Pith. sign in

REVIEW 3 major objections 5 minor 79 references

Brownian dynamics with soft constraints in soft matter systems

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read This paper establishes that for overdamped Brownian motion with stiff 'soft' constraints, the correct effective mobility in the long-time limit is obtained by first projecting the mobility tensor onto the constraint manifold and then averag

desk verdict A genuinely useful toolkit for constrained Brownian dynamics, but the new soft-soft mobility rule has an unproven integrability condition that the supporting simulations never exercise. read the letter →

arxiv 2601.09584 v2 pith:AYMIAXMZ submitted 2026-01-14 cond-mat.soft cond-mat.stat-mechmath-phmath.MP

Sophie Marbach , Adam Carter , Miranda Holmes-Cerfon This is my paper
classification cond-mat.softcond-mat.stat-mechmath-phmath.MP
keywords constrainedBrowniandynamicssoftconstraintssingularperturbationtheoryprojectedmobilityoverdampedLangevinequationcoarse-graininghydrodynamicinteractionseffectivediffusion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to put constrained Brownian dynamics on firm footing for soft-matter systems, where 'constraints' are really very stiff forces. It derives, by singular perturbation theory, the overdamped Langevin equation that governs a system on timescales long compared to the relaxation of the stiffly constrained degrees of freedom, recovering earlier formulas in a unified way. Its new result concerns 'soft-soft' constraints, where the mobility tensor varies on the same lengthscale as the confining potential; the paper shows the correct effective mobility is obtained by projecting first and then averaging over the equilibrium distribution of the confining potential. If correct, this gives practitioners a straightforward recipe — summarized in a table — to simulate tethered or confined colloids, polymers, and particles near walls, including hydrodynamic interactions.

What carries the argument

The carrying device is the singular-perturbation expansion of the backward generator in the ratio ε = L_z/L_s between the confinement width and the slow lengthscale. After changing variables to intrinsic coordinates s on the manifold and fast variables Z = c/ε, the generator splits as ε^{-2} L_0 + ε^{-1} L_1 + L_2; the Fredholm solvability condition at O(1) produces the reduced generator, with the projected mobility M̃ = M^y_ss - M^y_sz (M^y_zz)^{-1} M^y_sz, which in Cartesian form is M_P. The soft-soft extension adds an equilibrium average over the fast distribution π_0 ∝ e^{-U_c/k_BT}, yielding M̄_P.

What would settle it

Simulate a stiffly confined system with at least two slow variables (dimension d≥2) and a mobility whose off-diagonal term varies rapidly in the fast coordinate in a way that makes (M_y_zz)^{-1} M_y_sz non-curl-free; compare the numerically measured long-time effective mobility against the projected-then-averaged formula M̄_P. A concrete example: a 3D walker with two tangential coordinates and confinement z, with mobility M_12 depending on z via a function that violates the curl-free condition; measure the mean-squared displacement at long times.

Watch

Extended reading notes

Core claim

The central claim is that in the soft limit, the overdamped dynamics confined to a manifold M = {c(x)=0} are exactly given by d x/dt = -M_P ∇U_eff + k_B T ∇·M_P + sqrt(2k_B T) M_P^{1/2} η, with M_P the mobility projected by P_M = I - M C^T (C M C^T)^{-1} C and U_eff = U - k_B T log κ; and when mobility varies on the same scale as the confining potential, M_P must be replaced by its equilibrium average over the fast directions, M̄_P = ∫ M_P e^{-U_c/k_BT} dc / ∫ e^{-U_c/k_BT} dc. The order is projection before averaging. The paper proves this by singular perturbation on the backward Kolmogorov equation, using the Fredholm alternative to eliminate the stiff directions, and verifies the result n

Load-bearing premise

When the mobility varies rapidly in the confined direction, the proof requires that the matrix function (M_y_zz)^{-1} M_y_sz be curl-free; the authors state they were not able to show this in general, so the existence of the first-order corrector — and hence the derivation of the averaged mobility — must still be checked case by case.

Editorial extensions

If this is right

  • For any stiffly confined soft-matter system, long-time dynamics are captured by Eq. (6) with M_P and U_eff, valid on timescales τ ≫ k_BT/(D_0 k).
  • When mobility varies on the same scale as the confinement (e.g., near walls or with lubrication), the correct mobility is the projected-then-averaged tensor M̄_P; evaluating mobility at the mean height can be off by 10–20%.
  • The effective potential acquires an entropic term -k_BT log κ (and in intrinsic coordinates also +k_BT log|C|), which can be measured experimentally as vibrational entropy and changes stationary distributions.
  • The order of operations matters: averaging first and then projecting gives a different, incorrect mobility; this answers the practical question of which tensor to use in fast Brownian-dynamics solvers.
  • The Lagrange-multiplier route only yields the same equations if the noise is included in the divergence drift in a specific way; naive Itô or Stratonovich constraint prescriptions fail.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the project-then-average result holds, coarse-grained simulations of colloidal suspensions near boundaries should compute effective mobilities as Boltzmann-weighted averages over the confinement coordinate, which could be done with precomputed tables in existing Brownian-dynamics codes.
  • The paper's failure to prove the curl-free condition in general suggests a testable extension: for systems with more than one slow variable and strongly varying mobility, construct examples where the condition fails and check whether the final averaged equation still holds; if it fails, a generalized averaging procedure may be needed.
  • The result that constraints change the mobility even when the constrained coordinate is immobile (off-diagonal coupling) has a direct analog in single-particle systems with anisotropic shape; the same projection formulas could model rotational-translational coupling in optical traps.
  • The authors' open puzzle — why average mobility but restrict friction — might be resolved by examining the solvent coarse-graining: averaging over fast velocities (mobility) rather than forces (friction) suggests a fluctuation-dissipation perspective.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper derives effective constrained Brownian dynamics in the limit of stiff 'soft' constraints, starting from overdamped Langevin equations with a confining potential. The authors recover the known projected-mobility equations (Table I), including the effective potential with the log-kappa term, using a singular-perturbation expansion; they then extend the derivation to 'soft-soft' constraints, where the mobility varies on the same scale as the confining potential, and conclude that the correct effective mobility is obtained by projecting first and then averaging (Eq. (11)). The result is illustrated on several physical examples and validated numerically on toy models, including a two-dimensional model with y-dependent off-diagonal mobility (Figs. 8-9).

