Exact Calculation of the Fluctuation Spectrum for a Nonlinear Model - Van Kampen

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Exact Calculation of the Fluctuation Spectrum for a Nonlinear ModelSystemN. G. Van KampenCitation: J. Math. Phys. 2, 592 (1961); doi: 10.1063/1.1703743View online: http://dx.doi.org/10.1063/1.1703743View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v2/i4Published by theAmerican Institute of Physics.Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/Journal Information: http://jmp.aip.org/about/about_the_journalTop downloads: http://jmp.aip.org/features/most_downloaded

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JOURNAL OF MATHEMATICAL PHYSICS VOLUME 2, NUMBER 4 JULY-AUGUST, 1961

Exact Calculation of the Fluctuation Spectrum for a Nonlinear Model SystemN. G. VAN KAMPENPhysics Section, Convair San Diego, San Diego, California*

(Received January 23, 1961)Fluctuations in a circuit consisting of a diode and a condenser have been trea ted by means of approximatemethods [C. T. J. Alkemade, Physica 24,1029 (1958); N. G. van Kampen, ibid. 26, 585 (1960)J. In the

present paper, exact eigenfunctions and eigenvalues of the master equation for this system are obtained,and the spectral density of the equilibrium fluctuations is calculated. The most striking result is that thespectrum of the eigenvalues (which are reciprocal relaxation times) has an accumulation point, correspondingto the average time interval between two successive electron transitions.

1. INTRODUCTIONI N this paper an exact calculation is presented of thespectral density of the equilibrium fluctuations ina simple model of a nonlinear system. The system is anelectrical circuit consisting of a condenser and a diodein contact with a heat reservoir. I t has been shown1 thatthe electrical fluctuations in this system are governedby the master equationap(N)/at=p(N+1)-P(N)

+f{e-(N-l)P(N-1)-e- NP(N)}. (Ll)P(N) is the probability of having N electrons on the left-handcondenser plate (see Fig. 1); E=e2/kTC, e being the electroncharge, C the capacity of the condenser, T the temperature of thewhole system, k Boltzmann's constant;

W 1-W 2-e'/2Cl"=exp kT 'where W, and W2 are the work functions of the two electrodes;t is time measured in appropriate units.The reasons for this investigation are twofold.Firstly, the problem of fluctuations in nonlinearsystems has been studied by several authors,t-lo but noagreement has yet been reached. It therefore seemsworthwhile to obtain an explicit and rigorous result fora nontrivial example. In particular, it will be shownthat there are terms in the spectral density that cannotbe obtained by the usual expansion methods.Secondly, the model illustrates the distinction betweenslow fluctuations, which involve large numbers ofparticles, and rapid fluctuations connected withindividual particles. The relaxation times of the slowfluctuations are mainly determined by the RC timeof the circuit; they are the ones that are found by the* On leave of absence from the University at Utrecht,Netherlands.'N . G. van Kampen, Physica 26,585 (1960). D. K. C. MacDonald, Phil. Mag. 45, 63 (1954); D. Polder,ibid. 45, 69 (1954); D. K. C. MacDonald, Phys. Rev. 108, 541(1957).3 N. G. van Kampen, Phys. Rev. 110,319 (1958).4 C. T. J. Alkemade, Physica 24, 1029 (1958).6 R. O. Davies, Physica 24, 1055 (1958).6 W. Bernard and H. B. Callen, Revs. Modern Phys. 31, 1017(1959); Phys. Rev. 118, 1466 (1960).7 A. Marek, Physica 25, 1358 (1959).8 M. I_ax, Revs. Modern Phys. 32, 25 (1960).9 A. Siegel, J. Math. Phys. 1, 378 (1960).,. N. G. van Kampen, Can. J. Phys. 39,551 (1961).

592

usual approximate methods. The relaxation times of therapid fluctuations are determined by the average timebetween two successive individual electron transitions.The two kinds of fluctuations are separated by anaccumulation point in the spectrum of relaxation times.I t seems likely that such a distinction is a generalfeature of fluctuations in nonlinear systems. In linearsystems, of course, there is just one relaxation time.The mathematical problem consists of finding theeigenvalues and eigenfunctions of the difference equation(1.1), together with their relevant properties. Althoughthe mathematics is closely related to the theory of"q-difference equations",!l some of the results appearto be new and may have some interest by themselves.

