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A dl=


and the other is the Gauss law from two to three dimensions,

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where is a curve, S is a surface, and V is a volume in three-dimensional Euclidean space R3.

pv ds = ΟΣ

pv. ds =

Conservation Laws and Euler's Equation. First, we deal with the idea of continuity, or conservation of flow. If p is the density, or mass per unit volume, then the mass of the fluid in volume V (that is, Σ), is equal to f pdV. A two-dimensional surface in R3 has an outward normal vector n which is defined to be positive. The total mass of fluid flowing out of a volume Σ can be written as




Continuity of the flow implies a balance between the flow through the surface and the loss of fluid from the volume. That is, the decrease in mass in the volume must equal the outflow of fluid mass through the surface of the volume, which implies by the Gauss law that


(V. A)dV,

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One can generalize Euler's equation to a form more useful for a dissipative fluid. For this we look at the flux of momentum through a fluid volume. The momentum of fluid passing through an element dV is pv, and its time rate of change expressed in components is

Ə,(pv;) = (d,p)v; + p(Ə,v¡).

We can rewrite ap and a,v; as spatial derivatives by using Eqs. 1 and 2. Then
Ə1(pvi) = -ƏkПik,

where the momentum flux tensor Пik = pdik + pv¡Vk.

The meaning of the momentum flux tensor can be seen immediately by integrating
Eq. 3 and applying Stokes theorem.

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In tensor notation we have

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where the left-hand side is the rate of change of the ith component of momentum pv¡
in the volume and Пind is the ith component of momentum flowing through dS.
Therefore, II, is the ith component of momentum flowing in the kth direction. This
is more easily seen by writing

Пjk nk = рdiknk + pV¡Vkηk = pŋ+ pv(v · n).

Equations 1, 2, and 3 are the basic formalism for classical Newtonian ideal fluids (fluids
with no dissipation) and are also true for flows in general.

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Classical Dissipative Fluids-The Navier-Stokes Equations. The general Euler's
equation is (pv;) = -ƏkПik, where П; is now the momentum stress tensor. The form
of this tensor changes if the fluid is dissipative, for example, if viscous forces convert
the energy in the flow into heat. Traditionally, Ilik is modified in the following way.
Take ПI¡k = på¡k + pv¡v and introduce an unknown tensor of that describes the effects
of viscious stress. Then rewrite the momentum stress tensor as



П¡k = på¡k + pv¡Vk — po¦k = σik + pV¿Vk,

where σik = pdik pok is called the stress tensor and of the viscosity stress tensor.
The form of o can be deduced on general grounds. First we assume that the
gradient of the velocity changes slowly so ok is linear in Okv. Moreover, σ is zero
for v = 0, and under rotation it must vanish since uniform rotation produces no overall
transport of momentum. The unique form that has these properties is

σík = a(Ək V¡ + Ə¡Vk) +bå¡k©jVj,
where a and b are unknown coefficients. It is usually written in the form


σík = v(ƏkV¡ — Ə¡ Vk — 2/3ồi kƏjVj)+ÇÕikƏjVj,
where is the kinematic shear viscosity and is the kinematic bulk viscosity.
For an incompressible fluid (the density is constant so p = po) this tensor simplifies,
and Euler's equation goes over to the incompressible Navier-Stokes equations:


Ə1v + (v · V)v =



1Vp+vV2v and V · v = 0.

·Ə¡p + v· -Vi and Ok Vk = 0.


Ə1v¡ + (v¡Ə¡)v¡

In Part II we end the theoretical discussion of the lattice gas by giving the
incompressible limit of the lattice gas Navier-Stokes equations. ■

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continued from page 178

diate difficulty. We are unable to specify completely the initial state of the system or to follow its microdynamics. It follows that we cannot use a microdynamics that is this detailed. The obvious strategy is to make a smoothened model that reduces the number of degrees of freedom in the system to just a few. This reduction assumes maximum ignorance of the details of the system below some time and distance scale and replaces exact data on events by probabilistic outcomes. Measurements are assumed to be average values of quantities over large ensembles of representative systems. The assumption is that after a sufficiently long time these average observables are a close description of the fluid.

