Inelastic scattering models in transport theory and their small mean

Euro. Jnl of Applied Mathematics (1997), vol. 11, pp. 1{000. Printed in the United Kingdom
c 1997 Cambridge University Press
1
Inelastic scattering models in transport
theory and their small mean free path
analysis
J. BANASIAK 1, G. FROSALI 2 , and G. SPIGA 3
1
Department of Mathematics, University of Natal, Durban 4041 , South Africa
Dipartimento di Matematica Applicata\G.Sansone", Universita di Firenze
Via S.Marta 3, I-50139, Italy
3 Dipartimento di Matematica, Universit
a di Parma
Via M. D'Azeglio 85, I-43100 Parma, Italy
2
(Received 2 October 1998)
In this paper we perform an asymptotic analysis of a singularly perturbed linear Boltzmann
equation with inelastic scattering operator in the Lorentz gas limit, when the parameter
corresponding to the mean free path of particles is small. The physical model allows for
two-level eld particles (ground state and excited state), so that scattering test particles
trigger either excitation or de-excitation processes, with corresponding loss or gain of
kinetic energy.
After examining the main properties of the collision mechanism, the compressed Chapman-Enskog expansion procedure is applied to nd the asymptotic equation when the
collisions are dominant. A peculiarity of this inelastic process is that the collision operator has an innite dimensional null-space. On the hydrodynamic level this is reected
in the small mean free path approximation being rather a family of diusion equations
than a single equation, as is the case in classical transport theory. Also the appropriate
hydrodynamic quantity turns out to be dierent from the standard macroscopic density.
1 Introduction
In linear transport theory, where the collisions with the background are dominant and
drive the system towards equilibrium, the kinetic equation describing this process can
be approximated, in a suitable sense, by the hydrodynamic equation of diusion type.
This situation is commonly characterized by the existence, at the kinetic level, of a oneparameter familyof equilibrium distributions, which are the eigenfunctions of the collision
operator, corresponding to the isolated eigenvalue at zero. Moreover, the unknown of the
hydrodynamic equation turns out to be the total number density, which represents the
only macroscopic quantity conserved under scattering (i.e., under the dominant process).
In this paper we consider a linear Boltzmann equation for test particles in an absorbing
and inelastically scattering background, derived in the frame of extended kinetic theory
[12]. The adimensionalized transport equation singles out in a natural way the Knudsen number , which is small in all situations of hydrodynamic interest, and, in those
situations, labels the scattering term as the dominant one in the phase space balance.
2
J. Banasiak, G. Frosali, and G. Spiga
The corresponding asymptotic analysis for ! 0 gives rise again, as shown below, to a
hydrodynamic equations of the diusion type.
In a sense, the main practical conclusion extends thus from the elastic to the inelastic
case. However, the dierent physical environment requires a dierent appropriate mathematical setting, and raises interesting mathematical questions, as shown already in the
recent literature on the subject [3, 4, 5, 15].
The paper is conned, like the quoted bibliography, to the physical situation when the
test particle mass is negligible with respect to the eld particle mass (so-called Lorentz
gas limit). This simplies the collision integral from ve fold to the double one but, on the
other hand, introduces other mathematical problems related to the fact that the energy
interval gets partitioned into equivalence classes in such a way that each test particle
remains trapped in one of them for ever. This aects both the equilibrium distribution
at the kinetic level, and the hydrodynamic quantity, which must be appropriately dened,
as shown below. The frame looks very much similar to that arising in the eld of electron
transport in semiconductors [13, 15, 17], but with some signicant dierences pointed
out already in previous papers [4, 19].
This article generalizes the analysis of [5] to the more general case in which the excited
species can not be regarded as negligible, and the considered test particles may gain, and
not only loose, the kinetic energy by scattering i.e. up- and down-scattering are both
present. Indeed, in the particular case of down-scattering only, considered in [5], the
asymptotic limit turned out to be the free streaming rather than the diusion. At the
same time, the present article is also a generalization of [8], where the same problem was
studied by a simpler heuristic model, and the asymptotic analysis was addressed only
at a formal level. We also note that the results presented here are formally similar to a
part of those given in [9], the major dierence being that in the later paper the authors
assumed a bounded energy range which reduces the collision operator to a nite rank
operator and renders the analysis of the Appendix unnecessary.
In [7] the author provided a rigorous asymptotic analysis for the inelastic scattering
model but only for a particular case of Maxwell molecules and a specic scaling of the
equation. To avoid overlaps, in this paper we have decided to emphasise rather the
physical background and some consequences of the adopted assumptions. However, the
results given in the Appendix show that the analysis of [7] can be applied, with some
minor technical changes, for the general situation discussed here.
A discussion of the numerical results related to the analysis presented in this paper
will appear in [6].
The paper is organized as follows. In Section 2 we present and discuss in detail the
physical problem, while the mathematical setting, denitions and notations are given
in Section 3. The abstract asymptotic procedure, based on the rigorous "compressed"
Chapman-Enskog expansion [18], is performed in Section 4, which is then followed by
the explicit formal derivation of the diusive approximation, and by the analytic (even
though not in the closed form) evaluation of the diusion coecient. The crucial point
concerning conditions for invertibility of the collision operator has been shifted to the
Appendix in order to avoid too many technical details in the main text. Finally, Section 6
is devoted, for the readers' convenience, to a brief description of heuristic methods usually
applied in the classical transport theory in order to get the diusion approximation.
