Exact Evaluation of Collision Integrals for the Nonlinear
Boltzmann Equation
K. Kabin, Bernie D. Shizgal
Department of Chemistry, University of British Columbia
2036 Main Mall, Vancouver, B.C. V6T 1Z1 Canada
Abstract. In this work, we evaluate analytically the collision integrals for the nonlinear isotropic Boltzmann
equation for the hard spheres for several initial distribution functions. The initial distribution functions
include combinations of Maxwellians with different temperatures and delta-functions. We do not find any
preferential scattering into the high energy tail in the case when the initial distribution function is a sum
of two Maxwellians with different temperatures. Our results may be used to devise a new method for
the numerical integration of the nonlinear Boltzmann equation which avoids an approximate numerical
evaluation of the collision integrals.
I
INTRODUCTION
The evolution of the distribution function of a one component gaseous system is governed by the nonlinear
Boltzmann equation. The nonlinear Boltzmann equation is more difficult to analyze than the linearized version
upon which traditional transport theory is based [1]. The present paper is concerned with the approach to
equilibrium from some arbitrary nonequilibrium initial distribution. The detailed time dependence of the
distribution function depends on the interaction potential between the gas particles or equivalently on the
cross section for binary collisions. Most of the previous treatments of relaxation to equilibrium [2]- [9] are
based on the nonlinear Boltzmann equation use a hard sphere cross section or the cross section for "Maxwell
molecules" interacting by an inverse fourth power potential. Also, the analyses have generally been restricted
to relaxation of isotropic distributions. A great deal of interest in this problem was generated by Krook and
Wu [2] who discovered an exact solution for Maxwell molecules and particular initial conditions. This exact
solution was also independently reported by Bobylev [3]. There have been essentially two methods of solution
of the time dependent nonlinear Boltzmann equation. One method involves the expansion of the distribution
function in a basis set of polynomials and transforming the Boltzmann equation to a set of coupled ordinary
differential equations for the expansion coefficients [5,8]. An alternate approach considers the discretization
of the distribution function at a set of discrete points. The Boltzmann equation is thus reduced to a set of
ordinary differential equations in terms of the distribution function evaluated at the grid points [4,9].
The present paper presents several analytical results for the time dependence of initial values of collision
integrals that are useful in the interpretation of the time dependent calculations. We also present a new
numerical treatment of the nonlinear integral operators in the Boltzmann equation and an accurate study of
the relaxation to equilibrium. The motivation of this work is primarily to include self collisions in models of
energetic atoms in planetary exospheres [11] - [14]. Shizgal [11] presented a simple model for the calculation
of the distribution of hot oxygen atoms in the exosphere of Venus based on the Boltzmann equation. In this
model, the thermalization of hot atoms involves the assumption that there is a background of thermal oxygen
atoms and as a consequence the problem is linear. This may be true at low altitudes where the density of
energetic atoms is low relative of the density of thermal atoms. However, at very high altitudes the density
ratio is reversed and one must include collisions between energetic atoms. Therefore the nonlinear collision
operator describing self-collisions between energetic atoms must be included. Similarly, exospheric models for
the terrestrial hydrogen geocorona as well as the exospheres of Mars and Venus are based on a model of H
atoms dilutely dispersed in a background of thermal oxygen and the problem is thus linearized. However, at
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
35
very large altitudes the density of H is greater than O and the nonlinear collision operator describing H-H
self-collisions must be included. It is towards a resolution of these problems and others that the present study
was initiated.