Significance. If the main result holds, this is a useful contribution: it provides a practical table of constrained Brownian dynamics formulas, a self-contained singular-perturbation derivation that recovers earlier results of Morse and others, and a concrete answer to the project-then-average question for soft-soft constraints. The numerical checks in Sec. IV C are a genuine strength, as is the care taken to show that the effective dynamics depend on the physical form of the constraining potential. However, the general multi-dimensional soft-soft claim rests on an unproven integrability condition and on a few not-fully-justified averaging steps; these need to be resolved before the central formula can be considered established beyond the tested cases.

major comments (3)
  1. [Sec. V A, Eqs. (39)-(42)] For rapidly varying mobility, the first-order corrector f1 is constructed as a path integral of B = (M^y_zz)^{-1}M^y_sz, and its existence requires the stated curl-free condition. The authors explicitly say they were not able to prove this condition in general and that the final result is independent of it. This is load-bearing, not a side remark: the O(1) solvability condition (39) contains L1 f1, and the modified mobility (42) is obtained by substituting D_Z f1. If f1 does not exist, this Fredholm computation has no basis, and no proof of the alleged independence is given. Moreover, in every numerical test of the soft-soft case the slow coordinate s is one-dimensional (Sec. IV C), where the condition is automatic, or the off-diagonal block M_sz is absent (Sec. III A), so the nontrivial multi-dimensional case is never checked. This is the main unresolved point for Eq. (11).
  2. [Sec. V A, Eq. (42)] The averaging operator defined in Eq. (41) is not applied explicitly in Eq. (42). The preceding derivation suggests that the average acts on the product M^y_sz(M^y_zz)^{-1}M^y_sz, not on the factors separately; as written, \tilde M looks like a bare Schur complement evaluated at one point, which would not capture the rapid mobility variation. The display also has a dimensional inconsistency: D_Z f1 should be -(M^y_zz)^{-1}M^y_sz D_s a, not the expression printed before Eq. (42). Please state the averaged mobility tensor explicitly, with consistent transposes, and clarify what is averaged.
  3. [Sec. V B b] The equivalence between the extrinsic soft-soft mobility (11) and the intrinsic mobility \tilde M is asserted via the step: '∇s MP (∇s)^T |_x = ∇s|_x MP(∇s|_x)^T, since for the purpose of the averaging operator, ∇s|_{x∈M} is constant.' This is not self-evident: along a fibre of fixed s, ∇s(x(s,z)) generally changes with z, while the intrinsic derivation in Sec. V A averages J M_P J^T, which contains ∇s(z). The slow-variation assumption on the metric may be sufficient, but it is not invoked explicitly at this point. Without a justification, the equality between the extrinsic averaged mobility and the intrinsic averaged Schur complement is not fully established.
minor comments (5)
  1. [Sec. II, Eq. (11)] The notation in Eq. (11) is informal: the left side is a function of x, while the right side integrates over the constraint values c, which changes x. The footnote helps, but the main text should state that x is on the manifold and the integration runs along the fibre x(s,c).
  2. [Sec. V A] Typo: 'antatz' should be 'ansatz'. Also 'instrinsic' appears in Sec. V; 'softly softly' is hyphenated inconsistently throughout.
  3. [Sec. IV C 1] Equation references such as 'Eq. (IV C 1)' and 'Eq. (III B)' are confusing; please use numbered equation tags.
  4. [Fig. 9] The inset showing M12(y) and the probability densities lacks labeled axes and a legend; this makes it hard to see how the width of the density compares with the scale of variation of M12.
  5. [References] Reference 45 is cited as 'Reduced dynamics of stochastically perturbed gradient flows' with year 2010 but appears to be an unpublished note; please provide a preprint number or archival source.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the singular-perturbation derivation is self-contained, the soft-soft result is derived rather than assumed, and the self-citations are only motivational examples.