3
Small mean free path analysis for inelastic scattering
2 The physical problem
In this paper x and v represent the three-dimensional position and velocity vectors, v is
the modulus of v, ! stands for the unit vector in the direction of motion, and S 2 denotes
the unit sphere in R3. Hence we consider the linear Boltzmann equation which describes
the time evolution of a Lorentz gas of test particles in an inelastically scattering host
medium, as it can be obtained in the frame of extended kinetic theory [12]
@f (x; v; t) = ; v @f (x; v; t)
(2.1)
@t
c @x(x; v ) v+ Z
+ n1 S 4 + v 2 f(x; v+ !0 ; t)d!0 ; H(v ; )cS (x; v)f(x; v!; t)
S Z
c
(x;
S
+ n2 H(v ; ) 4 v) f(x; v; !0 ; t)d!0 ; vv+ cS (x; v+ )f(x; v!; t) ;
S2
1
with v = (v2 2) 2 , t 0 and
f(x; v; 0) =f (x; v);
(2.2)
where f = f(x; v; t) is the test particle distribution function (v = v!). The dimensionless
form of (2.1), keeping the same name to all dependent and independent variables, reads
as
@f (x; ; !; t) = ; 12 ! @f (x; ; !; t)
(2.3)
@t
@x
"
s
#
Z
1
(x;
+
1)
+
1
0
0
+
4
S 2 f(x; + 1; ! ; t)d! ; H( ; 1)(x; )f(x; ; !; t)
s
"
#
Z
(x;
)
+
1
0
0
+ H( ; 1) 4
f(x; ; 1; ! ; t)d! ; (x; + 1) f(x; ; !; t) :
S2
Here the adimensionalized spatial variable is measured in units of a typical macroscopic
length L, and , ranging in the interval [0; b), with 0 < b +1, is the dimensionless
p
quadratic speed v2 = 2, where in turn the characteristic speed is given by 2E=m,
and E represents the change of kinetic energy in each scattering collision, which is xed
and constant in our hypotheses. Time is correspondingly
measured in the units of the
characteristic time L=, and f(x; ; !; t) means f(x; 12 !; t). is a dimensionless parameter which will be specied later, and denotes the dimensionless scattering collision
frequency cS =^cS , where c^S is again a typical value, and for simplicity cS is assumed to be
angle independent. This collision frequency refers to the scattering of a test particle on a
eld particle in the ground state; the other collision frequency, relevant to the scattering
on excited eld particles, has been eliminated from the transport equation by use of the
microreversibility condition [12], so that this physical requirement is correctly accounted
for in (2.3). In addition, the background is assumed to be in thermal equilibrium at a
given temperature T , from which the parameter , representing the ratio of the excited
to the unexcited eld particle number density, is given by the Boltzmann factor
E n
2
= n = exp ; KT < 1:
(2.4)
1
4
J. Banasiak, G. Frosali, and G. Spiga
The host medium is allowed to be spatially inhomogeneous, so that the collision frequency
cS , and thus , may depend on the position x. Moreover, H is the Heaviside function
dened as H( ; 1) = 1 when 1 and zero otherwise. The function f represents the
expected number of test particles at time t in dx at x, and in dv at v, so that the test
particle density may be obtained as
(x; t) = 2
3
Zb
0
21 d
Z
S2
f(x; ; !; t)d!:
(2.5)
Clearly, and ! may be regarded as kinetic variables, whereas x and t are the hydrodynamic (macroscopic) variables.
The physical situation underlying equation (2.3) can be summarized as follows. A
rareed gas of test particles (t.p.) is imbedded in a xed, much denser background of
eld particles (f.p.), in equilibrium at a given temperature, in such a way that t.p.-t.p.
encounters are negligible if compared to t.p.-f.p. interactions, while at the same time the
f.p. distribution is not appreciably aected by collisions. Absorption collisions might be
easily added to the picture, but are not taken into account here since their eects can
be easily separated from the main problem by a simple ansatz, due to linearity. The
limiting case when the ratio m =m of f.p. mass to t.p. mass tends to innity is considered. Scattering is inelastic in the sense that the target eld particle is excited to
or de-excited from its rst excited state, with conservation of mass, momentum, and
total (kinetic plus internal) energy. This results in a loss or gain of kinetic energy of the
impinging test particle; by our adimensionalization, the amount of energy exchanged in
each collision is equal to one. Obviously, particles with energy < 1 can not trigger
a scattering collision (can not be down-scattered), and conversely particles in the same
interval can not be generated by up-scattering from lower energies. These threshold effects are accounted for by the Heaviside function H, which cuts o the loss term from
the down-scattering contribution, as well as the gain term from the up-scattering contribution. The physical conditions are the same as for the famous monoenergetic neutron
transport (or monochromatic radiative transfer) equation, with the only dierence that
elastic scattering is replaced by inelastic scattering. In the former situation, eect of
scattering collisions is isotropization in direction and conservation of energy, in the latter
isotropization is accompanied by a unit loss or gain for the energetic variable .
The dimensionless parameter represents the proper Knudsen number for the present
problem, and is given by n1 ^c L , with (n1c^S );1 representing a typical mean scattering
time.
In the sequel, will be considered as a small parameter, and the asymptotic analysis for
! 0 will be performed, to describe the physical fact that a typical mean free path n1c^
is small in comparison to L: We shall not impose any restriction on cS , like the Maxwell
molecule assumption of [5, 8], and cS (x; ) will be written as cS = c^S (x; ). Finally,
it is worthwhile to note that the present model turns out to be partially equivalent to
the semiconductor models investigated in [13], [14], [16], as rst observed in [19]. It
follows that all these models share the property of having innite number of collision
invariants, though their structure dier to large extent. Investigations of the relations
between various models of inelastic scattering are the subject of current research.
S
S
Small mean free path analysis for inelastic scattering
3 Preliminaries
5
To derive and understand the proper hydrodynamic quantity which in the considered
process will replace, as we shall see, the total number density, we have to analyse rst
the collision mechanism. The only conserved quantity in a collision, as typical in linear
transport, is the number of particles. In the case of elastic scattering, a monoenergetic
initial population remains monoenergetic, and the total number density (angle-integrated
one-speed distribution function) remains constant.
In the present context it proves convenient to dene, for 0 < < 1
fn (x; ; !; t) = f(x; + n; !; t); n = 0; 1; 2; : : :
(3.1)
which is motivated by the fact that, under the same assumption of no streaming and
absorption, a test particle initially at energy , can attain, during its life, only the
discrete sequence of energies which are equal to modulo 1, from ; [] < 1 on.
In the absence of streaming and absorption, and omitting the dependence on x to
simplify notation, we can rewrite the problem as
s
@fn = 1 ( + n + 1) + n + 1 Z f (; !0 ; t)d!0
(3.2)
@t 4
+ n S 2 n+1
Z
; 1 (1 ; n0)( + n)fn (; !; t) + (1 ; n0 ) (4+ n) 2 fn;1 (; !0 ; t)d!0
S
s
; ( + n + 1) + +n +n 1 fn (; !; t);
n = 0; 1; 2; : ::;
where nk is the Kronecker symbol, which is one when n = k and zero otherwise, and
with 2 (0; 1) as a mere parameter. A rst integral of clear physical meaning is easily
obtained by coupling together all equations in (3.2). Integrating formally over S 2 we
have
s
Z
1
Z
+
n
+
1
@
(3.3)
@t S 2 fn d! = ( + n + 1) + n S 2 fn+1 d!
Z
Z
; 1 (1 ; n0)( + n) 2 fnd! + (1 ; n0 )( + n) 2 fn;1 d!
