arXiv subjects

Joel Saucedo

Publications and source records attributed to Joel Saucedo.

5 recordsLinked to original sources

arXiv2026

Fractional Stochastic Navier-Stokes Equations: Local Well-Posedness and Enstrophy Balance

In this paper, we investigate a stochastic incompressible fluid on the three-dimensional torus in which power-law temporal memory, represented by a Caputo derivative of order $β\in(1/2,1)$, is coupled to non-local dissipation, represented by a fractional Laplacian of order $α\in(1,2)$, under spatially smooth trace-class noise with a superlinear coefficient. We first derive the equation from constrained Hamiltonian mechanics through a Mori-Zwanzig elimination of fast degrees of freedom, so that the memory kernel and the random force emerge together rather than being postulated; a Tauberian limit identifies the Caputo operator as the exact analytic image of a scale-free harmonic heat bath. Representing vorticity as a differential two-form, we prove that the vortex-stretching functional coincides with the sign-indefinite quadratic form of the strain tensor, and that the velocity is recovered from the vorticity with an exact one-derivative gain. Using a subordination identity for the Mittag-Leffler propagator, we show that the linear kernel is a probability density with an algebraic far-field tail. Within a Gelfand triple adapted to the Stokes operator we establish local existence and uniqueness in $H^s$ for $s>3/2$, with the data and the forcing transported by two distinct propagators, and we prove that the memory threshold $β>1/2$ is sharp for the existence of function-valued solutions under white-in-time forcing. We then derive an enstrophy balance, governed by a fractional differential inequality, read off the critical exponent $σ=β(1+3/α)$ separating the dissipation-dominated regime from possible stretching-driven growth, and close with a Beale-Kato-Majda continuation criterion together with a geometric condition for global existence.

math.PR↗

arXiv2025

Quadratic Volatility from the Pöschl-Teller Potential and Hyperbolic Geometry

This investigation establishes a formal equivalence between the generalized Black-Scholes equation under a Quadratic Normal Volatility (QNV) specification and the stationary Schrödinger equation for a hyperbolic Pöschl-Teller potential. A sequence of canonical transformations maps the financial pricing operator to a quantum Hamiltonian, revealing the volatility smile as a direct manifestation of diffusion on a hyperbolic manifold whose geometry is classified by the discriminant of the QNV polynomial. We perform a complete spectral analysis of the financial Hamiltonian, deriving its discrete and continuous spectra and constructing the pricing kernel from the resulting eigenfunctions, which are given by classical special functions. This analytical framework, grounded in a gauge-theoretic perspective, furnishes a non-trivial benchmark for derivative pricing and provides a fundamental geometric interpretation of market anomalies. Future research trajectories toward integrable systems and formal field-theoretic analogies are identified.

q-fin.PR↗

arXiv2025

Scaling Relations, Morphological Stability, and Asymptotic Freedom of Plasma-Surface Deposition Dynamics

Connecting plasma processing parameters to the resultant film microstructure remains a fundamental challenge in materials synthesis, one that has largely confined process design to empirical approaches. To bridge this gap, we develop a predictive analysis of coupling by applying a renormalization group (RG) analysis to an effective Hamiltonian for the stochastic dynamics of the plasma-surface interface, derived systematically from microscopic principles. The central result from this formalism is the system's exhibition of asymptotic freedom; the effective dimensionless coupling, $g$, between the plasma and the growing surface is found to weaken systematically at macroscopic length scales, a finding that provides a rigorous justification for the success of continuum-level models in describing large-scale film evolution. The RG framework yields a non-perturbative scaling relation for the mean grain area, $\langle A \rangle \propto \exp(κ/g)$, where $g$ itself is defined by fundamental parameters such as ion flux ($Φ$) and ion collision time ($τ_{\text{ion}}$). This relation reveals the origin of widely-observed empirical power-law scaling, showing it to be an effective behavior limited to specific process regimes. Crucially, the model furnishes sharp, testable predictions, including the pressure-independence of grain size within collision-dominated plasmas and a parameter-free criterion, $Λ_c = 1/(n^2-1)$, for the onset of morphological instability and faceting based on crystal symmetry. This work establishes a quantitative, parameter-sparse engine for predicting and ultimately controlling microstructural outcomes in thin film synthesis.

physics.plasm-ph↗

arXiv2025

Exact Solutions for Bimodal Distributions under Stochastic Plasma Irradiation in Thin Films

A persistent paradox complicates the study of plasma-irradiated thin films, where bimodal grain distributions and ambiguous scaling laws, roughly shifting between $Φ^{-1/2}$ and $Φ^{-1}$, or a general inverse dependence on plasma flux, are empirical yet remain theoretically unreconciled. Existing models fail to unify noise-driven evolution, defect saturation kinetics, and nucleation-loss balance within a single, self-consistent formalism. This work resolves these discrepancies by developing the first exact analytical theory for this system. We derive the closed-form steady-state grain area distribution, $P_{ss}(A)$, establish the precise dimensionless threshold for bimodality onset at $Π_c = 4/(3\sqrt{3})$, and demonstrate that defect saturation physics mandate a universal $\langle A \rangle \propto κ^2 Φ^{-1} e^{+2E_b/k_B T_s}$ scaling law. The framework reveals how competition between stochastic impingement and deterministic growth triggers microstructure fragmentation, resolving long-standing ambiguities in irradiation-induced surface evolution and providing a predictive foundation for materials processing.

cond-mat.mtrl-sci↗

arXiv2025

Non-Perturbative Solutions to the Vlasov-Boltzmann Equation for Weakly Ionized Plasmas

The electron energy distribution function (EEDF) in low-temperature plasmas exhibits features not fully captured by classical collisional models, particularly across the transition from kinetic to hydrodynamic regimes. This work attributes these phenomena to a dynamically broken scale invariance within the Vlasov-Boltzmann equation. By applying renormalization group (RG) techniques directly to the kinetic operator, we derive non-perturbative EEDF solutions valid across a range of collisionality. The formalism yields analytic scaling relations for electron heating and predicts the emergence of distinct EEDF forms - bimodal in the kinetic limit and generalized exponential in the hydrodynamic limit - separated by a critical pressure. It is shown that the stable RG fixed point governing the system's long-time behavior corresponds to a state of minimum entropy production, establishing a thermodynamic basis for plasma self-organization. The theory provides a quantitative explanation for the experimentally observed universal data collapse of rescaled EEDFs and resolves standing issues like Godyak's EEDF metamorphosis.

physics.plasm-ph↗