full rationale

The central derivation in Sec. V A starts from the original overdamped dynamics (1), changes to intrinsic coordinates, and obtains the limiting generator through a standard singular-perturbation expansion with a Fredholm solvability condition. The soft-soft averaged mobility (11) is a consequence of this expansion: the equilibrium averaging operator (41) and the stationary fast measure (40) emerge from the analysis, not from a prior assumption that the answer is a Boltzmann average of the projected mobility. The projected mobility M_P is defined independently in Eq. (9), and the averaging in Eq. (11) is derived, so the claim 'project-then-average is correct' is not self-definitional. The numerical validation in Sec. IV C compares the derived formula with simulations of the full original SDEs using the same prescribed M_12(y) and no fitted parameters, so there is no fitted input renamed as a prediction. The self-citations (e.g., Refs. 33, 34, 72) appear only as illustrative examples or background, not as load-bearing support for the derivation; the comparison with Morse [44] is an external benchmark, not a circular justification. The admitted limitation about the curl-free condition for f_1 (Sec. V A: 'we were not able to show it holds in general so in general it must be checked') is an unresolved mathematical generality condition, but it is not circularity: the final result is asserted to be independent of f_1 itself, and even if that assertion were wrong, the error would be a correctness gap rather than a reduction of the result to its inputs. No circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim does not fit data: it is a mathematical derivation from a stated model. No free parameters are fitted; all parameters in examples are physical inputs or chosen toy-model values. The soft-soft result rests on the unproved curl-free condition listed above. No new physical entities are introduced.

assumptions (6)
  • domain assumption The overdamped Itô Langevin equation (1) with mobility M(x), potential U_tot, and Boltzmann stationary distribution is the correct mesoscale model.
    Used as the starting point throughout; if inertial or memory effects matter, the constrained dynamics would need re-derivation.
  • domain assumption Constraints c_i are homogeneous to distances and C has full rank m≤n on M.
    Required for |C| to be dimensionless and for M to be a manifold; the paper notes distance constraints can violate full rank.
  • domain assumption Separation of timescales: L_z = sqrt(kBT/k) ≪ L_s, i.e., ε≪1, so the confined directions relax much faster than the slow manifold directions.
    Basis of the singular perturbation expansion; the stated validity regime of the constrained equations.
  • domain assumption The metric tensor and external potential vary slowly near the manifold; curvature of M is less than L_s^{-1}.
    Allows g0=g|_M and U(s;0) to replace the full functions in Sec. V A; fails for tight curvature or rapidly varying external forcing.
  • ad hoc to paper For rapidly varying mobility, the curl-free integrability condition on (M_y_zz)^{-1}M_y_sz holds.
    Introduced in Sec. V A to solve the O(ε) equation for f1; the authors state they were not able to prove it in general and that it must be checked.
  • standard math Standard mathematical tools: Itô calculus, Fredholm alternative, coarea formula, and Moore-Penrose pseudoinverse identities.
    Used throughout the derivations and appendices; assumed correct.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Brownian dynamics with soft constraints in soft matter systems." pith.science (2026). https://pith.science/paper/AYMIAXMZ

@misc{pith2026260109584,
  author       = {Pith},
  title        = {Pith review of: Brownian dynamics with soft constraints in soft matter systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYMIAXMZ}},
  note         = {Machine review of arXiv:2601.09584}
}
read the original abstract

Stiff forces, which bind objects together or otherwise confine motion, are found widely in soft-matter systems -- colloids with short range attractions, ligand-receptor contacts, particles in optical traps, fibres that resist stretching, etc. To assess the long-term effect of these stiff forces on dynamics and structure, it is useful to consider the limit where they are treated as constraints, so the system evolves strictly within allowed configurations. Efforts to derive equations involving both constraints, and the stochastic motion appropriate at the scales of soft matter, began around 50 years ago, yet, we are still lacking a straightforward way to extract the projected equations and apply them in modern formulations of mesoscale dynamics. Here, we address this gap with two key contributions: (1) a practical summary of the constrained Brownian dynamics equations with ``soft'' constraints, i.e. constraints imposed by stiff forces, which is illustrated through several representative examples, taking care to highlight the nontrivial effects of the constraints; and (2) a novel derivation using singular perturbation theory, establishing the validity of these equations over timescales exceeding the relaxation of stiffly constrained degrees of freedom. We further extend our approach to ``soft soft'' constraints, where mobility varies on lengthscales comparable to the restraining forces -- a scenario typical for particles in fluids experiencing hydrodynamic interactions. We hope our results will be useful for soft matter research, as a robust toolkit for studying tethered or confined systems.

Figures

Figures reproduced from arXiv: 2601.09584 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. e reports the effective, averaged, (diagonal) diffusion co￾efficients Di(z), and compares them to a “naive” picture where one would simply have evaluated the diffusion coefficients at the mean height Di(z). Because the confining potential U c (z) is indeed quite soft, Di(z) and Di(z) are markedly different, reaching up to 10 − 20% difference depending on parameter values (Fig. 2e). The coefficient that changes the m… view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Non-uniform constraint to a flat line; (b) Constraint to a [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Numerically estimated diffusivity (markers) for different [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Effective mobility decreases with off-diagonal contribution [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. A sketch of the change of variables used in the derivation [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

79 extracted references · 9 canonical work pages

  1. [1]

    Leibler , author M

    author author L. Leibler , author M. Rubinstein , \ and\ author R. H. \ Colby ,\ title title Dynamics of reversible networks , \ @noop journal journal Macromolecules \ volume 24 ,\ pages 4701--4707 ( year 1991 ) NoStop

