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arxiv: 2605.23803 · v1 · pith:UD3SZYNSnew · submitted 2026-05-22 · ❄️ cond-mat.stat-mech · cond-mat.mes-hall· math-ph· math.MP· physics.chem-ph

Chirality-sensitive mobility and dissipation of Brownian motion on a helical landscape

Debankur Bhattacharyya , Abraham Nitzan This is my paper

Pith reviewed 2026-05-25 02:39 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.mes-hallmath-phmath.MPphysics.chem-ph
keywords Brownian motionhelical landscapemobility tensorchiralitylinear responseOrnstein-Uhlenbeck processenergy dissipation
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0 comments X

The pith

The helical landscape produces equal cross-mobilities between axial forcing and angular motion, and between torque and axial transport, while also creating asymmetric energy dissipation under simultaneous drives.

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

The paper analyzes Brownian motion with inertia on a helical landscape wrapped on a cylinder. In the harmonic approximation the deterministic trajectories split into free motion along the screw coordinate and bounded oscillations perpendicular to it. This separation persists under isotropic damping in the stochastic equations but is broken by anisotropic damping, which mixes the axial and angular variables. After removing the diffusive zero mode along the screw direction, the remaining stable dynamics is cast as a linear Ornstein-Uhlenbeck process whose mobility tensor is obtained in closed form. The off-diagonal elements of that tensor are identical and serve as a direct dynamical marker of the helical geometry; combined axial and angular driving further produces an asymmetry in the rate of energy dissipation traceable to the same geometry.

Core claim

After projecting out the zero mode associated with diffusion along the screw coordinate, the stationary dynamics yields a mobility tensor whose off-diagonal elements are equal by time-reversal symmetry; these elements quantify the cross-response of angular velocity to axial force and of axial velocity to applied torque, furnishing a direct signature of the helical landscape, while simultaneous driving in both directions produces an asymmetry in the dissipation rate that likewise originates in the helical geometry.

What carries the argument

The linear Ornstein-Uhlenbeck process obtained after projecting out the diffusive zero mode along the screw coordinate, from which the full dynamical mobility tensor (including its off-diagonal cross terms) is derived in both time and frequency domains.

If this is right

  • The two cross-mobility coefficients are identical, as required by time-reversal symmetry.
  • These equal cross terms constitute a direct dynamical signature of the underlying helical geometry.
  • Simultaneous application of axial force and angular torque produces an asymmetry in the energy dissipation rate that is absent on a non-helical surface.
  • The construction applies to an infinite cylinder where the full phase-space distribution does not reach a normalizable stationary state without projection.

Where Pith is reading between the lines

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

  • The equality of cross-mobilities could be measured in colloidal or molecular systems to infer the presence of effective helical constraints without imaging the potential itself.
  • The dissipation asymmetry offers a possible route to chiral selectivity in driven transport at the nanoscale.
  • The same projection technique may extend to other systems possessing a single neutral diffusive mode, such as particles on periodic or screw-symmetric surfaces.

Load-bearing premise

The harmonic-well approximation remains valid so that deterministic motion cleanly separates into free propagation along the screw and harmonic motion in the transverse plane.

What would settle it

An experiment that records the full mobility tensor for a colloidal particle in a fabricated helical potential would falsify the claim if the measured off-diagonal elements are unequal or if the dissipation rate under combined axial and angular drives shows no helical asymmetry.

Figures

Figures reproduced from arXiv: 2605.23803 by Abraham Nitzan, Debankur Bhattacharyya.

Figure 1
Figure 1. Figure 1: FIG. 1: Plotted here for an example a helical potential [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Screw/Screw-Normal coordinate conversion. This figure illustrates how the helical coordinates ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Time correlation functions of the momenta [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Effect of damping anisotropy in momentum [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Nyquist plots for visualizing AC mobilities (Eq. (74)). Here we plot ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: DC mobilities. The scaled mobility prefactor [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Fourier domain time correlation functions. In this figure we show the real and imaginary parts of the one [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
read the original abstract

