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arxiv: 2605.06543 · v1 · submitted 2026-05-07 · ❄️ cond-mat.stat-mech · cond-mat.soft· math-ph· math.MP

Recognition: unknown

A Rayleigh criterion for mechanical instability: inducing activity by chemo-mechanical coupling

Aaron Beyen, Christian Maes, Francesco Casini

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:37 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.softmath-phmath.MP
keywords Rayleigh criterionmechanical instabilitychemo-mechanical couplingentropic contributionsfrenetic contributionsMarkov jump processesnonequilibrium systemsrotational motion
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0 comments X

The pith

Chemical driving induces sustained rotational motion in a Newtonian probe when entropic and frenetic contributions align in phase.

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

The paper derives Rayleigh-like criteria for the onset of mechanical activity in a slow Newtonian probe coupled to driven chemical reactions modeled as Markov jump processes. The criteria depend on the phase relation between entropic contributions tied to dissipation and frenetic contributions tied to the timing of jumps. When this phase produces positive feedback, chemical energy converts into persistent mechanical rotation without external torque. A reader would care because the same mechanism that creates unwanted instabilities can also generate useful nonequilibrium motion in coupled systems.

Core claim

We introduce a theoretical framework, inspired by Rayleigh's analysis of thermoacoustic instabilities, to study the emergence of mechanical activity. In particular, we derive Rayleigh-like criteria governing the onset of activity and the generation of rotational motion in a slow Newtonian probe coupled to driven chemical processes, described by Markov jump processes. These criteria are expressed in terms of the phase relation between entropic and frenetic contributions, providing a transparent condition for when chemical driving results in sustained rotational or active mechanical motion.

What carries the argument

Rayleigh-like criteria based on the phase relation between entropic and frenetic contributions in the chemo-mechanical coupling.

Load-bearing premise

Chemical processes follow Markov jump dynamics and the entropic and frenetic parts have well-defined, controllable phase relations in the coupled system.

What would settle it

In an experiment with a colloidal probe coupled to a controllable chemical cycle, sustained rotation appears when the measured phase relation violates the criterion or fails to appear when the criterion is satisfied.

Figures

Figures reproduced from arXiv: 2605.06543 by Aaron Beyen, Christian Maes, Francesco Casini.

Figure 1
Figure 1. Figure 1: FIG. 1: Probe (black dot) following the dynamics (1) for a 3 state model in a switching potential view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Plot of the friction coefficient view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Velocity statistics in time (a) average velocity (b) standard deviation with characteristic view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Steady state distributions for the (a) velocity normalized with view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Pseudocolor plot of the full steady state distribution view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Velocity statistics in time for positive and negative driving: (a) average velocity (b) view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Stationary velocity distributions for positive and negative driving with view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The stationary velocity distribution at larger driving has a shifted Maxwellian form with view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Stationary (a) velocity and (b) position distributions for a probe on the line. In (a), we view at source ↗
read the original abstract

Instabilities in thermodynamic systems are often undesirable, as they can lead to loss of control or even catastrophic behavior. Yet, the same mechanisms can also generate rich nonequilibrium behavior and may play a constructive role in living systems. We introduce a theoretical framework, inspired by Rayleigh's analysis of thermoacoustic instabilities, to study the emergence of mechanical activity. In particular, we derive Rayleigh-like criteria governing the onset of activity and the generation of rotational motion in a slow Newtonian probe coupled to driven chemical processes, described by Markov jump processes. These criteria are expressed in terms of the phase relation between entropic and frenetic contributions, providing a transparent condition for when chemical driving results in sustained rotational or active mechanical motion.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces a theoretical framework, inspired by Rayleigh's analysis of thermoacoustic instabilities, to derive Rayleigh-like criteria for the onset of mechanical activity and rotational motion in a slow Newtonian probe coupled to driven chemical processes modeled as Markov jump processes. The criteria are expressed in terms of the phase relation between entropic and frenetic contributions, providing a condition for when chemical driving produces sustained active mechanical motion.

