Measures on phase space as solutions of the one-dimensional neutron transport equation

Measures on Phase Space as Solutions of the One-Dimensional Neutron Transport Equation.

P.aUL 1N'ELS0~ Jr . - t I . DEA~N VICTORY Jr. (Lubbock, Texas, U.S.A.)

Summary. - The linear integral transport operator ]or slab geometry is ]ormulated and studied as a mapping on the set o] measures on the phase space o] the underlying system, with the expected number o] ne~ttrons emergent #ore a collision represented by a measure on the space o] outgoing velocities. Under appropriate assumptions it is shown that, i ] c represents the maximum number o] secondary particles per collision, then there exists e1>l such that the system is subcritical ]or c < c~. An example shows that c1>1 is sharp" in ge~teral, but further assumptions are given under which one can deduce c~ > 1. The idealized laws o] elastic and inelastic scattering are shown to satis]y our assu~n~tions.

1 . - I n t r o d u c t i o n .

In recent years there have appeared several contributions [1-10] s tudying various aspects of neut ron t ranspor t theory in a suitable space of integrable functions, as contrasted to the classical square-integrable setting. The physical rationale is t ha t the Ll -norm of the solution of a neutron t ranspor t problem has a physical interpre- tat ion, whereas the L~-norm does not. This fact is f requent ly reflected mathema- tically in the ease and naturalness with which results can be obtained, al though the absence of the inner product sometimes necessitates the use of techniques which tend to be unfamiliar to many physical scientists. Recent examples of work in an L 1 context which seem wor thy of specific note are those of LAt~SEN and ZWEISEn [9] and of LARSE~ [10]. The first of these contributions is in the general t radi t ion of the long line of research relating to the spectral theory of t ranspor t problems, while the second is somewhat more in the existence-uniqueness-properties spirit of the present article. Results which are more nearly antecedents of the present article are contained in references 1, 4, and 7. Briefly Ong0EST [1] used L 1 theory to establish suberit icali ty in the physically obvious non-multiplying case; CASE and ZWEI- s e n [4] extended this result to include some mult iplying t ranspor t for regions ha~Ting finite optical diameter, and :N~Lso~ [7] obtained similar results for the computat io- nally impor tan t case of t ranspor t in a slab.

As observed by VIDAV [5], it is physically reasonable to consider measures as solutions of the neut ron t ranspor t equation. A development in an L ~ setting cor- responds, via the wcll-kno~m Ra d o n -~ i k o d y m theorem, to restricting consideration

(*) Entrata in Redazione il 27 ottobre 1975.

70 PAVL IqELSO~ Jr . - I t . DEA~ VICT0~Y Jr . : Measures on phase space, etc.

to those measures which are absolutely continuous with respect to Lebesgue measure. Consideration of the t ransport equat ion in spaces of (not necessarily absolute ly continuous) measures would presumably permi t rigorous inclusion of discrete (i.e. delta-function-like) distribution functions in the description of particles emanating from a collisional event. Many collisional events (e.g. elastic and inelastic scattering) are ideally described by such singular dis tr ibut ion fnnctions, al though inclusion of greater realism (i.e. the Doppler effect) often yields a continuous distribution function for particles resulting from such a collision.

Although it is desirable to permi t measures as candidates for solutions to neu t ron t ranspor t problems, there tends to be significant technical difficulties. SU}tADOLC and VI~)Av [11] have considered this desideratum for t ime-dependent three-dimensional t ranspor t theory. These workers make the following comment regarding their work: (( The developed theory applies only to Bol tzmann operators in which the integral operator par t satisfies some ra ther restr ict ive conditions. These are not satisfied in many physically interesting cases ~) [11, pp. 514-5].

The basic purpose o] this paper is to extend to an appropriate space o] measures

the suberiticality results obtained previously by ~ELSO~ [7] in an L ~ ]ramework. In this regard the current work constitutes a considerable extension of the results initially nnnounced in [12]. The pr imary reason for choosing to ex tend [7] to a measure-space setting is the hope tha t technicnl difficulties wilt be minimized by the one-dimensional nature of the slab-transport problem, and also by the physically obvious nature of the question, thereby suggesting techniques which will be fruitful for more complex problems. Of course in any slab-transport problem there tends to be pathology associated with particles moving parallel to the slab faces, and nei ther the present work nor tha t in [7] is an exception; however this is more in the na ture of an interesting nuisance, t han a major barrier to the development

of a satisfactory theory. A brief outline of this paper follows. In Section 2 we collect assumptions and

notat ion, and formulate the s teady-s ta te slab-transport equation in an appropriate space of measures on the underlying phase space. Section 3 is devoted to the theorem (( nonmult iplying implies suberitical ,>, as obtained by estimating the norm of the integral t ranspor t operator defined in Section 2. In Sections ¢ and 5 we briefly discuss respectively our concept of subcritieality, with specific reference to various equivalent forms, and an example which shows tha t the result of Section 3 cannot be improved without stronger assumptions. Section 6 in concerned with such stronger conditions under which i t is possible to conclude tha t the spectral radius of the integral t ranspor t operator is strictly less than unity, and therefore subcrit icali ty is implied by (( not too multiplying. ,) One of these stronger conditions relates to eventual compactness of the integral t ranspor t operator. Sufficient conditions for this to hold are explored in Section 7. In Section 8 we discuss a restr icted form of the preceding theory in which the underlying measures are Bochner integrable functions from position space to the Banach space of signed Borel measures on veloci ty space. Section 9 is devoted to a proof tha t oar basic assumptions are satisfied by the idea-

PAUL ~ELSO~ Jr . - I-I. DEA~ VICTORY Jr . : Measures on phase spaee, etc. 71

lized laws of elastic and inelastic scattering. Finally, a few concluding comments are given in Section 10.

2. - D e f i n i t i o n s a n d n o t a t i o n .

As in [7] the slab faces are taken perpendicular to the z-axis and located at z--~ 0 and z ~ a, 0 < a < 0. Velocities are denoted generically by v - ~ vFt, v specifying the speed and ~ the direction ( ~ ~ unit vector in E3). The direction cosine of v relat ive to the positive z-axis will be denoted generically as #. I f v is a signed measure, then Ivl, ~+, and v- are the total, positive, and negative variations respectively.

