Appendix A: Operator formula We wish to determine the action of the propagator associated with the Liouville operator Eq. (39). This is equivalent to solving the evolution equation (recall i = k ) from t = 0 to t= τ . Integrating, we have giving The phase space compressibility of the phase space BK thermostat is Upon introducing the function one can easily find that so that the invariant measure in phase space reads as desired. 26 Appendix B: Invariant Measure of the BK phase space flows dp$_{i}$ dt = ( − p$_{k}$ m$_{k}$ p$_{i}$ + F$_{p}$$_{i}$ ) (A1) − m$_{k}$ p$_{k}$ ln ( − p$_{k}$ m$_{k}$ p$_{i}$ + F$_{p}$$_{i}$ )∣ ∣ ∣$^{τ}$ 0 = τ (A2) p$_{i}$ ( τ ) ≡ exp [ τ ( − p$_{k}$ m$_{k}$ p$_{i}$ + F$_{p}$$_{i}$ ) ∂ ∂p$_{i}$ ] p$_{i}$ (A3a) = p$_{i}$e − τp$_{k}$/m$_{k}$ + m$_{k}$ p$_{k}$ F$_{p}$$_{i}$ ( 1 − e − τp$_{k}$$^{/m$_{k}$}$) (A3b) = p$_{i}$e − τp$_{k}$/m$_{k}$ + τF$_{p}$$_{i}$e − τ p k 2 m k sinh [ τ p$_{k}$ 2 m$_{k}$ ] τ p$_{k}$ 2 m$_{k}$ . (A3c) κ$_{BK}$ = ∂ B BK ij ∂x$_{i}$ ∂H$_{BK}$ ∂x$_{i}$ = − 1 m$_{ζ}$ ∂G$_{1}$ ∂p ∂K$_{1}$ ∂p$_{ζ}$ − 1 m$_{ξ}$ ∂G$_{2}$ ∂q ∂K$_{2}$ ∂p$_{ξ}$ (B1) H BK T = H + K$_{1}$ m$_{ζ}$ + K$_{2}$ m$_{ξ}$ (B2) κ$_{BK}$ = 1 k$_{B}$T dH BK T dt (B3) dµ = dx exp [ − ∫$_{t}$ dtκ$_{BK}$ ] (B4a) = dx exp[ − βH $^{BK}$] exp[ ζ + ξ ] (B4c) = dx exp [ − βH BK T ] (B4b)