Appendix C: Invariant Measure of the BKNH phase space flows The phase space compressibility of the NH controlled Bulgac-Kusnezov thermostat is Upon introducing the function we have so that the invariant measure in phase space is The phase space compressibility of the Nos´e-Hoover-Bulgac-Kusnezov chain is Upon introducing the function we have so that the invariant measure in phase space reads κ$_{BKNH}$ = ∂ B BKNH ij ∂x$_{i}$ ∂H$_{BKNH}$ ∂x$_{i}$ = − 1 m$_{ζ}$ ∂G$_{1}$ ∂p ∂K$_{1}$ ∂p$_{ζ}$ − 1 m$_{ξ}$ ∂G$_{2}$ ∂q ∂K$_{2}$ ∂p$_{ξ}$ − 2 p$_{η}$ m$_{η}$ (C1) H BKNH T = H + K$_{1}$ m$_{ζ}$ + K$_{2}$ m$_{ξ}$ + p 2 η 2 m$_{η}$ (C2) κ$_{BKNH}$ = 1 k$_{B}$T dH BK T dt (C3) = dx exp[ − βH $^{BKNH}$] exp[ ζ + ξ + 2 η ] . (C4c) = dx exp [ − βH BKNH T ] (C4b) dµ = dx exp [ − ∫$_{t}$ dtκ$_{BKNH}$ ] (C4a) κ$_{BKNHC}$ = ∂ B BKNHC ij ∂x$_{i}$ ∂H$_{BKNHC}$ ∂x$_{i}$ = − 1 m$_{ζ}$ ∂G$_{1}$ ∂p ∂K$_{1}$ ∂p$_{ζ}$ − 1 m$_{ξ}$ ∂G$_{2}$ ∂q ∂K$_{2}$ ∂p$_{ξ}$ − p$_{η}$ m$_{η}$ − p$_{χ}$ m$_{χ}$ (D1) H BKNHC T = H + K$_{1}$ m$_{ζ}$ + K$_{2}$ m$_{ξ}$ + p 2 η 2 m$_{η}$ + p 2 χ 2 m$_{χ}$ (D2) κ$_{BKNHC}$ = 1 k$_{B}$T dH BK T dt (D3) dµ = dx exp [ − ∫$_{t}$ dtκ$_{BKNHC}$ ] (D4a) = dx exp [ − βH BKNHC T ] (D4b) = dx exp[ − βH $^{BKNHC}$] exp[ ζ + ξ + η + χ ] . (D4c) 27 Appendix D: Invariant Measure of the BKNHC chain phase space flows