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arxiv: 2509.13223 · v2 · submitted 2025-09-16 · 🧮 math.NA · cs.NA

Geometry, Energy and Sensitivity in Stochastic Proton Dynamics

Veronika Chronholm , Tristan Pryer This is my paper

Pith reviewed 2026-05-18 15:56 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords stochastic proton transportMilstein schemeLie-group integratorgeometric invariantspathwise sensitivitydose depositionnumerical SDE methods
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The pith

A logarithmic Milstein scheme paired with a Lie-group integrator preserves positivity and geometric structure in stochastic proton transport while producing consistent pathwise dose sensitivities.

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

The paper constructs numerical integrators for stochastic models that couple energy loss, range straggling, and angular diffusion in proton motion. For the energy process it proposes a logarithmic Milstein discretization that stays positive and attains strong order-one convergence. Angular dynamics are advanced with a Lie-group method so that the combined scheme respects the natural invariants of the system. Dose deposition is recast as a regularized path-dependent functional, and a pathwise sensitivity estimator is derived that remains consistent and directly computable from simulated trajectories. Numerical tests verify the expected convergence orders and the stability of the resulting sensitivity estimates.

Core claim

The logarithmic Milstein scheme for the energy equation guarantees positivity and achieves strong order one convergence; the combined method with Lie-group angular integrator maintains natural geometric invariants; the pathwise sensitivity estimator for regularised dose deposition is consistent and implementable.

What carries the argument

The logarithmic Milstein discretization of the energy SDE together with the Lie-group integrator for angular dynamics, which together enforce positivity and geometric conservation while supporting direct differentiation of the dose functional.

If this is right

  • The schemes produce particle trajectories that remain physically admissible over long integration intervals.
  • Strong order-one convergence holds for the energy process under the stated discretization.
  • The sensitivity estimator yields stable gradients of dose with respect to model parameters directly from path realizations.
  • Geometric invariants such as angular momentum constraints are preserved exactly by the combined integrator.

Where Pith is reading between the lines

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

  • The structure-preserving property could allow larger time steps in Monte Carlo proton therapy simulations without introducing artificial energy gain.
  • Pathwise sensitivities open the possibility of gradient-based optimization loops that adjust beam parameters while respecting the stochastic physics.
  • The same Lie-group plus logarithmic-Milstein pattern may extend to other transport problems that combine multiplicative noise with manifold-valued directions.

Load-bearing premise

The underlying stochastic differential equations accurately represent the coupled physical processes of energy loss, range straggling, and angular diffusion in proton transport.

What would settle it

Running the schemes on a simplified test case with known exact solution and observing either negative energy values or measured convergence rates below order one would refute the claimed properties of the integrator.

Figures

Figures reproduced from arXiv: 2509.13223 by Tristan Pryer, Veronika Chronholm.