We now wish to indicate notat ion and to describe the spaces of measures used in the analysis:

i) ~ , the a-Mgebra of the Lebesgue measurable sets of [0, a].

ii) 35~, the a-algebra of the Borcl sets in velocity space V (--~ E3).

iii) P ~ [0, a] × V ( ~ phase space).

iv) ~, x ~ , the smallest a-algebra in P which contains every measurable rectangle.

v) My, the Banach space of signed Borel measures defined on 35~, with norm given by the to ta l variat ion on V.

vi) Ms, the Banach space of signed measures defined on ~×35~, with norm given by the to ta l variat ion on :P.

Unless otherwise indicated, measurabil i ty properties of the functions encountered in our analysis will be with respect to one of the a-algebras listed here.

This paper considers the t ransport operator acting on Me. Such an operator is described by two quantities, k and ~, representing respectively the expected distri- but ion of particles emanat ing f rom a coltisional even t and the probabil i ty of a collision per uni t pa th length ( = cross section). We make the following basic ~ssump- tions regarding these quantit ies:

A.1) (z', v')--->k(z', v') is a mapping from P to M~:-, ( = finite Borel measures on 35,) such tha t the mapping

(2.1) (z'~ v ~) ---> k(z', v ') B ,

is measurable for every B e 33,.

A.2) a: [0, a] × V-+ nonnegative reals is measurable. a

A.3) fa(z, v ) d z < c~ for all v e V. 0

A.4) k ( z ' , v ' ) V o = O for all z ' e [0 , a] and all v ' e V , where V o = : { v e V : v # = 0 } .

A.5) sup{k(z', v~) V: (z', v ' ) e P } = 1.

72 PAUL NELSON Jr . - H. DEAN VICTORY Jr . : Measures o~ phase spaee~ etc.

Assumption A.2) is purely technicM in nature. Assumptions A.3) and A.4) seem physically reasonable and are essential to avoid possible pathologies associated respectively with velocities for which the slab has an infinite optical thickness and with neutrons traveling parallel to the slab faces.

We remark tha t A.1) is trivially satisfied in the classical ease of k(z ~, v') absolute ly continuous with respect to Lebesgue measure on V for all v ' e V almost everywhere in z 'E [0, a] with its l~adon-Nikodym derivat ive joint ly measurable relative to g~ × :g, × ~B,. However A.1) also holds for elastic and inelastic scattering functions, which are precluded in the classical formulation. This assertion is proved in Sec- tion 9.

Assumption A.1) can be used to prove the following result:

PRoPosI~IO~ 2.1. - Le t )/ denote the characteristic function of a set Q measurable relative to ~ × g, ×~g~. For any Borel set B c V,

(2.2) fBZ(Z, z', v)d~k(z', v')

is a measurable function of (z, z', v) relat ive to the a-algebra g~ × £~ × ~g~.

PROOF. -- Firs t we note explicitly tha t in (2.2) and frequent ly hereafter we use the notat ional device of indicating the variable (s) of integration by subscripts a t tached to the (~ differential ~> symbol. Le t C be the class of all Q ~ ~ × g~ × ~ for which the conclusion holds. We wish to show tha t C is a monotone class, in the sense of HALLOS [13], and tha t any measurable rectangle in g~ × g~ × :B~ is in C. As g~ × £~ × :g~ in the smallest monotone class containing these measurable rectangles (§§ 6 and 33 of [13]), the desired conclusion follows. The fact tha t the measurable rectangles are in C is an immediate consequence of A.1). Le t Q~ c @ c ... be a monotone increas- ing sequence of sets in C, and define the sections Qi(z, z ~) by

q~(z,z')= {v: (z,z', v) eq~}.

Let S = U Q~. By posit ivi ty of k(z' , v') as a measure on V, and the inclusions Ql(z, z r) c Q2(z, z ~) c ... we obtain

fZs(Z' , z, v) d~k(z', = l i m ~ z a z, v) t " (z', d~k(z', V t )

f rom the monotone convergence theorem. I t follows tha t S E C, as the limit of measurable functions is measurable. I f the Q~ are a decreasing sequence in C we conclude tha t (3QieC by essentially the same argument, except tha t posit ivi ty of k(z', v ') , A.5), and the dominated convergence theorem are used ra ther than the monotone convergence theorem. This shows C is a monotone class, and completes the proof.

PAuz ~ELS0~ J r . - I t . DEA~ VICTORY Jr . : Measures on phase spaee~ etc. 73

REMARK. - We can now assert the measurabi l i ty of

for any B e ~5~, because the in tegrand can be approx ima ted by a countable increas- ing sequence of simple functions measurable relat ive to ~, × g, x ~,..

To formula te the t r anspor t operator acting in Me, we need the following resul t :

P~0POSlTIO~ 2.2. - Le t E be ~ny measurable set in P----[0, a] x V, and define

(2.4) E(z)= {re V: (z, v) eE}.

Then

(2.5) :s(z) z"

is ~ measurable funct ion of (z, z', v').

P~ooF. - The proof is essentially the same as t ha t of Proposi t ion 2.1~ except

wi th the preceding r e m a r k replacing A.1) in assert ing the val id i ty of the conclusion for E a measurable rectangle. Details ~re omit ted.

Fo r E any measurable subset of P, we use the nota t ion (2.4) generically for the z-section of E. I f A c V we define

A + = { v E A : # > O } , A _ - ~ { v e A : # < O } .

Le t F 6 Mp. By Proposi t ion 2.2 we can define formal ly a set funct ion K F on ~ ×~5~, into the reMs b y

0 [0,~]×V B(z)+

#-l(r(z, v) exp - - (s, v)ds d~k(z , v )d(~, ~,)I dz f f f

I n the nex t section we show rigorously t ha t K F is a measure on phase space, [0, a] X V. Here we wish to point out the connection between (2.6) and t ranspor t in a slab. This connection is based on in te rpre t ing /~ as defining the distr ibution of collision densities,

i.e. if E ~ £z ×g~,, t hen /~ (E) is the collision ra te for neutrons with position and veloci ty in this subset of phase space. The quan t i ty K~P can be in terpre ted as defining

the dis tr ibut ion in phase space of the ~ nex t generat ion ~) collisions for neuti'o~ns

7~ PAUL NELSOn- Jr . - H. DEAN VICTORY Jr . : Measures on phase space, etc.

emanat ing f rom collisions described by F. In the case tha t k is absolutely continuous with respect to Lebesgue measm-e on velocity space and /" with respect to dzdv on phase space, then K F is absolutely continuous with respect to dzdv, and F and K F are then described by their Radon-Nikodym derivative. In this case, / ' is simply a(z', v') multiplied by the familiar angnlar flux. In this s i tuat ion KF(E) is ordinarily described by integrating in the order v' , z', v, z; however, in the classical situation, Fubini 's theorem will permit the order of integration to be changed to v, v ' , z', z, whieh is the only order meaningful in the current general setting. Thus (2.6) is a generMization of the classical expression for the next generation collision

density.

