Physics-Based Noise Formation Model

Updated 7 March 2026

Physics-Based Noise Formation Model is a framework that defines noise using microscopic and mesoscopic physics to improve parameter interpretability and calibration.
The approach utilizes stochastic differential equations, Fokker–Planck formulations, and spectral models to capture non-Gaussian and 1/f noise phenomena.
Applications include synthetic noise generation for imaging, system calibration, and cross-domain transfer, enhancing both theoretical insights and practical data processing.

A physics-based noise formation model is a theoretical and computational framework that expresses the statistical properties of noise in physical systems directly in terms of underlying microscopic or mesoscopic physics, rather than purely phenomenological or empirical fits. These models aim to faithfully reproduce the observed noise characteristics—such as power spectral densities, correlations, and non-Gaussian statistics—by incorporating first-principles descriptions of the relevant physical processes, interactions, and constraints at play. Across domains including condensed matter, electronic circuits, imaging sensors, and nonequilibrium systems, such models provide mechanistic insight, parameter interpretability, and transferability across processes and devices.

1. Principles of Physics-Based Noise Formation

Physics-based noise models begin with a precise identification of the primary physical mechanisms through which noise is generated and propagated in the system under study. Fundamental ingredients generally include:

Discrete or continuous stochastic forcing: Random fluctuations arising from quantized events, such as photon arrivals in detectors (shot noise), or from coarse-grained stochastic dynamics, e.g., additive α-stable processes in nonequilibrium baths (MacKay, 2024).
Dynamical evolution via stochastic differential equations (SDEs) or master equations: Noise is injected and transformed through the physical system, with its propagation structured by drift, diffusion, and feedback terms, as in Langevin, Fokker–Planck, or reaction–diffusion descriptions (Grant et al., 2017, Kaulakys et al., 2015, Hori et al., 2013).
Explicit coupling to microscopic or mesoscopic physical parameters: Model parameters (gain, quantum efficiency, trap densities, etc.) are tied to measurable device or material constants rather than abstract fitting coefficients (Liu et al., 30 Jan 2026, Wei et al., 2020).

This methodology supports calibration, interpretability, and synthesis of realistic noise for both theoretical analysis and data-driven applications.

2. Canonical Examples across Domains

Physics-based noise models appear in a range of contexts:

Domain	Noise Mechanisms Captured	Reference
CMOS/CCD Imaging	Photon shot, dark current, readout, quantization	(Wei et al., 2020, Wei et al., 2021, Liu et al., 30 Jan 2026)
Low-Light Denoising	Heteroscedastic Poisson-Gaussian, heavy-tailed read	(Feng et al., 2023, Zhang et al., 2023)
Nanostructure Formation	Deposition (shot) noise, phase fluctuation, surface	(Dhankhar et al., 2021)
TRNGs and Oscillators	Fractional Brownian motion, 1/f^α spectral densities	(Skorski, 2024)
Reaction-Diffusion Systems	Intrinsic chemical noise, diffusive transfer	(Hori et al., 2013)
Condensed Matter (Barkhausen)	Avalanche instability, domain artifacts, Hessian zeros	(Hentschel et al., 2014)

Central to these approaches is the linkage of noise sources to specific physical origins—phonon, photon, electronic, configurational, or environmental—and the precise manner in which these random processes are transformed by deterministic system dynamics.

3. Mathematical Formulation and Calibration

Physics-based models are marked by transparent derivations of noise statistics. Prototypical formulations include:

Stochastic SDEs and Fokker–Planck

Overdamped Langevin with α-stable noise:

$\dot{x}(t) = -F\,\text{sgn}(x(t)) + \sigma\,\xi_\alpha(t)$

where $\xi_\alpha$ is α-stable (Lévy), encompassing Gaussian as $\alpha\to2$ , and broader-tailed distributions for $\alpha<2$ (MacKay, 2024).

Fractional Fokker–Planck for α-stable baths:

$\frac{\partial f}{\partial t} = F\,\frac{\partial}{\partial x}[\text{sgn}(x)\,f] + \sigma^\alpha\,\frac{\partial^\alpha}{\partial|x|^\alpha}f$

capturing nonlocal stochastic transport and stationary PDFs normalizable for $1<\alpha\le2$ .

Spectral Models and Fluctuation Statistics

$1/f$ noise via broken-symmetry field:

$S(f)\propto\int\frac{dq_x}{q_x^2+f^2}\propto\frac{1}{|f|}$

derived from Goldstone-like fields in systems with continuous symmetry breaking (Grant et al., 2017).

