Scientific background

Introduced over a century ago, the Boltzmann equation remains the cornerstone of kinetic theory. To first order in Knudsen number (mean free path divided by hydrodynamic length scale), the Chapman--Enskog expansion leads to the Navier--Stokes equations [1]. Higher order expansions (i.e., Burnett and super Burnett) however, have had limited success, particularly with regard to convergence and boundary conditions [2]. The moment method, introduced by Maxwell, modified and extended by Grad [3] and by Waldmann [4], proves to be more successful in theoretical work but is not yet widely used as the basis for numerical calculations. While theoretical advances continue to be made, a general approach to solving the Boltzmann equation has not been established.

In this regard, numerical experiments have played an important role in modern kinetic theory. Historically, computational simulations of dilute systems have been done by two groups. The first is the statistical physics community [5], which has focused primarily on molecular dynamics simulations of gases and lattice-based schemes with simplified collision dynamics. The second branch is in the engineering community. Stochastic particle methods such as direct simulation Monte Carlo (DSMC) [6] and its variants were introduced by aerospace engineers during the 1960's and 70's, when space exploration was the primary application for rarefied flows.

The two branches developed separate numerical approaches since they focused on different problems. In statistical physics the work was primarily in equilibrium and simple non-equilibrium states, such as Couette and Poiseuille flow. The engineering community was interested in complex scenarios such as hypersonic flight and extreme non-equilibrium flows (e.g. plasma etching). In recent years, the two communities have discovered the utility of the algorithms developed by each, especially as their interests turn to meso-scale applications. Some of the new directions in computational kinetic theory include:

Microscopic flows:

Nanometer-scale flows play an increasingly important role in fluid dynamics, thanks to advances in miniaturization technology [7]. For example, in a computer hard disk the aerodynamic design of the read-write head elevates it to only 10 nm above the spinning platter [8]. When the characteristic length scale of a flow is comparable to the mean free path of the constituent molecules (about 60 nm in air at standard conditions), the traditional macroscopic description breaks down. Particle simulations, primarily molecular dynamics [9] and direct simulation Monte Carlo [10], are at present the main computational tools in the study of these transitional flows. In some cases the conventional hydrodynamic equations can be corrected by modifications, determined by kinetic theory, to the boundary conditions or transport properties. An example is the simple Poiseuille flow where these modifications have yielded the correct temperature and pressure profiles [11]. A related problem is that of sound propagation when the acoustic wavelength is comparable to the mean free path [12].

Fluctuations and instabilities:

Numerical simulations of dilute systems were the first to measure directly the long-ranged static correlations of non-equilibrium hydrodynamic fluctuations [13]. Dilute systems remain extremely useful in the study of hydrodynamic fluctuations and their influence on the onset of hydrodynamic instabilities. For instance, comparison with the numerical integration of stochastic hydrodynamic equations [14] will help to clarify the role of key parameters, such as the compressibility [15], the spatial dimension, and the influence of boundary conditions in the emergence of instabilities leading to vortex formations and to other complex behavior observed in laboratory experiments.

Dense gases:

The Boltzmann equation can be extended to dense hard sphere gases by Enskog's modification in which the separation of the particles at collision is explicitly included in the distributions [16]. Recently, this idea was incorporated into the direct simulation Monte Carlo algorithm by Frezotti [17] and by Montanero and Santos [18]. The Consistent Boltzmann Algorithm (CBA) is a related approach that evaluates the displacement between particles due to collisions [19]. This latter technique can be extended to arbitrary equations of state; one application has been the simulation of a van der Waals fluid in the gas-liquid coexistence region [20].

Granular flows:

Granular flows are ubiquitous in both nature and industry. In nature, rock and snow avalanches, the formation of sand dunes, and soil liquefaction during earthquakes are examples of flows involving assemblies of grains. Industrial applications include packing, segregation, mixing and drying, of granular materials, such as seeds, rocks, pellets, pills, etc. Granular fluids are composed of a large number of macroscopic elements, the grains, which undergo collisions very much like the molecules in a fluid. In contrast to regular fluids, at each collision a fraction of the grains' kinetic energy is dissipated due to a coupling with their internal energy. Granular media can thus be reasonably modeled as an ensemble of inelastic particles and numerical simulations, primarily molecular dynamics, have been widely used in the study of granular flows [21]. For dilute and semi-dense systems, stochastic particle algorithms, such as DSMC, CBA, and other dense gas variants, are promising alternatives.