S
s S Z
; ( + n + 1) + +n +n 1 2 fn d!; n = 0; 1; 2; : : ::
S
p
Multiplication by + n and summation over n yield for any 0 < < 1
"1 p Z
#
@ X
(3.4)
@t n=0 + n S 2 fn (; !; t)d! = 0 ;
where the series converges if the total number of particles at time t is nite, by (7.6). Of
course the series reduces to a nite sum when b < 1.
The angle independent expression in square brackets, though dependent on 2 (0; 1),
is thus the proper hydrodynamic quantity, instead of the total number density. From a
physical point of view, that expression comes from the fact that test particles which have
the same value of modulo 1 constitute a closed class which behaves independently from
6
J. Banasiak, G. Frosali, and G. Spiga
all the other classes, and that the particles are isotropically distributed by scattering.
In other words, during the whole evolution such a quantity remains unchanged under
scattering, for any 0 < < 1.
The above considerations illustrate the eects of inelastic scattering, which are considered dominant in the asymptotic analysis performed in this paper.
To simplify the notation we introduce the following abbreviations
p
i = s + i ; i = ( + i + 1) ;
pi = + +i +i 1 = i+1 ;
i
Z
1 f (x; ; !; t)d! ;
fi = 4
i
2
S
and we write the following system
8 @f0
@f0 ; p f + 1 p f
>
=
;
0! >
@t
@x 0 0 0 0 0 1
>
>
>
p f ; 1 f + 1 p f + f
>
1
< @f@t1 = ;1! @f
;
@x 1 1 1 0 1 1 1 2 0 0
:
.
.
>
..
..
>
>
@fn = ; ! @fn ; p f ; 1 f + 1 p f + f
>
n
>
@t.
@x n n n n;1 n n n n+1 n;1 n;1
>
.
: ..
..
(3.5)
This system is to be supplemented by the initial condition
fi (0; x; ; !) =f i (x; ; !); i = 0; 1; : : : :
(3.6)
After having introduced the vector (fi )i2Nwith innite components, the investigation
of system (3.5) will be carried out in the Banach space X of all summable sequences of
weighted functions of L1 (R3x [0; 1] S 2 ) with norm
0 0
1 1
1 Z Z Z
X
@ @ jfi(x; ; !)jdxA d!A p + i d < +1:
k(fi )i2NkX =
1
i=0 0
S2
R
3
(3.7)
Analytically speaking we introduced the operator P(f) = (fi )i2N such that P :
L1(R6x;v) ! X. Since
1 1
Z 0Z 0 Z 1
kf kL1 R6 = @ @ jf(x; v!)jv dvA d!A dx
R3 S20 0
1 1
Z Z X
1Z
@ @
jfi(x; ; !)ji d A d!A dx ;
=
+
(
2
x;v)
0
1
3
2
R3
S2
i=0 0
we see that 2P= 3 is an isometric isomorphism between L1 (R6x;v) and X.
(3.8)
Small mean free path analysis for inelastic scattering
7
Lemma 3.1 The solvability of the initial value problem (2.3) in L1 (R6x;v) is equivalent
to the solvability of the initial value problem (3.5), (3.6) in X and therefore there exists
a contraction semigroup (G(t))t0 in X solving (3.5).
Proof If f is a solution to the equation (2.3) subject to the initial condition f(0) =f,
then Pf is a solution to the Cauchy problem (3.5){(3.6). By [4] we know that, be-
cause we deal with eld free case, there exists a contraction semigroup (GT (t))t0 whose
generator is T = S + C= = ;v@x + C=, dened on D(T ) = D(S) \ D(C). Since
(G(t))t0 = fPGT (t)P;1 gt0 is a semigroup of contraction in X whose generator is
(PAP;1 ; PD(A)), it follows immediately that the solvability results for the initial value
problems (2.3) and (3.5) in the corresponding spaces are equivalent.
2
Now we want to study the spectral properties of the collision operator C, which, since
the collisions are dominant, are crucial for the asympotic analysis of system (3.5).
Using the \component" notation, the collision operator takes the form of a threediagonal matrix
0;0p0
0p0 0
0
::: ::: 1
BB 0 ;0 ; 1p1
1 p1 0
::: ::: C
C
B
C = B :::
:::
:::
:::
::: ::: C
C;
@ 0
n;1
;n;1 ; npn npn 0 : : : A
:::
:::
:::
:::
::: :::
where we have omitted the dependence on x and : It is easy to see that the null-space
N (C) consists of functions of X which are periodic for > 1; apart from the factor n,
i.e.
N (C) = ff0 = f0 (x; ) is arbitrary, but independent of !;
fn = fn (x; ) = nf0 (x; ); for n 1g:
Let us dene
1p
i
X
+ j (fj )(x; );
(3.9)
[P f ]i(x; ) = P1 j p
j =0 + j j =0
as an operator acting in L1 (R3x [0; 1] S 2) for each i, and P as an operator acting in X
0
BB 1
P1 p + j f BB
P f = Pj 1 j p + jj B
BB ...
j
BBn
@.
2
=0
=0
..
1
CC
CC
CC :
CC
CA
(3.10)
where is the classical projection in L1 (R3x) L1 (0; 1).
The complementary projection is dened now by Q = I ; P , where I is the identity
8
operator, given by
J. Banasiak, G. Frosali, and G. Spiga
0 f (x; ; !) ; P1 p P1
BB f (x; ; !) ; P1=0 p j Pj1
B
=0 j j
Qf = B
: : :: : :: : :P
BB
1
P
f
(x;
;
!)
;
@n
1 p
j
0
1
1
j
j
j
+
=0
j
+
=0
n
j
=0 j +j
=0
p + j f 1
p + j fj C
C
CC :
p + j f C
C
j A
j
: : :: : :: : :
It is easy to verify some properties of such operators, which follow easily from the definitions. We have C P = 0 and P C = 0; and consequently P C P = 0, P C Q = 0; and
QC P = 0:
The main peculiarity of the collision operators which have the structure similar to
C (other examples can be found in extended kinetic theory [12, 5] or in semiconductor
theory [14, 15]) is that their kernel is innite-dimensional and that the range is not
closed. These facts were pointed out in [13, 15, 5, 4], and due to them the attempts
to derive hydrodynamic approximation of the related Boltzmann equation were only
partially successfull [14, 15, 7].
One of the main technical point is the solution of the equation of the type
Cf = g
(3.11)
which amounts to nding the range R(C) of C. The rst step in this direction which,
in particular, gives the necessary condition of solvability of (3.11), is to determine R(C).