We study the Brownian dynamics and linear response of a particle with inertia moving in a 2-dimensional helical landscape imprinted on a cylindrical surface. In the harmonic well approximation, the deterministic motion separates into free propagation along the screw direction and harmonic motion in the transverse screw-normal direction. We show that for isotropic damping this simplification survives in the Langevin description, whereas anisotropic damping along the axial and angular directions couples the stochastic dynamics and destroys separability. The resulting anisotropic model is formulated as a linear Ornstein-Uhlenbeck process in phase space with a zero mode associated with diffusion along the screw coordinate, so that in an infinite system the full phase-space dynamics does not relax to a stationary distribution. To treat transport in this setting, we construct the stationary dynamics in the stable subspace obtained after projecting out the zero mode. This leads to a linear response theory for this system and yields closed analytical expressions for stationary time-correlation functions and the dynamical mobility tensor in both the time and frequency domains. The off-diagonal elements of the mobility tensor describe cross-response between axial forcing and angular motion, and between applied torque and axial transport. Consistent with time reversal symmetry, these cross mobilities are equal and provide a direct dynamical signature of the helical geometry. In addition, a simultaneous application of driving in both the axial and angular direction reveals asymmetry in energy dissipation rate due the helical landscape.

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 / 1 minor

Summary. The manuscript analyzes the Brownian dynamics and linear response of an inertial particle on a 2D helical landscape. In the harmonic well approximation, deterministic motion separates into free propagation along the screw direction and transverse harmonic oscillation. For isotropic damping this separation persists in the Langevin equation, while anisotropic damping couples the coordinates and is treated as a linear Ornstein-Uhlenbeck process possessing a zero mode. Projecting out the zero mode yields stationary dynamics in the stable subspace, from which closed analytical expressions are obtained for time-correlation functions and the dynamical mobility tensor in time and frequency domains. The off-diagonal mobility elements are equal by time-reversal symmetry and constitute a chirality signature; simultaneous axial and angular driving produces an asymmetry in the dissipation rate attributable to the helical geometry.

Significance. The closed-form mobility expressions and the explicit treatment of the zero mode via projection constitute a clear technical contribution, allowing linear response theory in a system without a global stationary distribution. If the central approximation holds, the equality of cross-mobilities supplies a symmetry-protected, experimentally accessible signature of helical chirality, while the predicted dissipation asymmetry offers an additional testable consequence of the geometry.

major comments (2)
  1. [Abstract] Abstract (harmonic well approximation paragraph): the separation into free screw propagation plus transverse harmonic motion is invoked as the starting point for both deterministic and stochastic analysis and is stated to survive only for isotropic damping. No quantitative estimate is supplied for the size of neglected anharmonic terms in the helical potential, nor for the transverse amplitude range over which the projected stationary dynamics and the resulting off-diagonal mobilities remain accurate. Because the equality of cross-mobilities and the dissipation asymmetry are direct consequences of the reduced linear dynamics, the lack of error bounds or an applicability regime is load-bearing for the central claims.
  2. [Abstract] Abstract (anisotropic damping and OU-process paragraph): the transition from the Langevin equation to the linear Ornstein-Uhlenbeck process after anisotropic damping is asserted without an explicit display of the coupled equations or the friction matrix that produces the zero mode. A concrete derivation of the projected stationary covariance or the mobility tensor from the OU generator would allow verification that time-reversal symmetry indeed forces the off-diagonal elements to be equal after projection.
minor comments (1)
  1. The abstract is information-dense; a short paragraph in the introduction that recalls the explicit form of the helical potential and the damping matrix would improve accessibility for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (harmonic well approximation paragraph): the separation into free screw propagation plus transverse harmonic motion is invoked as the starting point for both deterministic and stochastic analysis and is stated to survive only for isotropic damping. No quantitative estimate is supplied for the size of neglected anharmonic terms in the helical potential, nor for the transverse amplitude range over which the projected stationary dynamics and the resulting off-diagonal mobilities remain accurate. Because the equality of cross-mobilities and the dissipation asymmetry are direct consequences of the reduced linear dynamics, the lack of error bounds or an applicability regime is load-bearing for the central claims.

    Authors: We agree that an explicit applicability regime strengthens the presentation. The central claims rely on the linear dynamics, with the equality of cross-mobilities protected by time-reversal symmetry within that regime. In the revision we will add a quantitative discussion, including an expansion of a representative helical potential to estimate anharmonic corrections and bounds on transverse amplitude (e.g., displacement ranges yielding <5% error in the mobility tensor). revision: yes

  2. Referee: [Abstract] Abstract (anisotropic damping and OU-process paragraph): the transition from the Langevin equation to the linear Ornstein-Uhlenbeck process after anisotropic damping is asserted without an explicit display of the coupled equations or the friction matrix that produces the zero mode. A concrete derivation of the projected stationary covariance or the mobility tensor from the OU generator would allow verification that time-reversal symmetry indeed forces the off-diagonal elements to be equal after projection.