Multi component systems:

Instabilities occurring in reactive fluids and coagulation processes observed in colloidal systems are two examples where traditional molecular dynamic simulations become quite inefficient. For instance, the validity of the macroscopic rate equations describing the time evolution of the composition variables implies that one needs to have a large number of elastic collisions between successive reactive collisions in order to ensure mechanical and thermal equilibrium. As a consequence, only a fraction of the computation time will contribute effectively to the evolution of the chemical reactions. This results in much wasted bookkeeping with a corresponding waste of CPU time [22]. Similar difficulties arise in colloidal systems due to the vastly different time scales for collisions and coagulation. In this regard, stochastic particle simulations have proved useful in the study of reactive systems [23] and colloidal coagulation [24]. Mesoscopic algorithms, such as lattice Boltzmann [25], dissipative particle dynamics [26], or the highly efficient DSMC variant proposed recently by Malevanets and Kapral [27], are promising alternatives to model the dynamics of these dilute systems.

Plasma physics:

Molecular dynamic simulations of charged fluids are extremely time consuming, mainly because of the long-ranged Coulomb iterations. The question then arises as to the possibility of setting up alternative mesoscopic methods. Today, there are some engineering plasma problems that have been analyzed by stochastic particle simulations. For example, the exhaust plume from pulsed plasma thrusters has been computed by a combination of direct simulation Monte Carlo (DSMC) and Particle-In-Cell techniques [28]. In this workshop, we intend to discuss another possibility related to the DSMC simulation of dilute charged fluids, as described by the Landau-Vlassov equations. In fact, the Landau collision operator is nothing but a modified Boltzmann collision operator that can be handled by an appropriate modification of DSMC algorithm.

Particle/Continuum hybrids:

Many interesting flows that require the use of a microscopic simulation have not been studied due to their computational expense, which is several orders of magnitude greater than that of continuum hydrodynamic methods. Yet often one needs to simulate the fluid at the particle level in only a small fraction of the physical volume, for example at a shock or combustion interface. Hybrid schemes, which combine a microscopic algorithm with a continuum hydrodynamic calculation have proved useful is such scenarios. For example, the Adaptive Mesh and Algorithm Refinement (AMAR) scheme uses direct simulation Monte Carlo at the finest grid scale in an adaptive mesh refinement hierarchy and a Navier--Stokes solver at other scales [29]. An AMAR program adjusts grids, adding or removing finer levels, as the flow solution evolves, adaptively deciding where and when the calculation should switch from a continuum algorithm to a particle algorithm. Molecular dynamics-based hybrids have also been developed [30].


[1]     S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, (Cambridge University Press Cambridge, 1970).

[2]     C. Cercignani, Mathematical Methods in Kinetic Theory, (Plenum, New York, 1990).

[3]     H. Grad, Principles of the Kinetic Theory of Gases, in Handbuch der Physik, vol. XII, S. Flügge, ed. (Springer, Berlin 1958).

[4]     L. Waldmann, Transporterscheinungen in Gasen von mittlerem Druck, in Handbuch der Physik, vol. XII, S. Flügge, ed. (Springer, Berlin 1958).

[5]     The Microscopic Approach to Complexity in Non-Equilibrium Molecular Simulations, M. Mareschal ed., Physica A 240 (1997).

[6]     G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, (Clarendon, Oxford, 1994).

[7]     E.S. Oran, C.K. Oh, and B.Z. Cybyk, Direct Simulation Monte Carlo: Recent Advances and Applications, Annu. Rev. Fluid Mech., 30, 403 (1998).

[8]     F. Alexander, A. Garcia and B. Alder, Direct Simulation Monte Carlo for Thin Film Bearings, Phys. Fluids, 6 , 3854 (1994).

[9]     D. Risso and P. Cordero, Generalized hydrodynamics for a Poiseuille flow: Theory and Simulations, Phys. Rev. E 58, 546 (1998).