We have the following lemma
Lemma 3.2 The Banach space X admits the spectral decomposition
X = N(C) R(C)
(3.12)
with the spectral projection onto N(C) along R(C) given by (3.10). The closure of the
range of C is characterized by
g 2 R(C) if and only if
1 p
X
Z
j =0
S2
+ j gj (x; ; !)d! = 0;
where gj (x; ; !) = g(x; + j; !), j = 0; 1; : : : :
Proof The proof can be found in [4].
4 Abstract asymptotic procedure
(3.13)
2
In this paper we are concerned with the Cauchy problem for the singularly perturbed
kinetic equation in a Banach space X1 = L1 (R6x;v), which can be written in an abstract
form as
8 @f
<
= Sf + 1 Cf;
@t
(4.1)
:f(0) = f
where C and S are the collision and the transport operators dened by Eqn. (2.3).
9
In this Section we briey describe the compressed (modied Chapman-Enskog) asymptotic procedure as developed in [18]. We decompose the unknown function f 2 X1 , as
f = u + w = P f + Qf :
The principal assumption is that we don't expand the hydrodynamic part u of the solution
which is determined as the spectral projection P f of the solution f onto the null-space
of the collision operator C.
Operating with this projections on both sides of (4.1) and assuming that all the operations can be performed, we obtain the following problem
8 @u
>
= P S P u + P S Qw ;
>
@t
>
< @w = QSQw + QSP u + QC Qw ;
(4.2)
@t
>
u
u(0) =
:= P f ;
>
>
:w(0) = w := Q f ;
where we have taken into account that P C = C P = 0.
The solution of (4.2) is sought in the form
u(t) = (t) + u~() ;
w(t) = w(t)
+ w()
~ ;
(4.3)
where = t=. We approximate w,
u~ and w~ by the truncated power series in :
w = w0 + w1 + O(2) ;
u~ = u~0 + ~u1 + O(2) ;
w~ = w~0 + w~1 + O(2) :
(4.4)
According to the compressed asymptotic procedure the bulk part of the hydrodynamic
part is not expanded. These expansions are inserted into (4.2) and, truncated on the
2 level, yield the following set of equations (see [18]). For the bulk hydrodynamic part
we have
8 @
<
= P S P ; P S Q(QC Q);1 QS P ;
(4.5)
@t
:(0) = u ;P SQ(QC Q);1 w;
and for the bulk kinetic part we have
w0 0
w1 = ;(QC Q);1 QS P :
(4.6)
Then, for the initial layer part we have
@ w~0 = QC Qw~ ;
@~u1 = P SQw~ ;
u~0() 0;
(4.7)
0
0
@
@
and
8 @w~1
<
= QC Qw~1 + QS Qw~0 ;
(4.8)
@
:w~1(0) = (QC Q);1QSP u;
Small mean free path analysis for inelastic scattering
10
J. Banasiak, G. Frosali, and G. Spiga
whose solution can be easily obtained introducing the semigroup (GQC Q (t))t0 generated
by the operator QC Q in the subspace QX1. Thus the approximation fa = ua + wa of
the solution f = u + w to (4.2) takes the form
fa (t) = ua (t) + wa (t)
= (t) + ~u1 (t=) + w~0 (t=) + (w1 (t) + w~1(t=));
(4.9)
where all the terms have been dened above. The equation satised by the error, which
is dened as
y(t) = u(t) ; [(t) + ~u1(t=)];
z(t) = w(t) ; [w~0(t=) + (w1(t) + w~1 (t=))];
(4.10)
(4.11)
can be found by inserting (formally) the denitions of the error to equations (4.2). In
this way we obtain
8 @y
>
>
@t = P S P y + P S Qz + P S P u~1 + P S Qw~1
< @z
(4.12)
= QS P y + QS Qz + 1 QC Qz
>
@t
>
@
w
:
+ QS Qw~1 + QS P u~1 + QS Qw1 ; @t1 :
The error satises the following initial conditions:
y(0) = 0
z(0) = 2(QC Q);1QS P S Q(QC Q);1 w :
(4.13)
We note that z(0) is of order 2 and therefore can be discarded, provided it exists.
For a range of problems of classical linear kinetic theory the above procedure was shown
to yield the O(2 ) approximation uniformly in t on bounded intervals [0; T ], T < 1 (see
[18, 2, 11]). In the case of inelastic collisions the situation is more dicult due to the
noninvertibility of C. In [7] the author considered another scaling of (4.1):
@f = 1 Sf + 1 Cf;
@t 2
which is the same as in [15] and proved that the error of the approximation is of order
of (in L1 (R6x;v) and uniformly in t on bounded intervals), provided the rst moment
(i.e. in the discussed case the integral over the unit sphere S 2 ) of the kinetic part of the
initial value w=
Q f is equal to zero. Using the results of the Appendix one can use
the technique of [7] to prove the same error estimate under the same assumption for the
scaling (4.1). To get O(2) estimates one has to assume additionally that
Z
S2
see [6].
! @@xw d! = 0;
Z
S2
w ! @(x)
@x d! = 0;
11
Small mean free path analysis for inelastic scattering
5 Diusion approximation - formal considerations
According to Section 4 the formal diusion operator has the form
D = ;P S Q(QC Q);1 QS P ;
where the transport operator is given by
n
(Sf)n = ;n ! @f
@x :
Let us rewrite the spectral projection onto the kernel of C as
1
P
f (x; )
(P f)n (x; ) = n j =0
j
1
P
j =0
j
j j
= n H ;1()
1
X
j =0
j fj (x; ) = n H ;1;
(5.1)
(5.2)
1
1
P
P
where H() = j j and (f) = j fj (); the operator Q denotes the complemenj =0
j =0
tary projection: Q = I ; P .
Remark 5.1 It follows [3, 8] that such dened is, for each in the interval [0; 1[,
a collision invariant which represents the total number of particles that constitute the
closed class discussed in Section 3.
