Building a non-Markovian coarse-grained model
Keywords: Mechanical Engineering.

Many scientific problems deal with time-evolution equations that are computationally too costly to solve in full resolution of a system because of the system size and the timescale involved in the problems. One strategy to reduce the computational cost is to describe the system in reduced dimensions, which can be divided into two tasks: 1) identification of the coordinates that effectively describe the characteristics that we are interested in, 2) finding a governing equation for those selected coordinates, which should account for the effect of ignored degrees of freedom. Following the common naming convention in the context of biomolecular systems, the description in reduced dimensions will be referred to as a coarse-grained (CG) model; and the selected coordinates will be referred to as CG coordinates, and the governing equation for CG coordinates that only involves the CG coordinates will be referred to as a CG equation. In this dissertation, we build CG equations based on the Mori-Zwanzig (MZ) formalism. The MZ formalism gives an exact governing equation for CG coordinates, which we will call the MZ equation. The MZ equation consists of the three terms: the mean term, the memory term and the fluctuation term. The memory term depends on current and past values of CG coordinates. The fluctuation term is a function of fine-grain coordinates. So, to build a CG equation using the MZ equation, we need to 1) compute the memory term and 2) model the fluctuation term. The exact memory term given by the MZ formalism is prohibitively costly to compute since it includes a solution of a high dimensional partial differential equation. In the first part of the dissertation, we approximate the memory term so that we can evaluate the memory term without prohibitive computational cost. In the second part of the dissertation, we propose CG equations that are different from the standard MZ equation and thus have different forms of the memory term. Computing the memory term in our CG equations only requires trajectories of a full system without having to compute the high dimensional partial differential equation. With a certain type of approximation on the memory term, the MZ equation reduces to the the generalized Langevin equation (GLE). In the first part of the dissertation, we present a GLE approach when CG coordinates are multiple generalized coordinates, defined in general as nonlinear functions of microscopic coordinates. The CG equation for multiple generalized coordinates is described by the multi-dimensional GLE, which include the full memory matrix with nonzero off-diagonal entries. We first present a method to compute the memory matrix in a multi-dimensional GLE using trajectories of a full system. Then, in order to reduce the computational cost of computing the multi-dimensional friction with memory, we introduce a method that maps the GLE to an extended Markovian system. In addition, we study the effect of using a non-constant mass matrix in the CG model. In particular, we include mass-dependent terms in the mean force. We used the proposed CG model to describe the conformational motion of a solvated alanine dipeptide system, with two dihedral angles as the CG coordinates. We showed that the CG model can accurately reproduce two important kinetic quantities: the velocity autocorrelation function and the distribution of first passage times. In the second part of the dissertation, we propose two CG equations that include more complicated memory terms than the GLE. The first CG equation has the memory term that can be nonlinear in CG coordinates. We tested the CG equation using a simple heat-bath system. The second CG equation includes a coordinate dependent memory model. Only preliminary numerical results are presented for those two CG equations.