Reflections on the Boltzmann Equation
Berni J. Alder
Lawrence Livermore National Laboratory, Livermore, CA 94551
Abstract. Developments since the formulation of the Boltzmann equation will be discussed. This includes
assumptions in the derivation of the equation that make it valid only asymptotically. Merits of the solution
of the equation by various numerical techniques will be outlined including the lattice Boltzmann version.
Finally, attempts at extending applicability to higher density and embedding into a continuum will be
discussed.
Since the Boltzmann equation is the fundamental equation governing rarefied gas dynamics I thought I
would discuss what is new since the equation was formulated more than a hundred years ago. First of all
I should point out that the underlying approximation of uncorrelated binary collision (molecular chaos) was
never intended to apply to systems with long range forces such as plasmas. That is, of course, because with
long range forces all the particles always interact, even at low density. Furthermore, the Boltzmann equation is
also not a very good starting point for systems with inelastic collisions, such as sand or grains, because strong
correlations develop. The unexpected discovery that even for short range forces the Boltzmann approximation
of molecular chaos is not valid has led to a fundamental revision of the theory.
The discovery came through a study by molecular dynamics of the velocity autocorrelation function of a
hard sphere system under dense gas conditions [1]. It was found that the velocity of a particle persisted
after hundreds of collisions, that is, a typical particle remembered its initial velocity for that long a time, while
molecular chaos would have predicted an exponential decaying memory after a few collisions. The origin of this
long memory was traced to a hydrodynamic-like effect, namely a typical particle pushes the particles ahead of
it creating an above average positive pressure, that is relieved by a double vortex flow to the negative pressure
it creates behind itself. The particle, so to speak, kicks itself from behind and a simple dimensional argument
based on the conservation of momentum leads to a decay of the velocity as a power law in the reciprocal time
to the dimensionality of the system over two. Thus in two dimensions the transport coefficients diverge because
they are integrals of the reciprocal in time at long times. In three dimensions these integrals exist, however,
the Burnett coefficients diverge since each can be shown at long times to behave with one higher power of the
time. Thus the Chapman-Enskog expansion diverges.
These divergences persist even at low densities, so that the Boltzmann equation is only asymptotically valid,
that is, in the zero density limit. Any derivation of the Boltzmann equation from the Liouville equation must
cut off these correlations. To correct the Boltzmann equation two equivalent approaches have been pursued.
One is graph theoretical in which ring graphs are added corresponding to the physical effect that particle one
can collide with two, two with three and subsequently three with one. The other approach uses mode coupling.
There are two key conclusions to draw from this discovery, since the correlations are weak the quantitative
consequences are small and that hydrodynamics quantitatively applies on the microscopic level, namely at
nanometer and picosecond scales. We shall take advantage of that when we discuss hybrid methods.
On the mathematical level what is new since the Boltzmann equation was formulated is that computers
did not exist. This has freed us from the rather complex mathematical analysis that was required to solve
the non-linear integro-differential equation and led to a numerically stable, very efficient stochastic algorithm
represented by direct simulation Monte Carlo (DSMC) [2]. It is numerically stable because it is a particle
algorithm that obeys the H theorem and goes beyond the Boltzmann formulation in that fluctuations are
accounted for, however, the price you pay for that is that statistical averaging is required. DSMC is about
a hundred to a thousand times faster than molecular dynamics at low density because you can advance the
particles in a time step by the mean collision time rather than only by the shortest time to the next collision
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
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as in molecular dynamics. Another big advantage over molecular dynamics is that it can be paralleleized so
that with one thousand processors it is another 103 times faster than molecular dynamics. Since machines with
104 processors already exist and with 105 processors are on the drawing boards, DSMC will gain in speed by
another factor of one hundred. In fact, DSMC is the fastest correct particle method and has thus become the
first candidate to be used in hybrids. One builds hybrids because DSMC is still slow compared to Navier-Stokes
solvers.