2
Firstly, let us calculate formal expressions for P S P ; P S Q; QS P ; and QS Q. We have
1
1
X
X
j ;
(P Sf)n = nH ;1 j (Sf)j = ;nH ;1 j2 ! @f
@x
j =0
j =0
hence
@
1
X
(P S P f)n = ;n H ;1 j2 ! @x
(P f)j :
Because (P f)j is independent
then we obtain
j =0
of !, (! @@x (P f)j ) = 0, and consequently we
P S P f = 0;
(P S Qf)n = (P S(I ; P )f)n = (P Sf)n = ;n H ;1
Clearly,
hence
j :
j2 ! @f
@x
j =0
1
X
have
(5.3)
QS P f = (I ; P )S P f = S P f;
01
1
X
@ @ f A :
(QS P f)n = ;nH ; n! @x
j j
1
j =0
(5.4)
Let us consider the composition (QC Q);1QS P . To this end we must solve the system
QC Qf = QS P g :
(5.5)
12
J. Banasiak, G. Frosali, and G. Spiga
As we shall explain in detail in the appendix, it is sucient to restrict ourselves to the
diagonal part of system (5.5), because QS P f = S P f = 0. Introducing
8
0
>
for n = 0
<
0 (x; )1
(5.6)
bn = bn(x; ) = >
: (x; ) +n (x; ) for n > 1;
n
n+1
n;1
n
we can express the solution as
01
1
8
>
X
@
>
f (x; ; !) = b (x; )H ; ()! @x @ j (gj )(x; )A
>
>
j
>
>
.
.
< ..
..
01
1 :
>
X
@
>
fn (x; ; !) = nn bn(x; )H ; ()! @x @ j (gj )(x; )A
>
>
j
>
>
..
: ..
0
1
0 0
=0
(5.7)
1
=0
Now,
.
.
(P S(QC Q);1 S P f)n = ;n H ;1
1
X
@ ((QC Q);1S P f)
j2 ! @x
j
j =0
1
@ 1
X
@
3
j
3 A
n
;
2
;
1
@
0 b0 + j bj @x (f) ;
= ; H (4)
akl
l
j =1
k;l=1 @xk
X
3
R
0
where akl = !k !l d! = 43 kl , where kl is the Kronecker delta. Hence we obtain the
S2
following form of the \diusion" operator
0
1
1
j 4
4
n @
X
@ (f) :(5.8)
(P S(QC Q);1 S P f)n = ; 3H 2 @x @ 0 + +j A @x
0 1
j ;1 j
j =1 j j +1
The \diusion" equation (4.5) in components have the form
@(P f)n = ;(P S(QC Q);1 S P f) ; n = 0; 1; : : : :
n
@t
Recalling that (P f)n = n H ;1, it is possible to write the equation for as
@ = @ D @ ;
@t @x
@x
where the diusion coecient is given by
0
D
(5.9)
1
1
@ 3b + X
j j3 bj A :
= 3H()
0 0
j =1
(5.10)
It is to be noted that having solved (5.9) one obtains the approximating solution (see
Section 4) as in (5.2):
= fngn2N= fH ;1 ngn2N:
(5.11)
Small mean free path analysis for inelastic scattering
6 The classical Chapman-Enskog procedure
13
In this Section we want to comment on the application to the present context of the
heuristic approaches which are commonly used in standard transport theory (see e.g.
[11]) in order to deduce a macroscopic approximation of equation (3.5) with corrections
of rst order in . The traditional Chapman-Enskog recipe will work, of course, and yield
the same result as rigorously justied using the compressed asymptotic procedure and
working along the lines of the previous Sections. It proceeds along the following steps.
As we have seen in Section 3, the hydrodynamic quantity is given by
(x; ; t) =
1 p
X
n=0
+n
Z
S2
fn (x; ; !; t)d!;
(6.1)
where is only a parameter in [0; 1).
Integrating 1each equation of (3.5) over ! and summing over n, after having multiplied
times ( + n) 2 , we obtain the macroscopic equation
"1
#
Z
@ + @ X
(6.2)
@t @x n=0( + n) S 2 !fn (x; ; !; t)d! :
We expand fn in series of , as
1
X
fn =
k=0
k fn(k) ;
(6.3)
leaving unexpanded the hydrodynamic quantity , hence
Z
1 p
n
X
;
+ n 2 fn(k) d! = 0; k = 1; 2; : : : (6.4)
fn(0) = P1 j p + j 4
S
j =0
n=0
and fn takes the form
1
n
X
(6.5)
fn (x; ; !; t) = P1 j p + j (x;4; t) + k fn(k) (x; ; !; t):
j =0
k=1
R
With J n (x; ; t) = S 2 !fn d!, we have
Jn =
where
J nk
1
X
k=1
Z
k J (nk) ;
Jn = 0 ;
= 2 !fn(k) d!; k = 1; 2; : : : :
(6.6)
S
Now we want to determine the dominant term of J n, of order , to be inserted in the
macroscopic equation (6.2). Integrating equations (3.5) over !, after having multiplied
by !, we obtain
Z
p
@J
@
n
@t + + n @x 2 !!fnd! = ; [(1 ; n0 )n;1 + npn] J n : (6.7)
S
Expanding the previous equation in series of and taking account of
Z
n p
P
!!f
d!
=
I + O()
n
j
3 1
S2
j =0 + j
(0)
( )
14
J. Banasiak, G. Frosali, and G. Spiga
where I is the identity matrix, the contribution of J n at the order turns out to be
np + n
1
@ :
(1)
P
p
Jn = ; 1 j
3 j =0 + j (1 ; n0)n;1 + npn @x
Then we obtain
Z
1
1
X
X
@ + O(2 ):
n ( + n) 23
( + n) !fnd! = ; P1 j p
3 j =0 + j n=0 (1 ; n0)n;1 + n pn @x
S2
n=0
Now we are in the position to introduce such contribution of order into (6.2),
@ = @ D (x; ) @ (6.8)
@t @x
@x
where
1
n ( + n) 23
X
;
(6.9)
D (x; ) = P1 j p
3 j =0 + j n=0 (1 ; n0 )n;1(x; ) + n(x; )pn()
which, recalling the notations previously introduced, coincide, as expected, with (5.9)
and (5.10), respectively.
However, the most popular technique leading to a hydrodynamic diusive equation in
monoenergetic transport theory with elastic scattering is the so-called P1 approximation
[10], which amounts to assuming an almost isotropic distribution function, with small
linearly anisotropic correction; in addition it does not require the parameter to be small.
