Adaptive Constraint Cascade (ACC)_Case Study ACC_1.1: Three Illustrative Examples

We extend the Adaptive Constraint Cascade (ACC) framework, introduced in ACC\_1 for the double-well potential, to three additional physical systems: (1) a qubit in a thermal bath, (2) a quantum harmonic oscillator with damping, and (3) a particle in a generic single-well potential. In each case, the ACC\_1 framework equation – a Lindblad-type master equation with a $\lambda$-dependent dissipation rate – is applied, where $\lambda = k_B T / \Delta E$ (or an analogous parameter) controls the quantum–classical transition. We show how the same mathematical structure seamlessly interpolates between pure quantum dynamics (Schr"odinger equation) and classical thermal behavior (master equation), demonstrating the universality of the ACC approach. This work serves as an extension of ACC\_1, providing concrete, solvable examples for readers interested in applying ACC to their own systems.