Significance. If the derivation is sound, the work offers a transparent, phase-based criterion for chemo-mechanical instabilities that could help explain activity generation in biological systems and guide design of synthetic active matter. The extension of Rayleigh analysis to Markovian chemical driving is a clear conceptual advance, though its impact depends on whether the separation into entropic and frenetic terms remains robust under mechanical feedback.

major comments (2)
  1. [Derivation of the Rayleigh-like criterion] The central claim requires that entropic (force-like) and frenetic (kinetic) contributions to the chemical driving admit a well-defined, tunable phase relation even after coupling to the Newtonian probe. Because probe velocity modulates the jump rates, the frenetic term—which depends on the time-symmetric part of the path measure—acquires an implicit dependence on the mechanical coordinate; an explicit demonstration that this does not destroy or shift the assumed phase relation is needed (see the derivation of the instability criterion).
  2. [Model definition and Markov jump process setup] The model assumes that the chemical processes remain accurately describable by Markov jump processes whose entropic and frenetic parts can be separated and controlled independently. Under mechanical back-action this separation is not automatic; a concrete check (e.g., via an exactly solvable two-state example or a perturbative expansion) is required to confirm that the phase condition survives the coupling.
minor comments (2)
  1. [Introduction] Notation for the entropic and frenetic contributions should be introduced with explicit definitions and symbols at the first appearance rather than relying on later equations.
  2. [Abstract] The abstract states that criteria are 'expressed in terms of the phase relation' but does not indicate the final functional form; a compact statement of the criterion (e.g., an inequality involving the phase lag) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the scope and robustness of the proposed framework. We address each major comment below and describe the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Derivation of the Rayleigh-like criterion] The central claim requires that entropic (force-like) and frenetic (kinetic) contributions to the chemical driving admit a well-defined, tunable phase relation even after coupling to the Newtonian probe. Because probe velocity modulates the jump rates, the frenetic term—which depends on the time-symmetric part of the path measure—acquires an implicit dependence on the mechanical coordinate; an explicit demonstration that this does not destroy or shift the assumed phase relation is needed (see the derivation of the instability criterion).

    Authors: We appreciate the referee’s emphasis on this point. The derivation in the manuscript is performed in the limit of a slow Newtonian probe, where the mechanical coordinate enters the chemical rates parametrically and the phase relation is extracted from the time-averaged entropic and frenetic contributions along the chemical trajectories. Nevertheless, to make the robustness explicit, we will add a short perturbative expansion (to first order in the probe velocity) showing that the leading-order phase difference between the entropic and frenetic terms is unaffected by the mechanical back-action. This calculation will be placed in a new appendix and cross-referenced in the main derivation of the instability criterion. revision: yes

  2. Referee: [Model definition and Markov jump process setup] The model assumes that the chemical processes remain accurately describable by Markov jump processes whose entropic and frenetic parts can be separated and controlled independently. Under mechanical back-action this separation is not automatic; a concrete check (e.g., via an exactly solvable two-state example or a perturbative expansion) is required to confirm that the phase condition survives the coupling.

    Authors: We agree that an explicit verification strengthens the presentation. The standard decomposition of the path measure for Markov jump processes remains formally valid when the transition rates depend on the mechanical coordinate, because the rates are still functions of the instantaneous state. To provide the requested concrete check, we will include a minimal two-state chemical model in the revised manuscript. In this example the probe velocity enters the rates linearly; both analytic calculation and direct simulation confirm that the phase relation between entropic and frenetic contributions is preserved to the order required by the instability criterion. The example will be presented as a new subsection. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard Markov process decomposition to new coupling

full rationale

The paper derives Rayleigh-like instability criteria by expressing the onset of mechanical activity in terms of the phase difference between entropic and frenetic contributions in a Markov jump process coupled to a Newtonian probe. These contributions are defined independently via the standard decomposition of path weights into time-symmetric and time-antisymmetric parts; the phase relation is then obtained from the linear response of the jump rates to the probe velocity. No equation reduces to a fitted parameter renamed as a prediction, no self-citation supplies a uniqueness theorem that forces the result, and the ansatz is not smuggled in from prior author work. The framework remains self-contained against the external benchmark of standard nonequilibrium Markov process theory.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper's central claim depends on the validity of modeling the system with Markov jump processes for chemistry and Newtonian dynamics for the probe, plus the separation into entropic and frenetic parts which is standard in some nonequilibrium theories but applied here specifically.

