Experimental Test of Boltzmann-Lorentz Kinetic Equation*

VOL. 16, NO 2, 3

CHINESE JOURNAL OF PHYSICS

;uu;;;\ 1978

A Paper Contributed to the Special Issue of the Chinese Journal of Physics Celebrating Seventieth Birthday of Prof. Ta-You Wu

Experimental Test of Boltzmann-Lorentz Kinetic Equation* SOW-HUN CH E N (& 9 j$) Nuclear Engineering Department Massachusetts Institute of Technology Cambridge MA 02139

(Received July 8, 1978) Current status on experimental verification and assessment of the range of validity of the Boltzmann-Lorentz equation for neutral gases is briefly reviewed. Emphasis is made on importance of accurate determinations of various time correlation function and K-dependent correlation functions measurable by neutron scattering or by computer molecular dynamics simulation. Data are presented which show that the “Scaled Hard Sphere Boltzmann Theory” has a range bf validity up to the critical density.

I. INTRODUCTION T has been over a century since Ludwig Boltzmanncl) first wrote down his celebrated kinetic II equation for the low density gas in 1872. From current standard text books on statistical mechanic@’ one tends to get an impression that the Boltzmann equation is the last word on transport and dynamics in dilute gases. Aside from the fundamental contribution in clarifying questions on approach to equilibrium and establishment of the equilibrium velocity distribution function, the Boltzmann equation is a unique and important statistical mechanical equation which directly connects the dynamics of microscopic. as yell as macroscopic density fluctuations in a dilute gas to a given interparticle interaction. In spite of the tremendous literature which has accumulated over the last hundred years, an inspection of the two Proceedings of the International Centennial Boltzmann Symposiumscs’ reveals that a straight forward extension of the equation to higher orders in density have met with great difficulties. Even to the lowest order in density the Boltzmann equation is only approximate in the sense that it ignores the detail dynamics of density fluctuations in the spatial scale of the order of the particle dimension and in the time scale of the order ‘of the duration of particle collisions@b’. From the experimental standpoint the traditionally emphasized transport coefficient measurements”’ require only study of the long wavelength and slow fluctuations of the particle density and therefore the small-distance and short-time behavior of the. Boltzmann equation was never really tested. Two other experimental situations involving thesound dispersion and * Research supported by NSF grant CHE-7717350.

(1) L. Boltz&nn, “F urther &dies on the Thermal Equilibrium of Gas Molecules”, Wien. Ber. 66, 275

(1972); this article, as well as others by Boltzmann, is translated in.% G. Brush, Kinetic Theory, Vol. 2, Irreversible Processes, Pergamon Press (1966). (2) For example: (a) K. Huang, Statistical Mechanics, John Wiley (1963); (b) R. Balescu, Equilibrium and Non-equilibrium Statistical Mechanics, John Wiley (1975). (3) (a) The Boltzmann Equation-Theory and Applications, Ed. E.G. D. Cohen and W. Thirring, SpringerVerlag (1973); (b) Transport Phenomena, Ed. J. Kestin, AIP Conference Proc. 11, American Institute of Physics, New York (1973). (4) J. Kestin and E. A. Mason, “Transport Properties in Gases” in p. 137, Ref. 3(b). 9s

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EXPERIMENTAL TEST OF BOLTZMANN-LORENTZ KINETIC EQUATION

attenuation@’ in an equilibrium low density gas or the related Brillouin light scattering spectra”’ in gases; and the shockwave structurec7) in a non-equilibrium rarified gas, require solution of the Boltzmann equation for finite wavevectors and frequencies. However, even in these situations the distance scales involved are much larger than the diameter of gas atoms and the time variation of the density fluctuations much longer than the atomic collision time. With the advent of thermal neutron scattering@) one can measure the density fluctuation spectrum S(K, o) in the range where K u - l a n d or-l. Here u is the typical atomic diameter, t-o/V, the duration of atomic collisions and, V,=)/kiT/M, is the thermal speed of atoms. Thus by calculating S(K,o) from the Boltzmann equation and systematically comparing them with experimental measurements in dilute gases it is now possible to test the non-local (in space and time) aspect of the collision kernel in the equation. Another possibility of generating accurate correlation functions in a gas of atoms with a specific interatomic interaction is via the computer molecular dynamics (CMD) technique@). The versatility of this technique lies not only in its ability to generate various correlation functions, including those impossible to be measured in laboratory experiments, but also that it offers “controlled” experimental data generated with an aritifical interatomic potential function such as the hard sphere potential. This latter possibility greatly facilitates a more straightforward comparison of theoretical results with the “experimental” datacr”). In this paper we shall use data taken from these two categories of experiments to test the range of validity of the Boltzmann equation. We shall, however, limit ourselves to discussion of the simpler “self” Boltzmann equation which is usually called the Boltzmann-Lorentz Self-diffusion equation in the literature. The main reason is to keep the length of the paper reasonable. As for the experimental test of the full Boltzmann equation, we refer the reader to forthcoming articleclr’. In the next section we shall briefly introduce the Boltzmann equation and its recent extension to finite k domain. We shall then specialize to the self part of it for the subsequent discussions. In section III we shall discuss the velocity autocorrelation function and the kinetic energy autocorrelation function and in section IV we shall discuss the test particle density correlation function S,(K, 0). In the concluding section V we comment briefly on the next stage of theoretical as well as experimental activities needed for a complete test of the range of validity of the Boltzmann equation and its extensions.