CCD/CMOS imaging pipeline noise sum:

$D = K \mathcal{P}(I) + \text{Tukey}\,\Lambda(\lambda;\mu_c,\sigma_{TL}) + \mathcal{N}(0,\sigma_r^2) + \mathrm{Uniform}(-q/2, q/2)$

with each component linked to a physical subsystem (photon detection, electronic read, banding, digitization) (Wei et al., 2021, Wei et al., 2020).

Calibration

Model parameters are extracted from flat-field, bias, dark, or reference frames, with procedures specified to ensure reproducibility. Examples include linear fitting of variance-to-mean relationships for gain, quantile fitting for read noise, or Allan variance estimation for 1/f phase noise in TRNGs (Wei et al., 2020, Skorski, 2024).

4. Non-Gaussian and Multiscale Noise Phenomena

Physics-based models frequently capture non-Gaussianity via mathematically rigorous probability structures:

Heavy-tailed and α-stable statistics: Noise processes interpolate between Gaussian (α=2) and Lévy “L block” regimes, with power-law tails dominating rare, large fluctuations. Exact analytic stationary PDFs in potentials can be written in terms of Mittag–Leffler functions for these distributions (MacKay, 2024).
Multi-modal, mixed sources: In imaging, banding, row, and column patterning yield spatial correlations; in Barkhausen noise, magnetization jumps correspond to localized instabilities with explicit non-power-law distributions (Hentschel et al., 2014).
Superposed time scales: Entropy-bath models for 1/f noise introduce a structured hierarchy of time constants that, when superimposed, yield power-law noise spectra across orders of magnitude in frequency (Chamberlin et al., 2016).

5. Applications: Synthesis, Denoising, and Physical Inference

Physics-based noise models are essential for:

Realistic data synthesis and algorithm training: Synthetic noisy–clean pairs generated via calibrated physics models allow robust training of denoising networks without the need for laborious raw ground-truth data acquisition. These models have proven effective in imaging, low-light photography, and astronomical data processing, matching or surpassing performance of empirically trained models (Wei et al., 2020, Feng et al., 2023, Liu et al., 30 Jan 2026).
Interpretability and uncertainty quantification: Parameterization in terms of physical quantities allows experimentalists and devices engineers to track noise source contributions and propagate calibration uncertainty directly into performance bounds.
Generalization and cross-domain transfer: Unlike ad hoc or purely statistical models, physics-based frameworks maintain predictive accuracy across varying sensor designs, environmental conditions, and material parameters, supporting broad applicability in scientific workflows (Bartlett et al., 2021).

6. Extensions, Theoretical Connections, and Limitations

Research continues to generalize physics-based noise models along several axes:

Nonlinear transformations and 1/f^β universality: Nonlinear SDEs and variable mappings reveal how standard stochastic processes (Brownian motion, Bessel) can be reinterpreted through nonlinear observables to generate a broad spectrum of $1/f^\beta$ noises (Kaulakys et al., 2015).
Self-organized criticality and power-law fluctuations: Non-equilibrium noise models in V-shaped wells with α-stable forcing are intended for application in systems exhibiting avalanche statistics and self-organized criticality, where fat-tailed PDFs and 1/f noise are emergent (MacKay, 2024).
Noise-induced patterns and functional structures: In stochastic reaction–diffusion systems, intrinsic (internal) noise is essential to the emergence of spatial patterns even for linearly stable deterministic systems; control-theoretic analysis quantifies the amplification and selection of dominant spatial frequencies (Hori et al., 2013).
Parameter regimes and breakdowns: Each class of physics-based model has explicit domains of validity determined by linearity, symmetry, cutoff frequencies, and feature scales. Deviations, including strong nonlinearity, saturation, or finite system size, require model refinement or distinct physical extensions (Grant et al., 2017).

7. Table: Representative Classes of Physics-Based Noise Models

Model/System Type	Key Physical Noise Processes	Spectral/Formal Features	Example References
Imaging Sensors	Photon shot, readout, quantization	Poisson, heavy-tailed, row/column banded	(Wei et al., 2021, Wei et al., 2020, Liu et al., 30 Jan 2026)
Overdamped Langevin/V-well	α-stable Lévy flights, return force	Fractional Fokker–Planck, Mittag–Leffler solution	(MacKay, 2024)
Electronics/Transistors	Trapping/detrapping, mobility fluct.	$1/f$ SDE, bias-dependent, Verilog-A formulations	(Mavredakis et al., 2023)
Reaction–Diffusion	Stochastic birth/death, diffusion	Modal Lyapunov power spectrum, noise-induced pattern	(Hori et al., 2013)
True Random Number Generators	Fractional Brownian, 1/f phase noise	Riemann–Liouville fBM, Allan variance, min-entropy	(Skorski, 2024)

These models have established a foundation for both physical understanding and practical noise handling in a range of cutting-edge scientific and engineering applications.