# Beam-Plasma Collective Oscillations in Intense Charged-Particle Beams: Dielectric Response Theory, Langmuir Wave Dispersion, and Unsupervised Detection via Prometheus

arXiv: https://arxiv.org/abs/2603.10457 · HTML: https://mlsanity.com/paper/2603.10457/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 58 references that are in the map.
- First posted on arXiv 11 Mar 2026.

## Builds on (most cited first, 8 of 8)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Crystal Diffusion Variational Autoencoder for Periodic Material Generation](https://mlsanity.com/paper/2110.06197/) (2021)
- [An exact mapping between the Variational Renormalization Group and Deep Learning](https://mlsanity.com/paper/1410.3831/) (2014)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders](https://mlsanity.com/paper/1703.02435/) (2017)
- [PCA meets RG](https://mlsanity.com/paper/1610.09733/) (2016)

## Abstract (arXiv)

We develop a theoretical and computational framework for beam-plasma collective oscillations in intense charged-particle beams at intermediate energies (10-100 MeV). In Part I, we formulate a kinetic field theory governed by the Vlasov-Poisson system, deriving the Lindhard dielectric function and random phase approximation (RPA) polarization tensor for three beam distribution functions. We prove via the dielectric function epsilon(omega,q)=0 the existence of undamped Langmuir wave modes above a critical beam density n_c, obtain explicit beam-plasma dispersion relations, and show that Landau damping vanishes above the particle-hole continuum. The plasma frequency Omega_p^2 = ne^2/(m*epsilon_0) is fixed by the f-sum rule independently of distribution shape; higher dispersion coefficients depend on velocity moments. Space charge effects drive anomalous beam broadening with sqrt(n-n_c) onset and Friedel oscillations at q=2k_F. The beam-plasma transition belongs to the 3D Ising universality class via renormalization group analysis. In Part II, we validate these predictions using Prometheus, a beta-VAE trained on static structure factor data S(q) from particle-in-cell (PIC) beam simulations. Prometheus detects collective plasma oscillation onset in Gaussian and uniform distributions, confirms their absence in the degenerate Fermi gas (n_c -> 0), and resolves the Kohn anomaly at q=2k_F. Dispersion analysis of S(q,omega) from PIC simulations verifies the distribution-independent Omega_p predicted by the f-sum rule. All six validation checks pass. Predicted signatures -- density-tunable plasma resonances at omega_p proportional to sqrt(n), anomalous beam broadening with sqrt(n-n_c) onset, and Friedel oscillations -- are accessible at existing intermediate-energy beam facilities.