This is justied by the fact that the eect of scattering is isotropization, and then, after
an initial layer of the order of the mean collision time, one is dealing actually with an
almost isotropic distribution. The same is true, contrary to Ref. [5], in the present
situation, so that it is worth to investigate such an approach, expressed here by
1 (x; ; t) + 3 ! J (x; ; t) ;
fn (x; ; !; t) = 4
(6.10)
n
n
4
R
with n = S 2 fn d!: Integration of (3.5) over ! after multiplication by 1 and ! yields
8 @n p
@ J = (1 ; ) ; [(1 ; )
>
@t + + n @x
n
n0 n;1 n;1
n0 n;1
>
<
+ n pn]n + n pnn+1
(6.11)
Z
>
p
@J
@
>
: n + + n !!fnd! = ; [(1 ; n0)n;1 + npn] J n
@t
@x 2
S
R
and this coupled set gets closed on using (6.10) in order to eliminate S 2 !!fn d! as
1
I: The second equation becomes thus
3 n
n + p + n @n = ; [(1 ; )
@J
(6.12)
n0 n;1 + npn ] J n :
@t 3
@x
At this point physical arguments are usually invoked in order to drop the term with time
derivative (bound to be smaller than the others) and to get Fick's constitutive equation
n
J n = ; 3 (1 ; ) p ++nn+ p + n + 1 @
(6.13)
@x ;
n0 n;1
n
(retaining that term leads, in the elastic case, to the telegrapher rather than to the
15
diusion equation). In the present inelastic frame, such a procedure yields the following
closed coupled set for the n (n = 0; 1; : : :)
"
#
23
(
+
n)
@
@n = @ n
(6.14)
@t @x 3 (1 ; n0 )n;1p + n + np + n + 1 @x
+ 1 fn pnn+1 ; [(1 ; n0)n;1 + npn ] n + (1 ; n0 )n;1n;1g :
This shows a diusive nature, but still includes scattering contributions.
A way to get
rid of them is to sum over n after multiplication by ( + n) 12 , which reproduces the
hydrodynamic quantity on the left hand side, and yields
Small mean free path analysis for inelastic scattering
2
3
Z
1
2
(
+
n)
@
@ = @ 4 X
5
@t @x 3 j =0 (1 ; n0)n;1p + n + n p + n + 1 @x S 2 fn d! :
(6.15)
In any case, this method itself does not provide a selfconsistent diusive approximation
of the kind (6.7), (6.9), as is the case for elastic scattering. To achieve this, we have to
assume further that fn diers only by a small correction (to be discarded later) from the
equilibrium under scattering which corresponds to the density given by the rst summand on the right hand side of (6.5). This stronger assumption is of course completely
consistent with the physics of the problem, and shows how the P1-approximation argument must be augmented in order to let it work also in the inelastic frame. The reason
of its success in the presence of only one eld particle energy level is that, in such a case,
the \isotropic distribution" is exactly the same as the \equilibrium distribution", at the
only allowed energy. In our case, a discrete set of equally spaced energies is associated to
any 2 [0; 1), and equilibrium distributions at each energy must be additionally related
by powers of the parameter .
7 Appendix: Solvability of Cf = g
This appendix is devoted to study the solvability of Cf = g: It is evident from the
previous Sections that a crucial role in making the asymptotic procedure rigorous is
played by the invertibility of the collision operator in a suitable chosen subspace of
X1 = L1 (R6x;v) (without the innite dimensional null space of C). We note that x plays
a role of a parameter so that for a time being we will omit the notation of it; thus we
consider function v ! (v) and all the considerations will be carried in the space of
functions of variable v.
>From the physical point of view we assume that
min vs (v) max vs ;
(7.1)
for v > 1 and s 1. It follows for both upper and lower bounds that the case s = 1
corresponds to hard spheres, 0 < s < 1 to hard potentials, s = 0 to Maxwell molecules,
and s < 0 gives soft inverse power potentials. We note that in the case when s > 0 there
appears another singularity of the collision frequency due to its unboudedness at innity.
Moreover we note that it is unnecessary to make assumptions on (v) for v < , for the
presence of the Heavyside function in the collision terms.
16
J. Banasiak, G. Frosali, and G. Spiga
Therefore the operator C is dened on the domain D(C) which can be described as
the subset of L1 (R3) of functions satisfying
Z
R3
and, if s > 0,
jf(v)j(v;1 + 1)dv < +1;
Z
In other words, we have
R3
jf(v)j(vs + 1)dv < +1 :
D(C) = L1 (R3; (1 + v;1 + vS )dv)
(7.2)
where S = maxf0; sg.
In the sequence notation (with reduced energy variable = v22 ; 2 [0; 1[, see Eq. (3.1)
) this translates into the following condition
ZZ
1
0
S2
jf0 (; !)jdd! +
1 Z Z
X
1
j =1 0 S 2
( + j)
S
+1
2 jfj (; !)jdd! < +1:
(7.3)
Accordingly, we have to consider the following system of equations
8g0(; !) = ;0()p0()f0 (; !) + 0()p0()f1 ()
>
>
.
..
..
>
.
.
< ..
gn (; !) = ;(n ()pn () + n;1())fn (; !) + n()pn ()fn+1 () :
>
+n;1()fn;1 ()
>
>
..
..
: ...
.
.
R
Let gn = gn0 + gn1 where gn0() = gn() = 41 gn(; !)d! and gn1 = gn ; gn0
S2
(consequently gn1 = 0. Then, using the same notation for the unknown function f, we
see that we must have
8g00() = ;0()p0()f00() + 0()p0()f10()
>
>
.
..
..
>
.
.
< ..
(7.4)
g
=
;
(
n
0 ()
n ()pn() + n;1())fn0 () + n()pn ()fn+1;0 ()
>
>
+
()f
()
n;1
n;1;0
>
..
..
: ...
.
.
and in the same time
8g (; !) = ; ()p ()f (; !)
01
0
0
01
>
>
..
..
< ...
.
.
(7.5)
>
gn1(; !) = ;(n ()pn () + n;1())fn1 (; !) ;
>
..
..
: ...
.
.
so that the part annihilating constants is diagonal.
17
Firstly, we consider system (7.4), where 2 [0; 1[ is a parameter. The left-hand side is
derived from L1 (R3v), hence by (3.8) it must satisfy
Small mean free path analysis for inelastic scattering
1Z p
X
1
j =0 0
+ j jgj 0()jd < +1
(7.6)
and the compatibility condition for g is given by, [4],
1p
X
j =0
+ j gj 0() = 0 ;
(7.7)
for almost all .
The basic relation which enables us to solve system (7.4) is n = n;1pn;1(). With
this we rewrite the system (7.4) in the form
8 0g00() = ;00()p0()f00() + 00()p0()f10()
>
>
.
..
..
>
.
.
< ..
: (7.8)
n gn0() = ;(n n ()pn() + n;1n;1()pn;1())fn0 ()
>
+
()p
()f
()
+
()p
()f
()
n n
n
n+1;0
n;1 n;1
n;1
n;1;0
>
>
..
..
: ...
.
.