Figure 1
Figure 1. Figure 1: The three main interactions of a proton with matter. A nonelastic proton– nucleus collision, an inelastic Coulomb interaction with atomic electrons and elastic Coulomb scattering with the nucleus. Inelastic collisions with electrons lead to gradual energy loss, typically described deterministically through the Bragg-Kleeman or Bethe-Bloch equation [Int93], and are responsible for the sharp rise and falloff… view at source ↗
Figure 2
Figure 2. Figure 2: Angular diffusion on S 1 . Comparison of naive Euler, renormalised Euler, first￾order geometric and higher-order RKMK updates. The naive Euler schemes exhibit norm drift and bias in the angular distribution, while the exponential-map integrators (geometric Euler and higher-order RKMK) preserve the geometry. The higher-order scheme addition￾ally reduces statistical bias and improves convergence of long-time… view at source ↗
Figure 3
Figure 3. Figure 3: Example 4.8. Strong convergence of numerical schemes. Energy schemes achieve their expected rates (1/2 for Euler-Maruyama, 1 for Milstein), angular schemes achieve 1/2 (Geometric Euler) and 1 (Geometric RKMK with L´evy areas) and the spatial component achieves order 1 due to its deterministic structure. where φ is a spatial kernel that localises energy deposition near x and S(Et) is the stopping power. Thi… view at source ↗
Figure 4
Figure 4. Figure 4: Example 7.2. Impact of the energy-straggling amplitude κ on the dose map. Increasing κ broadens the Bragg peak and reduces its maximum, consistent with greater variance in energy deposition. In all panels the angular diffusion is held fixed at ϵ0 = 0.005. 7.3. Example (Effect of angular diffusion). We examine the role of the angular diffusion amplitude ϵ0 in Equation 1. Since angular diffusion governs the … view at source ↗
Figure 5
Figure 5. Figure 5: Example 7.3. Impact of constant angular diffusion on the dose map. A small ϵ0 yields a straight, narrow profile, while a large ϵ0 produces a broad fan-like spread with a characteristic dip before the Bragg peak. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x distance [cm] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y distance [cm] 0.00 1.35 2.70 4.05 5.40 6.75 8.10 9.45 10.80 12.15 (a) Constant angular diffusion, ϵ0 = 0.00… view at source ↗
Figure 6
Figure 6. Figure 6: Example 7.4. Comparison of constant and energy-dependent angular diffusion. Larger diffusion coefficients produce more fanned-out dose distributions, particularly near the distal edge. 7.5. Sensitivity computations. We now present a series of numerical experiments illustrating the com￾putation of pathwise sensitivities using the structure-preserving Milstein discretisation developed in the preceding sectio… view at source ↗
Figure 7
Figure 7. Figure 7: Example 7.7. Sensitivity with respect to α. The finite difference estimator underestimates the gradient magnitude and produces a more diffuse profile compared to the pathwise method. 7.8. Example (Parameter sensitivities under the full model). We now compute sensitivities using the full model with nonzero κ. In this case, the pathwise estimator for dose sensitivity is given by Equation 108. The parameters … view at source ↗
Figure 8
Figure 8. Figure 8: Example 7.7. Sensitivity with respect to p. The same qualitative features are captured by both methods, but the finite difference estimator underestimates the gradient magnitude and introduces artificial spreading. clearly for α, but also for p and κ. The bias arises from numerical diffusion in the finite difference method, which smooths sharp peaks in the sensitivity profile. Qualitatively, both estimator… view at source ↗
Figure 9
Figure 9. Figure 9: Example 7.8. Sensitivity with respect to α. In the presence of energy straggling, the sensitivity profile is more spread out than in [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example 7.8. Sensitivity with respect to p. As with α, the finite difference estimator smooths and underestimates the gradient magnitude. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x distance [cm] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y distance [cm] 900 720 540 360 180 0 180 360 540 (a) Pathwise sensitivity. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x distance [cm] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y distance [cm] 96… view at source ↗
Figure 11
Figure 11. Figure 11: Example 7.8. Sensitivity with respect to κ. The pathwise method captures sharp features, while the finite difference method underestimates the gradient magnitude. The profile reflects the widening and lowering of the peak caused by straggling. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x distance [cm] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y distance [cm] 15000 10000 5000 0 5000 10000 15000 20000 25000 (a) ∆α = 0.1… view at source ↗
Figure 12
Figure 12. Figure 12: Example 7.9: Bias–variance tradeoff in finite difference sensitivities. Larger stepsizes yield low variance but large bias. Smaller stepsizes reduce bias but increase vari￾ance. noise. For the central difference estimator, the optimal balance between ∆θ and sample size N yields a slow asymptotic rate of O(N −2/5 ) (see table 7.1 of [Gla04], and also [Gly89; FG89]). This is significantly worse than the O(N… view at source ↗
read the original abstract

We develop numerical schemes and sensitivity methods for stochastic models of proton transport that couple energy loss, range straggling and angular diffusion. For the energy equation we introduce a logarithmic Milstein scheme that guarantees positivity and achieves strong order one convergence. For the angular dynamics we construct a Lie-group integrator. The combined method maintains the natural geometric invariants of the system. We formulate dose deposition as a regularised path-dependent functional, obtaining a pathwise sensitivity estimator that is consistent and implementable. Numerical experiments confirm that the proposed schemes achieve the expected convergence rates and provide stable estimates of dose sensitivities.