3. - N o n m u l t i p l y i n g impl ies suberit ieal .

T ~ E o u ~ [ 1. - I f assumptions A.1) - A.5) hold and K F is defined by (2.6) for / ' e Me, then K I t e My and K is a bounded linear operator on Mp with I!Kt] < 1 .

PROOF. -- Wi thou t loss of generality, we take F to be positive, as any signed measure is the difference of two positive ones. Countable addi t ivi ty of K F is an easy consequence of the definitions (2.6), the ident i ty (W E i ) ( z ) = k ) E d z ) , and the Lebe- sgue monotone convergence theorem. As k a n d / ~ are both positive, K F is a measure on MF. We fur ther compute

(3.1) ]lKrll= KF[[O, a]× VJ= i ! f v). 0 [O,z] V V+

z

~' 0 [ ~ , a ] x V V -

Z'

As Proposit ion 2.2 assures us tha t

z

V+ z'

and

F'- z"

are measurable functions relative to £~ × £~ × ~Sv, Tonelli 's theorem [14, p. 140] enables

PAUL :NELSON Jr. - H. DEAN VIOTOI~Y Jr.:Meassures on phase space, etc. 75

us to interchange the order in (3.1). The result is

(3.2) l iKFi l =

The expressions

and

[O,a]xV z' V+ z'

[O,a]x V 0 V- z'

f~

are measurable relative to ~ × ~,, for fixed z'. Thus we can again interchange the orders of integration by Tonelli's theorem, which gives

(3.3) z

[0,a] x V V+ z ' z '

[ 0 , a ]×V V- o ~'

d(¢,¢)/'.

An easy calculation shows

(3.~)

and

a ~ a

z" ~t $,

0 z ' 0

, # > 0

, # < 0

both of which are measurable funct ions of z' and v and dominated by 1 for all z' and v. Thus (3.3) is dominated by

f [O,a] × V V -

76 PAVL NELSON Jr . - H. DEAN VICTOtCY Jr . : Measures on phase space, etc.

and by Assumption A.2), this is not greater than tlFI{. Hence t]K/'It < I1F]] and ]IK]t <1 . This completes the proof.

We write the equation of s teady-state neut ron t ransport in a slab in the form

(3.6) i f = cKF + Q

wh ere / " e M2 describes the distribution in phase space of collision densities, and given Q E Me defines the distribution of collision densities of particles emanating direct ly f rom internal or external sources. F r o m the normalization condition A.2) on k(z', v'), c ~ 0 represents physically the supremum over points in phase space of the expected number of neutrons emerging from a neut ron of velocity v ' undergoing a collision at location z'. We describe the situation c < 1 as nonmultiplying. If (3.6) has a positive solution in Mp for every nonnegative Q ~ M~, then we say the underlying system is subcritical. I f c < 1, then the inequali ty ]IK[I < 1 established in Theorem 1 shows the Neumann series of (3.6) converges in Me. As K maps the set of nonnegative measures into themselves, it follows tha t if Q e M +, then this Neumann series is composed entirely of nonnegative measure. We have therefore proved the following result:

TIIEORE~ 2. - - I f Assumptions A.1)- A.5) hold, then nonmult iplying t ranspor t in a slab is also subcritical.

In the subsequent analysis, all references to order in Me are relative to the cone of nonnegative measures in Mp, Such a cone is reproducing and normal in the sense of KA~LI~ [~5].

4. - Forms o f the criticality condit ion.

The definition of subcrit icali ty given in the preceding section is certainly physically reasonable. However, there are other forms of a definition for subcriticality which are equally reasonable f rom the physical viewpoint. Two specific examples are a definition based on the asymptot ic behavior of t ime-dependent t ransport , and the computat ional ly useful ~ k-effective ~ form in which the eigenparameter multiplies only tha t par t of the collision integral due to fission. There are subtle questions con- cerning the precise mathemat ica l relationship between these different forms of subcriticality. I t is beyond the scope of this paper to consider this part icular question. However , we would like to record the following theorem, which gives several different conditions which are nontrivially equivalent to our concept of subcriticality. The proof is omitted, as it is essentially the same as tha t of Le mma 1 in [7].

T~EORL~ 3. -- The following conditions are equivalent.

(a) ( c I - -K) -1 is a positive bounded linear opera tor on Mp,

(b) The t ranspor t problem (3.6) is subcritical.

PAVL NELSO~ Jr . - I t . DEA~ VICTORY Jr . : Measures on phase space, etc. 77

(c) The 2qeumann series ~c'~KnQ converges in Me for every Q e M p . n = 0

(d) c < c~, where c~ is the reciprocal of the spectral radius of K; i.e.

(4.1) (c~) -~ ~-- l im IIK tl~ ~, n - - ~ o o

5. - An example.

In view of Theorem 3 the conclusion of Theorem 2 can be res ta ted as the inequal i ty c1>1, where cl is defined by (4.1). In this section we give a simple example which shows this inequali ty is sharp; i.e. t h a t i t is possible to have the equali ty cl---- 1 under our basic assumptions A.1) - A.5).

The example consists of the monoenergetic neut ron t ranspor t problem with v = 1, a - - 1 , and k defined by

{ ~ ( / , - - # ' ) , # ' ¢ 0 k(z', v ') = 0 , t~' = O .

These functions do satisfy our conditions A.1) - A.5). (There is a legitimate philosophical question as to whether this scattering function is physically meaningful, in view of the apparent impossibility of determining it operationally. ~onetheless our viewpoint is t ha t it is mathematical ly meaningful, and it does satisfy our basic hypotheses. In any event an example is given in [7 ] for which cl = 1 and the above- ment ioned philosophical question does not arise).