Summing the rst i equations we get
i
X
l=0
l gl0 () = ;i i ()pi ()fi0 () + i i ()pi ()fi+1;0 ();
hence we obtain
Since
we obtain for j 1
Xi
1
fi+1;0() = fi0 () + ()p
() l gl0 ():
i i
i
l=0
1
0g00()
f10() = f00() + ()p
0 0
0 ()
fj 0 () = j f00() +
j;
i
X
j ; ;i X
i i ()pi () l gl ():
1
1
i=0
l=0
0
(7.9)
This solution is unique in the class of functions satisfying (7.7). The associated homogeneous equation admits only the zero solution; in fact any solution to equation (7.8) with
gn0 0 must have the form fj 0() = j f00(), whence
1
X
j =0
j fj 0() = f00 ()
1
X
j =0
j j = 0
implies f00() = 0 and, consequently, fj 0 () = 0, 8j:
Next we consider the second term on the right hand side of (7.9), and in order to give
a sucient condition on g0 for f0 , the solution to (7.4), to belong to D(C), we have to
18
J. Banasiak, G. Frosali, and G. Spiga
handle the sum in (7.9). Let us denote
ri () = iS
for i = 1; 2; : : : (where as before S = maxf0; sg, see Eq. (7.2)). By (7.7) we can rewrite
such a sum as
j ;1
j ;1
i
1
X
X
j ;1;i X
j ;1;i X
g
()
=
;
l gl0 ();
(7.10)
l l0
i=0 i i ()pi () l=i+1
i=0 i i ()pi () l=0
hence
1
X
j =1
0 1
1
j;
1
1 X
; ;i
j ; ;i X
X
X
l jgl ()j = @
j rj ()j A ()p () rj ()j ()p ()
i i
i l i
i i
i
i
j i
i
0
1
1
1
l;
1
; ;i
X
X
X
X
l jgl () = l jgl ()j @
j rj ()j A ;
1
1
1
0
=0
=0
= +1
= +1
1
0
l=i+1
0
l=1
1
i i ()pi ()
i=0 j =i+1
(7.11)
where the change of the order of summation is justied by the positivity of terms. By
[1], p. 263 (formula 6.5.32) we have
Z1
e;tt; dt ; x; e;x x; e;x j ; 1j ;
x
1
1
(7.12)
1
x
provided < 3. In our case ; 1 = S +1
> 0, hence, for x 1 we obtain
2
Z1
x
e;t t;1 dt x;1 e;x ;
and since for t > ;(ln );1 the function t ! t(+t)
we have by (7.13),
1
X
j =i+1
j rj ()j
c1
Z1
i
2 dt = c2
t ( + t) S+1
S
+1
2
(7.13)
is decreasing and using S +1
1,
2
Z1
;(+i) ln e; S
+1
2 d c3 iS +1 i
where i 1 and ck , k = 1; 2; 3 are constants. Thus, for any l 1
0 1
l;
X
@X
1
i=0 j =i+1
1
!
l;
; ;i
S
X
j
i
A
rj ()j ()p () c 1 + ()p () c bl ;
i i
i
i
i
1
1
4
with c4 ; c5 constant, where
8 l
< ;
bl = :1 + lP
i;s=
1
i=1
2
for
for
i=1
5
s0;
s<0;
by assumption (7.1). To make bl more compact we note that 1 l;s=2 and using
19
Small mean free path analysis for inelastic scattering
integration instead of the sum we see that we can use bl in the following form
bl = l1;=2 ;
where = minf0; sg (hence < 0) and l 1.
To get the convergence on both sides of (7.11) we must assume that
K() =
1
X
l=1
l1;=2 l jgl0()j 2 L1 ([0; 1]):
(7.14)
(7.15)
p
Let us now multiply j-th equation in (7.9) by + j and sum them to obtain by the
compatibility condition (7.7)
0 = f00 ()
where G is given by
1
X
j =0
j j + G()
j;
1 X
1
j ; ;i X
X
j ()p ()
l gl ()
i i
i
1
G() = ;
j =1
1
i=0
l=i+1
0
and satises jG()j K() for a.a. 2 [0; 1], by (7.11) and ri() 1. In particular, by
1
R1
P
(7.15), jG()jd < 1. Since j j is bounded away from zero, this determines f00
j =0
0
R
1
in terms of g0 in such a way that f00 ()d < +1 which shows that the condition at
0
v = 0 for f to belong to D(C) is satised.
1
P
Denoting H() = j j , we obtain
j =0
G() :
f00 () = ; H()
(7.16)
To estimate the solution in the norm of the domain D(C) we use Eq. (7.3) to obtain
jf00()j +
1
X
i=1
ri()i jfi0 ()j M1 jG()j + M1jG()j
1
X
i=1
ri ()i i + K() M2 K() ;
1
P
for some constants M1 ; M2 , where we have also used j jS +1 < 1: Then, using (7.15),
j =1
we obtain
jf00()j +
1
X
i=1
ri()i jfi0()j M2
1
X
l=1
p
l1;=2 + ljgl0 ()j:
Changing the description to the continuous one, replacing by the original variable v,
we see that f0 2 D(C) provided
kf0 kD(C ) M3
Z
R3nB
;
(0 1)
jg0(v)jv2; dv C3kv2; g0kL1 (R3) < +1:
v
p
(7.17)
In other words, g0 2 L1 (R3v; (1 + v2; )dv) = L1 (R+; (1 + 1;=2 ) d) L1 (S2 ; d!).
20
J. Banasiak, G. Frosali, and G. Spiga
Finally we consider the diagonal system (7.5). The orthogonal to constants part can
be determined as
8
f01(; !) = ; k0(0)1 g01(; !)
>
>
>
..
..
< ...
.
.
(7.18)
>fn1(; !) = ; k () +1+ ;1 () gn1(; !) ;
>
..
..
: ...
.
.
and we obtain
n
n
jf01(; !)j +
1
X
i=1
n
n
n
ri ()i jfi1 (; !)j M4 0 g01(; !) +
which translates into
!
1
X
;=
i=1
i
2
i jgi1(; !)j
kf1kD(C ) C4 k(1 + v; )g1 kL1(R3) :
(7.19)
0
Let us denote by X1 a subspace of functions of X which satisfy the compatibility
v
conditions (7.7):
X10 = ff 2 L1(R3v);
1p
X
Z p
j =0
S2
+ j f( + j!)d! = 0 for a:a: 2 [0; 1]g:
We can formulate the following theorem.