II. THE BOLTZMANN-LORENTZ EQUATION The basic kinetic equation is constructed for a distribution function f(r, v, t)dsrdavEnumber of particles in the phase space volume element dsrdsv. In the absence of external field the integrodifferential equation can be written in the formc12’

af -~,+v~vf-J[ffil.

(1)

The first term which expresses the rate of variation of particles in the volume element around position r and velocity v is given by contribution from two sources: direct streaming in (second term) Studies in Statisricul Mechanics, Vol. 5, Ed. J. de Boer and G. E. Uhlenbech, North Holland (1970). N. A. Clark, Phys. Rev. A12, 232 (1975); S. Yip, J. Acoust. Sot. Am. 49, 941 (1971). H. M. Mott-Smith, Phys. Rev. 82, 885 (1951); C. K. Chu, Physics Fluids, 8, 12, 1450 (1965). See for example, S.-H. Chen, H. C. Teh and S. Yip, “M olecular Interactions and Neutron Scattering from Moderarely Dense Gases” in Symposium on Neurron Inelastic Scattering, IAEA, Vienna (1978). See a review article by W. W. Wood, in Fundamental Problems in Starisrical Mechanics, Ed. E. G. D. Cohen, Vol. III, p. 331, North Holland (1975). B. J. Alder, D.M. Gass and T. E. Wainwright, J. Chem. Phys. 53, 3813 (1970). S.-H. Chen, K. Touqan and S. Yip, “E xperimental Test of the Generalized Enskog Equation for Moderately Dense Gases”, to be published. G. E. Uhlenbeck and G. W. Ford, “L ectures on Slatisrical Mechanics”, Chap. 4, American Mathematical Society, Providence RI (1963).

(5) See review article by T. D. Foch and G. W. Ford, in (6) (7) (8) (9) (10) (11) (12)

97

SOW-HSIN CHEN

and collisions (third term). Thus a particle has only two state of motions: repetitions of a welldefined free streaming interrupted by a brief binary collision. This physical situation occurs obviously only at low densities. What is not obvious is, how low? The collision integral J[/f,] can be written explicitly as:

We picture a pair of particles with velocities (v, vl) makes a collision and emerge with respective. velocities (VI, vi). g=_-/ v-v, I=/ v’-vi 1 is the relative velocities before and after the collision. The vector g is the same before and after the collision except it turns over an angle 8. I(g, 0) is the differential cross section for a collision into a solid angle dQ=sin Bd&f~, and is calculable once the interatomic potential function $(r) is given. f’/‘{ term gives the probability of restituting collision (u’, u+o, v,) (gain) and ffi term gives the probability of direct collision (v, v,)-+(v’, vi) loss. Since we are calculating the net gain of particles with velocity v we have to integrate over all possible velocities vl. Some observations are due: (a) this equation really describes collision and propagation of a “pair” of particles; hence it is akin to description of propagation of a density distrubance; (b) it takes into account only binary collisions; (c) it ignores spatial correlation between two colliding particles, i. e., ignores the finite size effect or excluded volume effect of atoms; (d) duration of collision is taken to be instantaneous; (e) it assumes the velocities of two atoms which are about to collide, are uncorrelated; this assumption excludes possibilities of repeated collisions or “correlated binary collisions” which affect long time behavior of J. At very low density it is reasonable to expect that the assumptions (b) and (e) are valid. However, there is no obvious reason why the effects indicated in (c) and (d), which will give rise to a non-local (in space and time) collision kernel, should be negligible even at low density. Since most experiments are done in equilibrium situations the deviation of f(r, v, t) from the equilibrium distribution r@(v) ought to be small. Here II is the equilibrium number density of the gas and @p(v) the Maxwell-Boltzmann distribution

@W=

(&$)3D exP(--G).

