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An Introduction to Computational Fluctuating Hydrodynamics

Alejandro L. Garcia, John B. Bell, Andrew Nonaka, Ishan Srivastava, Daniel Ladiges, Changho Kim

TL;DR

This work introduces fluctuating hydrodynamics (FHD) and computational fluctuating hydrodynamics (CFHD), focusing on finite-volume SPDE schemes to capture thermal and species fluctuations at mesoscopic scales. It derives the stochastic heat equation (SHE) from nonequilibrium thermodynamics as $\partial_t T = \partial_x (\kappa \partial_x T + \alpha T \tilde{Z})$ with $\kappa = \lambda/(\rho c_V)$ and $\alpha = \sqrt{2 k_B \lambda}/(\rho c_V)$, and shows how linearization yields an Ornstein–Uhlenbeck process with equilibrium fluctuations; a high-wavenumber cutoff is required for nonlinear well-posedness. The paper develops and analyzes CFHD finite-volume schemes for the SHE (FE and PC), demonstrates structure-factor validation, and surveys a broad family of FHD models including the Dean–Kawasaki form for diffusion, stochastic Burgers’ and reaction–diffusion systems, and multi-species and multi-phase fluids. It provides a set of exercises and a Python demonstration to solidify understanding and to enable practical mesoscale modeling. Overall, CFHD offers a computationally tractable framework to quantify fluctuations and emergent mesoscale behavior, informing nanoscale design and bridging fluctuations to macroscale fluid dynamics.

Abstract

These notes are an introduction to fluctuating hydrodynamics (FHD) and the formulation of numerical schemes for the resulting stochastic partial differential equations (PDEs). Fluctuating hydrodynamics was originally introduced by Landau and Lifshitz as a way to put thermal fluctuations into a continuum framework by including a stochastic forcing to each dissipative transport process (e.g., heat flux). While FHD has been useful in modeling transport and fluid dynamics at the mesoscopic scale, theoretical calculations have been feasible only with simplifying assumptions. As such there is great interest in numerical schemes for Computational Fluctuating Hydrodynamics (CFHD). There are a variety of algorithms (e.g., spectral, finite element, lattice Boltzmann) but in this introduction we focus on finite volume schemes. Accompanying these notes is a demonstration program in Python available on GitHub (https://github.com/AlejGarcia/IntroFHD).

An Introduction to Computational Fluctuating Hydrodynamics

TL;DR

This work introduces fluctuating hydrodynamics (FHD) and computational fluctuating hydrodynamics (CFHD), focusing on finite-volume SPDE schemes to capture thermal and species fluctuations at mesoscopic scales. It derives the stochastic heat equation (SHE) from nonequilibrium thermodynamics as with and , and shows how linearization yields an Ornstein–Uhlenbeck process with equilibrium fluctuations; a high-wavenumber cutoff is required for nonlinear well-posedness. The paper develops and analyzes CFHD finite-volume schemes for the SHE (FE and PC), demonstrates structure-factor validation, and surveys a broad family of FHD models including the Dean–Kawasaki form for diffusion, stochastic Burgers’ and reaction–diffusion systems, and multi-species and multi-phase fluids. It provides a set of exercises and a Python demonstration to solidify understanding and to enable practical mesoscale modeling. Overall, CFHD offers a computationally tractable framework to quantify fluctuations and emergent mesoscale behavior, informing nanoscale design and bridging fluctuations to macroscale fluid dynamics.

Abstract

These notes are an introduction to fluctuating hydrodynamics (FHD) and the formulation of numerical schemes for the resulting stochastic partial differential equations (PDEs). Fluctuating hydrodynamics was originally introduced by Landau and Lifshitz as a way to put thermal fluctuations into a continuum framework by including a stochastic forcing to each dissipative transport process (e.g., heat flux). While FHD has been useful in modeling transport and fluid dynamics at the mesoscopic scale, theoretical calculations have been feasible only with simplifying assumptions. As such there is great interest in numerical schemes for Computational Fluctuating Hydrodynamics (CFHD). There are a variety of algorithms (e.g., spectral, finite element, lattice Boltzmann) but in this introduction we focus on finite volume schemes. Accompanying these notes is a demonstration program in Python available on GitHub (https://github.com/AlejGarcia/IntroFHD).
Paper Structure (7 sections, 37 equations, 6 figures)

This paper contains 7 sections, 37 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Stochastic Heat Equation
  3. CFHD Schemes for SHE
  4. Stochastic Species Diffusion Equation
  5. A Menagerie of FHD Equations
  6. Suggested Exercises
  7. Conclusions

Figures (6)

  • Figure 1: (left) Space and time discretization for SHE; (right) Physical system simulated by the StochasticHeat program ($\ell = 20~\mathrm{nm}$, $\mathcal{A} = 4~\mathrm{nm}^2$).
  • Figure 2: Temperature variance $\langle \delta T_i^2 \rangle$ versus $x_i$ at equilibrium with Dirichlet boundary conditions for (left) FE scheme; (right) PC scheme. Theory (dashed line) is given by Eq. \ref{['eq:varcorrT_EQ']}.
  • Figure 3: Temperature correlation $\langle \delta T_i \delta T_j \rangle$ versus $x_i$ at equilibrium with Dirichlet boundary conditions with $x_j = \ell/4$ for (left) FE scheme; (right) PC scheme. Theory line given by Eq. \ref{['eq:varcorrT_EQ']}.
  • Figure 4: Temperature correlation $\langle \delta T_i \delta T_j \rangle$ versus $x_i$ at equilibrium with periodic boundary conditions using the PC scheme. Results are plotted both with (left) and without (right) the point at $x_j = \ell/4$.
  • Figure 5: Static structure factor, $S_k$, versus $k$ at equilibrium for (left) FE scheme; (right) PC scheme. Theory line given by Eq. \ref{['eq:SkTheory']}.
  • ...and 1 more figures