Mori–Zwanzig formalism as a practical computational tool

Abstract

An operational procedure is presented to compute explicitly the different terms in the generalized Langevin equation (GLE) for a few relevant variables obtained within Mori-Zwanzig formalism. The procedure amounts to introducing an artificial controlled parameter which can be tuned in such a way that the so-called projected dynamics becomes explicit and the GLE reduces to a Markovian equation. The projected dynamics can be realised in practice by introducing constraints, and it is shown that the Green-Kubo formulae computed with these dynamics do not suffer from the plateau problem. The methodology is illustrated in the example of star polymer molecules in a melt using their center of mass as relevant variables. Through this example, we show that not only the effective potentials, but also the friction forces and the noise play a very important role in the dynamics.

Statistical mechanics of Hamiltonian adaptive resolution simulations

“…In addition to the CG potential of interaction, the full CG dynamics requires the presence of friction and stochastic forces in order to fully account for eliminated degrees of freedom in the CG region. 19 As it is well-known, the equilibrium properties should not be affected by the presence of these additional forces that are, however, crucial in non-equilibrium or dynamic situations. This further development is left for future work.…”

Hamiltonian Adaptive Resolution Simulation for Molecular Liquids

“…In general, however, the employed CG potentials are only approximations to the the exact many-body CG potential [37,38]. As a consequence, there is usually a thermodynamic mismatch between the AA and the CG systems, which can have different chemical potentials and equations of state [3]. If compensations are introduced as time-independent functions of the position of the molecules, then the modified Hamiltonian H is expressed as…”

“…To handle this problem general strategies have been developed, which can be classified as either sequential or simultaneous. In the former class of methods, coarse-grained models are (usually) developed from microscopic input [1][2][3]; systems are then simulated separately at different levels of resolution. In the latter class, which we pursue here, systems are treated within a single simulation on different levels of resolution.…”

Parametric reduced models for the nonlinear Schrödinger equation

“…An elegant framework for deriving the effective model is the Mori-Zwanzig projection [2][3][4], which has recently become an extremely important tool to simplify complex dynamical systems [5][6][7][8][9][10][11][12][13][14][15]. In particular, this derivation led to a set of generalized Langevin equation (GLEs), a typical result of the Mori-Zwanzig procedure.…”

“…A simple approach proposed by [7,17,18] is to approximate the memory kernel in the GLE model with a delta function (but with a carefully chosen damping parameter). This certainly introduces additional modeling error that is difficult to quantify.…”