Pith. sign in

REVIEW 3 major objections 3 minor 1 cited by

This paper shows that the minimum-work trap protocol for moving an active particle near a wall is not the time-reversal of the return protocol: going away needs a large early jump and a slow middle, while going toward tracks the bulk soluti

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Near a no-slip wall, the minimum-work optical-trap protocol for transporting an active particle away from the wall breaks time-reversal symmetry with the return protocol.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection Solid numerical method paper with a real but fixable sign error in the activity discussion and a truncation-convergence gap that should be closed before the symmetry-breaking claim is published as stated. the 3 major comments →

arxiv 2603.17798 v2 pith:ATLMBPDP submitted 2026-03-18 cond-mat.soft cond-mat.stat-mech

Optimal transport and control of an active particle near a plane wall

Utkarsh Maurya , Kavya Swaminathan , Ejaz Ashraf , Rajesh Singh This is my paper
classification cond-mat.soft cond-mat.stat-mech
keywords optimal transport protocolsactive colloidsstochastic thermodynamicsChebyshev polynomial basisgenetic algorithm optimizationwall hydrodynamicstime-reversal symmetry breakingoptical tweezers
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Near a solid wall, the cheapest way to move a trapped active particle between two heights is not the same in the two directions. The paper finds that moving away from the wall demands a protocol with a large jump at the start, a nearly flat middle, and a steep finish, while moving toward the wall stays close to the known bulk linear-ramp solution until the last phase. This breaks the time-reversal symmetry that holds in bulk, where the optimal protocol for an outbound trip is exactly the time-reverse of the return trip. The result matters because near-wall transport is common in experiments, and using the bulk protocol in the away direction costs extra work—up to about 7% at the closest wall separation studied. The paper reaches this conclusion by representing the protocol as a short Chebyshev expansion and optimizing the coefficients with a genetic algorithm.

Core claim

The central discovery is that a no-slip boundary, through spatially reduced mobility and a wall-induced active drift, breaks the time-reversal symmetry of the optimal open-loop transport protocol. For a passive particle at H0=2, the away-from-wall optimal protocol has a pronounced jump at t=0, a near-plateau interior, and a steep terminal rise, whereas the towards-wall protocol follows the bulk prediction for most of the trajectory and deviates only near t=tf. The optimized protocols beat the bulk protocol under near-wall dynamics, with percentage work savings up to about 7% at H0=2, and activity modulates the effect in a direction-dependent way: pullers are helped moving away, pushers are h

What carries the argument

The key object is the open-loop trap-center protocol λ(t), expanded in a truncated Chebyshev basis (N=5) whose coefficients are the variables to optimize. Endpoint positions are enforced exactly, so jump discontinuities at the boundaries emerge naturally rather than being imposed. The objective is the mean thermodynamic work, evaluated by simulating stochastic trajectories with a Heun integrator and minimized by a genetic algorithm. The physics is carried by two position-dependent effects: the reduced mobility that suppresses diffusion near the wall, and the stresslet active drift that draws pullers toward and pushes pushers away from the wall. These effects make the problem analytically int

Load-bearing premise

The load-bearing assumption is that five Chebyshev polynomials can represent the true optimal protocol; if the real optimum needs sharper endpoint jumps or finer interior structure, the reported protocol shapes and work savings could be artifacts of the truncated ansatz rather than features of the physics.