Theorem 7.1 If g 2 X is such that g 2 L (Rv; (1 + v ; )dv) and g 2 L (Rv; (1 +
v; )dv), where = minf0; sg, then there exists a unique solution f 2 D(C) \ X to the
0
1
0
3
1
2
1
1
3
0
1
equation Cf = g. There exists a constant M such that for any g satisfying the above
conditions
;
kf kD(C ) M k(1 + v2; )g0kL1 (R3) + k(1 + v; )g1 kL1(R3) :
v
v
(7.20)
Remark 7.1 If we are interested in the L norm of the solution, we can repeat the
1
estimates of the theorem with the weight sequence ri equal identically 1. Then bl for
l 1 can be estimated by l1;s=2, s 1 and we obtain
kf kL1 (R3) M(k(1 + v2;s )g0 kL1 (R3) + k(1 + 1 v;s )g1kL1 (R3) );
(7.21)
where 1 is the characteristic function of [1; 1[.
2
v
v
v
Remark 7.2 Unfortunately, Theorem 7.1 gives only a sucient condition for g to belong
to the range of C and presently we shall show that this condition is not necessary.
For simplicity we consider (7.4) with the particular choice of n () = ns +1, for n 0
and 2 [0; 1[, and rewrite n-th equation in the following form
s+1
ns+1 s+1
+1
n gn0() = ; ns+1
(ns+1 fn0 ()) ; ns+1 fn0() + ns+1
+1 fn+1;0 () + s+1 (n;1 fn;1;0 ())
n
n;1
(7.22)
Now, if Fn () = ns+1 fn0() is summable for almost every and its sum is bounded
21
for 2 [0; 1], then the corresponding function f belongs to D(C). Denoting Gn() =
ngn0(), we write Eq. (7.22) as
Gn = ;An Fn ; Fn + Fn+1 + An;1Fn;1
Small mean free path analysis for inelastic scattering
+1
+1 is a bounded sequence. Let F = (;1)n =n2, then
where An() = +1
n
s
n
s
n
1 1
jGnj = An n + n + (n +1 1) + An; (n ;1 1)
2
2
1
2
2
1
n
2
and njGnj is not summable, therefore for p
f corresponding to this particular sequence Fn,
1;=2
Cf does not belong L1 (R+; (1 + ) d). Hence, we have constructed a function
belonging to the range of C, but that is not the image of a function in the subspace
described in Theorem 7.1.
2
This result raises question as to whether the result of Theorem 7.1 cannot be improved
to show that R(C) = X10 . For s < 0 this is clearly not the case because the inverse of
the diagonal part of C is not dened on the whole of L1 . In general, we can prove the
following partial inverse of Theorem 7.1.
Proposition 7.1 If g 2 X is such that for some v , g is of constant sign for v v ,
and if there exists f 2 X solving Cf = g, then g 2 L (R ; (1 + v ; )v dv), where
= minf0; sg.
0
1
0
1
0
0
1
3
2
2
0
Proof Using Eqs. (7.9) and (7.7) we see that
n
X
n
X
j ;1
n
1
j ;1;i X
X
X
j
rj ()j fj 0() = f00() rj ()j ; rj ()j ()p ()
l gl0 ():
i l=i+1
j =1
j =1
j =1
i=0 i i
(7.23)
and because the left-hand side and the rst term on the right-hand side are absolutely
convergent,as n tends to innity, we obtain that
Sn =
n
X
j =1
rj ()j
j;
1
X
j ; ;i X
l gl ()
i i ()pi ()
1
1
i=0
0
l=i+1
must be absolutely convergent. Assumption that g0 is, say, positive for large v amounts
to saying that gl0 0 for l > l0 for some l0 . Let us split (gl0 )l1 = (gl; )l1 + (gl+ )l1 ,
where gl; = gl0 ; for l l0 and 0 otherwise and consider
Ai () =
It follows that Ai =
1
X
l=i+1
l gl; ():
Pl0 g ; for i < l and A = 0 for i l . With this
l l
i
l i
X
j;
j ; ;i
jBj j = ()p () Ai () Kj
i i
i
i
0
= +1
1
=0
0
1
22
J. Banasiak, G. Frosali, and G. Spiga
l0P
;1
;1; Ai for j l0 . Therefor some constant K independent of j, since Bj = j
( )p ( )
i=0
fore
i
i
n
j
i
i
X
X
jS ; j = rj ()j Bj K j rj ()j
n
=1
n
j =1
is nite as n ! 1. Then for the convergence of Sn we must have the convergence of
j ;1
n
1
j ;1;i X
X
X
Sn+ = rj ()j ()p ()
l gl+ ():
i
i
i
j =1
i=0
l=i+1
However, this time all the terms of this series are nonnegative and we can repeat the
analysis of Theorem 7.1. Passing with n ! 1 in Sn+ , we have that such a convergence
depends on the asymptotic behaviour of
cl =
0 1
l;
X
@X
1
i=0 j =i+1
1
; ;i
j rj ()j A 1
i i ()pi ()
P c g+ . Using the reverse inequality implied by (7.12) i.e.
in the series 1
l=1 l l l
Z1
e;t t;1 dt 3 ;2 x;1 e;x ;
x
(7.24)
which is valid for x 2, where (3 ; )=2 = (3 ; S)=4 > 0, we have for i 1
1
X
j =i+1
j rj ()j
Z1
i+1
2 dt = C1
t( + t) S+1
Z1
;(+i+1) ln e; S
+1
+1 i
2 d C2 iS+1
where Ck , k = 1; 2 are constants. Discarding the term of cl with i =P
0, we see that for
large l, when s 0, cl behaves as l and, when s < 0, it behaves as li;=11 i;s=2 , which
can be estimated asymptotically by integration, giving
(l ; 1)1;s=2 :
1 ; s=2
Hence, for large l, cl behaves as l1;=2 with = minf0; sg and we see that to have the
convergence of the series Sn+ we must assume that
K() =
which proves the thesis.
1
X
l=l0 +1
l1;=2l jgl0 ()j 2 L1([0; 1]);
Acknowledgements
2
This work was partially supported by the CNR Special Project \Metodi Matematici in
Fluidodinamica Molecolare" and by the Italian Ministery of University (MURST National Project \Problemi non lineari nell'analisi e nelle applicazioni siche, chimiche e
biologiche: aspetti analitici, modellistici e computazionali") and was performed under
23
the auspices of the National Group for Mathematical Physics of the Italian Research
National Council (CNR-GNFM). One of the authors (J. B.) was also supported by the
Foundation for Research and Development of South Africa.
Small mean free path analysis for inelastic scattering
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