THE CONTINUUM 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 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 II¡k = på¡k + pv¡vk and introduce an unknown tensor of that describes the effects of viscious stress. Then rewrite the momentum stress tensor as where σ¡k = påik - pok is called the stress tensor and of the viscosity stress tensor. 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 П 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 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 Im, the mean free path of a particle, and much smaller than L, some global length, such as the edge of a container for the whole. gas. Thus lm « 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 (We use the convention that repeated indices are summed over.) Equation 3 defines the free-streaming operator, which represents the local change in ƒ per unit. time caused by the independent motion of particles alone. Now imagine a simple isolated gas with collisions. If C(f) is a function that models the rate of change of the distribution function f caused by collisions, then C(f) dVdr is the rate of change per unit time of the number of molecules in the phase-space volume element dVdг. The 1, 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 ğ and g2. Fixing attention on particle 1, assume that before colliding it occupies a phase-space region dr1, and after collision it occupies dĨ1; similarly, particle 2 occupies dг2 before colliding and dr1⁄2 afterwards. If particle 1 undergoes a collision, dr, 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 dr2, namely, g2d2; (3) the total volume of allowed outgoing phase space, dĺ1dÎ1⁄2; and finally (4) a probability for the collision process P {T}. 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 df In Part II we will use the same reasoning to construct the Boltzmann equation for the discrete lattice gas. The explicit form of the lattice gas collision operator is much simpler than in standard kinetic models. Note that the Boltzmann form for the (GL) collision term implicitly assumes only two-body collisions. It also assumes the collisions are pairwise statistically independent events occurring at a single point with detailed, or at most semi-detailed, balance symmetry for collision probabilities. 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) = ƒƒ(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(f)d and ƒ v 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.) dr (where B.E. stands for the Boltzmann equation). The first integral gives the continuity equation: 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 = pv¡Vk + dikP, where p is the pressure. This form of II 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), II 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 f 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 THE HILBERT CONTRACTION T 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. 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 to. Under some very mild assumptions, and assuming the existence of such a general perturbation expansion in some parameter 8, 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 f, 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 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. 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 |