2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163400We study the Langevin dynamics for the family of spherical spin glass models of statistical physics, in the presence of a magnetic field. We prove that in the limit of system size N approaching infinity, the empirical state correlation, the response function, the overlap and the magnetization for these N-dimensional coupled diffusions converge to the non-random unique strong solution of four explicit non-linear integro-differential equations, that generalize the system proposed by Cugliandolo and Kurchan in the presence of a magnetic field. We then analyze the system and provide a rigorous derivation of the FDT regime in a large area of the temperature-magnetization plane.Mathematical PhysicsDynamical Systems82C44; 82C31; 60H10; 60F15; 60K35text