As much of the mathematical work as seemed possiblehas been removed from the text to a series of appendixes.2. PRELIMINARIES

Equation (1.1) can be written more simplyap(N)/at= E - 1 ) P ( N ) + ~ ( E - l - 1 ) e - N P(N),

where E is the operator defined by Ef(N) = f(N+1).This can be further simplified by replacing N with anew variable sN=s+g, P ( N ) ~ P ( s ) ,

and choosing the constant g such that ~ e - g = 1. Then,ap(s)/at= (E-1)P(s)+ (E-l-1)e-P(s). (2.1)

Unless g happens to be an integer, s runs over a set ofnoninteger values. Let 'Y denote the distance of g to the

FIG. 1.

11 W. Hahn, Mathematische Nachrichten 2, 4 (1949).



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FLU C T U A T ION S PE e T RUM FOR A NON L IN EAR M O D E L S Y S T E M 593next higher integer12 :

g+'Y=integer, 0:::;'1'(s) = if; (s)/'l1o(s), if; (s)EJC'Y'with the scalar product

(1/>1,1/>2) = (if;1,if;2) =S'l1or!>I1/>2.In particular,

+Scp(E-l-1)e-'''or!>

= S'l1or!>(E-L l)I/>+SI/>(E-l_1) (E'l1o1/>= S'l1or!>(E-l_1)I/>+SI/>(1-E)'l1oE-11/>= -S'l10[(1-E-l)I/>J2. (2.5)

12 The introduction of 'Y as distinct from g is convenient butnot strictly necessary.13 For the theta functions, we follow the notation of E. T.

Whittaker and G. N. Watson, A Course of Modern Analysis(Cambridge University Press, New York, 1946), Chap. 21. The

This is clearly negative, unless I/> is a constant and henceproportional to Fif;=Sif;FI/>,F'l1 or!> = 'l1of />.

These relations show that F is the operator thatcorresponds to F in the dual space.The solution of the time-dependent equation (2.1)is equivalent to solving the eigenvalue problem

Fif;=-Aif;,or alternatively,

(2.6)We shall indeed solve this last equation, but first anauxiliary function has to be introduced.

3. AUXILIARY FUNCTION '7C(z)Definition:

00 1-e- k1r(z)= I I .k=1 1 - e-(k+ z )

Fundamental property:1r(Z) = ( l -e - EZ )1r(z-l).

From this follows, as 1r(0) = 1,m1r(m)= I I ( l -e- k ) for m=1,2 , .

k=1

(3.1)

1r(Z) is periodic with period 21ri/ f. I t has simple polesat z= -m (m= 1, 2, . . . ), the residues being given by

lim (z+m)1r(z) (_l)m+l exp( -!fm2+!fm)

z--+-m

There are of course additional poles at z= - m+ 21rim'/ f,where m'=l , 2, . . . . Otherwise 1r(z) is regular andit has no zeros. For all Z and all integral m,1r(z-m)1r(-z-l+m)

1r(z)1r( - z-1)= (- l )m exp[-!em2+e(z+!)m]. (3.2)

As Re z ---+ 00 , 1r(z) tends to a constant G,00lim 1r(Z) = I I ( l -e- k )=G. (3.3)k=1

parameter q of all theta functions in this paper has the value e - ~ '(and will, therefore, not be written explicitly). Accordingly, thequasi-period is !i .



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594 N. G. VAN KAMPENThis is the same constant G that occurs in the theoryof theta functions.1 3 As Rez - 00 avoiding the poles,1r(z) tends to zero. More specifically, if m ~ 00 throughintegral values,

1r(Z)1r( - z -1)1r(z-m)'"'"' (-1)mG

Finally,

I t is convenient to define as a separate function1rl(Z) =1r(Z+1ri/E),

with the fundamental property

(3.5)

(3.6)1rl(Z) has no singularities on the real axis. Equation(3.3) yields1rl(z-m)1rl( - z - 1 +m)

1rl(Z)1rl( - z - 1 )

while (3.4) becomes1rl(Z)1rl (- z - 1 )1 r l ( z -m) ' " ' " ' : - - - -

GXexp[-!Em2+E(z+!)m]. ( m ~ 00) . (3.8)

The so-called q-binomial coefficientsl4 l5 can bedefined in terms of 1r(z) by

This is an entire analytic function of y, which equals1 for y= 0, and vanishes for y= -1 , - 2, . . . . From thefundamental property (3.1) of 1r(z), the following threeidentities are obtained.