    Authors: We will revise the manuscript to display the coupled Langevin equations under anisotropic damping, the explicit friction matrix, and a step-by-step derivation of the projected stationary covariance together with the mobility tensor from the OU generator. This will make transparent how time-reversal symmetry enforces equality of the off-diagonal elements after projection. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from Langevin via projection and linear response is self-contained

full rationale

The paper begins with the standard inertial Langevin equation on a helical potential, applies the stated harmonic-well approximation to separate screw and transverse motion, recasts the anisotropic case as a linear Ornstein-Uhlenbeck process, projects out the zero mode to obtain a stationary dynamics on the stable subspace, and then invokes standard linear-response theory to produce closed-form expressions for the mobility tensor and time-correlation functions. The equality of off-diagonal cross-mobilities is a direct consequence of time-reversal symmetry applied to the resulting linear dynamics, not a fitted parameter or self-citation. No equation reduces a claimed prediction to an input by construction, and no load-bearing step relies on prior work by the same authors. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard stochastic-process assumptions and the harmonic-well model; no new entities are introduced and no parameters are fitted to data within the abstract.

axioms (3)
  • domain assumption The deterministic motion separates into free propagation along the screw direction and harmonic motion in the transverse direction under the harmonic well approximation
    Invoked at the start of the abstract to simplify both deterministic and stochastic dynamics.
  • standard math Time-reversal symmetry implies equality of the off-diagonal mobility elements
    Used to conclude that cross-mobilities are equal and serve as a chirality signature.
  • domain assumption Projection onto the stable subspace after removing the zero mode yields a well-defined stationary process for linear response
    Required to obtain time-correlation functions and mobility in an infinite system with free diffusion along the screw coordinate.

pith-pipeline@v0.9.0 · 5790 in / 1709 out tokens · 33841 ms · 2026-05-25T02:39:53.004345+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    We start with rewriting Eq

    Time evolution in the stable and zero-mode subspaces Since we are primarily interested in measuring re- sponses in the original cylindrical coordinate frame, we analyze the anisotropic system in the (ϕ, L ϕ, z, pz) phase space and use spectral projectors to separate the zero- mode. We start with rewriting Eq. (16) in a matrix stochastic differential equat...

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    Time correlation functions in the stable subspace To quantify the time correlations in the fluctuation of the system state at timet:δ ⃗X(t) = ⃗X(t)− ⟨ ⃗X(t)⟩and a future timet+τ, we introduce the time correlation matrix51 S(t+τ|t) =⟨δ ⃗X(t+τ)δ ⃗X T (t)⟩ =e CτS(t), (54) where,S(t) is the phase space covariance matrix defined in Eq. (40). In the long time l...

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    (37) respectively and d ⃗W(t) represent a vector Wiener process as column vector with two independent components

    Continuous time Sylvester-Lyapunov Equation Consider a generic Ornstein-Uhlenbeck(OU) process is given by the matrix stochastic differential equation (SDE) d ⃗X(t) =C ⃗X(t) dt+Σd ⃗W(t) (D1) whereC,Σare given in Eq.(36) and Eq. (37) respectively and d ⃗W(t) represent a vector Wiener process as column vector with two independent components. The evolution eq...

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  4. [4]

    Projection operator approach for solving the continuous time Sylvester-Lyapunov Equation with a zero mode In this appendix we calculate asymptotic (t→ ∞) expressions for the covariance matrix by applying projection operator machinery to Sylvester-Lyapunov equation44 to remove the zero-eigenmode. The SylvesterLyapunov equation is given as dS(t) dt =CS(t) +...

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    Spectral projector and construction of the evolution operator of Sylvester-Lyapunov equation Starting from the drift matrixCgiven in Eq. (36), we write the characteristic polynomial as det (C−λI) = 3Y i=0 (λ−λ i) =λp(λ) =λ(λ 3 +a 2λ2 +a 1λ+a 0) (D17) where{λ i|i= 0,1,2,3}are the eigenvalues ofCwithλ 0 = 0 and a2 = (γϕ +γ z) a1 = γϕγz + 4π2 + 1 R2 = γϕγz +...