axioms (3)
  • domain assumption The probe is a slow Newtonian object
    Stated in abstract as 'slow Newtonian probe'
  • domain assumption Chemical processes are described by Markov jump processes
    Explicitly stated in abstract
  • ad hoc to paper Entropic and frenetic contributions can be separated and their phase relation determines stability
    This is the core of the derived criteria

pith-pipeline@v0.9.0 · 5422 in / 1472 out tokens · 58252 ms · 2026-05-08T04:37:34.599962+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

84 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    J. W. Strutt (3rd Baron Rayleigh). The Explanation of Certain Acoustical Phenomena.Proc. R. Inst. GB., 8:536–542, 1878

  2. [2]

    R. I. Sujith and S. A. Pawar.Thermoacoustic Instability: A Complex Systems Perspective. Springer Series in Synergetics. Springer, 2021

    2021

  3. [3]

    J. W. Strutt (3rd Baron Rayleigh). XXXIII. On maintained vibrations.Philos. Mag., 15(94):229–235, 1883

  4. [4]

    Sun, H.-X

    S.-Y. Sun, H.-X. Zhang, W. Fang, X.-D. Chen, B. Li, and X.-Q. Feng. Chapter three - bio-chemo- 22 mechanical coupling models of soft biological materials: A review. volume 55 ofAdvances in Applied Mechanics, pages 309–392. Elsevier, 2022

    2022

  5. [5]

    X.-Q. Feng, B. Li, S.-Z. Lin, M.-Y. Wang, X.-D. Chen, H.-X. Zhang, and W. Fang. Mechano-chemo- biological theory of cells and tissues: review and perspectives.Acta Mech. Sin., 41(7), 2025

    2025

  6. [6]

    C. Maes. Local detailed balance.SciPost Phys. Lect. Notes, page 32, 2021

    2021

  7. [7]

    Maes and K

    C. Maes and K. Netoˇ cn´ y. Time-reversal and entropy.J. Stat. Phys., 110:269–310, 2003

    2003

  8. [8]

    S. Katz, J. L. Lebowitz, and H. Spohn. Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors.J. Stat. Phys., 34:497–537, 1984

    1984

  9. [9]

    H. Tasaki. Two theorems that relate discrete stochastic processes to microscopic mechanics, 2007. arXiv:0706.1032 [cond-mat.stat-mech]

    work page arXiv 2007

  10. [10]

    Fodor, C

    ´E. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. van Wijland. How Far from Equilibrium Is Active Matter?Phys. Rev. Lett., 117(3), 2016

    2016

  11. [11]

    Zimmermann and U

    E. Zimmermann and U. Seifert. Efficiencies of a molecular motor: a generic hybrid model applied to the F1-ATPase.New J. Phys., 14(10):103023, 2012

    2012

  12. [12]

    Pietzonka, A

    P. Pietzonka, A. C. Barato, and U. Seifert. Universal bound on the efficiency of molecular motors.J. Stat. Mech.: Theory Exp., 2016(12):124004, 2016

    2016

  13. [13]

    J¨ ulicher, A

    F. J¨ ulicher, A. Ajdari, and J. Prost. Modeling molecular motors.Rev. Mod. Phys., 69(4):1269–1281, 1997

    1997

  14. [14]

    U. Basu, C. Maes, and K. Netoˇ cn´ y. How Statistical Forces Depend on the Thermodynamics and Kinetics of Driven Media.Phys. Rev. Lett., 114(25), 2015

    2015

  15. [15]

    Fruchart and V

    M. Fruchart and V. Vitelli. Nonreciprocal many-body physics, 2026. arXiv:2602.11111 [cond-mat.stat- mech]

    work page arXiv 2026

  16. [16]

    te Vrugt, B

    M. te Vrugt, B. Liebchen, and M. E. Cates. What exactly is ‘active matter’?, 2025. arXiv:2507.21621 [cond-mat.soft]

    work page arXiv 2025

  17. [17]

