ouville operators as follows: where In L$_{A}$ there appears an operator with the form where ( k, i ) = ( χ, ξ ) for L$_{A}$ . The action of the propagator associated with this operator on p$_{i}$ is derived in Appendix A, and is given by The apparently singular function is in fact well behaved as p$_{k}$ → 0, and can be expanded in a Maclaurin series to suitably high order [37]. In our implementation we used an eighth order expansion. The propagators for the BKNH dynamics can now be defined as 15 U BKNH α ( τ ) = exp [ τ ˜ L BKNH α ] , (42) ( τ p$_{k}$ 2 m$_{k}$ ) − 1 sinh [ τ p$_{k}$ 2 m$_{k}$ ] (41) e $^{τ L$_{i}$}$p$_{i}$ = p$_{i}$e − τ p k m k + τF$_{p}$$_{i}$e − τ p k 2 m k ( τ p$_{k}$ 2 m$_{k}$ ) − 1 sinh [ τ p$_{k}$ 2 m$_{k}$ ] . (40) L$_{i}$ = ( − p$_{k}$ m$_{k}$ p$_{i}$ + F$_{p}$$_{i}$ ) ∂ ∂p$_{i}$ , (39) F$_{p}$$_{η}$ = p 2 ζ m$_{ζ}$ + p 2 ξ m$_{ξ}$ − 2 k$_{B}$T . (38d) F$_{p}$$_{ζ}$ = p 2 m − k$_{B}$T (38c) F$_{p}$$_{ξ}$ = q ∂V ∂q − k$_{B}$T (38b) F ( q ) = − ∂V ∂q (38a) L BKNH A ≡ ˜ L BKNH 1 + ˜ L BKNH 4 + ˜ L BKNH 7 = F ( q ) ∂ ∂p + p$_{η}$ m$_{η}$ ∂ ∂η − p$_{η}$ m$_{η}$ p$_{ζ}$ ∂ ∂p$_{ζ}$ + ( − p$_{χ}$ m$_{χ}$ p$_{ξ}$ + F$_{p}$$_{ξ}$ ) ∂ ∂p$_{ξ}$ (37a) ˜ L BKNH B ≡ ˜ L BKNH 2 + ˜ L BKNH 3 = p m ∂ ∂q + F$_{p}$$_{ζ}$ ∂ ∂p$_{ζ}$ (37b) L BKNH C ≡ ˜ L BKNH 5 + ˜ L BKNH 6 + ˜ L BKNH 8 = − p$_{ζ}$ m$_{ζ}$ p ∂ ∂p − p$_{ξ}$ m$_{ξ}$ q ∂ ∂q + p$_{ζ}$ m$_{ζ}$ ∂ ∂ζ + p$_{ξ}$ m$_{ξ}$ ∂ ∂ξ + F$_{p}$$_{η}$ ∂ p$_{η}$ , (37c)