    1991

  2. [2]

    Schallamach ,\ title title A theory of dynamic rubber friction , \ @noop journal journal Wear \ volume 6 ,\ pages 375--382 ( year 1963 ) NoStop

    author author A. Schallamach ,\ title title A theory of dynamic rubber friction , \ @noop journal journal Wear \ volume 6 ,\ pages 375--382 ( year 1963 ) NoStop

    1963

  3. [3]

    Meng , author N

    author author G. Meng , author N. Arkus , author M. P. \ Brenner , \ and\ author V. N. \ Manoharan ,\ title title The free-energy landscape of clusters of attractive hard spheres , \ @noop journal journal Science \ volume 327 ,\ pages 560--563 ( year 2010 ) NoStop

    2010

  4. [4]

    author author H. N. \ Lekkerkerker , author R. Tuinier , \ and\ author M. Vis ,\ @noop title Colloids and the depletion interaction \ ( publisher Springer Nature ,\ year 2024 ) NoStop

    2024

  5. [5]

    author author D. G. \ Grier ,\ title title Optical tweezers in colloid and interface science , \ @noop journal journal Current opinion in colloid & interface science \ volume 2 ,\ pages 264--270 ( year 1997 ) NoStop

    1997

  6. [6]

    Franosch \ and\ author S

    author author T. Franosch \ and\ author S. Jeney ,\ title title Persistent correlation of constrained colloidal motion , \ @noop journal journal Physical Review E—Statistical, Nonlinear, and Soft Matter Physics \ volume 79 ,\ pages 031402 ( year 2009 ) NoStop

    2009

  7. [7]

    Franosch , author M

    author author T. Franosch , author M. Grimm , author M. Belushkin , author F. M. \ Mor , author G. Foffi , author L. Forr \'o , \ and\ author S. Jeney ,\ title title Resonances arising from hydrodynamic memory in brownian motion , \ @noop journal journal Nature \ volume 478 ,\ pages 85--88 ( year 2011 ) NoStop

    2011

  8. [8]

    author author C. A. \ Mirkin , author R. L. \ Letsinger , author R. C. \ Mucic , \ and\ author J. J. \ Storhoff ,\ title title A dna-based method for rationally assembling nanoparticles into macroscopic materials , \ @noop journal journal Nature \ volume 382 ,\ pages 607--609 ( year 1996 ) NoStop

    1996

Show all 79 references
  1. [9]

    Feng , author L.-L

    author author L. Feng , author L.-L. \ Pontani , author R. Dreyfus , author P. Chaikin , \ and\ author J. Brujic ,\ title title Specificity, flexibility and valence of dna bonds guide emulsion architecture , \ @noop journal journal Soft Matter \ volume 9 ,\ pages 9816--9823 ( ...

    2013

  2. [10]

    Wang , author Y

    author author Y. Wang , author Y. Wang , author X. Zheng , author \'E . Ducrot , author J. S. \ Yodh , author M. Weck , \ and\ author D. J. \ Pine ,\ title title Crystallization of dna-coated colloids , \ @noop journal journal Nature communications \ volume 6 ,\ pages 1--8 ( y...

    2015

  3. [11]

    author author W. B. \ Rogers , author W. M. \ Shih , \ and\ author V. N. \ Manoharan ,\ title title Using dna to program the self-assembly of colloidal nanoparticles and microparticles , \ @noop journal journal Nature Reviews Materials \ volume 1 ,\ pages 1--14 ( year 2016 ) NoStop

    2016

  4. [12]

    Cui , author S

    author author F. Cui , author S. Marbach , author J. A. \ Zheng , author M. Holmes-Cerfon , \ and\ author D. J. \ Pine ,\ title title Comprehensive view of microscopic interactions between dna-coated colloids , \ @noop journal journal Nature communications \ volume 13 ,\ pages...

    2022

  5. [13]

    author author E. W. \ Gehrels , author W. B. \ Rogers , author Z. Zeravcic , \ and\ author V. N. \ Manoharan ,\ title title Programming directed motion with dna-grafted particles , \ @noop journal journal ACS nano \ ( year 2022 ) NoStop

    2022

  6. [14]

    author author R. J. \ Macfarlane , author B. Lee , author M. R. \ Jones , author N. Harris , author G. C. \ Schatz , \ and\ author C. A. \ Mirkin ,\ title title Nanoparticle superlattice engineering with dna , \ @noop journal journal science \ volume 334 ,\ pages 204--208 ( ye...

    2011

  7. [15]

    author author P. K. \ Jana \ and\ author B. M. \ Mognetti ,\ title title Translational and rotational dynamics of colloidal particles interacting through reacting linkers , \ @noop journal journal Physical Review E \ volume 100 ,\ pages 060601 ( year 2019 ) NoStop

    2019

  8. [16]

    Varilly , author S

    author author P. Varilly , author S. Angioletti-Uberti , author B. M. \ Mognetti , \ and\ author D. Frenkel ,\ title title A general theory of dna-mediated and other valence-limited colloidal interactions , \ @noop journal journal The Journal of chemical physics \ volume 137 ,...