    Gompper, R

    G. Gompper, R. G. Winkler, T. Speck, A. Solon, C. Nardini, F. Peruani, H. L¨ owen, et al. The 2020 motile active matter roadmap.J. Phys. Condens. Matter, 32(19):193001, 2020

    2020

  18. [18]

    Fodor and M

    ´E. Fodor and M. C. Marchetti. The statistical physics of active matter: From self-catalytic colloids to living cells.Phys. A: Stat. Mech. Appl., 504:106–120, 2018. Lecture Notes of the 14th International Summer School on Fundamental Problems in Statistical Physics

    2018

  19. [19]

    Ramaswamy

    S. Ramaswamy. Active matter.J. Stat. Mech.: Theory Exp., 2017(5):054002, 2017

    2017

  20. [20]

    U. Seifert. Stochastic thermodynamics of single enzymes and molecular motors.Eur. Phys. J. E, 34(3):26, 2011

    2011

  21. [21]

    W¨ urthner, F

    L. W¨ urthner, F. Brauns, G. Pawlik, J. Halatek, J. Kerssemakers, C. Dekker, and E. Frey. Bridging scales in a multiscale pattern-forming system.Proc. Natl. Acad. Sci. U.S.A., 119(33), 2022

    2022

  22. [22]

    Burkart, M.C

    T. Burkart, M.C. Wigbers, L. W¨ urthner, and E. Frey. Control of protein-based pattern formation via guiding cues.Nat. Rev. Phys., 4:511–527, 2022

    2022

  23. [23]

    Maes and K

    C. Maes and K. Netoˇ cn´ y. Nonequilibrium corrections to gradient flow.Chaos, 29(7):073109, 2019. 23

    2019

  24. [24]

    Beyen, C

    A. Beyen, C. Maes, and J.-H. Pei. Coupling an elastic string to an active bath: The emergence of inverse damping.Phys. Rev. E, 112:L042103, 2025

    2025

  25. [25]

    Pei and C

    J.-H. Pei and C. Maes. Induced friction on a probe moving in a nonequilibrium medium.Phys. Rev. E, 111(3):L032101, 2025

    2025

  26. [26]

    Pei and C

    J.-H. Pei and C. Maes. Transfer of Active Motion from Medium to Probe via the Induced Friction and Noise.Phys. Rev. Lett., 136(3), 2026

    2026

  27. [27]

    C. Maes. Frenesy: Time-symmetric dynamical activity in nonequilibria.Phys. Rep., 850:1–33, 2020

    2020

  28. [28]

    Basu and C

    U. Basu and C. Maes. Nonequilibrium Response and Frenesy.J. Phys. Conf. Ser., 638:012001, 2015

    2015

  29. [29]

    Gaspard.The Statistical Mechanics of Irreversible Phenomena

    P. Gaspard.The Statistical Mechanics of Irreversible Phenomena. Cambridge University Press, 2022

    2022

  30. [30]

    Maes.Non-Dissipative Effects in Nonequilibrium Systems

    C. Maes.Non-Dissipative Effects in Nonequilibrium Systems. Springer International Publishing, 2018

    2018

  31. [31]

    L. Y. Chen, N. Goldenfeld, and Y. Oono. Renormalization Group Theory for Global Asymptotic Analysis.Phys. Rev. Lett., 73(10):1311–1315, 1994

    1994

  32. [32]

    L. B. Arosh, M. C. Cross, and R. Lifshitz. Quantum limit cycles and the Rayleigh and van der Pol oscillators.Phys. Rev. Res., 3:013130, 2021

    2021

  33. [33]

    Lefebvre and C

    B. Lefebvre and C. Maes. Frenetic steering: Nonequilibrium-enabled navigation.Chaos, 34(6), 2024

    2024

  34. [34]

    Baerts, U

    P. Baerts, U. Basu, C. Maes, and S. Safaverdi. Frenetic origin of negative differential response.Phys. Rev. E, 88(5), 2013

    2013

  35. [35]

    N. G. Van Kampen.Stochastic Processes in Physics and Chemistry. North Holland, 3rd edition, 2007

    2007

  36. [36]