II
NONLINEAR ISOTROPIC BOLTZMANN EQUATION
We consider the time dependent isotropic speed distribution function, f(y\, t), of an ensemble of particles of
mass m, as a function of reduced speed, y = VI/VQ where VQ = y^fcT/m. For the hard sphere cross section,
(To, we use the nondimensional time t in units of r = ^/Trm/WKT/ncrQ where n is the uniform particle density
and T is the equilibrium temperature. The nonlinear, isotropic Boltzmann equation for a hard sphere cross
section is given by
df(yi, t)/dt = Fin (2/1, t) - Fout (2/1, t)
where
r>00
F0ut(y\,t) = f ( y i , t ) I Sout(yl,y2)f(y2,t)dy2
Jo
(1)
and
Fin(Vi,t)= t
Jo
I
Jo
Sin(y'l^yl,y2)f(y'l,t)f(y2,t)dy'ldy'2
(2)
For a hard sphere cross section, the scattering kernels can be written as follows [8],
min
[ ^2/1,2/2) = \hj-TT
(?/i'2/2 ? yi,2/2
V z 2/i 2/2
where H(x) is the usual Heaviside step function and energy conservation gives y| — 2/{2 + 2/22 ~~ 2/iOwing to the conservation of particle number, we have the relationship between the in- and out-scattering
terms, that is,
/•oo
i,2/2) = /
Jo
Sin(y
We also have the symmetry property [6],
Sin(y[ -> 2/1,2/2) = ^n(2/2 -> 2/1,2/i)
(3)
The in-scattering collision integral, Fin(y\^t) defined by the double integral (2) can be rewritten in a more
convenient form in terms of the functions
/*yi
Fi(yi,t) = \ f(y,t)dy
Jo
and
In terms of these functions,
36
Fin(yi,t) = j| [2yiFl(yl,t)F2(yl,t)+y21F%(yl,t)+I(yl,t)]
(4)
where
So
The double integral above is over the area 5o, with boundaries defined by the circle y® + y® = y± and the
straight lines y[ = y± and y^ = y\. This is a significant simplification of original expression (2) because we have
reduced most of the double integrals to the products of single integrals. With the substitution, £1 = (y[/yi)2
and &> = (2/2/2/i)2 into /(yi,t), we get,
,t) = V-j- I I ^l+f*~lf(y[,t)fM,t)d£ld£2
4 J JSl
(5)
?i?2
This double integral is now over area $1, which is a triangle with the vertices at (1,0), (1,1), and (0,1). This
integral can be efficiently evaluated using, for example, Gaussian integration rule for a simplex [16].
Ill
INITIAL MAXWELLIAN DISTRIBUTIONS
We are concerned with the analytic evaluation of the initial time derivative dfMax(y\)/dt
Maxwellian distribution at a specified temperature, that is,
for an initial
where f) = T/T(0). We begin by evaluating -Tv^t (2/1 ? 0) denoted by F0ut(y\) and hence we require the integral
o
_
which is the usual collision frequency [1], and we have that,
yi
it2
o
—
(6)
1
¥ ~ | _
yi
—V —
J
where a = f3y\. To evaluate the integral /(yi,0) denoted by I(y\) occurring in Fin(yi) in Eq. (4), we consider
the transformation a? = £1 + £2 — 1? z = £2 (Jacobian =1). We then evaluate /(z/i), by reducing the double
integral to a repeated one,
yG
/•
/•
_________
^.6
^ J Jso
4 70
,.6
4
A
/•!
_
p-a
/•!
7x
f /T
; - — -_ ——
4 a3/2 2
For a Maxwellian distribution, we also have that,
o
2
o;
and ^2(2/1) = y^e~a/2a. After some algebra, it can be verified with these results that the collision integral for
the Maxwellian distribution dfMax(yi)/dt = ^ n (z/i) — Fout(y\) is indeed zero.
37
Now consider an initial distribution that is a sum of two Maxwellians at two different temperatures, that is
/(yi,0) = a$e-Plv* + by^e~^^ where fa = T/Ti(0) and fa = T/T2(Q). The form of the collision integral,
Eq. (5), is then
I(yi) = ^[a2Iaa(yi) + b2Ibb(yi) + 2afc/ab(yi)]
where Iaa(yi) and Ibbfyi) are analogous to (7). Since the terms Iaa(yi) and Ibbfyi) will cancel with the
corresponding terms from Sout(yi)^ it is only the cross term, that gives a non-vanishing contribution to
dfMax(yi)/dt. This term is,
2 ___
/
/.i
y/x=le-aixdx I
e~^-a^zdz
(8)
Jx-l
e""1
®2 - ai
e""2
x^rerf(x/c^)
/2
2a%
x^rerf(x/aT)
a
i ~ ®2
2al/2
Taking the limit OL\ —» a^ we can recover expression (7) for I(yi).
Now it is easy to evaluate ^i n (yi),
*S~in(yi) — 0,0 ——•= ———————————————-———————-
4\/2
"I "2(0:1 -0:2)
Finally, using [1] we can write
Thus, we have evaluated analytically the time derivative dfMax(y\)/dt
for the sum of two Maxwellians at
the initial time. The variation of this initial time derivative is shown in Fig. 1 for a sum of two Maxwellians.
In the figure, we keep the amplitude and temperature ratio of the first Maxwellian fixed at unity and vary
separately the temperature ratio (Fig. 1A) and the amplitude (Fig. IB) of the second Maxwellian. It is clear
from the figures that the integral of these functions is zero owing to particle conservation. It is interesting to
notice that the time derivative is always negative for the largest values of the reduced velocity, and so we see an
initial downscattering and not a preferential upscattering into the high-energy tail. This holds for any f a / 'fa
ratio. Apparently, for this kind of initial distributions, the high-energy tail is always initially over-populated
compared with the equilibrium distribution. These results are in contrast to the findings of [9] who obtained
a transient overshoot of the high-energy tail. However, the work in [9] concentrated on the time evolution of
the initial distribution given by the two delta- functions.