    2012

  9. [17]

    Melio , author S

    author author J. Melio , author S. E. \ Henkes , \ and\ author D. J. \ Kraft ,\ title title Soft and stiff normal modes in floppy colloidal square lattices , \ @noop journal journal Physical Review Letters \ volume 132 ,\ pages 078202 ( year 2024 ) NoStop

    2024

  10. [18]

    Alon \ and\ author S

    author author R. Alon \ and\ author S. Feigelson ,\ title title From rolling to arrest on blood vessels: leukocyte tap dancing on endothelial integrin ligands and chemokines at sub-second contacts , \ in\ @noop booktitle Seminars in immunology ,\ Vol. volume 14 \ ( organizatio...

    2002

  11. [19]

    Ley , author C

    author author K. Ley , author C. Laudanna , author M. I. \ Cybulsky , \ and\ author S. Nourshargh ,\ title title Getting to the site of inflammation: the leukocyte adhesion cascade updated , \ @noop journal journal Nature Reviews Immunology \ volume 7 ,\ pages 678--689 ( year ...

    2007

  12. [20]

    Korn \ and\ author U

    author author C. Korn \ and\ author U. Schwarz ,\ title title Dynamic states of cells adhering in shear flow: from slipping to rolling , \ @noop journal journal Physical review E \ volume 77 ,\ pages 041904 ( year 2008 ) NoStop

    2008

  13. [21]

    Mammen , author S.-K

    author author M. Mammen , author S.-K. \ Choi , \ and\ author G. M. \ Whitesides ,\ title title Polyvalent interactions in biological systems: implications for design and use of multivalent ligands and inhibitors , \ @noop journal journal Angewandte Chemie International Editio...

    1998

  14. [22]

    Sakai , author S

    author author T. Sakai , author S. I. \ Nishimura , author T. Naito , \ and\ author M. Saito ,\ title title Influenza a virus hemagglutinin and neuraminidase act as novel motile machinery , \ @noop journal journal Scientific reports \ volume 7 ,\ pages 1--11 ( year 2017 ) NoStop

    2017

  15. [23]

    Sakai , author H

    author author T. Sakai , author H. Takagi , author Y. Muraki , \ and\ author M. Saito ,\ title title Unique directional motility of influenza c virus controlled by its filamentous morphology and short-range motions , \ @noop journal journal Journal of virology \ volume 92 ,\ p...

    2018

  16. [24]

    M\" u ller , author D

    author author M. M\" u ller , author D. Lauster , author H. H. \ Wildenauer , author A. Herrmann , \ and\ author S. Block ,\ title title Mobility-based quantification of multivalent virus-receptor interactions: New insights into influenza a virus binding mode , \ @noop journal...

    2019

  17. [25]

    Maxian , author A

    author author O. Maxian , author A. Mogilner , \ and\ author A. Donev ,\ title title Integral-based spectral method for inextensible slender fibers in stokes flow , \ @noop journal journal Physical Review Fluids \ volume 6 ,\ pages 014102 ( year 2021 ) NoStop

    2021

  18. [26]

    a utler , author W. F. \ Van Gunsteren , \ and\ author P. H. \ H \

    author author V. Kr \"a utler , author W. F. \ Van Gunsteren , \ and\ author P. H. \ H \"u nenberger ,\ title title A fast shake algorithm to solve distance constraint equations for small molecules in molecular dynamics simulations , \ @noop journal journal Journal of computat...

    2001

  19. [27]

    velocity

    author author H. C. \ Andersen ,\ title title Rattle: A “velocity” version of the shake algorithm for molecular dynamics calculations , \ @noop journal journal Journal of computational Physics \ volume 52 ,\ pages 24--34 ( year 1983 ) NoStop

    1983

  20. [28]

    Leimkuhler \ and\ author C

    author author B. Leimkuhler \ and\ author C. Matthews ,\ title title Efficient molecular dynamics using geodesic integration and solvent–solute splitting , \ 10.1098/rspa.2016.0138 journal journal Proceedings of the Royal Society A: Mathematical, Physical and Engineering Scien...

    2016

  21. [29]

    Maxian , author B

    author author O. Maxian , author B. Sprinkle , \ and\ author A. Donev ,\ title title Bending fluctuations in semiflexible, inextensible, slender filaments in stokes flow: Toward a spectral discretization , \ @noop journal journal The Journal of Chemical Physics \ volume 158 ( ...

    2023

  22. [30]

    Maxian \ and\ author A

    author author O. Maxian \ and\ author A. Donev ,\ title title A simulation platform for slender, semiflexible, and inextensible fibers with brownian hydrodynamics and steric repulsion , \ @noop journal journal Physics of Fluids \ volume 36 ( year 2024 ) NoStop

    2024

  23. [31]

    author author W. T. \ Funkenbusch , author K. S. \ Silmore , \ and\ author J. W. \ Swan ,\ title title Approaches for fast brownian dynamics simulation with constraints , \ @noop journal journal Journal of Computational Physics \ volume 509 ,\ pages 113043 ( year 2024 ) NoStop

    2024

  24. [32]

    Delmotte \ and\ author F

    author author B. Delmotte \ and\ author F. B. \ Usabiaga ,\ title title Modeling complex particle suspensions: Perspectives on the rigid multiblob method , \ @noop journal journal Physical Review Fluids \ volume 10 ,\ pages 100701 ( year 2025 ) NoStop