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    Markovian dynamics in the stable subspace Suppose we project the state vector ⃗X(t) the stable subspace as ⃗YQ(t) =Q 0 ⃗X(t), then the equation of motion for ⃗YQ(t) is given as d⃗YQ(t) =Q 0 d ⃗X(t) =Q 0C ⃗X(t) dt+Q 0Σd ⃗W(t) =CQ 0 ⃗X(t) dt+Q 0Σd ⃗W(t) =C ⃗YQ(t) dt+Q 0Σd ⃗W(t), (D33) which is also a multivariate OU process in the stable subspace of the dyn...

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    Using Eq

    Evaluation of the bounded part of the covariance matrix at long time limit Here we discuss details of the calculatingS ∞. Using Eq. (D26) and Eq. (D28) we write eCt =Π 0 + (b0(t)I+b 1(t)C+b 2(t)C2)Q0, eCT t =Π 0 T +Q 0(b0(t)I+b 1(t)CT +b 2(t)(CT )2) (D35) where ⃗b(t) = (b0(t)b 1(t)b 2(t))T are time-dependent functions and can be written as ⃗b(t) =e At⃗b(0...

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    Reduction of anisotropic case to isotropic case The Fourier-space coefficients are given as   ˜b0(ω) ˜b1(ω) ˜b2(ω)   =− 1 p(iω)   ω2 −a 1 −ia 2ω −a2 −iω −1   , p(iω) =−iω 3 −a 2ω2 +ia 1ω+a 0 = (a0 −a 2ω2) +iω(a 1 −ω 2), (D47) For isotropic case we havea 0 =γω 2 0, a1 =γ 2 +ω 2 0, a2 = 2γ. Thus we get p(λ) = (λ+γ)(λ 2 +γλ+ω 2 0).(D48) Substituting ...

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    Stationary measure Here we show that the dynamics in the stable subspace relaxes to a Gaussian stationary state, clarify the inter- pretation of the integration using this stationary measure and derive an integration by parts formula which will be useful for the general derivation of response relations . We choose a orthonormal basis vectors{⃗ u 1, ⃗ u2, ...

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    We define a functionf(⃗ z) =F(U⃗ z), and write the vector ⃗Az =U T ⃗Ay

    Integration under measure and integration by parts formula We express the integration of a test functionF(⃗ y) with the stationary measureπ Q(d⃗ y) on the stable subspace (i.e., where⃗ y=Q 0⃗ y) in the phase space coordinates as Z πQ(d⃗ y)F(⃗ y) = Z d⃗ z πz Q(⃗ z)F(U⃗ z) (E7) Given the 4 dimensional directional derivative operator projected along the stab...

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    We can obtain the expected value of O( ⃗Y) at timeTby averaging over the initial conditions sampled from some probability measureµ(d⃗ y) in the sable subspace

    Review of response relation calculations Whenϵ= 0, for a test functionϕ(⃗ y) we write 35 ˆLϕ(⃗ y) = (Q0C⃗ y)· ∇ϕ(⃗ y) +1 2 Q0ΣΣTQ0 T :∇∇ϕ(⃗ y),(F4) where ˆLis the infinitesimal generator 35 of the unperturbed process in the stable subspace projection which describes the time evolution of a functionut(⃗ y) =E h O( ⃗Y(T))| ⃗Y(t) =⃗ y i , i.e., the condition...

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    Special case: forcing perturbation in momentum equations and response in average velocities For the forcing perturbation presented in Eq. (F3) the generator due to the perturbation can be written as ˆL1(t) = Q0B ⃗f(t) · ∇(F20) We sets= 0 to represent the represent a time where the system has reached a stationary state in the stable subspace; then we haveR...

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    Chiral induced spin selectivity,

    Proof ofQ 0B=βS ∞ Q BVT Given ⃗YQ =Q 0 ⃗X, the stationary state in the stable space relaxes to a stationary state, and in the orthogonal representation ⃗YQ =U⃗ z, it relaxes to a the distribution πz Q ∝exp −1 2 ⃗ zTS−1 z ⃗ z (F30) Also, the harmonic Hamiltonian Eq. (12) can be written as the matrix relation H= 1 2 ⃗X TK ⃗X(F31) with the Hessian matrixKdef...

    work page doi:10.1021/acs.chemrev.3c00661 1950