Inertial effects in the Brownian dynamics with rigid constraints

Abstract

SUMMARY: To investigate the influence of inertial effects on the dynamics of an assembly of beads subjected to rigid constraints and placed in a buffer medium, a convenient method to introduce suitable generalized coordinates is presented. Without any restriction on the nature of the soft forces involved (both stochastic and deterministic), pertinent Langevin equations are derived. Provided that the Brownian forces are Gaussian and Markovian, the corresponding Fokker-Planck equation (FPE) is obtained in the complete phase space of generalized coordinates and momenta. The correct short time behavior for correlation functions (CFs) of generalized coordinates is established, and the diffusion equation with memory (DEM) is deduced from the FPE in the high friction limit. The DEM is invoked to perform illustrative calculations in two dimensions of the orientational CFs for once broken nonrigid rods immobilized on a surface. These calculations reveal that the CFs under certain conditions exhibit an oscillatory behavior, which is irreproducible within the standard diffusion equation. Several methods are considered for the approximate solution of the DEM, and their application to three dimensional DEMs is discussed.

Thermodynamics of a subensemble of a canonical ensemble

“…In general, the explicit evaluation of the bath-induced potential ∆ S beyond the weak systembath coupling limit is a difficult task [11,27,28], except in the case of a harmonic bath (modelling, e.g., a Gaussian solvent) bilinearly coupled to the system. In the latter case, the integrations over x i in Eq.…”

Effects of the Bead‐Solvent Interaction on the Dynamics of Macromolecules, 1

“…However, before we analyze the dependence of the friction tensors for a few particular choices of the bead‐solvent interaction potential, let us first consider the (normalized) correlation functions (CF) for the center‐of‐mass, $C_{\alpha \beta }^{\,(C)} (t)$ , and the internal momenta, $C_{\alpha \beta }^{\,(Q)} (t)$ , respectively. These (auto‐)correlation functions are defined by:14 where the angular brackets, 〈…〉, refer to the average over the equilibrium state of the macromolecule. If, for example, we take Γ to denote all the coordinates and momenta of the macromolecule, this bracket can be written as: where $h = \int {d\Gamma f_{eq} (\Gamma ;\,t)}$ accounts for the proper normalization of the average.…”

“…As known from molecular dynamic simulations (MDS), however, the discrete (atomistic) structure of the solvent leads to clear deviations from a pure Brownian behaviour of the beads9–12 and, hence, may play an important role in studying the macromolecular dynamics. In the last few years, therefore, several attempts were made to derive a Fokker‐Planck equation (FPE) for the phase‐space distribution of the interacting beads, starting from the first principles of statistical Hamiltonian mechanics 13–18. Until today, however, such Fokker‐Planck (‐type) equations are difficult to apply to any particular system in practice, since they usually contain terms (projection operators) which refer to the total Hamiltonian of the system and which are known as the generalized friction tensors in the literature.…”