    Born and R

    M. Born and R. Oppenheimer. Zur Quantentheorie der Molekeln.Ann. Phys., 389(20):457–484, 1927

    1927

  37. [37]

    Szabo and N.S

    A. Szabo and N.S. Ostlund.Modern Quantum Chemistry: Introduction to Advanced Electronic Struc- ture Theory. Dover Books on Chemistry. Dover Publications, 1996

    1996

  38. [38]

    Maes and K

    C. Maes and K. Netoˇ cn´ y. Heat Bounds and the Blowtorch Theorem.Ann. H. Poincar´ e, 14(5):1193– 1202, 2013

    2013

  39. [39]

    H. Mori. Statistical-Mechanical Theory of Transport in Fluids.Phys. Rev., 112:1829–1842, 1958

    1958

  40. [40]

    H. Mori. Transport, Collective Motion, and Brownian Motion.Prog. Theor. Phys., 33(3):423–455, 03 1965

    1965

  41. [41]

    R. Zwanzig. Ensemble Method in the Theory of Irreversibility.J. Chem. Phys., 33(5):1338–1341, 1960

    1960

  42. [42]

    Widder, J

    C. Widder, J. Zimmer, and T. Schilling. On the generalized Langevin equation and the Mori projection operator technique.J. Phys. A: Math. Theor., 58(40):405001, 2025

    2025

  43. [43]

    Generalised Langevin Dynamics: Significance and Limitations of the Projection Operator Formalism

    C. Widder and T. Schilling. Generalised Langevin Dynamics: Significance and Limitations of the Projection Operator Formalism, 2026. arXiv:2604.20453 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  44. [44]

    Grabert.Projection Operator Techniques in Nonequilibrium Statistical Mechanics

    H. Grabert.Projection Operator Techniques in Nonequilibrium Statistical Mechanics. Communications and Control Engineering. Springer-Verlag, 1982

    1982

  45. [45]

    C. Maes. Response Theory: A Trajectory-Based Approach.Front. Phys., 8, 2020

    2020

  46. [46]

    Baiesi, C

    M. Baiesi, C. Maes, and B. Wynants. Nonequilibrium Linear Response for Markov Dynamics, I: Jump Processes and Overdamped Diffusions.J. Stat. Phys., 137(5–6):1094–1116, 2009

    2009

  47. [47]

    Baiesi, E

    M. Baiesi, E. Boksenbojm, C. Maes, and B. Wynants. Nonequilibrium Linear Response for Markov 24 Dynamics, II: Inertial Dynamics.J. Stat. Phys., 139(3):492–505, 2010

    2010

  48. [48]

    Tanogami

    T. Tanogami. Violation of the Second Fluctuation-dissipation Relation and Entropy Production in Nonequilibrium Medium.J. Stat. Phys., 187(3), 2022

    2022

  49. [49]

    Nakayama, K

    Y. Nakayama, K. Kawaguchi, and N. Nakagawa. Unattainability of Carnot efficiency in thermal motors: Coarse graining and entropy production of Feynman-Smoluchowski ratchets.Phys. Rev. E, 98(2), 2018

    2018

  50. [50]

    M. M. Dygas, B. J. Matkowsky, and Z. Schuss. A singular perturbation approach to non-Markovian escape rate problems with state dependent friction.J. Chem. Phys., 84(7):3731–3738, 1986

    1986

  51. [51]

    A. N. Tikhonov. Systems of differential equations containing small parameters in the derivatives.Mat. Sb., 31 (73)(3):575–586, 1952. in Russian

    1952

  52. [52]

    S. A. Lomov.Introduction to the General Theory of Singular Perturbations, volume 112 ofTranslations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1992

    1992

  53. [53]

    Statistical physics approaches to subnetwork dynamics in biochemical systems

    B Bravi and P Sollich. Statistical physics approaches to subnetwork dynamics in biochemical systems. Phys. Biol., 14(4):045010, 2017

    2017

  54. [54]