    2025

  25. [33]

    Marbach \ and\ author M

    author author S. Marbach \ and\ author M. Holmes-Cerfon ,\ title title Mass changes the diffusion coefficient of particles with ligand-receptor contacts in the overdamped limit , \ @noop journal journal Physical Review Letters \ volume 129 ,\ pages 048003 ( year 2022 ) NoStop

    2022

  26. [34]

    Marbach , author J

    author author S. Marbach , author J. A. \ Zheng , \ and\ author M. Holmes-Cerfon ,\ title title The nanocaterpillar's random walk: diffusion with ligand--receptor contacts , \ @noop journal journal Soft Matter \ volume 18 ,\ pages 3130--3146 ( year 2022 ) NoStop

    2022

  27. [35]

    author author P. C. \ Bressloff \ and\ author J. M. \ Newby ,\ title title Stochastic models of intracellular transport , \ @noop journal journal Reviews of Modern Physics \ volume 85 ,\ pages 135 ( year 2013 ) NoStop

    2013

  28. [36]

    author author S. A. \ McKinley , author A. Athreya , author J. Fricks , \ and\ author P. R. \ Kramer ,\ title title Asymptotic analysis of microtubule-based transport by multiple identical molecular motors , \ @noop journal journal Journal of theoretical biology \ volume 305 ,...

    2012

  29. [37]

    author author J. P. \ Lee-Thorp \ and\ author M. Holmes-Cerfon ,\ title title Modeling the relative dynamics of dna-coated colloids , \ @noop journal journal Soft matter \ volume 14 ,\ pages 8147--8159 ( year 2018 ) NoStop

    2018

  30. [38]

    Fogelson \ and\ author J

    author author B. Fogelson \ and\ author J. P. \ Keener ,\ title title Transport facilitated by rapid binding to elastic tethers , \ @noop journal journal SIAM Journal on Applied Mathematics \ volume 79 ,\ pages 1405--1422 ( year 2019 ) NoStop

    2019

  31. [39]

    Fogelson \ and\ author J

    author author B. Fogelson \ and\ author J. P. \ Keener ,\ title title Enhanced nucleocytoplasmic transport due to competition for elastic binding sites , \ @noop journal journal Biophysical journal \ volume 115 ,\ pages 108--116 ( year 2018 ) NoStop

    2018

  32. [40]

    author author N. G. v. \ Kampen ,\ title title Statistical mechanics of trimers , \ 10.1007/bf00382618 journal journal Applied Scientific Research. An International Journal on Thermal, Mechanical, and Electromagnetic Phenomena in Continua \ volume 37 ,\ pages 67 -- 75 ( year 1...

    1981 doi

  33. [41]

    Fixman ,\ title title Simulation of polymer dynamics

    author author M. Fixman ,\ title title Simulation of polymer dynamics. i. general theory , \ @noop journal journal The Journal of Chemical Physics \ volume 69 ,\ pages 1527--1537 ( year 1978 ) NoStop

    1978

  34. [42]

    author author E. J. \ Hinch ,\ title title Brownian motion with stiff bonds and rigid constraints , \ 10.1017/S0022112094001746 journal journal Journal of Fluid Mechanics \ volume 271 ,\ pages 219--234 ( year 1994 ) NoStop

    1994 doi

  35. [43]

    author author H. C. \ \"O ttinger ,\ title title Brownian dynamics of rigid polymer chains with hydrodynamic interactions , \ @noop journal journal Physical Review E \ volume 50 ,\ pages 2696 ( year 1994 ) NoStop

    1994

  36. [44]

    author author D. C. \ Morse ,\ title title Theory of constrained brownian motion , \ in\ 10.1002/0471484237.ch2 booktitle Advances in Chemical Physics \ ( publisher John Wiley & Sons, Ltd ,\ year 2003 )\ Chap. chapter 2 , pp.\ pages 65--189 NoStop

    2003 doi

  37. [45]

    Fatkullin , author G

    author author I. Fatkullin , author G. Kovacic , \ and\ author E. Vanden-Eijnden ,\ title title Reduced dynamics of stochastically perturbed gradient flows , \ @noop \ ( year 2010 ) NoStop

    2010

  38. [46]

    Lelievre , author M

    author author T. Lelievre , author M. Rousset , \ and\ author G. Stoltz ,\ title title Langevin dynamics with constraints and computation of free energy differences , \ @noop journal journal Mathematics of computation \ volume 81 ,\ pages 2071--2125 ( year 2012 ) NoStop

  39. [47]

    Sprinkle , author E

    author author B. Sprinkle , author E. B. \ Van Der Wee , author Y. Luo , author M. M. \ Driscoll , \ and\ author A. Donev ,\ title title Driven dynamics in dense suspensions of microrollers , \ @noop journal journal Soft Matter \ volume 16 ,\ pages 7982--8001 ( year 2020 ) NoStop

    2020

  40. [48]

    Mao , author A

    author author X. Mao , author A. Souslov , author C. I. \ Mendoza , \ and\ author T. C. \ Lubensky ,\ title title Mechanical instability at finite temperature , \ @noop journal journal Nature communications \ volume 6 ,\ pages 5968 ( year 2015 ) NoStop

