the Hamiltonian and associated Liouville operators ˜ H BKNHC = H ( q, p ) + p 2 ζ 2 m$_{ζ}$ + p 2 ξ 2 m$_{ξ}$ + p 2 η 2 m$_{η}$ + p 2 χ 2 m$_{χ}$ + k$_{B}$T ( ζ + ξ + η + χ ) (50) ˜ L BKNHC 1 = − ∂V ∂q ∂ ∂p + q ∂V ∂q ∂ ∂p$_{ξ}$ (51a) ˜ L BKNHC 2 = p m ∂ ∂q + p 2 m ∂ ∂p$_{ζ}$ (51b) ˜ L BKNHC 3 = − k$_{B}$T ∂ ∂p$_{ζ}$ (51c) ˜ L BKNHC 4 = − k$_{B}$T ∂ ∂p$_{ξ}$ (51d) ˜ L BKNHC 5 = p$_{ζ}$ m$_{ζ}$ ∂ ∂ζ − p$_{ζ}$ m$_{ζ}$ p ∂ ∂p + p 2 ζ m$_{ζ}$ ∂ ∂p$_{η}$ (51e) ˜ L BKNHC 6 = − p$_{ξ}$ m$_{ξ}$ q ∂ ∂q + p$_{ξ}$ m$_{ξ}$ ∂ ∂ξ + p 2 ξ m$_{ξ}$ ∂ ∂p$_{χ}$ (51f) ˜ L BKNHC 7 = p$_{η}$ m$_{η}$ ∂ ∂η − p$_{η}$ m$_{η}$ p$_{ζ}$ ∂ ∂p$_{ζ}$ (51g) ˜ L BKNHC 8 = − k$_{B}$T ∂ ∂p$_{η}$ (51h) ˜ L BKNHC 9 = p$_{χ}$ m$_{χ}$ ∂ ∂χ − p$_{χ}$ m$_{χ}$ p$_{ξ}$ ∂ ∂p$_{ξ}$ (51i) ˜ L BKNHC 10 = − k$_{B}$T ∂ ∂p$_{χ}$ . (51j) L BKNHC A ≡ ˜ L BKNHC 1 + ˜ L BKNHC 4 + ˜ L BKNHC 9 = F ( q ) ∂ ∂p + p$_{χ}$ m$_{χ}$ ∂ ∂χ + ( − p$_{χ}$ m$_{χ}$ p$_{ξ}$ + F$_{p}$$_{ξ}$ ) ∂ ∂p$_{ξ}$ (52a) L BKNHC B ≡ ˜ L BKNHC 2 + ˜ L BKNHC 3 + ˜ L BKNHC 7 = p m ∂ ∂q + p$_{η}$ m$_{η}$ ∂ ∂η + ( − p$_{η}$ m$_{η}$ p$_{ζ}$ + F$_{p}$$_{ζ}$ ) ∂ ∂p$_{ζ}$ (52b) L BKNHC C ≡ ˜ L BKNHC 5 + ˜ L BKNHC 6 + ˜ L BKNHC 8 + ˜ L BKNHC 10 = − p$_{ζ}$ m$_{ζ}$ p ∂ ∂p − p$_{ξ}$ m$_{ξ}$ q ∂ ∂q + p$_{ζ}$ m$_{ζ}$ ∂ ∂ζ + p$_{ξ}$ m$_{ξ}$ ∂ ∂ξ + F$_{p}$$_{η}$ ∂ p$_{η}$ + F$_{p}$$_{χ}$ ∂ p$_{χ}$ , (52c) 20 We combine commuting Liouville operators as follows: