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

Abstract

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.

Figures (5)

• Figure 1: Prometheus $\beta$-VAE architecture for dense beam data. The 256-dimensional structure factor $S(q)$ is encoded to a 2-dimensional Gaussian latent code $(\boldsymbol{\mu},\boldsymbol{\sigma}^2)$, then decoded by mirrored transposed convolutions. The $\beta$-weighted KL penalty encourages the latent code to capture the order parameter.
• Figure 2: Data generation pipeline. For each of three momentum distributions (Fermi, Gaussian, uniform), 1000 configurations are generated at 20 densities by PIC simulation, equilibrated for five plasma periods, and stored as 256-dimensional $S(q)$ vectors.
• Figure 3: Latent order parameter $\Phi(n)$ versus density for three momentum distributions. The Gaussian and uniform distributions exhibit a monotonic decrease of approximately 0.7 units across the density range, indicating a phase transition. The Fermi distribution remains flat ($\Delta\Phi < 0.04$), consistent with the theoretical prediction $n_c \to 0$ for the degenerate Fermi gas (Theorem \ref{['thm:existence']}).
• Figure 4: Mean KL divergence versus density for three momentum distributions. The Gaussian and uniform distributions show a monotonic decrease in KL divergence, reflecting the encoder's increasing certainty as the system moves deeper into the single-particle phase. The Fermi distribution maintains a constant KL divergence, consistent with the absence of a phase transition.
• Figure 5: Peak order parameter magnitude versus $\beta$ for the uniform distribution. The optimal value $\beta^* = 0.1$ produces the strongest phase transition signal with peak magnitude 9.75. Higher $\beta$ values over-regularise the latent space, suppressing the order parameter signal.

Theorems & Definitions (6)

• Theorem 1: No new counterterms
• Theorem 2: Existence of collective modes
• proof
• Proposition 3: Dispersion relation
• Theorem 4: Dispersion constraints from Ward identities
• Theorem 5: Selection rules