A simple extension of the Nosé thermostat

We introduce a one-parameter generalization of Nosé's thermostat that rescales coordinates and momenta in the virtual system by a tuning parameter a. The resulting real-time equations of motion preserve a stationary extended density whose marginal over the physical coordinate and momentum variables (q, p) is canonical and independent of a. Thus, a tunes the dynamics without altering the target ensemble. For a harmonic oscillator with angular frequency ω, the symmetric case (a=12) is Liouville-integrable (a second invariant confines trajectories) and, therefore, non-ergodic. A local linear analysis of the (q, p) block shows that 0 ≤ a ≤ 1 yields only node/spiral types and, thus, precludes chaos. By contrast, a < 0 or a > 1 creates genuine saddle sectors whenever the thermostat variable satisfies |ζ|>ζc=ω/-a(1-a), furnishing a minimal stretch-squeeze mechanism for robust chaotic mixing. Numerical tests on harmonic and double-well models corroborate these predictions. Under fixed-volume periodic boundary conditions, inserting the virial identity with laboratory velocities imposes a hidden constraint. We remove it either by adopting a state-dependent gauge parameter or by centering the virial-work term, thereby eliminating the hidden "pressure lock." We also outline compatibility with Nosé-Hoover chains and a continuous, Hamiltonian-like construction for the grand canonical (μVT) ensemble. Overall, this framework provides a minimal, deterministic thermostat whose single parameter controls chaotic mixing while preserving the desired ensemble.