What would settle it

Run the same optimization at H0=2 with N=8 or with explicit endpoint-jump variables; if the mean work falls by more than the statistical error or the away protocol loses its initial-jump/plateau/terminal-rise shape, the central symmetry-breaking claim is an artifact of the truncation. Alternatively, in an optical-trap experiment measure the distribution of particle positions under the optimized versus the bulk protocol near a wall; if the optimized away protocol does not reduce mean work relative to the bulk ramp, the predicted asymmetry is absent.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Near-wall transport needs direction-specific protocols; the bulk linear-ramp protocol is measurably suboptimal in the away direction, and the work saving grows as the wall is approached.
  • Activity reverses its role with direction: pullers reduce the work of moving away while pushers reduce the work of moving toward the wall, so the sign of activity alone does not determine cost.
  • The optimization method requires only the ability to simulate stochastic trajectories, so it transfers to other stochastic systems where exact protocols are unavailable.
  • In the bulk limit the method reproduces the known analytical protocol and the theoretical minimum work, so the numerical machinery is validated against exact results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetry breaking should appear in any spatially inhomogeneous environment—near interfaces, in gradients, or in confined geometries—because opposite journeys sample the position-dependent drag and drift in reversed order.
  • A sharper test of the truncation would be to repeat the optimization at H0=2 with a larger Chebyshev basis or with endpoint jumps as free parameters; if work drops materially or the shape changes, the N=5 ansatz is the limiting factor.
  • Adding feedback or jointly optimizing trap stiffness would likely lower the work further, so the reported savings are a floor for what control could achieve.
  • The one-dimensional wall-normal setup neglects lateral motion and orientational fluctuations; experiments with bottom-heavy swimmers could reveal whether rotational noise enhances or weakens the asymmetry.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper presents a numerical method, the Ritz method with a Chebyshev basis and a genetic algorithm, for optimizing open-loop trap protocols that minimize the mean thermodynamic work of a trapped stochastic particle. The method is applied to an active Brownian particle modeled as a stresslet near a no-slip wall, with position-dependent mobility given by the Brenner formula. The method is validated against the Schmiedl–Seifert analytic solution in the bulk limit (H_0=1000), recovering both the linear-ramp protocol and the exact work W*=6.25. The central physical claim is that the presence of the wall breaks the time-reversal symmetry of the optimal protocol: the away-from-wall protocol acquires a large initial jump, a near-plateau interior, and a steep terminal rise, while the towards-wall protocol stays close to the bulk solution until the final phase. The paper also claims that the activity (pusher vs. puller) modulates these effects in a direction-dependent way.

Significance. If the reported results are correct, the paper offers a flexible numerical framework for optimal transport in complex, spatially inhomogeneous stochastic environments, requiring only the ability to simulate trajectories. The method is validated against an external analytic benchmark (not circular), the code is publicly available, and the ensemble diagnostics (CV<0.14%, low pairwise L2 RMSE) indicate reliable convergence of the optimizer within the chosen ansatz. The claimed symmetry-breaking is a potentially new physical effect in near-wall active transport. However, the central shape claim is not yet backed by a convergence analysis of the Chebyshev truncation, and the interpretation of the activity effect directly contradicts the model equation and the paper's own Table I.