We shall show tha t c~ = 1 for this part icular model. This will be accomplished by displaying, for an arb i t rary positive integer n and arb i t rary e > 0, an element q of Me, depending on n and e, such tha t IIK~qIl/Hql[ > 1 - e . This implies tha t IlK' ]l > 1 for each positive integer n and therefore, by (4.1), c~<l . As Theorem 2 gives c~>l ,

the equali ty c1= 1 follows. I t remains only to construct q as promised. For given/z0 > 0 let q(z, v) = (5(# -- #o).

F r o m (2.6), with k and a as just described~ i t is easy to compute

Kq(z, v) : (1 -- exp [-- z/#o]) (~(# --/~o) > (1 --/&) U(z -- 5)8(# -- #o),

where U is the Heaviside step function defined b y

0 , x < 0 , U(x) = 1 , x > O ,

and 5 = 5(/~o) is given by

(5.i) ----- -- #o In/zo.

B y induction we obtain the est imate

K ' q ( z ) > (1 - - ~o)~ U ( z - - n~) ~(/~ - - ~o) •

78 PAUL NELSO:N Jr . - g . DEA~ VICTory Jr . : Measures on phase space, etc.

Fro m this inequali ty we readily obtain

It qlt_ (5.2/ 1 - - - .

Now (5.1) shows tha t 5-->0 as ~o--~0, which wi th (5.2) implies tha t ilK'qlt/ilq[i can be made arbi trar i ly close to 1 by choosing #o sufficiently near zero. This completes the construction of the desired example.

6. - Sufficient conditions for cl > 1.

In the example of Section 5 the fact t ha t e l = 1 is closely associated with the fact tha t the slab appears to have an arbitrari ly large optical thickness for particles traveling nearly parallel to the slab faces. One suspects tha t this behavior and its consequence can be avoided by assuming tha t some of the scat tered neutrons must go into velocities associated with optical thicknesses having some bound. Theorem 4 below provides a rigorous formulat ion of this fact.

For any positive real number y and z e [0, a] we define a Borel measurable set

where

z

O, g > O ,

a , # < 0 .

The integral in the definition of W(y, z) is the opticM distance to the exterior of the sl~b as seen at z by a neut ron of velocity v. The additional assumption needed is the following:

A.6) For some y > 0 the inequali ty

(6.1) sap I f d~k(z ' , v ' ) }<l (~',v')eP ~~ W(v , ~')

is satisfied. The hypothesis A.6) essentially says tha t the expected number of neutrons which

are emi t ted per collision so as to have distance y to t ravel before reaching the slab face must be bounded below c, uniformly over all possible conilgurations of the col-

a liding particle. For the case of the optical thickness, fo'(~s, v)ds, a bounded func-

o tion of v, this means tha t the expected number of particles emit ted per collision into velocities such t ha t lttl < e mus t be bounded below e, uniformly in (z', v'), for some s > 0. Obviously A.6) will be satisfied in most problems of physical interest, and it leads to the following result.

PAuL 5TELSO~ Jr . - I t . DEA~ VICTORY Jr. : Measures on phase space, etc. 79

T ~ v . o ~ 4. - I f conditions A.1) - A.6) are satisfied, then ItK[t ~ i for K considered as a linear operator on My, and consequently cl > 1.

P~oo~. - The proof consists basically of a slight sharpening of the estimates involved in the proof of Theorem 1. I t is modeled on the proof of Theorem 2 of [7]. By A.6) there exists y satisfying the inequali ty (6.1). F ro m (3.3) we obtain the est imate

(6.2) ]lKrll <f f p(~,, ~)eok(~,, ~,)d~,,~,f t" V n W(v,z ' )

+ f f 1" V~ W ( v # )

where we in t roduced the nota t ion p(z ~, v) for the expressions in (3.4). In the first of the integrals on the right of (6.2), p is overest imated by 1 - exp [ - -y ] , and in the second it is overest imated by 1. Upon subst i tut ing these estimates into (6.2) and using A.5), the result can be wri t ten as

IIKI"II < [1 -- exp [-- y] (1 -- F(y))] ]lrl],

where F(y) is the quant i ty on the left of (6.1). As y is arbi trary, and A.6) says/?(y) < 1 for sufficiently large y, the desired result follows.

A second subcriticality condition for systems which are not excessively multiplying is a corollary of the foliowing result.

Pl~oyosI~Io~ 6.1. - I f assumptions A .1 - A.5) obtain, then 2 ~ 1 c a n n o t be an eigenvalue of K .

PnooF. - Suppose contrariwise tha t KF-~ 1' for some nonzero / ' ~ Me. We shall establish the contradictory inequal i ty tlKFII < I]Fl[. For this purpose the mini- mal i ty proper ty of the Jo rdan decomposition [14, p. 127] allows us to assume ~P has values in M + (-~ finite measures on P) wi thout loss of generality. Fu r the rmore we know tha t

(6.3) o = K_P([o, a] × Yo) = r ( [o , a] × Yo),

the first equal i ty following f rom A.4) and (2.6), and the second f rom K F ~ IF. sequently we have

f f p(~', ~)dok(~', ~')~(o,o,)r, (6.4) ]lKrjI 1" V~ Vo

Con-

where p is as in t roduced in the proof of Theorem 4. For arbi t rary positive integer m let

B~ = {(z', v) e [0, a] × V~-- 17o: p(z', v) <1 -- 1Ira},

and for a rb i t rary z '~ [0, a] denote the z'-seetion of B~, by Bm(z') as in (2.4).