This approach seems very familiar and obvious from elementary courses in statistical mechanics. But it is unclear how to go from a statistical-mechanical description of an atomic system to the prediction of the details of collective motions that come from the evolution of that system. Fidelity to the atomic picture brings with it considerable mathematical difficulties. As we will see below and in "The Hilbert Contraction," the success of the derivation of the Navier-Stokes equations from the kinetic theory picturethat one derives the Navier-Stokes equations with the correct coefficients and not some other macrodynamics-is justified after the fact.

Kinetic Theory and the Boltzmann Transport Equation. Complete information on the statistical description of a fluid or gas at, or near, thermal equilibrium is assumed to be contained in the one-particle phase-space distribution function f(t, r, I) for the atomic constituents of the system. The variables t and r are the time and space coordinates of the atoms and I stands for all other phase-space coordinates (for example, momenta). In this rapid overview of kinetic transport theory, we will not dwell on the many and difficult questions raised

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defines the density function N(t, r) for the particles in the system over all space. Therefore NdV is the mean number of particles in the volume dV. Here dV is a physical volume x L3 whose characteristic length L is much larger than lm, the mean free path of a particle, and much smaller than Lg, some global length, such as the edge of a container for the whole gas. Thus Im <L« Lg.

The basic equation of kinetic theory is the evolution equation for f(t, r, г) in the presence of gas collisions. Imagine first that the system has no collisions. Conservation of phase-space volumes, or Liouville's theorem, tells us that

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The Boltzmann Form of the Collision Term. Let the particles in a two-body collision process have incoming distribution functions g1 and g2 and outgoing distribution functions g and g2. Fixing attention on particle 1, assume that before colliding it occupies a phase-space region dÃ1, and after collision it occupies dĨ1; similarly, particle 2 occupies dг2 before colliding and d2 afterwards. If particle 1 undergoes a collision, dĨ, will not in general be in dr1, and particle 1 is said to be lost from dr1. From these considerations we can compute the functional structure of the general loss term for a binary collision.

The probability of loss will be proportional to the product of four terms: (1) the number of particles of type 1 already in the volume, namely g1; (2) the number of type-2 particles that enter the volume from some phase-space range dг2, namely, g2dг2; (3) the total volume of allowed outgoing phase space, dà ̧dÎ1⁄2; and finally (4) a probability for the collision process P {I}. Now we sum over all possible allowed volumes of phase space. So the total number of losses L in the volume dV and from dr due to binary collision processes is

L = dv dr Pg {r}g182 dг1⁄2dÏ¡dÏ1⁄2.


Similarly, particle gain into the phase

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in velocity space to a local MaxwellBoltzmann form. This means that p, v, and T will depend on space as well as time. These local distribution functions are solutions to the Boltzmann transport equation. For the non-uniform case, one gets a picture of the full solution as an ensemble of local Maxwell-Boltzmann distributions covering the description space of the fluid, with some gluing conditions providing the consistency of the patching.

Recovering Macrodynamics-The Euler Equations. If we assume a simple fluid and neglect all dissipative processes (viscosity, heat transfer, etc.), we can quickly derive the Euler equations (presented in "The Continuum Argument") from the Boltzmann transport equation. But first we need the notion of average quantities and some observations about collisions in a dissipation-free system.

As before, let p(t,r) = ff(t,r,г) dг be the density field of the gas. Then a mean gas velocity v=fv'f(t, r, r) dr, where v' is a microvelocity. We will use v as a macroscopic variable that character

izes cells whose length L in any direction is much, much greater than the mean free path in the gas, Im; that is, L » Im.

Since, by assumption, collisions preserve conservation laws exactly, the moments of C(f), in particular the integrals C(ƒ)d and ƒv С(ƒ) dг, are equal to zero (similarly for any conserved quantity). We use this fact by integrating the Boltzmann equation in two ways: f(B.E.) dr and f v(B.E.) dT (where B.E. stands for the Boltzmann equation). The first integral gives the continuity equation:


Ə1p+Ə; (pv) = 0.