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

1 major / 2 minor

Summary. The manuscript develops numerical schemes for stochastic models of proton transport coupling energy loss, range straggling, and angular diffusion. It introduces a logarithmic Milstein scheme for the energy equation that preserves positivity and achieves strong order one convergence, a Lie-group integrator for the angular dynamics that maintains geometric invariants, and a pathwise sensitivity estimator for a regularised path-dependent dose deposition functional. Numerical experiments are reported to confirm the expected convergence rates and to demonstrate stable dose sensitivity estimates.

Significance. If the central claims hold, the work provides structure-preserving integrators and an implementable sensitivity method for SDE models in proton transport. These features could improve the fidelity of dose calculations in radiotherapy simulations by respecting physical invariants such as positivity and angular geometry while supplying consistent pathwise gradients. The emphasis on geometric integration and regularised functionals represents a targeted advance within numerical analysis for stochastic particle dynamics.

major comments (1)
  1. [Section on the pathwise sensitivity estimator] Section on the pathwise sensitivity estimator (formulation of the regularised dose deposition functional): the consistency result is established for the smoothed functional at fixed positive regularization parameter. No error bounds or convergence analysis are supplied for the bias as the regularization parameter tends to zero, which is required to recover sensitivities of the original (non-regularised) dose functional. This gap is load-bearing for the abstract claim that the estimator is consistent and practically useful for dose sensitivities.
minor comments (2)
  1. [Abstract] The abstract states strong order one convergence without specifying the norm (e.g., L^p or almost-sure) or the precise statement of the theorem; a short clarification would aid readers.
  2. [Section on the combined method] The description of the combined scheme could include an explicit statement of which geometric invariants (energy positivity, angular norm, etc.) are preserved at the discrete level, with a brief reference to the relevant theorem or proposition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the pathwise sensitivity estimator. We address the point in detail below.

read point-by-point responses

  1. Referee: Section on the pathwise sensitivity estimator (formulation of the regularised dose deposition functional): the consistency result is established for the smoothed functional at fixed positive regularization parameter. No error bounds or convergence analysis are supplied for the bias as the regularization parameter tends to zero, which is required to recover sensitivities of the original (non-regularised) dose functional. This gap is load-bearing for the abstract claim that the estimator is consistent and practically useful for dose sensitivities.

    Authors: We thank the referee for this observation. The manuscript formulates dose deposition explicitly as a regularised path-dependent functional and establishes consistency of the pathwise sensitivity estimator for this regularised version at any fixed positive regularization parameter; this scope is stated in both the abstract and the relevant section. We do not assert consistency for the non-regularised functional in the absence of a limiting argument. We agree that error bounds on the bias incurred as the regularization parameter tends to zero would be needed to recover sensitivities of the original dose functional and that such bounds are not supplied. Providing a full convergence analysis for the vanishing-regularization limit would require additional approximation-theory estimates that lie outside the primary focus of the present work on structure-preserving integrators and implementable sensitivity estimators for the regularised case. In practice a small but fixed regularization parameter is selected to ensure differentiability while approximating the physical quantity, and the numerical experiments demonstrate stability of the resulting sensitivity estimates. In the revised version we will insert a short clarifying remark on the scope of the consistency result together with a note on the regularization limit as a natural direction for future analysis. revision: partial

Circularity Check

0 steps flagged

No significant circularity; methods derived from standard SDE and geometric integration techniques

full rationale

The paper introduces a logarithmic Milstein scheme for the energy SDE and a Lie-group integrator for angular dynamics, then formulates dose deposition as a regularised path-dependent functional to obtain a pathwise sensitivity estimator. These constructions follow from established stochastic calculus (Milstein expansion) and Lie-group methods for preserving invariants, without any quoted reduction of the claimed convergence or consistency results to fitted parameters, self-definitions, or load-bearing self-citations. The abstract and described content present the schemes and estimator as derived outputs rather than tautological renamings or inputs relabeled as predictions. A score of 2 reflects the normal allowance for minor self-citation that does not carry the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Ledger populated from abstract claims only; paper assumes standard SDE modeling of proton physics and applicability of Milstein/Lie-group techniques without introducing new entities or many free parameters.

axioms (1)
  • domain assumption Stochastic differential equations coupling energy loss, range straggling and angular diffusion correctly describe proton transport.
    Invoked when developing the schemes and dose functional.

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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