80 PAUL ~ELSO~ Jr . - g . DEA~ VICTORY Jr. : Measures on phase space, etc.

The function

(6.5) (z', v') ~ fp(z', v) d,~k(z', v')

is measurable relative to £~ × 3~, by Proposition 2.1, measurabili ty of p, and the equi- valence of the two obvious definitions of measurable rectangle in a product space (p. 140 of [13]). In view of this it is permissible to split the integral over V ~ ]7o in (6.4) into two integrals over B~(z') and ( V ~ Vo)~B,~(z') respectively. There easily follows the estimate

(6.6) IlKrli < iIrtl - ~ ( k(z', v') B~(z') d r . 1)

The function

(z', v') -~ k(z', v') B~(z')

is measurable by the same type of argument as outlined above for (6.5). Further- more assumption A.4) implies t ha t k(z', v')B,,(z') converges monotonically upward to I~:(z', v ' )V everywhere in (z ' ,v ' )eP. The monotone convergence theorem then implies the integral in (6.6) converges monotonely upward to

fk (z', v')VdF P

as m -+ co. :But comparison with (6.4) shows this integral is not less than [IKFII = [If'l!, and therefore is positive. Consequently the integral on the right of (6.6) is positive for sufficiently large m, whence the desired inequality IIKFI] < ]]FI]. This completes the proof of Proposition 6.1.

The following theorem is an immediate consequence of the inequa~ty c1>~1, Proposition 6. t , and the two facts tha t the spectral radius of a positive operator must be in the spectrum of the operator [15] ~m4 the spectrum of an eventually compact operator consists entirely of eigenvalues.

THEO~X 5. - I f conditions A.1)- A.6) hold and K is an eventually compact operator (i.e. some iterate of K is compact) on Mp, then cl > 1.

We now turn to the problem of determining conditions on the data, k and o, under which K is eventually compact.

7. - Eventual compactness of K.

The basic aim in this section is to establish the theorem tha t K 2 is a compact operator on Me under conditions A.1)- A.5) and the following hypothesis:

PAUL NELSO~ Jr . - H. DEA~ VICTORY Jr . : Measures on phase space, etc. 81

A.7) There exists a a-finite measure ~ e M + such t h a t

f¢(z, v)d,,adz< oo P

and fur thermore the following two conditions hold:

i. Fo r any s > 0 there exists 5 > 0 such tha t A e 3~ and gA < 5 imply

fd~k(z', v') < A

for all (z', v ' ) eP . (We refer to this p roper ty as absolute cont inui ty of with respect to ~, uniformly in (z', v').)

if. The limit

l im f d,,k(z', v ~)= 0

holds uniformly in (z', v ' ) e P .

k(z', v')

I f the measure e has the proper ty

l im c~{v E V: i # l < ~}, a-->O

then proper ty ii above is implied by proper ty i. This is the case, for exampl% when is Lebesgue measure on V = unit sphere in E S (i.e. monoenergetic t ransport) .

I f condition A.7) holds, then k(z', v') has a I¢adon-Nikodym derivat ive with respect to ~ everywhere in (z', v ' )e [0, a] × V. We denote such a derivat ive by ~(z', v ' , • ). Fo r any e > 0 we define

0, I~I<~ or ~(z',v',v)>l/~, Icdz' , v ' , v) = k(z ' , v', v) otherwise.

Let kdz', v') be the a-finite measure on V whose Radon-Nikodym derivative with respect to ~ is kdz', v', • ). Final ly let K~ be the operator defined by replacing k(z', v') with G(z', v') in (2.6). I t is clear tha t Theorem 1 applies to K~ as well as to K, and therefore Ks is a bounded linear operator on Me with norm not exceeding one.

PROPOS~TIO:N 7.1. - I f hypotheses A.1) - A.5) and A.7) hold, then for each s > 0 the operator K~ is compact on M e.

PROOF. - As in (2.4) let A(z) denote the z-section of the measurable subset A of P . Le t E be the unit ball in Mp. We wish to prove tha t K~E is uni/ormly abso-

6 - . 4 n n a l i d i M a t e m a ~ t c a

82 PAtvL NELSO~ Jr. - H. DnA~ VICTORY Jr.': Measures on phase space, etc.

lutely continuous with respect to the element y of M~ defined by

yA = f f a(z, v)d=~dz. o .4(~)

F o r / ~ e Me we can apply Tonelli's theorem to (2.6), as in the proof of Theorem 1, to write

KS.~<A) = f { ; f ~--10'(~', +~) exp [--/~-1 i o'(s, v, ds] ~g(~/, vl, v} dv~ ~z} ~(,, v.)~ P ~' A(~)+ ~'

I" o A(~). ~'

If F e E we obtain the estimate a

O A

from the preceding equality and the fact tha t 0 < $~/t#[ <1/e*. This inequality estab- lishes the desired uniform absolute continuity.

:From the uniform absolute continuity just established, i t follows tha t K~E is a weakly sequentially compact subset of Mp (problem IV. 13. 22 of [16]). Conse- quently K~ is a weakly compact operator on Mp. Thus there is a sequence in K~E, say {],}, which converges weakly in Mp. Fur thermore {/~} is in LI(y) by what was just proved, and it is readily seen tha t {],} is weakly convergent in Ll(y). By what was just shown above, the restriction of K to Ll(y) is a weakly compact operator from Ll(y) to Me. I f follows from a theorem of Dunford and Pett is (Theorem VI. 8.12 of [16]) tha t {K~],.} converges relative to the norm in Me. This completes the proof of Proposition 7.1.

PRoeosI~Io~ 7.2. - I f hypotheses A.1) - A.5) and A.7) are satisfied, then K~ converges to K in the operator norm (of Me) as e -~ 0.

P~oo~'. - I f we estimate ILK--Kell as was done for IIKII in the proof of Theo- rems 2 and 4, there results

ILK-- K~[I < sup I~p(z', v)d~[k(z', v')--k~(z', v ' ) ] l , (~',.')e_P V

( . t J

where again p(z', v) is defined by (3.4). I f we use the notation

Z~(z', v')= {v: $(z', v', v) > l/e},

PAUL ~ELSON Jr . - H. DEAN VICTORY Jr. : Measures on phase space, etc. 83

then this est imate can be wri t ten

(W,v')eP Z~ t~l<~

Bu t parts i and ii of A.7) imply respectively tha t the two integrals on the right tend to 0 as s - ~ 0 , uniformly in (z,, v , ) ~ P . This completes the proof of Propo- sition 7.2.

TEE0~E~ 6. - I f conditions A.1) - A.5) ~nd A.7) are satisfied, then K~ is a compact operator on My.

Pl¢OO~. - This is an immediate consequence of Propositions 7.1 and 7.2, and the closure of the class of compact linear operators under uniform convergence.