    Polettini and M

    M. Polettini and M. Esposito. Effective Thermodynamics for a Marginal Observer.Phys. Rev. Lett., 119(24), 2017

    2017

  55. [55]

    Bo and A

    S. Bo and A. Celani. Multiple-scale stochastic processes: Decimation, averaging and beyond.Phys. Rep., 670:1–59, 2017

    2017

  56. [56]

    Balakrishnan.Elements of Nonequilibrium Statistical Mechanics

    V. Balakrishnan.Elements of Nonequilibrium Statistical Mechanics. Springer International Publishing, 2020

    2020

  57. [57]

    C. Maes. On the Second Fluctuation-Dissipation Theorem for Nonequilibrium Baths.J. Stat. Phys., 154(3):705–722, 2014

    2014

  58. [58]

    J. A. McLennan. Statistical Mechanics of the Steady State.Phys. Rev., 115:1405–1409, 1959

    1959

  59. [59]

    Khodabandehlou and C

    F. Khodabandehlou and C. Maes. Close-to-equilibrium heat capacity.J. Phys. A: Math. Theor., 57(20):205001, 2024

    2024

  60. [60]

    Nicoletti and D

    G. Nicoletti and D. M. Busiello. Tuning Transduction from Hidden Observables to Optimize Information Harvesting.Phys. Rev. Lett., 133:158401, 2024

    2024

  61. [61]

    Dechant and E

    A. Dechant and E. Lutz. Fundamental limits on nonequilibrium sensing.Nat. Commun., 16(1), 2025

    2025

  62. [62]

    A. Beyen. Inducing activity by chemo–mechanical coupling.https://github.com/AaronBeyen/ Inducing-Activity-by-Chemo-Mechanical-Coupling

  63. [63]

    Ramaswamy

    S. Ramaswamy. The Mechanics and Statistics of Active Matter.Annu. Rev. Condens. Matter Phys., 1(Volume 1, 2010):323–345, 2010

    2010

  64. [64]

    te Vrugt and R

    M. te Vrugt and R. Wittkowski. Metareview: a survey of active matter reviews.Eur. Phys. J. E, 48(2):12, 2025

    2025

  65. [65]

    A. Dhar, A. Kundu, S. N. Majumdar, S. Sabhapandit, and G. Schehr. Run-and-tumble particle in one-dimensional confining potentials: Steady-state, relaxation, and first-passage properties.Phys. Rev. E, 99:032132, 2019

    2019

  66. [66]

    Metson and R

    J. Metson and R. Golestanian. Emergent single-species non-reciprocity from bistable chemical dynam- 25 ics, 2026. arXiv:2603.21863 [cond-mat.soft]

    work page arXiv 2026

  67. [67]

    Dolai, C

    P. Dolai, C. Maes, and K. Netoˇ cn´ y. Calorimetry for active systems.SciPost Phys., 14:126, 2023

    2023

  68. [68]

    Dean Astumian and M

    R. Dean Astumian and M. Bier. Fluctuation driven ratchets: Molecular motors.Phys. Rev. Lett., 72(11):1766–1769, 1994

    1994

  69. [69]

    P. Reimann. Brownian motors: noisy transport far from equilibrium.Phys. Rep., 361(2–4):57–265, 2002

    2002

  70. [70]

    H¨ anggi and F

    P. H¨ anggi and F. Marchesoni. Artificial Brownian motors: Controlling transport on the nanoscale. Rev. Mod. Phys., 81(1):387–442, 2009

    2009

  71. [71]

    Golestanian, T

    R. Golestanian, T. B. Liverpool, and A. Ajdari. Propulsion of a Molecular Machine by Asymmetric Distribution of Reaction Products.Phys. Rev. Lett., 94:220801, Jun 2005. 26 Appendix A. Coupling diffusion to reaction There exists a multitude of ways by which coupling to jumpers produces motion, [67]. A recurrent mechanism in small-scale systems is to couple...

    2005

  72. [72]

    equal and opposite forces

    Semi-reciprocity and local detailed balance The naive statement of Newton’s third law (“equal and opposite forces”) is not generally invariant under arbitrary coordinate transformations. More to the point is that interactions should not create or destroy total momentum,i.e.the total stress-energy tensor should be divergence-free. Yet, in the overdamped li...