[10]     M. Malek Mansour, F. Baras and A. Garcia, On the validity of hydrodynamics in plane Poiseuille flows, Physica A, 240, 255 (1997).

[11]     S. Hess and M. Malek Mansour, Temperature profile of a dilute gas undergoing a plane Poiseuille flow, Physica A, 272, 481 (1999).

[12]     N. Hadjiconstantinou and A. Garcia, Molecular simulations of sound wave propagation in simple gases, preprint.

[13]     M. Malek Mansour, A. Garcia, G. Lie and E. Clementi, Fluctuating Hydrodynamics in a Dilute Gas, Phys. Rev. Lett. 58, 874 (1987).

[14]     A. Garcia, M. Malek Mansour, G. Lie and E. Clementi, Numerical Integration of the Fluctuating Hydrodynamic Equations, J. Stat. Phys., 47, 209 (1987); P. Espanol, Stochastic differential equations for non-linear hydrodynamics, Physica A 248, 77 (1998).

[15]     I. Bena, M. Malek Mansour and F. Baras, Hydrodynamic Fluctuations in the Kolmogorov Flow: Linear Regime, Phys. Rev. E, 59, 5503 (1999) ; ibid Hydrodynamic Fluctuations in the Kolmogorov Flow: Non Linear Regime, Phys. Rev. E, (in press)

[16]     H. van Beijeren and M. H. Ernst, "The modified Enskog equation", Physica, 68, 437-456, 1973 ; P. Résibois and M. De Leener, Classical Kinetic Theory of Fluids, (Plenum, New York, 1976)

[17]     A. Frezzotti. A particle scheme for the numerical solution of the Enskog equation, Phys. Fluids, 95, 1329-1335, 1997.

[18]     J. M. Montanero and A. Santos, Simulation of the Enskog equation à la Bird, Phys. Fluids, 9, 2057-2060, 1997.

[19]     F. Alexander, A. Garcia and B. Alder, A Consistent Boltzmann Algorithm, Phys. Rev. Lett. 74, 5212 (1995).

[20]     N. Hadjiconstantinou, A. Garcia, and B. Alder, The Surface Properties of a van der Waals Fluid, Physica A 281, 337-347 (2000).

[21]     H. J. Herrmann and S. Luding, Modeling granular media on the computer, Contin. Mech. Thermodyn., 10(4), 189-23 (1998).

[22]     F. Baras and M. Malek Mansour, Particle Simulations of Chemical Systems, Adv. Chem Phys., 100, 393-474 (1997).

[23]     M. Kraft and W. Wagner, Numerical study of a stochastic particle method for homogeneous gas phase reactions, preprint.

[24]     A. Kolodko, K. Sabelfeld and W. Wagner, A stochastic method for solving Smoluchowski's coagulation equation, Mathematics and Computers in Simulation. vol.49 (1999), N 1-2, p. 57-79; A. Eibeck and W. Wagner, Stochastic particle approximations for Smoluchowski's coagulation equation, preprint.

[25]     A. J. C. Ladd, Numerical Simulations of Particulate Suspensions via a discretized Boltzmann Equation. Part I. Theoretical Foundation, J. Fluid Mech., 271, 285, 1994.

[26]     P. Espanol and M. Serrano, Dynamical regimes in the dissipative particle dynamics model, Phys. Rev. E, 59, 6340 (1999).

[27]     A. Malevanets and R. Kapral, Mesoscopic Model for Solvent Dynamics, J. Chem. Phys., 110, 8605 (1999).

[28]     I.D. Boyd, M. Keidar, and W. McKeon, Modeling of a Pulsed Plasma Thruster From Plasma Generation to Plume Far Field, AIAA Paper, 99-2300, (June 1999).

[29]     A. Garcia, J. Bell, Wm. Y. Crutchfield and B. Alder, Adaptive Mesh and Algorithm Refinement using Direct Simulation Monte Carlo, J. Comp. Phys. 154, 134 (1999).

[30]     N. Hadjiconstantinou and A. Patera, Heterogeneous Atomistic--Continuum Representations for Dense Fluid Systems, Int. J. Mod. Phys. C, 8, 967 (1997).

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