2008

2008

[40] [40]

Lock and key colloids

Stefano Sacanna, William TM Irvine, Paul M Chaikin, and David J Pine. Lock and key colloids. Nature, 464(7288):575–578, 2010

2010

[41] [41]

Folding dna to create nanoscale shapes and patterns

Paul WK Rothemund. Folding dna to create nanoscale shapes and patterns. Nature, 440(7082):297–302, 2006

2006

[42] [42]

Dna origami nano-mechanics

Jiahao Ji, Deepak Karna, and Hanbin Mao. Dna origami nano-mechanics. Chemical Society Reviews, 50(21):11966–11978, 2021

2021

[43] [43]

Bioinspired micro- robots

Stefano Palagi and Peer Fischer. Bioinspired micro- robots. Nature Reviews Materials, 3(6):113–124, 2018

2018

[44] [44]

Micro/nanorobotic swarms: from fundamentals to functionalities

Junhui Law, Jiangfan Yu, Wentian Tang, Zheyuan Gong, Xian Wang, and Yu Sun. Micro/nanorobotic swarms: from fundamentals to functionalities. ACS nano, 17(14):12971–12999, 2023

2023

[45] [45]

Stochastic thermodynamics, fluctuation the- orems and molecular machines

Udo Seifert. Stochastic thermodynamics, fluctuation the- orems and molecular machines. Reports on progress in physics, 75(12):126001, 2012

2012

[46] [46]

The Fokker-PlanckEquation Methods of Solution and Applications

Hannes Risken. The Fokker-PlanckEquation Methods of Solution and Applications. Springer, 1989

1989

[47] [47]

Multiscale methods: averaging and homogenization

Grigoris Pavliotis and Andrew Stuart. Multiscale methods: averaging and homogenization. Springer Sci- ence & Business Media, 2008. 9 Appendix A: Methods

2008

[48] [48]

Langevin dynamics and steady-state probability distribution invariance We consider a minimal stochastic model that isolates non-reciprocal mechanical response. Each particle ex- periences a force composed of three contributions: (i) a reciprocal part derived from a potentialE, (ii) a trans- verse, non-reciprocal odd-elastic component, and (iii) ad- ditive...

[49] [49]

Model definitions We consider several models which are all consistent with Eqs. (A2,A3). The number of particles isM. In 10 some cases it is useful to consider the particles as vertices of a polygon whose signed area is A= MX i=1 xiyi+1 −y ixi+1 2 .(A11) in which indices are added moduloM, so (xM+1 , yM+1) = (x1, y1). a. Two particle model.The model of Fi...

[50] [50]

A 2, the two-particle model is de- scribed in polar co-ordinates (detailed calculations are given in Supplementary Information)

Analysis of two-particle model As explained in Sec. A 2, the two-particle model is de- scribed in polar co-ordinates (detailed calculations are given in Supplementary Information). The dynamics of the particle separation decouples from the centre of mass; converting the associated Langevin dynamics for (r(t), θ(t)) to a Fokker-Planck equation for the prob...

[51] [51]

as ⟨ ˙θ⟩ ≈k a s 2T πkr 2 0 exp − kr2 0 2T .(A17)

[52] [52]

Dynamics of the odd square For the odd square model, we explain in the main text that the low-temperature dynamics of the internal an- gleβis described by Eq. (2). We outline the derivation of this result, with details in Supplementary Information. The central assumption is that the configuration remains always close to a rhombus shape, with small deviati...

[53] [53]

Equations of motion.All stochastic dynamics were integrated using an Euler-Maruyama scheme

Numerical simulations a. Equations of motion.All stochastic dynamics were integrated using an Euler-Maruyama scheme. The particle positions were updated according to: ri(t+ ∆t) =r i(t) +F i(t)∆t+ √ 2T∆tξ i(t),(A26) withF i(t) as in Eq. (A3), the time step is ∆tand the components of the noise vectorsξ i were sampled as inde- pendent standard normal variate...

[54] [54]

Flip ratesRand first-passage timesτestimation Flip rates and first-passage times are defined in terms of basins arounds target states, defined using thresholds on suitable order parameters, as appropriate for each model (see below). For any given initial condition, the mean first passage time is obtained by measuring first time at which the system enters ...

[55] [55]

Supplementary Video Supplementary Movies 1 and 2 provide representa- tive stochastic trajectories illustrating the dynamics dis- cussed in the main text. Movie 1 (based on Fig.2b) shows folding of the four-particle square network from a square to a line state under passive (k a = 0) and non-reciprocal (ka = 1) dynamics, together with the time evolution of...

[56] [56]

= (−0.5,0),x 0 2 = (x 0 2, y0

[57] [57]

= (0, √ 3 2 ),x 0 3 = (x 0 3, y0

[58] [58]

= (0.5,0) andX 0 = (x0 1,x 0 2,x 0 3). Considering small deformations of the (passive) spring network about this configuration, the (normalised) eigenmodes are ξx = 1√ 3 (1,0,1,0,1,0), ξy = 1√ 3 (0,1,0,1,0,1), z1 = 1 2 √ 3 1,− √ 3,−2,0,1, √ 3 , z2 = 1 2 √ 3 √ 3,1,0,−2,− √ 3,1 , z3 = 1 2 √ 3 √ 3,1,− √ 3,1,0,−2 , z4 = 1 2 √ 3 −1, √ 3,−1,− √ 3,2,0 . (A40) wh...

[59] [59]

= cos β 2 ,0 , (x0 2, y0

[60] [60]

= 0,sin β 2 , (x0 3, y0

[61] [61]

= −cos β 2 ,0 , (x0 4, y0

[62] [62]

= 0,−sin β 2 . (A45) and we write s0 = x0 1, y0 1, x0 2, y0 2, x0 3, y0 3, x0 4, y0 4 (A46) Considering small deformations about this state, the eigenmodes of the square spring network are ξx = 1 2 (1,0,1,0,1,0,1,0), ξy = 1 2 (0,1,0,1,0,1,0,1), za = 1√ 2 0,cos β 2 ,−sin β 2 ,0,0,−cos β 2 ,sin β 2 ,0 , zb = 1√ 2 sin β 2 ,0,0,−cos β 2 ,−sin β 2 ,0,0,cos β 2...

[63] [63]

Initialize the assembly in the rectangular configuration (a special case of the ladder state) and measure the time required to reach the chevron configuration, as defined in MethodsA.6

[64] [64]

Repeat forNstatistically independent realizations

[65] [65]

This protocol assumes that once the chevron configuration is reached, the double-square is immediately stabilized by a bound cargo and does not revert to the ladder state

Collect and sort the transition times to construct the density plot in Fig.3e, representing the distribution of first-passage times from ladder to chevron. This protocol assumes that once the chevron configuration is reached, the double-square is immediately stabilized by a bound cargo and does not revert to the ladder state. The density plot therefore qu...