(3)

For the study of density fluctuations in a equilibrium gas we therefore put f(r, v, t) = ti@(v) + F(r, v, t>1.

(4)

Substituting (4) into (1) and linearizing it we get the linearized Boltzmann equation, describing density fluctuations. 3F

-at +v.VF= J[F, Fl]

(5)

J[F, ~,~=nJdgv,j- dG’g Z(g, 0) [F’O; + F;O’- FO, - F,‘B].

(6)

where

In order to compute the density fluctuation spectrum S(K,w) from Eq. (6) it was realized by Nelkin and YipcL3) in 1964 that one should solve Eq. (6) as an initial value problem with an initial condition F(r, v, 0) -J(r)@(v) + n&)@(v)

(7)

where g(r) is the equilibrium pair correlation function. This is equivalent to saying that at t=O we locate a pair of particles one at the origin and the other at a distance r away with probability ng(r), with both velocities Maxwellian distributed. If we observe the subsequent propagation of this pair of particles, the associated density disturbance will be governed by the Eq. (6). This interpretation of the:solution of the Boltzmann equation opens up a tremendous possibility in terms of experimental (13) S. Yip

and M. Nelkin, Phys. Rev. 135, A 1241 (1964).

, ..,

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test of the equation. problem as

Because now we can immediately identify the solution of the initial value

G(r, r)=Jd% F(r, v, 1)

(8)

where G(r, t) is the well known van Hove correlation function (I*) observable by the neutron scattering. Since S(K, w) is nothing but the double Fourier transform of G(r, t), if we introduce a Fourier-Laplace transform &vz) =

Jd+eik-r Jam dte-ztF(r, v, r)

(9)

We can then transform Eq. (5) into _ _ (z--k*v)&kvz)=@(v).S(k)+J[F, F2-j.

(10)

The density fluctuation spectrum S(k, w) can now be obtained from solution of Eq. (10) via

J

S(K, o) - -;5 R e d%F(Kvz = iw).

(11)

Extension of the Boltzmann Eq. (1) to include the non-local effects has recently been made(“). The expression is simple and tractable in the case of a hard sphere gas. Because collisions between hard spheres are instantaneous the non-locality in time does not exist and one has to worry about only non-locality in space. The correct low density collision integral JLD is given as

[,r]=pl~,:+eik.;~F:~~_~~l_e-ik.;.~~~.

(13).

We notice that the bracket in (13) is similar to the bracket in (6) except for the spatial Fourier transform and phase factors efik*;s in two terms. Here P is a unit vector joining centers of the two colliding hard spheres. A further extension of the low density non-local Boltzmann equation to higher densities by taking into account only the excluded volume effects, in the same spirit as the At higher densities there, well known Enskog approximation cl@, has been made by various authorsc17’. are two effects to be considered: (A) the streaming term in the Boltzmann equation has to-be modified because now the pair of particles does not propagate freely, but they move in the mean field produced by all the other atoms. This mean field term is given to all order in density as ik*vr@(v)[C(k)-g(o)C,(k)F(kz)

( 1 4 )

This term is to be added to the left hand side of Eq. (10). C(k) and C,(k) are the direct correlation functions at finite and low densities, respectively, and F(kz) is fl(kvz) integrated over v. In the low density limit the pair correlation function at contact g(u)-+l, C(k)-C,(k) and the term (14) tends to zero, as it should be; (B) the collision frequency is going to be enhanced because higher order collisions such as three particle collisions, repeated two particle collisions, and a two-particle collision with a third particle overlapping, etc.cX8). The so-called generalized Enskog approximationclg’ takes into account only a certain class of these events more akin to the excluded volume effects. It amounts to multiplying the non-local low density collision integral (12) by a density dependent constant g(u), i. e., --___(14) L. van Hove, Phys. Rev. 95, 249 (1954). (15) G.F. Mazenko, T.Y. C. Wei and S. Yip, Phys. Rev. A6, 1981 (1972). (16) See for example, S. Chapman and T.G. Cowling, The Marhematical Theory of Non-Uniform Gases, Chap. 16, 3rd Ed., Cambridge Univ. Press (1970). (17) See G. F. Mazenko and S. Yip, “R enormalized Kinetic Theory of Dense Fluids”, in Statistical Mechanics, Pt. B, Ed. B. J. Berne, Plenum Press (1977) and references therein. (18) See J. V. Sengers, “The Three-Particle Collision Term in the Generalized Bolrzmann Equation”, in Ref. 3(a). ‘_(19) P.M. Furtado, G. F. Mazenko and S. Yip, Phys, Rev. 12A, 1653 (1975);

SOW-HSIN CHEN

J&T p11= g@)Jm[E Fll.