The key to ~pplying Theorem 6 is clearly finding a suitable measure ~ which s~tis- ties condition A.7). In [7] this role was played by Lebesgue measure. I f k(z', v') consists of ~n ~bsolutely continuous par t (modulo Lebesgue measure) plus a singular par t concentrated at only eountably ma n y points, over ~11 (z ~, v ' ) e [0, a] × V, then one clearly can find a suitable ~. In other c~ses (e.g. elastic scattering), it is unclear

how to construct such ~.

8. - T h e B o c h n e r i n t e g r a b l e c a s e .

In [12] the authors presented, wi thout proofs, ~ restricted form of the results of Sections 1-6 above. The fundamenta l aim of this section is to indicate how the results presented in [12] are consequences of those proved above. The underlying space in this section is the class of Boehner integr~ble functions from [0, a] to M~, the B~nach sp~ee of signed Borel measures on V. This space will be denoted B(M~). As the norm on M~ is given b y the variat ion norm

it follows from [17, pp. 81-82] tha t B(M+) is a Banach space under the norm

(8.21 IIril = f f eolr< ll . 0 V

I t is clear t ha t B(M+) can be regarded ~s a subsp~ce of Mp, and we adopt this view. Section I I I . 1 of [17] will be our s tandard reference on abstract integration theory. I n this section Assumption A. t ) ~bove is usually replaced by the following:

A. I ' ) k is ~ strongly measurable mapping from [0, a] × V to M + = nonnegative measures in M+, in the sense that there exists a sequence of functions {k~}, each mapping [0, a] × V into M, and taking on at most countably ma n y values, each

84 PAUL NELS0~; Jr . - H. DEAN VICTOICY Jr . : Measures on phase space, etc.

value being achieved on a measurable subset of [0, a] × V, and such tha t k~(z, v') converges to k(z, v') relative to the var ia t ion-norm (8.1) for all v ' e V almost everywhere in z e [0, a].

RE~A~K. -- We note tha t the sense of strong measurabil i ty in A.I ' ) agrees with tha t of [17] relat ive to the measure on £, × 5~, defined b y

(8.3) o A(~)

where y is <( counting measure ~> on V and A(z) is the z-section of A. (The domain of fl as an extended real-valued function includes £~ × 3~ because, in the termino- logy of HALlos [13], this domain is a C-ring and it contains all measurable rectangles A × B with A e £~ and B ~ 55~. Fur the rmore fl is readily shown to be countably addit ive on its domain, and therefore it is a measure.) However fi is not a a-finite measure, and consequently one must be somewhat cautions about using the results of [17] regarding strong measurabil i ty, as the blanket assumption of g-finiteness is made in this reference.

I t is clear tha t A.I ' ) implies A.1), as A.1) is certainly satisfied for each of the k~; fur thermore, A. I ' ) implies k~(z', v ' )B-+k(z ' , v ' )B for every B e 5~ and a l l v ' e V almost everywhere in z e [0, a], and the class of measurable functions is closed under pointwise convergence. However , as detailed in [12], A.I ' ) does not include the impor- t an t case of elastic scattering, whereas, as shown in the next section, this t y p e of event is included under A.1). Thus A.1) is str ict ly more general than A.I ' ) , and the above results are a considerable improvement on those announced in [12]. However the following result shows t ha t A. I ' ) includes the cl~ssicM case.

P~OPOSITIO~ 8.1. -- I f k: [0, a ] × V - + M + is such t h a t k(z', v') is absolutely con- t inuous with respect to Lebesgue measure on V for all v ' e V almost everywhere in z' e [0, a], and the associated l~.adon-~ikodym derivat ive k(z', v ' , v) is measurable relative to £ ~ × ~ × ~ , then k satisfies A.I ' ) above.

P ~ o o ~ . - As L~(V) is separable, there exists a dense sequence {h.}. Given a positive integer ~a we define

(8.4) Q~: {(z', v'): f l$(z', vr, v) -- h~(v)ldv < l/m} . v

Because k(z', v', v) -- h~(v) is measurable relative to £~ × 2~ × ~ the integral in (8.3) is a measurable funct ion of (z', v'), and therefore Q~ is a measurable subset of [0, a] × V. I f we inductively define R I = Q1, R~-Q~,.~R~_I, then the {Ri} are disjoint measurable sets whose union is {[0, a] --~ A} × V, where A is a null subset of [0, a]. Le t

PAUL i~ELSON Jr . - I t . DEAN VICTO]~.Y Jr . : Measures on phase space~ etc. 85

the function k~: {[0, a],~A} × V-* M + be defined by

f k..(z', v') B = ~ Z~(z', v') h~(v) d r , B

where Z~ is the characteristic function of Ri. Then (8.4) shows the countably-valued sequence {k~(z', v')} converges to k(z', v') in the M~-norm (8.1)7 except possibly for (z', v') c A × V. This completes the proof.

REMA~. - The conclusion of Proposit ion 8.1 holds if the assumption tha t $(z', v', v) is joint ly measurable is replaced by the requirement tha t the mapping (z', v ' ) -~ k(z', v'~ . ) eL~(V)be measurable in the s tandard sense tha t the inverse of an open set in LI(V) is measurable relative to £~ ×:B~. This measurabil i ty condition suffices to assure the Q~ defined by (8.4) are measurable sets, which is the only use made in the proof of the joint measurabil i ty of ~.

By the measurabil i ty of (2.3), the quant i ty

(8.5) [O,~] × V A+ ~"

z

[~,a] x V A - z '

exists as an extended real almost everywhere in z for any A e ~B~. We can actually prove mor% as follows:

PROP0SI~ION 8.2. -- I f A .1 ) -A .5 ) hold, then for every F e M p and A E ~ the quant i ty on the r ight-hand side of (8.5) is finite almost everywhere in z E [0, a]. Fur thermore the set function I(F(z) so defined on ~ is actually a measure for almost every z in [0, a].

P~ooF. - As in the proof of Theorem 1 we estimate

a

f lf£F(zl AIdz <~ ltl'II , 0

which implies the claimed finiteness. Countable addi t iv i ty of KF(z) for almost all z in [0, a] follows from applying the monotone convergence theorem twice to each of the terms on the r ight-hand side of (8.5). This completes the proof.

