On-the-Fly Emission–Absorption Approximation

Updated 19 November 2025

On-the-fly emission–absorption approximation is a set of computational methods for simulating time-dependent energy transport in complex, non-stationary media.
It employs local discretization and trajectory-marching to compute emission and absorption events, achieving high angular and spectral resolution in turbulent environments.
The approach supports end-to-end differentiability and gradient-based inversion, facilitating integration with hydrodynamic solvers and ab initio spectroscopy models.

The on-the-fly emission–absorption approximation encompasses a set of methodologies used across computational astrophysics, atmospheric radiative transfer, and molecular spectroscopy for efficiently simulating time-dependent energy transport under emission and absorption, often in high-dimensional, non-stationary, or turbulent systems. It rests on the principle of updating radiative or spectroscopic quantities by direct local evaluation (“on the fly”) along characteristic paths or dynamical trajectories, circumventing the need for global solution of integro-differential equations at every time step. While the detailed implementations are field-specific, core features include local discretization or trajectory-marching, batch computation of emission and absorption events, and, where relevant, coupling to real-time evolving media. The approach is especially prevalent where high angular or frequency resolution is needed and memory or computational cost is limiting.

1. Mathematical Foundations and Generic Formulation

The on-the-fly emission–absorption approximation is fundamentally grounded in the time-dependent radiative transfer equation

$\frac{1}{c}\frac{\partial I(\mathbf{x}, \hat{\mathbf{n}}, t)}{\partial t} + \hat{\mathbf{n}} \cdot \nabla_{\mathbf{x}} I(\mathbf{x}, \hat{\mathbf{n}}, t) = -\kappa(\mathbf{x}) I(\mathbf{x}, \hat{\mathbf{n}}, t) + j(\mathbf{x}),$

where $I$ is the specific intensity, $\kappa$ is the absorption coefficient, $j$ is the emissivity, and $c$ is the signal speed (Branca et al., 12 Nov 2025). This form prescribes the balance of emission and absorption without explicit scattering redistribution; isotropic scattering is usually folded into an effective emissivity term. The equation is representative for photon, phonon, or neutron transport in various media.

In molecular spectroscopy, analogous treatment appears in the propagation of spectroscopic correlation functions or wavepackets, with semiclassical or quantum–semiclassical propagation including both emission- and absorption-type events “on the fly” as a trajectory or wavepacket traverses the relevant potential energy surfaces (Begušić et al., 2019, Begušić et al., 2020).

2. Numerical Implementations and Algorithmic Structure

Numerical realization of the on-the-fly emission–absorption approximation relies on tracing rays, particle characteristics, or semiclassical trajectories, discretizing emission and absorption locally along these paths.

Astrophysical radiative transfer, as in Ray-trax (Branca et al., 12 Nov 2025), proceeds by:

Sampling $N_{\Omega}$ discrete ray directions per source (Fibonacci sphere sampling).
Marching each ray via substeps of length $\Delta s$ up to a total macroscopic timestep $\Delta t$ ; at each substep $k$ ,

$\begin{aligned} \mathbf{x}_{k+1} &= \mathbf{x}_k + \hat{\mathbf{n}} \Delta s \ \tau_{k+1} &= \tau_k + \kappa(\mathbf{x}_k) \Delta s \ I_{k+1} &= I_k \exp(-\kappa(\mathbf{x}_k) \Delta s) + j(\mathbf{x}_k) \exp(-\tau_k) \Delta s \end{aligned}$

Trilinear interpolation for $j$ and nearest-neighbor lookup for $\kappa$ ensure smoothness and differentiability.
The intensity is deposited back onto the spatial grid after each march, maintaining a consistent mapping between discrete rays and voxelized fields.
Extension to multiple frequency bins treats the frequency axis as a batch dimension, supporting spectral resolution without kernel changes.

In nonlinear spectroscopy, on-the-fly methods propagate ab initio trajectories or wavepackets and evaluate emission, absorption, and stimulated emission terms at each time step. Integral and dispersed pump–probe signals are constructed via accumulation and averaging over quasi-classical trajectories, with quantities such as transition dipoles and energy gaps computed on demand along each path (Vasquez et al., 4 Sep 2025, Xu et al., 2021).

3. Computational Scaling, Efficiency, and Comparison to Other Methods

Efficiency is a cornerstone of the on-the-fly approach:

In Ray-trax (Branca et al., 12 Nov 2025), per-timestep computational work scales as $\mathcal{O}(N_\text{src} N_\Omega N_s)$ , where $N_\text{src}$ is the number of sources, $N_\Omega$ is directions per source, and $N_s \approx c \Delta t / \Delta s$ the number of steps per ray. Memory footprint is $\mathcal{O}(N_\text{src} N_\text{cells})$ , with $N_\text{cells}$ being the number of spatial voxels.
Moment methods (such as M1) reduce memory to $\mathcal{O}(N_\text{cells})$ but lose angular fidelity, while Monte Carlo methods scale with photon packets and can require very large ensembles for low noise.
CPU-based ray tracers are not fully vectorized and incur high per-ray cost; Ray-trax leverages full GPU vectorization for efficient amortization across rays, sources, and frequency bins.

In on-the-fly molecular spectroscopy, scaling is dictated by the number of classical trajectories, the cost of ab initio calculations along each path, and the number of field-matter events (doorway–window pairs) that must be sampled (Vasquez et al., 4 Sep 2025). For practical convergence (≲10% error in pump–probe features), typically 500–2000 trajectories suffice, and post-processing overhead is minimal compared to electronic structure evaluations.

