A Deterministic Limit for Sonoluminescence: Deriving the Critical Ignition Radius via Thermal Diffusion in Acoustic Cavitation

The prediction of peak core temperatures and photon emission during Single-Bubble Sonoluminescence (SBSL) relies heavily on adiabatic gas laws coupled with the Rayleigh-Plesset hydrodynamic equations. While these thermodynamic frameworks effectively estimate the bulk probability of extreme localized heating during inertial void collapse, they fail to deterministically define the exact spatial boundary where the collapsing core transitions from a compressed gas into a light-emitting plasma. This paper introduces a strict continuum framework for acoustic thermodynamics. By modeling the collapsing void as a dynamic kinematic balance between the spatial capacity for heat dissipation (thermal diffusion) and the localized rate of kinetic compression (adiabatic convergence), we derive a universal critical ignition radius (Rignition). We demonstrate that sonoluminescence is not a statistical thermodynamic anomaly, but an exact deterministic limit where localized adiabatic heating strictly overpowers the advective thermal-diffusing capacity of the surrounding fluid. We propose a blueprint for Acoustic Inertial Confinement Fusion (AICF), providing the exact geometric scaling required to reliably cross the Lawson criterion.