~V~E/~IAEK. -- U p o n comparing (2.6) and (8.5) one readily sees the above proposi- t ion asserts t ha t anything in the range of K can be interpreted as a measure on V

defined pointwise for z e [0, a].

TttEOREM 7. -- I f A.I ' ) and A.2) - A.5) are satisfied, then the operator K defined

by (2.6) maps Mp into B(M~).

86 PAUL NELSON Jr. - I t . DEAN VICTOI~Y Jr. : Measures on phase space, etc.

PR00F. - By the preceding remark it is sufficient to prove the assertion for defined by (8.5). Firs t we remark tha t Corollary t to Theorem 3.5.3 of [17] shows we may assume tha t the sequence k~(z', v') of condition A.1 ~) converges uniformly (in M~) to k(z', v') for all v' ~ V almost evm'ywhere in z' ~ [0, a]. (Note tha t a-finiteness of the underlying measure space is not used in the proof of Theorem 3.5.3 of [17].) Consequently we may assume tha t /~(z', v') is uniformly bounded in M,, for all v ' e V almost everywhere in z ' s [0, a]. For FeB(M,) , let ~,F(z) be defined by (8.5) in ~nalogy with KF(z), except with k replaced by k~. The conclusion of Proposition 8.2 holds with /~ replaced b y / ~ . By using Tonelti's theorem as in the proof of Theorem 1, we obtain the estimate

a 0~

0 O V V

a

< 0 V

< e s s sup [sup IIk(~', ~') - k~(~', ~')I/~J" IIrll,(~) • z ' e [0 ,a ] v ' ~ V

Now the first factor in the last expression tends to zero as n - * 0% by the preceding condition on /¢~. I f the /~=F were known to belong to B(M~), the desired conclusion _#;T'eB(M,) would follow from Theorem 3.7.7 of [17].

I t remains only to establish tha t ~,FeB(M~) for arbitrary F e Me. By Theo- rem 3.7.4 of [17], this is equivalent to showing tha t ~ / " is strongly measurable from [0, a] to kL, and satisfies the inequality

(s.6) q

]l &Fl[~o,o) = f IlxtJ'(z)I]~d~ < oo. 0

The strong measurabil i ty is the only real issue, as the inequality in (8.6) can be obtained precisely us was the inequality I!KFi! < ilFi! in the proof of Theorem 1. To establish this strong measurabil i ty we use the fact tha t

k~(z', v') = ~ Xdz', v') ~ , i = 1

where each 2.~e M ~ , and the Z~ are characteristic functions of disjoint measurable sets in [0, a] × V. Without loss of generality we m a y assume F(z)~ M +. If this is substi tuted into the defining expression for I~F(z) , there results

a ~ o o g

PAUL ~TELSON Jr. - g . DEAN VICTORY Jr . : Measures on phase space, etc. 87

Now the part ial sums of the infinite series here are majorized respectively by

exp .4:~ ftr

I~urthermore, just as in the proof of Proposit ion 8.2, for almost all z ~ [0, a]

f I+(z', v') d~,F(z') v

exists as a measurable function of z' on [0, zJ, and

0 V

exists. Similar s ta tements hold for I_ , with obvious changes in the limits. Conse- quent ly by two applications of the dominated convergence theorem for each t e rm on the right of (8.7), we can take the infinite summation outside the integwals, a t least almost everywhere in z e [0, a]. Fm' thermore, we can use Tonelli 's theorem to just ify successively interchanging the order of integration on d .~ with d,,F(z') and dz', which gives

c o o o

I~nF(z ) A = ~ t+(z) ~A+ ~- ~ t~(z) 2 ig_ , i=1 i = 1

where 1+, f~- are measurable functions. Now the set functions A - + ~,A+, 2~A_ are measures on ~ . , and the rat ional linear combinations of finitely many of these form a countable set. Because 1+ and ]7 are nonnegat ive and measurable, it is clear tha t we can use convergence of the above two series to approximate ~f~F arbitrari ' ly closely in the norm of M~, almost everywhere in z e [0, a], by a function f rom [0, a] to M~ which only takes on values in this countable subset of M,, and fur thermore such tha t each such value is ~chieved on a measurable subset of [0, a]. This says ~f,F(z) is, almost everywhere in z e [0, aJ, the limit of a sequence of countably valued functions, which is to say ~ F is strongly measurable. This completes the proof of Theorem 7.

I t is an a ]ortiori consequence of Theorem 7 tha t K maps B(M~) into itself under the hypotheses of the theorem. For K regarded ~s an operator in B(M~) the results described in Theorems 1-6 clearly carry over mutatis mutandis. Alternat ively these results can be obtained as an application of those given in [7] by simply noting tha t A.I ' ) implies k(z', v') is absolutely continuous everywhere in v almost everywhere in z' relat ive to the measure

o o

i = 1

where the {~} are the countable set of values taken on by the (k.} of A.I ' ) .

88 PAUL 5TELSO~ Jr . - H. DEAI~ VICTORY Jr . : Measures on phase space, etc.

9. - The ideal scat ter ing laws .

A general inelastic scattering law can be wri t ten in the form

(9.1) kt(z', v') = --~-~l]~(z', v ' ) 6 ( v - vi) ,

where 6 denotes the unit atomic measure concentrated at zero (Dirac delta function), and/~(z ' , v') is the fraction of the total macroscopic cross section at location z' and neut ron velocity v ' for inelastic scattering into speed vi. I t is reasonable to assume ]~ is measurable relative to the a-algebra £z x S~, in which case the scattering law (9.1) clearly satisfies A. I ' ) of Section 8. By vir tue of the discussion in Section 8 it then also satisfies A.1) of Section 2.

As shown in [12], A.I ' ) is not satisfied by the idealized elastic scattering law. The next theorem shows tha t A.1) sufficiently generalizes A.I ' ) to include elastic scattering. We write the general elastic scattering law in the form [18, Chap. 6]

(9.2) k~(z', v ' ) : co

~,g~(z ~, v')p~(v', ~2.g~') ~{v - v'(A~ ÷ 1)-1[(A~÷ ~ . ~ ' - - 1)~÷ ~ - g t ' ] } , n = l

where g~(z', v') is the fraction of the total macroscopic cross section at location z' and neut ron velocity v' for elastic scattering from the n- th nuclide, p~(v ' ,~ .Ft ' ) is the differentiM probabil i ty of scattering through polar angle cos -~ ~t.~t ' in the laboratory system for a neut ron of velocity v' undergoing elastic scattering with a nucleus of the n- th nuclide type, and A~ is the mass number of the n- th nuclide.

