Some formal properties of the MTGLE representation of generalized Brownian motion theory

The molecular time scale generalized Langevin equation (MTGLE) representation of generalized Brownian motion theory [S. A. Adelman, Adv. Chem. Phys. 44, 143 (1980)] is reformulated to yield a more rigorous and general theory. Key results include: (i) The theory is extended so as to be valid for an arbitrary real scalar process ṙ0(t) whose dynamics are governed by an arbitrary Hermitian Liouville operator L and whose statistics are governed by an arbitrary inner product 〈 〉; (ii) a rigorous development of the equivalent harmonic chain representation of ṙ0(t) is presented; (iii) the dynamical and statistical properties of the equivalent chain are rigorously shown to be formally identical to the dynamical and statistical properties of physical harmonic chains obeying classical Boltzmann statistics; (iv) the concept of clamping of equivalent chain atoms is developed for both dynamics and statistics in a rigorous manner. The clamping concept plays a role in the MTGLE theory somewhat analogous to the role played by successive projection operations in the Mori theory; (v) the MTGLE random forces Ṙp(t) are shown to be nonstationary processes if the inner product is 〈 〉. A new inner product 〈 〉r0,r1,⋅⋅⋅rp−1 is developed which renders the random forces stationary; (vi) the normal modes of the equivalent chain are shown to be simply related to the eigenfunctions of L; (vii) the MTGLE fluctuation–dissipation theorems are shown to be direct consequences of the orthonormality of the eigenfunctions of L; (viii) the connection between the MTGLE and Mori representations is made.