Generalized Langevin dynamics of a nanoparticle using a finite element approach: Thermostating with correlated noise

Abstract

A direct numerical simulation (DNS) procedure is employed to study the thermal motion of a nanoparticle in an incompressible Newtonian stationary fluid medium with the generalized Langevin approach. We consider both the Markovian (white noise) and non-Markovian (Ornstein-Uhlenbeck noise and Mittag-Leffler noise) processes. Initial locations of the particle are at various distances from the bounding wall to delineate wall effects. At thermal equilibrium, the numerical results are validated by comparing the calculated translational and rotational temperatures of the particle with those obtained from the equipartition theorem. The nature of the hydrodynamic interactions is verified by comparing the velocity autocorrelation functions and mean square displacements with analytical results. Numerical predictions of wall interactions with the particle in terms of mean square displacements are compared with analytical results. In the non-Markovian Langevin approach, an appropriate choice of colored noise is required to satisfy the power-law decay in the velocity autocorrelation function at long times. The results obtained by using non-Markovian Mittag-Leffler noise simultaneously satisfy the equipartition theorem and the long-time behavior of the hydrodynamic correlations for a range of memory correlation times. The Ornstein-Uhlenbeck process does not provide the appropriate hydrodynamic correlations. Comparing our DNS results to the solution of an one-dimensional generalized Langevin equation, it is observed that where the thermostat adheres to the equipartition theorem, the characteristic memory time in the noise is consistent with the inherent time scale of the memory kernel. The performance of the thermostat with respect to equilibrium and dynamic properties for various noise schemes is discussed.

Computational Models for Nanoscale Fluid Dynamics and Transport Inspired by Nonequilibrium Thermodynamics1

“…[41], see also Ref. [42]. It is clear that in the limit of the characteristic memory times s 1 ; s 2 !…”

Velocity autocorrelation of a free particle driven by a Mittag-Leffler noise: Fractional dynamics and temporal behaviors

“…From (26), clearly the VACF of a free particle driven by a Mittag-Leffler noise has derivative zero at the origin, which is a property required according to [41]. In contrast the respective derivative for the power-law model diverges in the superdiffusive range [24], violating the recognized condition that the VACF must have zero initial slope [42].…”

“…This correlation behaves as a power law * paissan@cab.cnea.gov.ar for large times, but is nonsingular at the origin due to the inclusion of a characteristic time and as a consequence has finite variance. Some applications of a generalized Langevin equation with a Mittag-Leffler noise can be seen in [26][27][28]. In [28], the anomalous diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise is studied, and the results for free particle given in [25] are recovered as the limit case.…”

Modeling of a Nanoparticle Motion in a Newtonian Fluid: A Comparison Between Fluctuating Hydrodynamics and Generalized Langevin Procedures

“…The fluctuating hydrodynamics method essentially consists of adding stochastic stresses to the stress tensor (random stress) in the momentum equation and stochastic fluxes to the heat flux where an energy equation is present in the formulation [1]. In generalized Langevin approach, the thermal fluctuations from the fluid are incorporated as random forces and torques in the particle equation of motion where the power spectrum for the variance of the random force and torque terms are in terms of a correlated or colored noise with a well defined characteristic memory time [2]. In this paper, we have considered both the procedures to evaluate the suitability and efficacy of any particular procedure for future comprehensive evaluations.…”

“…Over the years, numerical simulations based on these procedures have been carried out employing the finite volume method [3], lattice Boltzmann method (LBM) [4,5], stochastic immersed boundary method [6], smoothed profile method [7, 8] and the finite element method [1, 2]. In this paper, a direct numerical simulation adopting an arbitrary Lagrangian-Eulerian (ALE) based finite element method (FEM) is employed to simulate the Brownian motion of a nanoparticle in an incompressible Newtonian fluid contained in a horizontal micron sized circular tube.…”