Adaptive colored-noise thermostats and piston barostats with exact <i>NPT</i> sampling

We introduce two algorithmic components for molecular dynamics that combine rigorous ensemble preservation with practical adaptivity: (i) an adaptive generalized Langevin (AGLE) thermostat based on a low-rank Ornstein-Uhlenbeck embedding and (ii) a generalized Langevin piston barostat stabilized by sparse metropolized isothermal-isobaric volume moves. Both methods preserve the target equilibrium distribution by construction in linear settings, reduce correlation times without manual parameter tuning, and remain robust at finite timesteps. The AGLE selectively damps under-thermostatted spectral bands while maintaining exact fluctuation-dissipation balance. The barostat suppresses resonances of traditional Parrinello-Rahman schemes, adapts automatically to a target log-volume relaxation time, and, when combined with sparse Monte Carlo moves, guarantees exact sampling of the isothermal-isobaric ensemble. Validation requires only linear-algebraic tests: all moment identities, autocorrelation functions, and volume statistics can be obtained without running interacting trajectories. Both schemes are black-box, require minimal tuning, and guarantee ensemble exactness at a finite timestep. These developments provide flexible building blocks for large-scale molecular simulations, unifying colored-noise thermostats with modern barostatting strategies.