TtIEORElVI 8. -- I f the functions g~ and (v', v ) ~ p , ( v ' , Ft.gt') are measurable relative to the a-algebras £~ X 3~ and ~ x ~ respectively, then an elastic scattering law of the form (9.2) satisfies hypothesis A.1) of Section 2.

P~ooF. - I t clearly suffices to consider one t e rm of the form on the r ight-hand side of (9.2). For this purpose we drop the nuclide subscript. The problem is then to show tha t

g(z', v')fp(v', gt-~2') Z~(v)~{v - v'(A ÷ 1)-~ [ ( A ~ ÷ gt.gt' - - 1 ) ~ ÷ ~2.gt']} dv V

is measurable in (z', v'), where Z~ is the characteristic function of the set B e ~ . I t clearly suffices to show

f / (z ' , v '~ ' , vgt)~{v - v '(A ÷ I) -I [ (A~÷ -- ~2.gt']} gt 1)½÷ dv V f~

v,a,, v'(A + 1)-1[(A2 + -- .rt]a)

. [v'(A ÷ 1)]~ (A~÷ ~2.~ ' -- 1) agt

PAU~L NELSO~ Jr . - H. DEA~- VICTORY Jr . : Measures on phase space, etc. 89

is measurable for ] a measurable funct ion of (z', v' , v) relat ive to ~ × ~o × ~ , , where S is the uni t sphere in E ~. Bu t this follows immedia te ly f rom Fubini ' s theorem, as the measurab i l i ty of ] and cont inui ty of (v', ~t, ~ ' ) ~ v ' (Aq-1) -~ [(APff-~ 4 2 ' - - 1 ) ½ q - ~ . ~ ']

imply the iu tegrand is measurable relat ive to ~ × 33~ × 53 s, where :5 s denotes the Borel sets on S.

1 0 . - C o n c l u d i n g c o m m e n t s .

I n analogy with section 8 of [7] the above results can be extended to ~ set t ing in which V is t aken as a measurable space, with each v ~ V being associated with

veloci ty v ~ e E ~. Fu r the rmore it is necessary to assume the following:

A.8) The mapp ing v - + v ~ is measurable f rom V to Borel sets in E ~.

Wi th veloci ty as above replaced b y the s ta te vectors v in such an abs t rac t (~ s ta te space >), the above results go th rough mutatis mutandis. Via this generalization the results are ex tended not only to monoenerget ic and mul t igroup models, bu t also to situations in which the s ta te vector carries such informat ion as spin s ta te or type of part icle (e.g. as in coupled neut ron g a m m a - r a y t ranspor t problems).

The results presented here would seem to provide a suitable existential basis for developing a convergence theory for one-dimensional mul t i -group approximat ions . The extension jus t ment ioned would be quite impor t an t in this context . I t is intended

to pursue this m a t t e r elsewhere.

REFERENCES

[1] J. E. 0LHOEFT, The Doppler e]]ect /or a non.uni]orm temperature distribution in reactor ]uel elements, Report WCAP-2048, University of Michigan (1962).

[2] G. BIRKItOFF, Positivity and criticality, Proc. l l - th Syrup. in Appl. 5'Iath., pp. 116-126, Am. Math. Soc., Providence, R . I . (1961).

[3] V. S. VLADI~I~OV, Mathematical problems in the one-velocity theory o] particle transport, Trudy Mat. Inst. Steklov., 61 (1961). English translation report no. AECL-1661, Atomic Energy of Canada Limited, Chalk River, Ontario (1963).

[4] K. M. CASE - P. F. ZW~IF:EL, Existence and uniqueness theorems/or the ne~ttron transport equation, J. Math. Phys., 4 (1963), pp. 1376-1385.

[5] I. VIDAV, Existence and uniqueness o] non-negative eigen]~netions o] the Boltzmann operator, J. Math. Anal. Appl., 22 t1968), pp. 144-155.

[6] A. SU~ADOLC, Linearized Boltzmann equation in L ~ space, J. Math. Anal. Appl., 35 (1971), pp. 1-13.

[7] P. NELSON, Suberiticality ]or transport o~ multiplying particles in a slab, J. Math. Anal. Appl., 35 (1971), pp. 90-104.

[8] Y. ROX:aN, On the estimation o] errors in reactor theory, Nuct. Sci. Eng., 54 (1974), pp. 467-470.

90 PAUL NELSOh- J r . - H . DEAN VICTORY J r . : Measures on phase space, etc.

[9] E. W. LA~SEN - P. F. ZW~IF~L, On the spectrum o] the linear transport operator, J. Math. Phys. , 13 (I974), pp. 1987-1997.

[10] E. W. LARsv.~-, Solution o] neutron transport problems in L 1, to appear in J. Math. Anal. Applic.

[11] A. SUHADOLC - I. VDAV, Zinearized Boltzmann e~uation in spaces o] measnres, Mathe- mat ica Balkanica, 3 (1973), pp. 514-529.

[12] P. NELSON - H. D. V I C T O R Y , The slab transport equation in a space o] measures, to appear in Proc. 4th Blaeksburg Conference on Transport Theory.

[13] P. R. HALMOS, Measure Theory, D. Van Nostrand, Princeton, N . J . (1950). [14] W. RVDIN, Real and Complex Analysis, McGraw-Hill , New York (1966). [t5] S. KARLI~', Positive operators, J. Math. Mech., 8 (1959), pp. 907-937. [16] N. DUNFOI~D - J. T. SCR~WA~TZ, I~inear Operators. Par t I : General Them'y, Interseience

(1957). [17] E. HILLE - R. S. PHILLIPS, I'unctional Analysis and Semigroups, Am. Math. Soc., Pro-

vidence, R . I . (1957). [ 1 8 ] S. GLASSTOI~-E - M. C. E D L U N D , The Elements o/NuetearReaetor Theory, D. Van Nostrand,

Princeton, N . J . (1952).

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Measures on phase space as solutions of the one-dimensional neutron transport equation