DSMC discretizes space by partitioning particles into cells of mean free path length but does not discretize
velocities. A potentially even faster approach than DSMC would be to discretize velocities as well. That was
tried some years ago in the form of lattice gas [3]. The idea was to have only a single velocity or stopped particle
on each lattice site so that one could use integer arithmetic. This leads to a cellular automata that when hard
wired on a chip could process 1015 particles for seconds; still not Avogadro's number of particles for macroscopic
times but impressive. You must, of course, when executing collisions make sure that conservation laws and
detailed balance are obeyed. Unfortunately that algorithm has a number of diseases that can be overcome with
more different velocity particles but at a price that the calculation is no longer a simple cellular automata.
The diseases are Gallilean invariance caused by the exclusion principle that allow only one distinct particle per
site, lack of isotropy in three dimensions, extra conserved variables, a high viscosity due to restricted collision
rules so that you cannot go to high Reynolds number flows and noisy results. The latter problem lead to the
formulation of the lattice Boltzmann approach where probability densities are followed on lattice sites [4]. In
that process you lose the integer algebra advantage as well as numerical stability. Furthermore you need to
specify 26 moments of the distribution function to reproduce the full Navier-Stokes equation including the
energy conservation equation [5]. The algorithm turned out to be neither fast or numerically more stable than
Navier-Stokes.
One of the problems with the Boltzmann equation and the DSMC algorithm for general use for fluid flow
is that it is applicable only to dilute gases which have a finite viscosity. The dilute gas limitation has been
overcome by displacing the pair of colliding particles in accordance with the stochastic collision that they
undergo instead of leaving them in place as if they were point particles. This leads to the consistent Boltzmann
algorithm (CBA) [6] and it is so-called because it replaces the perfect gas equation of state in the Boltzmann
equation by the correct second virial coefficient equation consistent with the order of the density in the transport
coefficients. In fact, at higher densities such displacements allow for an arbitrary equation of state to be
introduced (consistent universal Boltzmann algorithm, CUBA) [7] and for hard spheres leads to an Enskog-like
transport coefficient theory for which the corresponding kinetic equation has been derived. CUBA has been
used to generate the van der Waals equation of state in the two phase coexistence region which can be utilized
to study drop dynamics [8], for example. It should be emphasized that CUBA extends DSMC to higher density
at very small computational cost.
In particle methods the viscosity cannot be arbitrarily set as in Navier Stokes solvers, but is determined by
the collision rules. Thus it would be nice to find a way to lower the viscosity to study high Reynolds number
flow. Unfortunately that has not been possible. It is possible to raise the viscosity by making a random
displacement in the CBA algorithm without changing the equation of state and that might be useful for some
applications [9]. In the process of investigating these problems a way was found to determine exact finite cell
size [10] and finite time step [11] corrections for DSMC. The analysis corrects for the fact that collisions occur
at a distance. These corrections allow running larger cells in DSMC and correct for the errors incurred.
We intend to use the DSMC method by embedding it in a Navier-Stokes solver to study the onset of
instabilities, for example, in Poiseuille flow or flow in a pipe at high Reynolds number. The idea is to have
realistic boundary conditions with correct fluctuations in the region where gradients are steep. The object is
to introduce particle methods in the flow field on the "fly" wherever linear Navier-Stokes breaks down. The
algorithm is thus numerically stable and the hope is to learn how microscopic fluctuations lead to macroscopic
instabilities. This is opposite to what has been traditionally done in turbulence, namely how long wavelength
phenomena dissipate to smaller wavelength. The approach extends Navier-Stokes to microscopic scales by
introducing particles methods at the finest grid level in an already adaptive mesh refinement (AMR) algorithm
and is called the adaptive mesh and algorithm refinement (AMAR) method [12]. AMAR can easily cover four
or five orders of magnitude in distance scales and so can AMAR but more physically correct. In synchronizing
the microscopic and continuum levels one must do it in a seamless way and while matching fluxes one must
make sure not to violate the conservation laws. This approach cannot only be viewed as extending continuum
methods to microscopic scales and thus removing many of the restrictions of Navier-Stokes hydrodynamics but
also extending particle methods to mesoscopic scales.
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