major comments (3)
  1. [III D, Eq. (3), Table I] The text states that for pullers (α>0) the stresslet drift is directed away from the wall and is 'thermodynamically beneficial' during away transport, and that for pushers (α<0) the drift is towards the wall and opposes the trap motion. This is the opposite of Eq. (3): v_A(h)=-(s0/8)[(b/h)^2-(b/h)^4], so for h>b and s0>0 the drift is towards the wall. The paper's own Table I agrees with Eq. (3): at H0=2 and away transport, ⟨W⟩_GA is 10.03 for α=+25, 8.90 for α=0, and 7.82 for α=-25, i.e., pullers incur the highest work because their drift opposes the away motion, while pushers are assisted. The same reversal appears in Appendix A ('pushers are assisted during towards transport'). This is a load-bearing error in the interpretation of the activity effect and must be corrected throughout Sections III D, IV, and the appendix.
  2. [II D, Fig. 8, Appendix B 3] The central claim of Section III E is a claim about the shape of the optimal protocol (large initial jump, plateau, steep terminal rise for away transport). The only convergence evidence is that mean work plateaus for N≥3 and that Fig. 8 shows generation convergence at fixed N=5. Neither demonstrates convergence of the protocol shape with respect to the truncation order. At H0=1000 the bulk linear ramp is exactly representable with N=2, so that validation does not exercise the higher-order modes needed to resolve a near-wall boundary layer. The text explicitly says 'expanding the basis size further strains the fixed generation budget', meaning N=5 was selected not on the basis of shape convergence but because of computational limits. Thus the reported symmetry-breaking shapes in Fig. 5 could be artifacts of the N=5 ansatz. A systematic N-convergence study (e.g., N=5,6,7,8) at a represent
  3. [II B, Eq. (7)] The Langevin equation (1) is written without specifying the stochastic calculus convention (Itô vs. Stratonovich). The numerical integrator in Eq. (7) is the standard Heun scheme for Stratonovich SDEs with multiplicative noise. Since the diffusion coefficient D(h) varies strongly near the wall through the Brenner mobility, the two conventions differ by a spurious drift proportional to D'(h). The bulk validation (D constant) is insensitive to this ambiguity, but the near-wall work values and protocol shapes in Figs. 3–7 can depend on the convention. The authors should state which convention is intended, justify it physically (e.g., as a vanishing-correlation limit), or demonstrate that the reported symmetry breaking is unchanged under the alternative convention. Without this clarification, the simulated model is not uniquely defined.
minor comments (3)
  1. [Section III C, Appendix A] The paper refers to a '5×3 parameter grid' in Section III C and Appendix A, but the parameter grid defined in Section II F is 4×3 (H0 ∈ {2,3,10,1000} and α ∈ {-25,0,25}). Please correct the inconsistent grid dimensions.
  2. [Fig. 4 and Section III B] The text says 'Fig. 4b shows ∆W% for the passive case', but Fig. 4 is a heatmap over both H0 and α, implying all α values are shown. Please clarify whether Fig. 4b is restricted to α=0 or displays the full parameter range.
  3. [Eq. (B2) and text after it] For the stated parameters k=1, ΔH=5, tf=2, Eq. (B2) gives W* = 1·25/(1·2+2) = 6.25, which is correct. No change needed, but consider adding a short derivation or reference for completeness.

Circularity Check

0 steps flagged

No circularity: the near-wall protocols and symmetry breaking are free outputs of a numerical optimization benchmarked against an external analytic solution; self-citations supply only model ingredients.

full rationale

The paper's derivation chain is not circular. The equations of motion (Eq. 1), Brenner mobility (Eq. 2), and active drift (Eq. 3) are physical inputs taken from the literature. The Chebyshev/Ritz ansatz (Eq. 6) is an explicit numerical representation, and the coefficients are optimized by a genetic algorithm using stochastic trajectory simulations to minimize the mean work (Eq. 5). The protocol shapes, including the striking away/towards asymmetry in Figs. 3, 5, and 6, are outputs of this optimization, not inputs. The method is validated against the analytic Schmiedl–Seifert bulk solution (Eqs. B1–B2) in the H0 = 1000 limit, where the GA recovers both the linear-ramp protocol and the exact minimum work W* = 6.25; this is an external, parameter-free benchmark. The paper explicitly states the truncation at N=5 and reports that the mean work plateaus for N≥3, but that is a numerical convergence choice, not a derivation that assumes the result. The only self-citations are Ref. [27] for the stresslet active drift and Ref. [30] for the Ritz method; both are model/method ingredients from prior published work by some of the authors and are not used to justify the central symmetry-breaking claim. No step reduces to its own input, no fitted parameter is renamed a prediction, and no uniqueness theorem is imported from the authors. Thus there is no significant circularity; the minor self-citation does not carry the conclusion.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The model uses standard stochastic thermodynamics plus wall corrections from the cited literature. The only hand-chosen quantity in the optimization pipeline is the basis truncation N=5 and the GA hyperparameters, which affect the completeness of the search but are not physics fit to data.