    2015

  41. [49]

    Kallus \ and\ author M

    author author Y. Kallus \ and\ author M. Holmes-Cerfon ,\ title title Free energy of singular sticky-sphere clusters , \ @noop journal journal Physical Review E \ volume 95 ,\ pages 022130 ( year 2017 ) NoStop

    2017

  42. [50]

    Mannattil , author J

    author author M. Mannattil , author J. M. \ Schwarz , \ and\ author C. D. \ Santangelo ,\ title title Thermal fluctuations of singular bar-joint mechanisms , \ 10.1103/PhysRevLett.128.208005 journal journal Physical Review Letters \ volume 128 ,\ pages 208005 ( year 2022 ) NoStop

    2022 doi

  43. [51]

    note These fibres are defined by keeping internal parameterization variables constant, as shown in our derivation in Section sec:generalderivation , however they could have an expression as a limit of a gradient flow, as in Sharma.2021 -- we leave such a construction for futur...

    2021

  44. [52]

    Sprik \ and\ author G

    author author M. Sprik \ and\ author G. Ciccotti ,\ title title Free energy from constrained molecular dynamics , \ @noop journal journal The Journal of chemical physics \ volume 109 ,\ pages 7737--7744 ( year 1998 ) NoStop

    1998

  45. [53]

    Ciccotti , author R

    author author G. Ciccotti , author R. Kapral , \ and\ author E. Vanden-Eijnden ,\ title title Blue Moon Sampling, Vectorial Reaction Coordinates, and Unbiased Constrained Dynamics , \ 10.1002/cphc.200400669 journal journal ChemPhysChem \ volume 6 ,\ pages 1809 -- 1814 ( year 2...

    2005 doi

  46. [54]

    Ciccotti , author T

    author author G. Ciccotti , author T. Lelievre , \ and\ author E. Vanden-Eijnden ,\ title title Projection of diffusions on submanifolds: Application to mean force computation , \ @noop journal journal Communications on Pure and Applied Mathematics: A Journal Issued by the Cou...

    2008

  47. [55]

    Gompper , author H

    author author G. Gompper , author H. A. \ Stone , author C. Kurzthaler , author D. Saintillan , author F. Peruani , author D. A. \ Fedosov , author T. Auth , author C. Cottin-Bizonne , author C. Ybert , author E. Cl \'e ment , et al. ,\ title title The 2025 motile active matte...

    2025

  48. [56]

    Levitz , author L

    author author P. Levitz , author L. Michot , author N. Malikova , author M. Scheel , \ and\ author T. Weitkamp ,\ title title Probing particle dynamics in a fully opaque porous network using x-ray differential dynamic radiography (xddr) , \ @noop journal journal Soft Matter \ ...

    2025

  49. [57]

    author author H. Brenner ,\ title title The slow motion of a sphere through a viscous fluid towards a plane surface , \ @noop journal journal Chemical Engineering Science \ volume 16 ,\ pages 242--251 ( year 1961 ) NoStop

    1961

  50. [58]

    Fax \'e n ,\ @noop title Einwirkung der Gefasswande auf den Widerstand gegen die Bewegung einer kleinen Kugel in einer zahen Flussigkeit \ ( publisher F \"o rf

    author author H. Fax \'e n ,\ @noop title Einwirkung der Gefasswande auf den Widerstand gegen die Bewegung einer kleinen Kugel in einer zahen Flussigkeit \ ( publisher F \"o rf. ,\ year 1921 ) NoStop

    1921

  51. [59]

    Fish , author A

    author author R. Fish , author A. Carter , author P. Diez-Silva , author R. Delgado-Buscalioni , author R. P. \ Pelaez , \ and\ author B. Sprinkle ,\ title title libmobility: A python library for hydrodynamics at the smoluchowski level , \ @noop journal journal arXiv preprint ...

    2025

  52. [60]

    Perkins \ and\ author R

    author author G. Perkins \ and\ author R. Jones ,\ title title Hydrodynamic interaction of a spherical particle with a planar boundary: Ii. hard wall , \ @noop journal journal Physica A: Statistical Mechanics and its Applications \ volume 189 ,\ pages 447--477 ( year 1992 ) NoStop

    1992

  53. [61]

    author author A. L. \ Thorneywork , author J. Gladrow , author Y. Qing , author M. Rico-Pasto , author F. Ritort , author H. Bayley , author A. B. \ Kolomeisky , \ and\ author U. F. \ Keyser ,\ title title Direct detection of molecular intermediates from first-passage times , ...

    2020

  54. [62]

    author author R. W. \ Perry , author M. C. \ Holmes-Cerfon , author M. P. \ Brenner , \ and\ author V. N. \ Manoharan ,\ title title Two-dimensional clusters of colloidal spheres: Ground states, excited states, and structural rearrangements , \ @noop journal journal Physical r...

    2015

  55. [63]

    Holmes-Cerfon , author S

    author author M. Holmes-Cerfon , author S. J. \ Gortler , \ and\ author M. P. \ Brenner ,\ title title A geometrical approach to computing free-energy landscapes from short-ranged potentials , \ @noop journal journal Proceedings of the National Academy of Sciences \ volume 110...

