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Exact Cap Response and Aperture Selection for Molecular Communication under Aligned Drift

Yen-Chi Lee, Chun-An Yang, Nan Yang

arXiv:2610.20233v1cs.ITcs.ETeess.SPmath.IT

TL;DR

Spatial reception under drift must balance timely molecule capture against ISI, while accounting for where molecules arrive. The paper derives and evaluates exact cap responses from the joint hitting law, then analyzes SID-based aperture selection; for the studied geometry at Ts = 0.15 s, the optimal aperture widens from 29.13° at zero drift to 61.86° at |v| = 40 µm/s.

  • Problem

    Drift changes both arrival timing and absorption location, while prior cap analysis used an approximate zero-drift response and full-surface responses did not address spatial selection.

    Source: §I · p. 1

  • Method

    The paper derives exact cap-restricted impulse, cumulative, and eventual responses from the joint arrival-time and absorption-location law, then analyzes SID-based aperture selection.

    Source: §I · p. 1, §IV · p. 3

  • Results

    At Ts = 0.15 s, the SID-maximizing aperture rises from 29.13° at zero drift to 61.86° at |v| = 40 µm/s.

    Source: C. SID Loss from Retaining the… · p. 5

  • Takeaways & Limitations

    Under the stated single-crossing condition, opposite drifts of equal magnitude can favor the same aperture while yielding different SID payoffs.

    Source: §VI · p. 5

    The findings concern the tested geometry and SID objective; they do not establish BER or capacity improvements.

    Source: D. Sensitivity to Symbol Duration · p. 5

Abstract

from arXiv · show

Spatial reception under directed molecule transport requires balancing useful molecule capture against intersymbol interference (ISI). To characterize this balance, we study a fully absorbing spherical receiver that counts absorption events only within a prescribed cap. Using the known joint first-hitting law under aligned drift, we derive exact cap impulse, cumulative, and eventual responses. We perform stable cumulative evaluation. We then assess the derived responses through particle simulations and time-step comparisons, and demonstrate that our conditional-timing analysis explains when opposite drifts of equal magnitude can favor the same wider aperture. We also show that for the studied geometry and a symbol duration of $0.15$ s, the aperture that maximizes the signal-to-interference difference (SID) increases from $29.13^\circ$ at zero drift to $61.86^\circ$ at a drift magnitude of $40$ $\mu$m/s. These results (i) connect joint arrival-time and location statistics to spatial reception design, (ii) characterize the tradeoff between useful molecule capture and ISI, (iii) support SID-based aperture selection for a given drift and symbol duration, and (iv) quantify the SID loss incurred by retaining the zero-drift optimal aperture for the same symbol duration.

I. INTRODUCTION

Drift redistributes useful and late-arriving molecules across the receiver, so cap design must balance capture against ISI using both arrival time and surface location. The model counts hits in a transmitter-facing cap while the entire spherical surface remains absorbing.

I. INTRODUCTION

  • A suitable cap can improve the signal-interference balance by excluding surface regions that contribute more late-arriving than useful molecules.Drift affects both delivery timing and where molecules are absorbed.Source: §I · p. 1
  • Exact cap-restricted responses from the joint arrival-time and absorption-location law address aperture selection under signed aligned drift.Earlier cap analyses used an approximate zero-drift response, while existing full-surface responses did not address spatial selection.Source: §I · p. 1
  • The analysis derives cap impulse and cumulative responses, evaluates the cumulative response stably, and validates predictions with particle simulations and shared-path time-step comparisons.It also studies conditional SID/aperture behavior and the cost of retaining zero-drift designs across symbol durations.Source: §I · p. 1

II. SYSTEM MODEL

  • The receiver is a fully absorbing sphere centered at the origin, with a point transmitter outside it in an unbounded three-dimensional medium.The transmitter position is x0 = r0k, with r0 > r.Source: §II · p. 2
  • The model counts hits within a transmitter-facing cap, but every surface hit terminates a trajectory, including hits outside the cap.The cap selects recorded events without changing the absorbing boundary condition.Source: §II · p. 2

III. EXACT CAP RESPONSE UNDER ALIGNED DRIFT

The joint time-location law for drift-free diffusion outside a sphere is reweighted for finite-time aligned drift. This retains the temporal and spatial dependence needed for cap counting.