free parameters (1)
  • Chebyshev truncation order N = 5
    Number of basis functions used for the protocol. Chosen by hand after sweeps over N∈{2,…,8} showed a plateau for N≥3; near-wall convergence is not explicitly shown. This truncation defines the search space and therefore the admitted protocol shapes.
axioms (5)
  • domain assumption Overdamped Langevin dynamics with multiplicative noise (Eq. 1) describes the active colloid near a wall
    Standard model for colloidal transport; acknowledged that inertia is neglected.
  • domain assumption Brenner mobility formula (Eq. 2) gives the wall-modified mobility
    Leading-order far-field correction; the authors note lubrication corrections are not captured near H→1.
  • domain assumption Active drift vA(h) from stresslet/Blake image (Eq. 3) is the only activity-induced velocity, with fixed orientation
    Far-field force-dipole approximation from Ref. 27; assumes bottom-heavy alignment along the wall normal.
  • domain assumption 1D wall-normal reduction; lateral motion and rotational coupling neglected
    Explicit limitation stated in Section IV.
  • standard math Fluctuation–dissipation relation D(h)=μ(h)kBT holds locally near the wall
    Einstein relation invoked in Eq. (1).

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimal transport and control of an active particle near a plane wall." pith.science (2026). https://pith.science/paper/ATLMBPDP

@misc{pith2026260317798,
  author       = {Pith},
  title        = {Pith review of: Optimal transport and control of an active particle near a plane wall},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ATLMBPDP}},
  note         = {Machine review of arXiv:2603.17798}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

The control of active colloidal particles via optical traps is a cornerstone for research of matter at the micron and nanometer scale. A central challenge in this domain is the derivation of optimal transport protocols that minimize the mean work required to move a particle over a finite-time interval. Here, we present the Ritz method in which open-loop protocols are constructed from a global basis of Chebyshev polynomials. The protocols are optimized using either a genetic algorithm or a gradient-based method. We apply the method to study optimal transport of an active particle, which is modeled as a force-dipole (or a stresslet) near a no-slip wall. The methodology is validated in the limits of zero activity and infinite wall separation, where it successfully recovers the known analytical protocols and the theoretical minimum work. Crucially, we demonstrate that the presence of the activity breaks the time-reversal symmetry of the optimal protocol found. This symmetry breaking is shown to be a complex function of the transport direction and the particle's intrinsic activity. Because the presented approach requires only the capability to simulate stochastic trajectories, it offers a robust, principled framework for optimizing transport protocols in complex fluid environments that remain inaccessible to exact analytical treatment.

Figures

Figures reproduced from arXiv: 2603.17798 by Ejaz Ashraf, Kavya Swaminathan, Rajesh Singh, Utkarsh Maurya.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic of the system. An active colloidal [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Flowchart of the genetic algorithm. Each [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Optimal trap protocols for transport away from the wall, plotted as [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Heatmaps of GA optimisation results for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Symmetry breaking in the optimal protocol for [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Optimal trap protocols for transport [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Convergence of the GA over 200 generations [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Heatmaps of GA optimisation results for [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Streamlines of fluid flow around a puller active [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Self-propulsion protocols for swift non-equilibrium state transitions and enhanced cooling in active systems

    cond-mat.stat-mech 2026-04 unverdicted novelty 6.0

    Self-propulsion noise statistics define speed limits on non-equilibrium transitions in active matter, with non-stationary initials allowing faster cooling than passive protocols.

Reference graph

Works this paper leans on

40 extracted references · 2 linked inside Pith · cited by 1 Pith paper

  1. [1]

    This direction-reversal of the activity effect, together with the protocol-shape asymmetry, completes the characterisation of the broken time-reversal symmetry

    face additional resistance. This direction-reversal of the activity effect, together with the protocol-shape asymmetry, completes the characterisation of the broken time-reversal symmetry. TABLE II: Ensemble convergence diagnostics for away-from-wall transport, computed from 30 independent trials per grid point. CV: coefficient of variation100×σ/µof⟨W⟩ GA...

  2. [2]

    The bulk limit In the limitH 0 → ∞, we haveµ→µ 0 andv A →0, andthus, Eq.(1)reducestothemotionofanoverdamped colloidal particle in a harmonic trap in the bulk fluid, which was studied by Schmiedl and Seifert [5]. By variational calculus, the minimum-work protocol for0< t < tf is the linear ramp λ∗(t) =λ i + (λf −λ i)t tf ,(B1) withλ ∗(0+) =λ i andλ ∗(t− f ...

