This will be called the Bulgac-Kusnezov-Nos´ e-Hoover chain (BKNHC), and corresponds to ‘massive’ NH thermostatting of the demon variables [32]. 

The ability to derive numerically stable measure-preserving time-reversible algorithms [19] for Nos´ e controlled BK dynamics is very encouranging for future applications to ther- mostatted spin systems. 

This paper is organized as follows. In Sec. II we briefly sketch the unified formalism for non-Hamiltonian phase space flows and measure-preserving integration. The BK dynamics is formulated in phase space and a measure-preserving integration algorithm is derived in Sec. III. The BKNH and BKNH-chain thermostats are treated in Secs. IV and V respectively. Numerical results for the one-dimensional harmonic oscillator using these thermostats are presented and discussed in Sec. VI. Section VII reports our conclusions. 

In addition we include several Appendices. A useful operator formula is derived in Ap- pendix A, while invariant measures for the BK, BKNH, and BKNHC phase space flows are derived in Appendices B, C, and D, respectively. 

