An idealized model for nonequilibrium dynamics in molecular systems

Abstract

The irreversible generalized Langevin equation (iGLE) contains a nonstationary friction kernel that in certain limits reduces to the GLE with space-dependent friction. For more general forms of the friction kernel, the iGLE was previously shown to be the projection of a mechanical system with a time-dependent Hamiltonian. [R. Hernandez, J. Chem. Phys. 110, 7701 (1999)] In the present work, the corresponding open Hamiltonian system is further explored. Numerical simulations of this mechanical system illustrate that the time dependence of the observed total energy and the correlations of the solvent force are in precise agreement with the projected iGLE.

Derivation of the generalized Langevin equation in nonstationary environments

“…(7) which was obtained by the heuristic consideration. [28][29][30][31][32][33][34] The generalized fluctuation-dissipation theorem proposed [28][29][30][31][32][33][34] has also been proved on general footing here. Note also that, compared to the equilibrium case, we need to reinterpret the "temperature" of the nonstationary process as the average of the square of the velocity.…”

“…29 and its numerical evidence was provided in Ref. 33 within the framework of bilinearly coupled harmonic oscillators Hamiltonian systems.…”

“…For the last decade there have been several attempts at establishing the nonstationary representation termed as the irreversible generalized Langevin equation (iGLE). [25][26][27][28][29][30][31][32][33][34] In particular, the idea of generalized friction kernel 28, 34 dependent on both the time values at the past and the present was proposed. As the (nonstationary) bath relaxes to the equilibrium state as time proceeds, the friction kernel is shown to converge to the usual equilibrium one, which depends only on the time difference.…”

“…As the (nonstationary) bath relaxes to the equilibrium state as time proceeds, the friction kernel is shown to converge to the usual equilibrium one, which depends only on the time difference. The generalization of the fluctuation-dissipation theorem was also proposed: the autocorrelation of the random force averaged over a given distribution of the initial condition of the bath dof corresponds to the generalized time-dependent friction kernel [28][29][30][31][32][33][34] or with an exponentially dumping correction term, [25][26][27] depending on how to formulate the iGLE. The iGLE can be applied not only to phenomena subject to an outside force that changes the solvent response, 28 but also to systems whose nonstationarity is induced by the dynamics of the system itself.…”

“…Especially, they require a special form for the total-system Hamiltonian that depends on the history of the system dynamics, 29,33 in deriving the form of iGLE in a heuristic fashion. 28 In this paper, we present a rigorous and general derivation of iGLE that does not depend on any specific form of the original Hamiltonian, by using the projection-operator formalism.…”