III. EXACT CAP RESPONSE UNDER ALIGNED DRIFT

B. Cap Impulse and Cumulative Responses

The exact cap responses follow by integrating the drift-weighted angular coefficient and cumulative radial modes. Bounds on the time-integrated modes justify convergence of the cumulative and eventual series.

B. Cap Impulse and Cumulative Responses

  • Substituting the joint-density expansion and integrating over the cap yields exact impulse and cumulative responses from drift-weighted angular and radial modes.The exact series require truncation and quadrature for numerical evaluation.Source: B. Cap Impulse and Cumulative Responses · p. 3
  • The infinite-time cumulative value is obtained from the Laplace transform of each radial time mode evaluated at the transmitter radius and parameter κ.The transform solves an exterior modified-Helmholtz problem with a radial factor normalized at the receiver and decaying at infinity.Source: B. Cap Impulse and Cumulative Responses · p. 3
  • For ρ > 1, geometric summability bounds the time-integrated modes and justifies uniform convergence of cumulative and eventual response series and their first aperture derivatives.The same bounds do not provide a uniform truncation-error bound for the instantaneous CIR.Source: B. Cap Impulse and Cumulative Responses · p. 3

C. Stable Evaluation of the Cumulative Response

Stable cumulative evaluation avoids the conditionally convergent integral produced by integrating the spectral impulse response directly. It integrates a damped tail instead, with checks on response properties and the zero-drift limit.

C. Stable Evaluation of the Cumulative Response

  • Writing the cumulative mode as its eventual value minus a damped tail preserves exponential damping and separates eventual-mass accuracy from the finite-time residual.The tail integral starts at zero and retains damping at large q.Source: C. Stable Evaluation of the Cumulative… · p. 3
  • At zero drift, the angular coefficients have a nonsingular limit, and a zero aperture gives zero response.For a full sphere at zero drift, only mode m = 0 remains.Source: C. Stable Evaluation of the Cumulative… · p. 3
  • Counted probabilities are nondecreasing in time and aperture and satisfy 0 ≤ Hv(t; α) ≤ Hv(∞; α) ≤ 1.These properties also check the evaluated finite modal sum.Source: C. Stable Evaluation of the Cumulative… · p. 3

IV. SID-BASED APERTURE SELECTION

The section defines SID as a difference between useful and late-arriving counted molecules, then uses it to select a counting aperture for a given drift and symbol duration. Conditional arrival timing explains when opposite drift signs can favor the same aperture.

IV. SID-BASED APERTURE SELECTION

  • SID_v(α) = S_v(α) − I_v(α) = 2H_v(T_s; α) − H_v(∞; α) balances useful capture against late-arriving molecule capture.It is an unnormalized probability difference, not an interference ratio or BER.Source: §IV · p. 3
  • The selected aperture maximizes SID_v(α) over α ∈ [0, π], while capture or detector constraints would define a different optimization problem.The analysis makes no claim of BER-optimality.Source: §IV · p. 3
  • The derivative of SID balances useful and late-arriving contributions from an infinitesimal ring, with a positive spatial factor separating drift direction from timing.The section introduces modal coefficients and conditional arrival timing to analyze this balance.Source: §IV · pp. 3–4
  • η_κ(θ; T_s) is the fraction arriving within T_s conditional on eventual arrival at angle θ, and the signed spatial factor cancels from this ratio.The counted channel taps are defined for symbol intervals of duration T_s.Source: §IV · pp. 3–4
  • Under the stated single-crossing condition, η_κ yields a unique SID-maximizing aperture for either drift sign, nondecreasing in κ and T_s among profiles satisfying that condition.Opposite signs with equal drift magnitude have identical interior stationary angles and local SID increase/decrease regions.Source: §IV · p. 4
  • Increasing |v| downweights late times conditional on arrival at an angle, but does not imply greater absolute capture; the single-crossing result need not hold for arbitrary geometries or objectives.The authors compare all stationary roots and both endpoints numerically rather than assuming the condition holds.Source: §IV · p. 4

A. Simulation Setup and Response Validation

Particle simulations assess the exact restricted-cap cumulative responses under the stated geometry and velocities. The largest discrepancy on the validation grid is within the prespecified comparison tolerances.