99 (15)

Range of validity of the generalized Enskog equation combining modifications (A) and (B) has recently been tested with extensive hardsphere CMD simulationP). A test against neutron scattering data from low density rare gases is also reported@‘. It has been shown by Konijnendijk and Van LeeuweP) that the transport coefficients calculated from the generalized Enskog equation agree with those from the classical Enskog equatioP’. This shows that we cannot learn anything about non-locality of the collision kernel from studies of transport coefficients. In order to specialize to the Boltzmann-Lorentz equation of self-diffusion, we go back to the original non-linear Boltzmann equation (1) and (2). In a self-diffusion problem we are interested in propagation of a tagged particle, or a test particle in the gas. Let us call the distribution function of the test particle fs(r, v, t) and of the other (bath) particles f,(r, vl, t). Then since all mechanical properties of the test particle and bath particles are identical, their scattering cross sections are all identical too. We thus have

The collision integral J[/sfi,]=O since we have only one test particle. The collision integral between the test particle and a bath particle should have the bracket part equal to

[“I-fpq-fs&

(17)

where replacement fb-+& has been made because we are interested only in propagation of the test particle and not a pair of particles as in the case of the full Boltzmann equation. We notice from (17) and (16) that the Boltzmann-Lorentz equation is inherently a linear equation. A finite k and finite density extension of the Boltzmann-Lorentz equation (16) and (17) has been made(z1’ for the case of hard sphere system, in the same spirit as the generalized Enskog equation in the case of density fluctuations. First of all the mean-field term is zero in the test particle case because the mean force on ‘one’ particle in a homogeneous system is always zero. Second, the nonlocal effect in the collision integral (e. g., equation (13)) does not arise at the level of Enskog approximation; the only modification to the low density collision integral is simply to multiply it by a factor g(u). This is an extremely interesting result which should be amenable to experimental tests. The complete Boltzmann-Lorentz-Enskog equation is therefore

where F,(kvz,)=J d3reik*r~om dte-z’f,(r, v, t)

(20)

and we also use the initial condition

f&, v, 0) =@v)@r).

(21)

We note that in order to get equation (19) we merely have to drop the two terms with phase factors in equation (13). This is equivalent to ignoring the propagation of the bath particle. III. VELOCITY AND KINETIC ENERGY AUTOCORRELATION FUNCTIONS The simplest case to discuss about solution of the Boltzmann-Lorentz-Enskog equation is to take (20) B. J. Alder, W. E. Alley and S. Yip (to be published). (21) H. H. U. Konijnendijk and J. M. J. Van Leeuwen, Physica 64, 342 (1973). (22) See for example, J. 0. Hirschfelder, C. F. Curt& and R. B. Bird, Mofecular Theory of Gases and Liquids, John Wiley & Son (1954).

100


the spatially uniform case. This is equivalent to taking the k-0 case in equation (18). Solution of this equation therefore directly depends only on properties of the collision integral JEs. For the convenience of the subsequent discussions it is more advantageous to recast equation (18) in time domain. We start by introducing a space-time test particle correlation function, or better, its Space-Fourier transform: (22) At t-0 we have F,(kvv’O) - (-A- ,$lJ(V-Vi(0))J(V’-Vt(O)))=d(V-V’~(V)

(23)

In can be shown@3 that because of linear response of the system to fluctuations from thermal equilibrium the test particle correlation function F,(kvv’t) obeys the same kinetic equation as the test particle distribution function j(kvt). It is, however, more transparent in reasoning to work with F,(kvv’t) because the intermediate scattering function Fs(kt), which is the spatial Fourier transform of the van Hove Self-correlation function G,(r, t), can be obtained from it by contraction with respect to velocities v and v’, i. e., FM -

s

d3v dWF,(kvv’t). s

(24)

Furthermore the velocity and kinetic energy correlation functions can also be obtained from various other contractions of F,(k=Ovv’t). For example