Position-dependent memory kernel in generalized Langevin equations: Theory and numerical estimation

Abstract

Generalized Langevin equations with non-linear forces and position-dependent linear friction memory kernels, such as commonly used to describe the effective dynamics of coarse-grained variables in molecular dynamics, are rigorously derived within the Mori-Zwanzig formalism. A fluctuation-dissipation theorem relating the properties of the noise to the memory kernel is shown. The derivation also yields Volterra-type equations for the kernel, which can be used for a numerical parametrization of the model from all-atom simulations.

Understanding and Modeling Polymers: The Challenge of Multiple Scales

“…Using the Mori−-Zwanzig formalism and considering a single CG particle, they showed that it is possible to derive a GLE that complies with the GENERIC structure, if one allows for position dependent memory kernels. 288 One may be tempted to set aside these problems and design CG models that primarily target static equilibrium properties.…”

“…Luckily, recent work by Vroyland and Monmarché suggests a possible way out of this dilemma. Using the Mori–-Zwanzig formalism and considering a single CG particle, they showed that it is possible to derive a GLE that complies with the GENERIC structure, if one allows for position dependent memory kernels …”

“…The form (eq ) has been derived by Kinjo and Hyodo based on Mori–Zwanzig projections and additional approximations. , The fluctuation–dissipation relation can also be derived from the Mori–Zwanzig projection operator formalism, but one can show that it is a general feature of GLEs which satisfy an orthogonality condition for the relation between random force and velocity . Several methods have been developed and analyzed that allow to determine memory kernels from FG simulations and thus construct GLE-based CG models in a bottom-up fashion. ,,− …”

Pair-Reaction Dynamics in Water: Competition of Memory, Potential Shape, and Inertial Effects

“…Different analytical rate theories based on the GLE have been developed but necessarily rely on various approximations, the effects of which are difficult to disentangle. ,,, This is where numerical solutions of accurately parametrized GLEs become instrumental. The extraction of memory kernels from general time series data is an active field of research, ,− in particular in the context of reaction kinetics. ,,− With recent methodological advances, it is possible to extract memory kernels from trajectories in the presence of arbitrary, not necessarily harmonic, potentials and to numerically solve the resulting GLE by Markovian embedding techniques. ,,, While the one-dimensional GLE may in principle contain nonlinear friction contributions, the approximate linear friction GLE, which only includes a linear coupling of the velocity to a friction kernel with no further dependencies on position or velocity, becomes valid for a broad class of systems under well-defined conditions; this explains why it accurately describes the dynamics of very different physical systems. , In this connection, it is important to note that most existing reaction-rate theories are in fact based on the approximate linear friction GLE.…”

“… 22 , 32 , 38 , 39 This is where numerical solutions of accurately parametrized GLEs become instrumental. The extraction of memory kernels from general time series data is an active field of research, 33 , 40 − 45 in particular in the context of reaction kinetics. 10 , 30 , 46 − 49 With recent methodological advances, it is possible to extract memory kernels from trajectories in the presence of arbitrary, not necessarily harmonic, potentials and to numerically solve the resulting GLE by Markovian embedding techniques.…”