4. Accuracy, Validation, and Benchmarks

The approximation has been robustly validated in its principal domains:

In Ray-trax (Branca et al., 12 Nov 2025), the method recovers analytic angle-integrated solutions for constant $\kappa$ (reference intensity $J_\text{ref}(\mathbf{x}) = \sum_i \frac{L_i}{4\pi r_i^2} \exp(-\kappa r_i)$ , with $r_i = \|\mathbf{x} - \mathbf{x}_i\|$ ), reproducing $1/r^2$ scaling and exponential attenuation. Average relative error is ~3%, with monotonic convergence as $\Delta s \to 0$ and $N_\Omega$ increases.
In turbulent astrophysical media, setting $\kappa(\mathbf{x}) \propto \rho(\mathbf{x})$ (where $\rho$ is a turbulent density field) gives rise to realistic shadowing, channeling, and illuminated bubbles, with the light travel front propagating through complex structures, suitable for direct hydrodynamic coupling (Branca et al., 12 Nov 2025).
In nonlinear molecular spectroscopy, on-the-fly doorway–window simulations recapitulate benchmark results for canonical systems such as pyrazine, delivering quantitatively accurate transient absorption and time-resolved fluorescence signals in line with fully quantum mechanical models (Vasquez et al., 4 Sep 2025, Xu et al., 2021).
Hybrid on-the-fly PRD treatments for moving atmospheres (e.g., for Mg II k and Ly $\alpha$ lines) match full angle-dependent redistribution results within 1–2% but are 30–130× faster than the brute-force method (Leenaarts et al., 2012).

5. End-to-End Differentiability, Inverse Problems, and Coupling to Dynamics

A major methodological advance is the preservation of end-to-end differentiability:

Ray-trax is implemented within JAX, using trilinear interpolation and deposition primitives, yielding well-defined gradients $\partial I/\partial j$ and $\partial I/\partial \kappa$ everywhere (Branca et al., 12 Nov 2025).
This property enables gradient-based inversion, e.g., optimizing a scaling parameter $A$ in a template $e_0(\mathbf{x})$ by analytic backpropagation through the entire ray-marching sequence, with matching to closed-form minimizers.
When coupled to differentiable hydrodynamics solvers, the joint system supports gradient-based calibration of subgrid models, physics-constrained learning, and inference of parameters from radiative observables (enabling, e.g., physics-informed machine learning and optimal control in coupled PDE settings).

In molecular spectroscopy, semiclassical trajectory-based approaches and thawed Gaussian wavepacket propagation schemes similarly support backpropagation of gradients with respect to initial conditions, molecular parameters, or functional forms of transition dipoles, facilitating rigorous structure–spectrum and dynamics–signal linking (Begušić et al., 2019, Begušić et al., 2020).

6. Limitations, Extensions, and Domain of Applicability

While highly effective in its design domain, the on-the-fly emission–absorption approximation has intrinsic limitations:

In radiative transfer, the method assumes the emission–absorption form (no explicit angular redistribution beyond local re-emission), limiting accuracy in cases where scattering is non-isotropic or polarization/magneto-optical effects are present (Branca et al., 12 Nov 2025, Leenaarts et al., 2012).
Memory scaling is set by the number of source–grid pairs; although amortized across modern accelerators, this may become prohibitive for full-sky, high-resolution simulations with many sources.
In nonadiabatic molecular spectroscopy, classical treatment of nuclear motion neglects zero-point energy and tunneling, and pulse-separation approximations (doorway–window factorization) omit finite-pulse overlap effects (Xu et al., 2021).
Extensions have been proposed, for example, to handle moving atmospheres with Doppler shifts by frame transformation, inclusion of finite temperature via thermo-field dynamics, or augmentation with linear and nonlinear vibration–transition dipole coupling (Leenaarts et al., 2012, Begušić et al., 2020, Begušić et al., 2019).

Possible improvements involve explicit handling of cross-redistribution among lines, angular or polarization dependence in collisional redistribution fractions, and expansion to 3D short- and long-characteristic solvers.

7. Applications Across Physics, Astrophysics, and Spectroscopy

The on-the-fly emission–absorption approximation is central to:

GPU-accelerated radiative transfer embedded in turbulent hydrodynamic simulations, including time-dependent light propagation, shadowing, and variable source embedding (Branca et al., 12 Nov 2025).
Real-time formation of astrophysical and atmospheric spectral lines, including rapid simulation of PRD effects in dynamic, moving atmospheres (Leenaarts et al., 2012).
On-the-fly ab initio evaluation of vibronic absorption and emission spectra, incorporating Herzberg–Teller and non-Condon effects, anharmonicity, finite temperature, and mode–mode coupling (Begušić et al., 2019, Begušić et al., 2020).
Simulation of time-resolved nonlinear electronic spectroscopy (pump–probe, 2D, TRF) via the quasi-classical doorway–window factorization using trajectory surface hopping and ab initio on-the-fly evaluation of all field-matter couplings (Vasquez et al., 4 Sep 2025, Xu et al., 2021).

In each setting, the critical advantage is the combination of angular, temporal, and spectral resolution with computational scalability, enabling direct, coupled treatment of source, medium, and observation regimes.