It is quite reasonable to expect that at any later time, the time derivative will be only smaller. We note, that
the results of this section may be used to develop a new method for numerical integration of the Boltzmann
equation. If the initial distribution can be expanded into a linear combination of the Maxwellians with different
temperatures (or Maxwellians and delta- functions, using the results of the following sections), then the above
formulas give the exact expressions for the time derivative of this distribution function. After a time-step, which
has to be smaller than the collisional time, the updated distribution function has to be expanded again into a
new linear combination of the Maxwellians. This method may have some advantages for certain applications
because it completely eliminates the need for a numerical evaluation of collision integrals. In a sense, this
numerical method is similar to Godunov's methods of the gas-dynamics.
IV
DELTA FUNCTION INITIAL DISTRIBUTIONS
With an initial delta function distribution, f ( y i ) = o,6(yi — ?/o)? we have with Eq. (2) that
Fout(yi) = a2<%i - 2/o) Sout (2/1, 2/o), and since ^Jirf(yi) = 0 unless 3/1 = y0, we have that
t (2/o,2fo)
=
3
is determined with Eq. (2) and energy conservation which with y[ = y^ = y® reduces to y| —
o — 2/i- Hence ^^(2/1,2/0) — 0? if 2/1 > V^Z/o- Conservation of momentum and energy requires that no
38
0.20
0.04
0.15
0.10
0.02
0.05
0.00
0.00
-0.05
-0.02
-0.10
-0.15
-0.04
-0.20
0
FIGURE 1. The initial time derivative for a sum of Maxwellians, /(yi,0) = ay^e~
+ byle'^; a = 1, ft = 1:
(A) b = 1; Values of ft are (a) 3, (b) 5 (c) 7 (d) 10 and (e) 12. (B) ft = 7; Values of b are (a) 1 (b) 2 (c) 4 and (d) 8.
35
1.4
30
1.2
25
1.0
^ 20
§
10
15
|O.S
0.6
0.4
10
0.2
5
0.0
0
0
1
0
1
2
3
y
FIGURE 2. The initial time derivative for a sum of delta functions /(yi, 0) = a<5(yi — MI
= 0, u2 = 3 (B) a = 1, ui= 1, u 2 = 3; Values of b are (a) 1 (b) 2 (c) 3 and (d) 4
— ^2); (A) a = 1, b
particles after the first collision will have a velocity larger than \/22/o- The cut off at \/22/o appears because at
the initial moment there are no particles with relative velocities larger than \/22/o- For y$ < y\ < \/22/o and for
2/1 < 2/0? the min function in Eq. (2) is y<2 and yi, respectively. We therefore have that
/Fl f y* t_____
m (2/1,2/o) = « W o1— < 2/i \A/5 - 2/i
V
y
° I 0
2
r
if
?/i<2/o
if 2/o < 2/i <
if 2/1 > \/22/0
(9)
•^/n (2/1? 2/o) is a continuous, although not a smooth function. It is easy to check by an elementary integration
that
f°°
Jo
_
°
Ut
f°°
6
_
1
Jo
2^^
3
consistent with particle number conservation. Similarly, energy conservation gives,
Jo
Jo
We now consider the time derivative of the initial distribution consisting of a sum of two delta-functions,
/ = aS(yi —Ui)-\- bS(yi — u^). The evaluation of the out-scattering integral follows the calculation for a single
39
FIGURE 3. The initial time derivative for a sum of a delta function and a Maxwellian, f(yi, 0) = a6(y\ —yv)-\-byie &yi ;
yo = 2, b = 1, (A) /? = 4; Values of a are (a) 1, (b) 2 (c) 4 and (d) 8. (B) a = 1; Values of /? are (a) 4 (b) 0.5 (c) 0.375
and (d) 0.2
delta function and we have that,
+ bu2) + bS(yi - u2)——(aui + bu2)
The in-scattering part is explicitly given by the expression
+a
Figure 2A shows the variation of the initial time derivative for a single delta function at y$ = 3 and unit
amplitude. Figure 2B shows the case for a sum of two delta functions at u\ = 1 and u2 = 3 versus the amplitude
b. In this Figure, one can see the relevance of this calculation to the results of [9] regarding the transient overpopulation of the high-energy tail. For the interaction of the two initial S-functions we see the overpopulation
of the high-energy part of the distribution which analytically confirms the numerical calculation of [9]. Of
course, our results are applicable only for times smaller than the mean collision time, so the overpopulation of
the tail did not have a chance to propagate all the way to infinitely large speeds. Instead, it stops at -\/u{ + u2,
the limit imposed by the conservation of energy, but the trend to overpopulate the tail is definitely there.