  3. [3]

    ⟨W⟩ GA is the mean thermodynamic work of the best GA- evolved protocol, evaluated retrospectively atN traj = 100,000trajectories using the near-wall Heun integrator (Eq

    Evaluated quantities Two work values are reported for each grid point. ⟨W⟩ GA is the mean thermodynamic work of the best GA- evolved protocol, evaluated retrospectively atN traj = 100,000trajectories using the near-wall Heun integrator (Eq. 7).W Seifert is the mean work of the Seifert linear- rampprotocol(Eq.B1)evaluatedthroughthesamenear- wall simulator ...

  4. [4]

    8 shows the convergence history for the representative grid pointH 0 = 2,α= 25

    Convergence of results Fig. 8 shows the convergence history for the representative grid pointH 0 = 2,α= 25. The best protocol from each trial is retrospectively evaluated at 100,000trajectories per generation using a shared fixed noise seed, a technique known as Common Random Numbers (CRN), which ensures that work values across trials and generations are ...

  5. [5]

    Peliti and S

    L. Peliti and S. Pigolotti,Stochastic thermodynamics: an introduction(Princeton University Press, 2021)

    2021

  6. [6]

    Shiraishi,An Introduction to Stochastic Thermodynamics(Springer, 2023)

    N. Shiraishi,An Introduction to Stochastic Thermodynamics(Springer, 2023)

    2023

  7. [7]

    Seifert,Stochastic Thermodynamics(Cambridge University Press, 2025)

    U. Seifert,Stochastic Thermodynamics(Cambridge University Press, 2025)

    2025

  8. [8]

    Ashkin, Nobel lecture: Optical tweezers and their application to biological systems (2018)

    A. Ashkin, Nobel lecture: Optical tweezers and their application to biological systems (2018)

    2018

  9. [9]

    Schmiedl and U

    T. Schmiedl and U. Seifert, Optimal finite-time processes in stochastic thermodynamics, Phys. Rev. Lett.98, 108301 (2007)

    2007

  10. [10]

    Ramaswamy, The Mechanics and Statistics of Active Matter, Annu

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

    2010

  11. [11]

    M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013)

    2013

  12. [12]

    M. t. Vrugt, B. Liebchen, and M. E. Cates, What exactly is’ active matter’?, arXiv preprint arXiv:2507.21621 (2025)

    Pith/arXiv arXiv 2025

  13. [13]

    Garcia-Millan, J

    R. Garcia-Millan, J. Schüttler, M. E. Cates, and S. A. Loos, Optimal closed-loop control of active particles and a minimal information engine, Phys. Rev. Lett.135, 088301 (2025)

    2025

  14. [14]

    S. A. Loos, S. Monter, F. Ginot, and C. Bechinger, Universal symmetry of optimal control at the microscale, Phys. Rev. X14, 021032 (2024)

    2024

  15. [15]

    Monter, S

    S. Monter, S. A. Loos, and C. Bechinger, Optimal transitions between nonequilibrium steady states, PProc. Natl. Acad. Sci.122, e2510654122 (2025)

    2025

  16. [16]

    Zhong and M

    A. Zhong and M. R. DeWeese, Beyond linear response: Equivalence between thermodynamic geometry and optimal transport, Phys. Rev. Lett.133, 057102 (2024)

    2024

  17. [17]

    L. K. Davis, K. Proesmans, and É. Fodor, Active matter under control: Insights from response theory, Phys. Rev. X14, 011012 (2024)

    2024

  18. [18]

    Cocconi, J

    L. Cocconi, J. Knight, and C. Roberts, Optimal power extraction from active particles with hidden states, Phys. Rev. Lett.131, 188301 (2023)

    2023

  19. [19]

    Casert and S

    C. Casert and S. Whitelam, Learning protocols for the fast and efficient control of active matter, Nature Communications15, 9128 (2024)

    2024

  20. [20]

    Gupta, S

    D. Gupta, S. H. L. Klapp, and D. A. Sivak, Efficient control protocols for an active ornstein-uhlenbeck particle, Phys. Rev. E108, 024117 (2023)

    2023

  21. [21]

