Collective Mpemba-Type Relaxation in Degenerate Bosonic Modes Coupled to a Common Thermal Reservoir
We investigate collective Mpemba-type relaxation in a degenerate family of bosonic modes coupled to a common thermal reservoir. Starting from a fully symmetric M-mode description and employing a representative mean-field reduction, we derive effective master equations for weak-coupling Markovian, strong-coupling Caldeira-Leggett, and weak-coupling non-Markovian regimes. In the weak-coupling Markovian limit, relaxation separates into an incoherent thermal channel decaying at rate gamma and a collective coherent channel decaying at rate M gamma, yielding an explicit Mpemba crossing time determined by the initial-state preparation. In the strong-coupling Caldeira-Leggett regime, transient quadrature dynamics enriches the relaxation pattern, delaying crossings in the overdamped sector and generating multiple crossings in the underdamped sector. In the non-Markovian regime, reservoir memory reshapes the same channel-competition mechanism through time-dependent decay rates and a Lamb shift, producing delayed and clustered crossing windows. Numerical results based on the energy and the Kullback-Leibler divergence reveal Mpemba and anti-Mpemba behavior, criterion dependence, and multiple transient reorderings induced by collective coherence, quadrature dynamics, and reservoir memory.