The derivation and approximation of coarse-grained dynamics from Langevin dynamics

Abstract

We present a derivation of a coarse-grained description, in the form of a generalized Langevin equation, from the Langevin dynamics model that describes the dynamics of bio-molecules. The focus is placed on the form of the memory kernel function, the colored noise, and the second fluctuation-dissipation theorem that connects them. Also presented is a hierarchy of approximations for the memory and random noise terms, using rational approximations in the Laplace domain. These approximations offer increasing accuracy. More importantly, they eliminate the need to evaluate the integral associated with the memory term at each time step. Direct sampling of the colored noise can also be avoided within this framework. Therefore, the numerical implementation of the generalized Langevin equation is much more efficient.

From generalized Langevin equations to Brownian dynamics and embedded Brownian dynamics

“…Equation of this form has been derived in our previous work 32 , where the full model is the Langevin dynamics.…”

“…The memory term often comes from a coarse-graining step, e.g., 2,[8][9][10][11]15,16,19,22,[25][26][27]29,30,32,34,40,43,46 , which has also been an recent emerging area of interest in molecular modeling. Among the many approximation schemes [6][7][8]12,20,23,32,34 for the GLE model, the BD model (3) is clearly the simplest. To understand how the BD approximation can naturally come about, we first present a derivation which eliminates momentum variables from the GLE, so that the resulting model only involves the coordinates of the molecular variables.…”

Data-driven parameterization of the generalized Langevin equation

“…In principle, GLEs can be derived using the Mori-Zwanzig projection formalism (1,2). Examples of such derivations can be found for a variety of applications (3)(4)(5)(6)(7)(8)(9)(10), for example, climate modeling (11,12). The GLE approach eliminates a large number of irrelevant degrees of freedom, reducing system dimensions to make direct computation feasible.…”

“…For example, the memory functions obtained in past studies (8,9,13,14) have involved functions of high-dimensional matrices. Darve et al (14) proposed a more efficient algorithm to compute the memory kernel by solving an equation for the orthogonal dynamics derived from the Mori-Zwanzig formalism.…”

“…In addition to the direct derivation of memory kernels (8,9,13,14), there have been numerous attempts to compute the memory kernel from full molecular dynamics (MD) simulations (18)(19)(20), especially for systems with zero net mean force. Such analyses lead to integral equations of the first kind, which are numerically unstable without additional regularization.…”