  73. [73]

    That implies there is no reciprocity, except when interpreting the chemistry to take place at infinite temperature

    No feedback as thermal disequilibrium When the probe dynamics forx(t) is externally driven by switching potentialsU(x,σ), as specified by the stateσ(t) of the jumper, and the jump ratesk(σ,σ′) forσ(t) arex-independent, then there 28 is no feedback from mechanics to chemistry. That implies there is no reciprocity, except when interpreting the chemistry to ...

  74. [74]

    29 As a more realistic illustration, we briefly discuss the case of (early) models of molecular motors (flashing ratchets with binding), [13, 71]

    Feedback without semi-reciprocity Consider yet another model for the diffusion processx(t)∈R, Γ ˙x(t) =σ(t)f(x(t)) + √ 2ΓkBT ξ(t) where the rates (or the chemistry) for the processσ(t) =±1 now depend on the mechanical state kx(1,−1) =αe−Υx, k x(−1,1) =αeΥx +δ Hence, there is feedback, but quite generally, for a givenf(x), there is no joint (dimensionless)...

  75. [75]

    Mean force We start by analyzing the Born-Oppenheimer distribution solving (8). Forσ∈Z3, the solution is ρx(σ) =e−βλU(x,σ) Nx gx(σ) (B1) gx(σ) =ψx(σ,σ+ 1)ψx(σ,σ+ 2)e βλ 2 (U(x,σ+1)+U(x,σ+2)) +ψx(σ,σ+ 1)ψx(σ+ 1,σ+ 2)e βλ 2 (U(x,σ+2)+U(x,σ))e−βw +ψx(σ+ 1,σ+ 2)ψx(σ,σ+ 2)e βλ 2 (U(x,σ+1)+U(x,σ))eβw (B2) with normalizationNx = ∑ σ∈Zne−βλU(x,σ)gx(σ) and where t...

  76. [76]

    Noise amplitude Writing outB(x) in (12) explicitly yields B(x) =λ2N ∫ ∞ 0 dτ   ∑ σ,σ0∈Zn ∂U ∂x(x,σ)ρx(σ,τ|σ0,0) ∂U ∂x(x,σ0)ρx(σ0)−   ∑ σ∈Zn ∂U ∂x(x,σ)ρx(σ)   2  Hereρx(σ,τ|σ0,0) is the transition probability for theσ-process at fixedxand starting from configurationσ0 att= 0. Under weak coupling, sinceB(x) is already of orderO(λ 2), we only need ...

  77. [77]

    Friction coefficient The covariance⟨·;·⟩BO in the formula (11) forνcan be rewritten as a single expectation value ν(x) =−λN ∫ ∞ 0 dτ ⟨∂U ∂x(x(t),σ(τ))·∂logρx ∂x (σ) ⟩ BO x +λN ∫ ∞ 0 dτ ⟨∂U ∂x(x(t),σ(τ)) ⟩ BO x · ⟨ ∂logρx ∂x (σ) ⟩ BO x =−λN ∫ ∞ 0 dτ ⟨∂U ∂x(x(t),σ(τ))·∂logρx ∂x (σ) ⟩ BO x (B6) 32 where we have used the normalization ofρx to eliminate the se...

  78. [78]

    Large driving limit In the large driving limitw→∞, (B1) reduces to limw→∞ρx(σ)∝e βλ 2 (U(x,σ+1)−U(x,σ))limw→∞ψx(σ+ 1,σ+ 2)ψx(σ,σ+ 2) which is what is needed for (26). A similar analysis shows that forw→−∞ lim w→−∞ ρx(σ)∝e βλ 2 (U(x,σ+2)−U(x,σ))limw→∞ 1 ψx(σ+ 2,σ) For the friction and noise, using the property⟨X;Y⟩BO x = ⣨( X−⟨X⟩BO x ) Y ⟩BO x , the expres...

  79. [79]

    Compute the accelerationa i in (C1) from the current state (x i,⃗ σi)

  80. [80]

    Update the velocity:v i+1 =v i +ai ∆t

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