    Shankar, V

    S. Shankar, V. Raju, and L. Mahadevan, Optimal transport and control of active drops, PProc. Natl. Acad. Sci.119, e2121985119 (2022)

    2022

  22. [22]

    G. K. Batchelor, The stress system in a suspension of force-free particles, J. Fluid Mech.41, 545 (1970)

    1970

  23. [23]

    Lauga,The fluid dynamics of cell motility, Vol

    E. Lauga,The fluid dynamics of cell motility, Vol. 62 (Cambridge University Press, 2020)

    2020

  24. [24]

    J. R. Blake, A note on the image system for a Stokeslet in a no-slip boundary, Proc. Camb. Phil. Soc.70, 303 (1971)

    1971

  25. [25]

    M. C. Engel, J. A. Smith, and M. P. Brenner, Optimal control of nonequilibrium systems through automatic differentiation, Phys. Rev. X13, 041032 (2023)

    2023

  26. [26]

    Whitelam, Demon in the machine: learning to extract work and absorb entropy from fluctuating nanosystems, Phys

    S. Whitelam, Demon in the machine: learning to extract work and absorb entropy from fluctuating nanosystems, Phys. Rev. X13, 021005 (2023)

    2023

  27. [27]

    Whitelam, C

    S. Whitelam, C. Casert, M. Engel, and I. Tamblyn, Benchmark control problems in nonequilibrium statistical mechanics, arXiv preprint arXiv:2506.15122 (2025)

    Pith/arXiv arXiv 2025

  28. [28]

    C. W. Gardiner,Handbook of stochastic methods (Springer Berlin, 1985)

    1985

  29. [29]

    Kubo, The fluctuation-dissipation theorem, Rep

    R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys.29, 255 (1966)

    1966

  30. [30]

    Brenner, The slow motion of a sphere through a viscous fluid towards a plane surface, Chem

    H. Brenner, The slow motion of a sphere through a viscous fluid towards a plane surface, Chem. Eng. Sci. 16, 242 (1961)

    1961

  31. [31]

    G. Turk, R. Adhikari, and R. Singh, Fluctuating hydrodynamics of an autophoretic particle near a permeable interface, J. Fluid Mech.998, A34 (2024)

    2024

  32. [32]

    Sekimoto, Langevin equation and thermodynamics, Prog

    K. Sekimoto, Langevin equation and thermodynamics, Prog. Theor. Phys. Suppl.130, 17 (1998)

    1998

  33. [33]

    I. M. Gelfand and F. SV,Calculus of variations(Courier Corporation, 2000)

    2000

  34. [34]

    Kikuchi, R

    L. Kikuchi, R. Singh, M. E. Cates, and R. Adhikari, Ritz method for transition paths and quasipotentials of rare diffusive events, Phys. Rev. Res.2, 033208 (2020)

    2020

  35. [35]

    J. P. Boyd,Chebyshev and Fourier spectral methods (Dover, 2000)

    2000

  36. [36]

    P. E. Kloeden and E. Platen,Numerical solution of stochastic differential equations(Springer, 1992)

    1992

  37. [37]

    Paszkeet al., Pytorch: An imperative style, high- performance deep learning library, inAdvances in Neural Information Processing Systems 32(2019) pp

    A. Paszkeet al., Pytorch: An imperative style, high- performance deep learning library, inAdvances in Neural Information Processing Systems 32(2019) pp. 8024– 8035

    2019

  38. [38]

    Alvarado, E

    J. Alvarado, E. G. Teich, D. A. Sivak, and J. Bechhoefer, Optimal control in soft and active matter, Annu. Rev. Condens. Mat. Phys.17(2026)

    2026

  39. [39]

    R. E. Goldstein, Green algae as model organisms for biological fluid dynamics, Ann. Rev. Fluid Mech.47, 343 (2015)

    2015

  40. [40]

    Pozrikidis,Boundary Integral and Singularity Methods for Linearized Viscous Flow(Cambridge University Press, 1992)

    C. Pozrikidis,Boundary Integral and Singularity Methods for Linearized Viscous Flow(Cambridge University Press, 1992)

    1992

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.