algorithms derived by means of a symmetric Trotter factorization of the Liouville propagator: are not only time-reversible but also measure-preserving. exp[ τL ] = n$_{s}$ − 1 ∏ α =1 exp [ τ 2 L$_{α}$ ] exp exp [ τL$_{n}$$_{x}$ ] × n$_{s}$ − 1 ∏ β =1 exp [ τ 2 L$_{n}$$_{s}$$_{−}$$_{β}$ ] (13) 6