    2013

  56. [64]

    author author W. B. \ Rogers \ and\ author J. C. \ Crocker ,\ title title Direct measurements of dna-mediated colloidal interactions and their quantitative modeling , \ @noop journal journal Proceedings of the National Academy of Sciences \ volume 108 ,\ pages 15687--15692 ( y...

    2011

  57. [65]

    Pavliotis \ and\ author A

    author author G. Pavliotis \ and\ author A. Stuart ,\ @noop title Multiscale methods: averaging and homogenization \ ( publisher Springer Science & Business Media ,\ year 2008 ) NoStop

    2008

  58. [66]

    Zwanzig ,\ title title Diffusion past an entropy barrier , \ @noop journal journal The Journal of Physical Chemistry \ volume 96 ,\ pages 3926--3930 ( year 1992 ) NoStop

    author author R. Zwanzig ,\ title title Diffusion past an entropy barrier , \ @noop journal journal The Journal of Physical Chemistry \ volume 96 ,\ pages 3926--3930 ( year 1992 ) NoStop

    1992

  59. [67]

    author author M. H. \ Jacobs ,\ 10.1007/978-3-642-86414-8 title Diffusion Processes \ ( publisher Springer Berlin Heidelberg ,\ address Berlin, Heidelberg ,\ year 1967 ) NoStop

    1967 doi

  60. [68]

    Reguera \ and\ author J

    author author D. Reguera \ and\ author J. M. \ Rubí ,\ title title Kinetic equations for diffusion in the presence of entropic barriers , \ 10.1103/PhysRevE.64.061106 journal journal Physical Review E \ volume 64 ,\ pages 061106 ( year 2001 ) NoStop

    2001 doi

  61. [69]

    Kalinay \ and\ author J

    author author P. Kalinay \ and\ author J. K. \ Percus ,\ title title Corrections to the Fick - Jacobs equation , \ 10.1103/PhysRevE.74.041203 journal journal Physical Review E \ volume 74 ,\ pages 041203 ( year 2006 ) NoStop

    2006 doi

  62. [70]

    author author J. M. \ Rubi ,\ title title Entropic diffusion in confined soft-matter and biological systems , \ @noop journal journal EPL (Europhysics Letters) \ volume 127 ,\ pages 10001 ( year 2019 ) NoStop

    2019

  63. [71]

    note It could be the case that they do not form a global parameterization of M . While it is possible to handle this case using an atlas of such local parameterizations, combined with a smoothing function to move between parameterizations, this is highly technical and not a si...

  64. [72]

    Holmes-Cerfon ,\ title title Stochastic disks that roll , \ @noop journal journal Physical Review E \ volume 94 ,\ pages 052112 ( year 2016 ) NoStop

    author author M. Holmes-Cerfon ,\ title title Stochastic disks that roll , \ @noop journal journal Physical Review E \ volume 94 ,\ pages 052112 ( year 2016 ) NoStop

    2016

  65. [73]

    author author H. C. \ \"O ttinger ,\ title title Preservation of thermodynamic structure in model reduction , \ @noop journal journal Physical Review E \ volume 91 ,\ pages 032147 ( year 2015 ) NoStop

    2015

  66. [74]

    author author L. D. \ Landau \ and\ author E. M. \ Lifshitz ,\ @noop title Mechanics ,\ Butterworth-Heinemann\ ( publisher Butterworth-Heinemann ,\ year 1976 ) NoStop

    1976

  67. [75]

    note Here are the intermediate steps to show this: align* & (P_ N (P_ N M P_ N ^ T ) )_ i \\ & 1em = (P_ N )_ i, j _ k (P_ N M P_ N ^ T )_ j, k \\ & 1em = _ k ( (P_ N )_ i, j (P_ N M P_ N ^ T )_ j, k )- (P_ N M P_ N ^ T )_ j, k _ k (P_ N )_ i, j \\ & 1em = _ k (P_ N M P_ N ^ T...

  68. [76]

    Leimkuhler , author T

    author author B. Leimkuhler , author T. Vlaar , author T. Pouchon , \ and\ author A. Storkey ,\ title title Better training using weight-constrained stochastic dynamics , \ @noop journal journal arXiv preprint arXiv:2106.10704 \ ( year 2021 ) NoStop

    2021 arXiv

  69. [77]

    Sch \"o nle , author D

    author author C. Sch \"o nle , author D. Carbone , author M. Gabri \'e , author T. Leli \`e vre , \ and\ author G. Stoltz ,\ title title Efficient monte-carlo sampling of metastable systems using non-local collective variable updates , \ @noop journal journal arXiv preprint ar...

    2025 arXiv

  70. [78]

    Holmes-Cerfon ,\ @noop title Applied Stochastic Analysis ,\ Vol

    author author M. Holmes-Cerfon ,\ @noop title Applied Stochastic Analysis ,\ Vol. volume 33 \ ( publisher American Mathematical Society, Courant Institute of Mathematical Sciences ,\ year 2024 ) NoStop

    2024

  71. [79]

    Sharma \ and\ author W

    author author U. Sharma \ and\ author W. Zhang ,\ title title NonReversible Sampling Schemes on Submanifolds , \ 10.1137/20m1378752 journal journal SIAM Journal on Numerical Analysis \ volume 59 ,\ pages 2989--3031 ( year 2021 ) NoStop

    2021 doi

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.