A. Simulation Setup and Response Validation

  • The simulations used D = 80 µm^2/s, r = 5 µm, r_0 = 10 µm, and T_s = 0.15 s, with 250,000 trajectories per velocity through 0.30 s.Velocities were −40, −20, 0, 20, and 40 µm/s; aperture-design curves were evaluated from −40 to 40 µm/s in 2 µm/s steps.Source: A. Simulation Setup and Response… · p. 4
  • 1.61 × 10^-3 was the largest discrepancy on the 80-point grid between exact cap CDFs and particle estimates for both drift signs.Comparisons met prespecified tolerances allowing three sampling standard errors plus a fixed numerical margin; time-step comparisons used shared Brownian paths.Source: A. Simulation Setup and Response… · p. 4

B. Comparison with Approximate Zero-Drift Model

At T_s = 0.15 s, the section compares an approximate zero-drift response with the exact model. The approximate model's optimal aperture is close to, but differs from, the exact optimum.

B. Comparison with Approximate Zero-Drift Model

  • Optimizing 2FA(Ts; α) − FA(∞; α), including the normalized infinite-time limit and endpoints, gives an approximate-model optimum of 27.62°, versus 29.13° for the exact response.Source: B. Comparison with Approximate… · p. 5
  • Subsequent comparisons use the exact zero-drift optimum to isolate drift mismatch from approximation errors.The section compares against the normalized approximate zero-drift response from.Source: B. Comparison with Approximate… · pp. 4–5

C. SID Loss from Retaining the Zero-Drift Aperture

The study quantifies how drift and symbol duration alter SID-optimal apertures and the loss from retaining zero-drift designs, then interprets the trend through conditional timing. Its results support SID-based aperture selection within the tested fully absorbing model, without establishing BER or capacity gains.

C. SID Loss from Retaining the Zero-Drift Aperture

D. Sensitivity to Symbol Duration

VI. CONCLUSION

APPENDIX A NUMERICAL IMPLEMENTATION AND VALIDATION

The appendix specifies numerical procedures for response evaluation, aperture optimization, particle detection, and validation. Across the reported comparisons, the analytical predictions and time-step checks met their prespecified tolerances, while residual detector bias was not bounded.

A. Response Evaluation and Aperture Optimization

  • Response evaluation truncates the modal series at m = 40 and uses quadrature checks, while aperture maxima are checked through stationary candidates, endpoints, and independent optimization.Refinement checks include probability bounds, cap nesting, and time monotonicity; the stated agreement checks are not uniform error guarantees.Source: A. Response Evaluation and Aperture… · p. 6

B. Particle Simulation and Absorption Detection

  • Particle simulations use exact Gaussian drift-diffusion increments at three nested time steps with shared Brownian paths for paired absorption comparisons.The detector shifts the radius to reduce missed crossings, removes each trajectory at its first detected absorption, and assesses remaining cap-indicator error through nested-step comparisons.Source: B. Particle Simulation and Absorption… · p. 6

C. Validation Criteria and Sampling Uncertainty

  • Validation compares particle CDFs and SID gains with analytical predictions and uses paired estimates from shared trajectories to characterize sampling uncertainty.The paired gain interval covers sampling variation under the discretized detector, excluding time-step bias.Source: C. Validation Criteria and Sampling… · pp. 6–7
  • At the baseline, maximum CDF error was 1.61×10^-3; across three durations, all 45 CDF cases and 30 SID-gain estimates met their prespecified comparison tolerances.The largest paired gain change between the two smallest steps was 8.32 × 10−4; these comparisons assess sensitivity without bounding residual detector bias.Source: C. Validation Criteria and Sampling… · pp. 6–7

APPENDIX B PROOF OF PROPOSITION 1

The proof links conditional arrival timing to the SID derivative, then uses a single-crossing condition to establish how the maximizing aperture changes with drift magnitude and symbol duration.

APPENDIX B PROOF OF PROPOSITION 1

  • The fraction of arrivals by T_s is nondecreasing in drift parameter κ and in T_s, with the proof requiring no hitting-time moment assumption.For fixed angle, ηκ is the early-arrival mass Aκ divided by total mass Zκ = Aκ + Bκ.Source: Appendix B · p. 7
  • Because the SID derivative’s interior prefactor is positive and its timing fraction depends on drift through κ = v^2/(4D), opposite drift signs share stationary angles and local increase/decrease regions.Source: Appendix B · p. 7
  • Under the single-crossing condition, the crossing angle is the unique global SID-maximizing aperture for either drift sign.Pointwise monotonicity of ηκ means the maximizing aperture is nondecreasing in κ or T_s when the other parameter is fixed.Source: Appendix B · p. 7
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