The second integral gives the momentum tensor equation:

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the momentum flux tensor, we need to assume that each region in the gas has a local Maxwell-Boltzmann distribution. With this assumption one can show that the momentum flux tensor in Eq. 7 has the following form:

Пik = pvi Vk + SikP,

where p is the pressure. This form of Ilik gives the same Euler equation that we found by general continuum arguments. (We will see in Part II that the form of II for the totally discrete fluid is not so simple but depends upon the geometry of the underlying lattice. Again by assuming a form for the local distribution function (the appropriate form will turn out to be Fermi-Dirac rather than Boltzmann), ПI¡k will reduce to a form that gives the lattice Euler equation.)

Recovering the Navier-Stokes Equation. The derivation of the Navier-Stokes equation from the kinetic theory picture. is more involved and requires us to face the full Boltzmann equation. Hilbert accomplished this through a beautiful argument that relies on a spatial-gradient perturbation expansion around some singleparticle distribution function fi assumed to be given at to. In "The Hilbert Contraction" we discuss the main outline of his argument emphasizing the assumptions involved and their limitations. Here we will summarize his argument. Hilbert was able to show that the evolution of f for times t> to is given in terms of its initial data at to by the first three moments of f, namely the familiar macroscopic variables p (density), v (mean velocity), and T (temperature). In other words, he was able to contract this many-degree-offreedom system down to a low-dimensional descriptive space whose variables are the same as those used in the usual hydrodynamical description. The beauty of Hilbert's proof is that it is constructive. It explicitly displays a recursive closed tower of constraint relations on the moments of ƒ that come directly from the



he Boltzmann equation is a microscopic equation for collidinggas evolution valid in a very tight regime. It is first order in time and so requires a complete description of the one-particle distribution function at one time, say t = 0, after which its functional form is completely fixed by the Boltzmann transport equation.

it is constructive. It explicitly displays a recursive closed tower of constraint relations on the moments of ƒ that come directly from the Boltzmann equation. The proof also shows that such a contracted description is unique-a very powerful result.

Describing the one-particle distribution function completely is a hopeless procedure, since the amount of information is too large. However, one wants to recover hydrodynamics, which is essentially a partial differential equation for a macroscopic description of the fluid at long times and distances compared to molecular scales. So there must exist a contraction mechanism that reduces the number of degrees of freedom required to describe the solution to the Boltzmann transport equation at such long times and distances. It is not obvious how that can happen, but Hilbert gave a proof that is central to understanding that it must happen and in a rather surprising way. We will call this process the Hilbert contraction. All analyses of the Boltzmann equation are based on this contraction. We would like to give it in detail because it is a beautiful argument, but space forbids this, so we outline how Hilbert reasoned.

Since we don't know what else to do when faced with such a highly nonlinear system, we construct a perturbation expansion in a small variable around some distribution function f, assumed to be given to us at fo. Under some very mild assumptions, and assuming the existence of such a general perturbation expansion in some parameter 6, Hilbert was able to show that the evolution of ƒ for t> to is given in terms of its initial data at to by the first three moments of ƒ, namely p, v, and T. The system has contracted down to a low-dimensional descriptive manifold whose coordinates are the same variables used by the hydrodynamic description. The beauty of Hilbert's proof is that

It must be pointed out that Hilbert's construction is on the time-evolved solution to the Boltzmann transport equation, not on the equation itself, which still requires a complete specification of f. It amounts to a hard mathematical statement on an effective field-theory description for times much greater than elementary collision times, but with space gradients still smooth enough to entertain a serious gradient perturbation expansion. As such, it says nothing about the turbulent regime, for example, where all these assumptions fail.

In standard physics texts one can read all kinds of plausibility arguments as to why this contraction process should exist, but they lack force, for, by arguing tightly, one can make the conclusion go the other way. This is why the Hilbert contraction is important. It is really a powerful and mathematically unexpected result about a highly nonlinear integrodifferential equation of very special form. Beyond Hilbert's theorem and within the Boltzmann transport picture, we can say nothing more about the contraction of descriptions.

The construction of towers of moment constraints, coupled to a perturbation expansion that Hilbert developed for his proof of contraction, was used in a somewhat different form by Chapman and Enskog. Their main purpose was to devise a perturbation expansion with side constraints in such a way as to pick off the values of the coupling constants-which are called transport coefficients in standard terminology-for increasingly more sophisticated forms of macrodynamical equations.

One makes the usual kinetic assump

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