Coupling a nano-particle with isothermal fluctuating hydrodynamics: Coarse-graining from microscopic to mesoscopic dynamics

Abstract

We derive a coarse-grained description of the dynamics of a nanoparticle immersed in an isothermal simple fluid by performing a systematic coarse graining of the underlying microscopic dynamics. As coarse-grained or relevant variables, we select the position of the nanoparticle and the total mass and momentum density field of the fluid, which are locally conserved slow variables because they are defined to include the contribution of the nanoparticle. The theory of coarse graining based on the Zwanzing projection operator leads us to a system of stochastic ordinary differential equations that are closed in the relevant variables. We demonstrate that our discrete coarse-grained equations are consistent with a Petrov-Galerkin finite-element discretization of a system of formal stochastic partial differential equations which resemble previously used phenomenological models based on fluctuating hydrodynamics. Key to this connection between our "bottom-up" and previous "top-down" approaches is the use of the same dual orthogonal set of linear basis functions familiar from finite element methods (FEMs), both as a way to coarse-grain the microscopic degrees of freedom and as a way to discretize the equations of fluctuating hydrodynamics. Another key ingredient is the use of a "linear for spiky" weak approximation which replaces microscopic "fields" with a linear FE interpolant inside expectation values. For the irreversible or dissipative dynamics, we approximate the constrained Green-Kubo expressions for the dissipation coefficients with their equilibrium averages. Under suitable approximations, we obtain closed approximations of the coarse-grained dynamics in a manner which gives them a clear physical interpretation and provides explicit microscopic expressions for all of the coefficients appearing in the closure. Our work leads to a model for dilute nanocolloidal suspensions that can be simulated effectively using feasibly short molecular dynamics simulations as input to a FEM fluctuating hydrodynamic solver.

Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians

“…Using the fact that the Mori product is antisymmetric with respect to the Liouvillian, 13 i.e., [3] (…”

“…We write Q instead of Q(s), since Q is time-independent in this approximation 13. Equation (133) holds also when using the Liouvillian L H instead of L S .…”

Hydrodynamic fluctuations in quasi-two dimensional diffusion

“…This means that even at the level of linearized FHD there are collective memory effects that arise due to renormalization of the transport coefficient by fluctuations. In particular this means that the equations (40) and (44) are not exact even for an ideal gas mixture, even at the linearized level, i.e., at the level of a Gaussian approximation for the fluctuations. Table I).…”

“…(Left) Numerical results from BD-q2D for dynamic structure factors for several wavenumbers (see legend), compared to the theoretical prediction(40) with χ c (k) given in(24). The parameters used are taken from Setup B (seeTable I), φ = 1, and the results are averaged over 1000 simulations, each of length 2 · 10 4 time steps.…”

Everything you always wanted to know about SDPD⋆ (⋆but were afraid to ask)

“…On the other hand, in Eulerian grid based methods, it is necessary to couple the hydrodynamic description of the solvent using a given discrete approach with a different treatment of the solid-liquid interface and fluid-structure-interaction (e.g., by coupling with molecular-dynamics-like approaches for suspended objects by using immersed boundary, smooth profile methods, etc.) for which the exact enforcement of thermodynamic consistency at the discrete level remains a challenge [103] . In conclusion, the different simulation approaches have pros and cons in regard to efficiency and accuracy which are difficult to assess based on formal discussions.…”