V
INITIAL DELTA FUNCTION AND MAXWELLIAN DISTRIBUTION
Next we can calculate the integrals for the initial distribution function f ( y i ) = a<5
the substitution of this distribution in Eq. (1) for ^(?/i,0) we find that,
J(yi)
—yo) -\-by2e~@yi . With
(10)
The first term in a2 in Eq. (10) arises from the delta function analogous to the result in Section II. The term
in J(yi) is the result of integral operator on the Maxwellian and the last term is the result of the integral
operator on the delta function.
We get four contributions to the in-scattering kernel one proportional to a2 resulting from the delta function
and gives Fin(yi) as obtained in Section II. The term in b2 is, owing to detailed ballance, identical to the
analogous term in Eq. (10). The integration over the delta functions in the "cross terms" proportional to
40
0.6
0.8
x 0.4
0.4
0.2
0.2
0.0
0.0
FIGURE 4. Time evolution of the distribution function; the dimensionles times t = t' /r are equal to (a) 0 (b) 0.1
(c) 0.4 (d) 0.7 and (e) 1.4; the dashed curves are the equilibrium distributions; (A) f ( y , 0) =
= y22~
e 5y +
ab can be done, and with an interchange of y[ and y'2 and the use of the symmetry property (3), the two
"cross-terms" are equal and hence we get,
/>oo
/
Jo
Sin(y0
The integrals over the delta functions yield Sin(yo —» 2/152/2)- The expression for ^in(yi) is given by (9).
It is easy to see that if y$ > y\
and if yQ <
if 0 < 2/2 ^ y 2/1 ~ 2/0
if ^2/1-2/0 < 2/2
if 2/2 > 2/i
This makes the remaining integration over y2 easy and gives
if 2/o
4
t
As it could have been expected Fin(y\) is continuous although not a smooth function at y± = y$.
Figure 3 shows the initial time derivative of the distribution function for the sum of a delta function and a
Maxwellian, that is /(yi,0) = a6(yi — ?/o) + by^e~^2yi. Figure 3A shows the variation of the time derivative
versus the amplitude a of the delta function at y$ = 2 whereas Fig. 3B shows the variation versus the
temperature ratio (3. An important aspect of the results is whether y$ < -y/3/2/3 (Fig. 3A) or y$ > -y/3/2/3,
which in Fig. 3B is equal to 0.612, 1.732, 2 and 2.74 for curves a-d, respectively.
VI
TIME DEPENDENT SOLUTIONS
We solve the time dependent nonlinear Boltzmann equation with a finite difference technique. The reduced
speed variable is discretized on the finite interval [0,ymaa;], that is y^i = ^+Ay. Similarly, the time variable is
discretized according to £ n+ i = tn + At. We use an Euler time integration algorithm such that the discretized
analog of the Boltzmann equation is
41
F™ut(yi) is determined from Eq. (1) with a Simpson's rule integration for the integration over 7/2- The
discretization of the in-scattering integral presents a major difficulty that we have overcome with an application
of Gaussian integration discussed by Stroud [16]. The double integral over ^ and £2 m Eq. (5) over the triangle
Si is divided up into several triangle integrations. Further details of these numerical procedures will appear in
a subsequent publication [15].
The results of this time integration of the Boltzmann equation is shown in Fig. 4. Figure 4A shows the
time evolution of the an initial distribution, f(y) = y2e~6y + e^y~3^ . The times are in units of r defined
earlier. The considerable upscattering that occurs in the time evolution for this initial distribution function
is consistent with the results in Fig. 1. The present method of discretization and evaluation of the collision
integrals is very efficient and computationally fast. The distribution functions that are calculated are smooth
functions and do not give rise to nonphysical oscillatory solutions that can arise with polynomial expansions.
Figure 4B shows the time evolution of the an initial distribution, /(y,0) = e~6(y~1^ -\-e~6(y~3} ? which is more
localized than the initial distribution in Fig. 4A.
VII
SUMMARY
In the present paper, we have introduced a new discretization scheme for the collision integrals in the isotropic
nonlinear Boltzmann equation. The new scheme was found to be efficient computationaly and easily coded. We
have also evaluated collision integrals for several different initial distributions involving delta functions and/or
Maxwellians. These results have been used to interpret the time dependent behaviour of the distribution
functions for different initial conditions. Future work is concerned with the determination of energy dependent
relaxation times and a comparison between the linear and nonlinear Boltzmann equations. There is also a
considerable interest to compare the energy dependendence of these relaxation times for a hard sphere gas with
the Coulomb case in a plasma. In addition, the extension of this work to anisotropic distributions is also being
considered.
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