A Gauge-Inspired Coupled Fokker--Planck System for Multistate Stochastic Dynamics: Formalism and 1-D Numerical Implementation
Abstract
Extending the Fokker--Planck equation to a coupled system of equations through non-Abelian symmetry groups offers a powerful theoretical framework for coupled multi-state stochastic systems. However, translating this abstract gauge field formulation into viable numerical predictions remains an unaddressed challenge. Motivated by our previous work on ice crystal habit evolution, in this work we bridge this gap by presenting the first numerical solver for an SU(2) gauge-coupled Fokker--Planck system in (1+1) dimensions. Using an operator-splitting scheme combined with a non-oscillatory advection solver, our algorithm strictly ensures probability conservation, non-negativity, and numerical stability. We demonstrate that non-Abelian gauge coupling acts as a cross-reactive feedback loop that fundamentally alters system dynamics, driving continuous probability exchange between states, suppressing localised density peaks, and forcing long-time spatial delocalisation. Validated across linear and quadratic coupling regimes up to strong interactions (g <= 1.0), this work establishes the foundational computational infrastructure needed to integrate complex multi-state physics. Overall, while our approach provides a key step toward improving ice cloud representation in atmospheric and climate models, the underlying physical framework and computational engine remain fundamentally general and multidisciplinary, offering a versatile paradigm for any domain requiring coupled multi-state stochastic equations.
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BibTeX
@misc{cosmo2026gaugeinspired,
title = {{A Gauge-Inspired Coupled Fokker--Planck System for Multistate Stochastic Dynamics: Formalism and 1-D Numerical Implementation}},
author = {Francesco Pio De Cosmo and Gianluca Di Natale and Leandro Cieri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14052},
eprint = {2610.14052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14052},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Gauge-Inspired Coupled Fokker--Planck System for Multistate Stochastic Dynamics: Formalism and 1-D Numerical Implementation
AU - Francesco Pio De Cosmo
AU - Gianluca Di Natale
AU - Leandro Cieri
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.14052
UR - https://arxiv.org/abs/2610.14052
AB - Extending the Fokker--Planck equation to a coupled system of equations through non-Abelian symmetry groups offers a powerful theoretical framework for coupled multi-state stochastic systems. However, translating this abstract gauge field formulation into viable numerical predictions remains an unaddressed challenge. Motivated by our previous work on ice crystal habit evolution, in this work we bridge this gap by presenting the first numerical solver for an SU(2) gauge-coupled Fokker--Planck system in (1+1) dimensions. Using an operator-splitting scheme combined with a non-oscillatory advection solver, our algorithm strictly ensures probability conservation, non-negativity, and numerical stability. We demonstrate that non-Abelian gauge coupling acts as a cross-reactive feedback loop that fundamentally alters system dynamics, driving continuous probability exchange between states, suppressing localised density peaks, and forcing long-time spatial delocalisation. Validated across linear and quadratic coupling regimes up to strong interactions (g <= 1.0), this work establishes the foundational computational infrastructure needed to integrate complex multi-state physics. Overall, while our approach provides a key step toward improving ice cloud representation in atmospheric and climate models, the underlying physical framework and computational engine remain fundamentally general and multidisciplinary, offering a versatile paradigm for any domain requiring coupled multi-state stochastic equations.
ER -