Gaseous microflow modeling using the Fokker-Planck equation

Singh, Shubham; Thantanapally, Chakradhar; Ansumali, Santosh

doi:10.1103/physreve.94.063307

Cited by 16 publications

(14 citation statements)

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“…It is important to note that numerical implementations of LBM and DSMC have distinct benefits over their hydrodynamic counterparts [12–15,47,48]. For example, computer programs based on LBM and DSMC can be efficiently parallelized; they can also take advantage of a large number of cores available in graphical processing units (GPUs).…”

Section: Non-equilibrium Behaviourmentioning

confidence: 99%

Boltzmann equation and hydrodynamic equations: their equilibrium and non-equilibrium behaviour

Verma

2020

Phil. Trans. R. Soc. A.

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This short article summarizes the key features of equilibrium and non-equilibrium aspects of Boltzmann and hydrodynamic equations. Under equilibrium, the Boltzmann equation generates uncorrelated random velocity that corresponds to k 2 energy spectrum for the Euler equation. The latter spectrum is produced using initial configuration with many Fourier modes of equal amplitudes but with random phases. However, for a large-scale vortex as an initial condition, earlier simulations exhibit a combination of k −5/3 (in the inertial range) and k 2 (for large wavenumbers) spectra, with the range of k 2 spectrum increasing with time. These simulations demonstrate an approach to equilibrium or thermalization of Euler turbulence. In addition, they also show how initial velocity field plays an important role in determining the behaviour of the Euler equation. In non-equilibrium scenario, both Boltzmann and Navier–Stokes equations produce similar flow behaviour, for example, Kolmogorov’s k −5/3 spectrum in the inertial range. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

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Section: Non-equilibrium Behaviourmentioning

confidence: 99%

Boltzmann equation and hydrodynamic equations: their equilibrium and non-equilibrium behaviour

Verma

2020

Phil. Trans. R. Soc. A.

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“…This Prandtl number problem could be fixed with the introduction of a cubic drift model, 2,19 which was demonstrated for various challenging test cases. 20 This issue is also discussed in the recent work by Singh et al, 21 in which they proposed a way to fix the Prandtl number resulting from the Fokker-Planck equation. They achieved that by introducing an extra streaming in the particle position to fix the transport properties including the Prandtl number.…”

Section: Introductionmentioning

confidence: 91%

Controlling the bias error of Fokker-Planck methods for rarefied gas dynamics simulations

2019

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Direct simulation Monte-Carlo (DSMC) is the most established method for rarefied gas flow simulations. It is valid from continuum to near vacuum, but in cases involving small Knudsen numbers (Kn), it suffers from high computational cost. The Fokker-Planck (FP) method, on the other hand, is almost as accurate as DSMC for small to moderate Kn, but it does not have the computational drawback of DSMC, if Kn is small [P. Jenny, M. Torrilhon, and S. Heinz, “A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion,” J. Comput. Phys. 229, 1077–1098 (2010) and H. Gorji, M. Torrilhon, and P. Jenny, “Fokker–Planck model for computational studies of monatomic rarefied gas flows,” J. Fluid Mech. 680, 574–601 (2011)]. Especially attractive is the combination of the two approaches leading to the FP-DSMC method. Opposed to other hybrid methods, e.g., coupled DSMC/Navier-Stokes solvers, it is relatively straightforward to couple DSMC with the FP method since both are based on particle solution algorithms sharing the same data structure and having similar components. Regarding the numerical accuracy of such particle methods, one has to distinguish between spatial truncation errors, time stepping errors, statistical errors and bias errors. In this paper, the bias error of the FP method is analyzed in detail, and it is shown how it can be reduced without increasing the particle number to an exorbitant level. The effectiveness of the discussed bias error reduction scheme is demonstrated for uniform shear flow, for which an analytical reference solution was derived.

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“…We believe that leveraging standard methods of stochastic calculus, while harnessing the potential of quantum computers, may lead to the development of new quantum tools in the field of complex system simulations. This can possibly pave the way to important advances in chemistry, biochemistry and all the other connected fields in which stochastic analysis plays a relevant role such as quantitative finance, epidemiology, and computational fluid dynamics [11][12][13]. Here we propose a new VHA to solve the Fokker-Planck-Smoluchowski (FPS) eigenvalue problem based on the isomorphism existing between the FPS operator and the quantum Hamiltonian [14,15].…”

Section: Introductionmentioning

confidence: 99%

Quantum computing for classical problems: Variational Quantum Eigensolver for activated processes

Pravatto,

Castaldo,

Gallina

et al. 2021

Preprint

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The theory of stochastic processes impacts both physical and social sciences. At the molecular scale, stochastic dynamics is ubiquitous because of thermal fluctuations. The Fokker-Plank-Smoluchowski equation models the time evolution of the probability density of selected degrees of freedom in the diffusive regime and it is therefore a workhorse of physical chemistry. In this paper we report the development and implementation of a Variational Quantum Eigensolver procedure to solve the Fokker-Planck-Smoluchowski eigenvalue problem. We show that such an algorithm, typically adopted to address quantum chemistry problems, can be applied effectively to classical systems paving the way to new applications of quantum computers. We compute the conformational transition rate in a linear chain of rotors experiencing nearest-neighbour interaction. We provide a method to encode on the quantum computer the probability distribution for a given conformation of the chain and assess its scalability in terms of operations. Performance analysis on noisy quantum emulators and quantum devices (IBMQ Santiago) is provided for a small chain showing results in good agreement with the classical benchmark without further addition of any error mitigation technique.

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Gaseous microflow modeling using the Fokker-Planck equation

Cited by 16 publications

References 50 publications

Boltzmann equation and hydrodynamic equations: their equilibrium and non-equilibrium behaviour

Boltzmann equation and hydrodynamic equations: their equilibrium and non-equilibrium behaviour

Controlling the bias error of Fokker-Planck methods for rarefied gas dynamics simulations

Quantum computing for classical problems: Variational Quantum Eigensolver for activated processes

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