From (3.2) follows

Xexp[!Em2- E(Z- y+!)m]. (3.13)From (3.4) one finds for m + 00

[Z l ~ 1r(z) (-1)m

y+mJ=1r(z-y)1r(y-z-1)Xexp[!Em2-E(z-y+!)m]; (3.14)

and for m ~ - 00

[ Z], , - , 1r(z) (-1)my+m 1r(Y)7r(-y-l)Xexp[!Em2+E(y+!)m]. (3.15)

I t is convenient to define separately

[Z] 1r1(Z) [Z+1ri/E]Y 1 1r(y)1rl(z-y) Y4. SOLUTION OF EQUATION (2.6)

In order to find solutions of the difference equation(2.6),(4.1)

we put+00cf>(s) = ~ a.eE(.,.f-U)8.

a. and u are yet to be determined. Clearly u may berestricted byReu< 1, Imu(s) into the equation, one obtains atwo-term recursion formula for the a.,e- ET { e-(.,.f-u-r) -1 }a.+e(.,.f-o+l) {1-e-(,+u+I)}a.+l=0.

[ Z]= 1-e-EZ[Z-1],y 1-e- 'Y y -1 (3.10) This may be written in the form

(3.11)

(3.12)14 c. F. Gauss, Werke II (GOttingen, 1863) 11, in particularp.16.15 G. Szego, Orthogonal Polynomials (American Mathematical

Society, New York, 1959), p. 33.

1r(V+u) 1r(v+u+l)a.+e(.,.f-u+r+!) a'+1 = o.1r(v+u- r -1) 1r(v+u- r)I t then follows immediately that

1r(v+u-r-l)a,=c exp[-!EV2-E(u+r+!)v] ,1r(v+u)16 The value X= 1 would have to be studied separately. However, in Sec. 8 is shown that it cannot contribute to the densityspectrum of the fluctuations.



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FLU C T U A T ION S P E e T RUM FOR A NON L IN EAR MOD E L S Y S T E M 595with c independent of v. On account of (3.2), one also has

11"(7)a.= c' ( -1) ' exp( - ~ v 2 - 2 ~ O " v ) ,11" (v+0")1I" (T-O"- v)

=C'(-l)'[ T ] e x p ( - ~ v 2 - 2 ~ O " v ) .v+O"We thus have obtained the following general solutionof (4.1):

+00 [ T ]cf>(S;O",T)= L (-1) '. ~ o o v+O"

Xexp[ ~ v 2 - 2 ~ O " v + ~ ( v + 0 " ) s J , (4.4)T being related to Aby (4.3). I t follows from (3.3) and(3.4) that this series converges for all 0" and T, unless Tis a negative integer. I f 0" is an integer, all terms withV T-O" are zero.

cf>(s; O",T) has a number of obvious periodicityproperties with respect to 0" and T, but they are immaterial because of the restrictions (4.2) and (4.3).However, there is also a quasi-periodicity in T, which isexpressed by the following important recursion relations.Firstly, using (3.11) one finds1I"(T)( l -e - ET E)cf>(s;O",T) cf>(s;0",T-1). (4.5)1I"(T-1)

Secondly, introducing the dual functionw(s; O",T)=WO(S)cf>(s; O",T),

one finds with the use of (3.12)( l-e- T E-l)it(s; O",T) =it(s; 0", T+ 1). (4.6)

Thirdly, either by using (3.10) or by using (4.5) andthe difference equation (4.1) itself,(l-E-l)cf>(s; O",T)

= - ( l-e- T )e(-I)E-2cf>(s; 0", T-1). (4.7)5. NORMALIZATION CONDITION

In order to determine the eigenfunctions of F, thosevalues of 0" and T have to be selected for which thenormalization conditionS'l10(s)[cf>(Sj O",T)]2< 00 (5.1)

is satisfied. I f 0" is an integer (necessarily zero), andT=n (=0,1,2, .), the series (4.4) breaks off andreduces to a polynomial in e" of degree n. Hence, (5.1)is satisfied. This gives us a first type of eigenfunctions:cf>n(l) (s)=(s; O",T) for s 00 has now to beinvestigated.

Behavior for s ~ 00I f " ~ O , the series does not break off for negative v.Thus, for s - 00 , the terms with large negative v willpredominate, so that the asymptotic expression (3.15)may be inserted. Thus,

c f > ( s ~ - 00 ; O",T)+00'" L exp[ ! ~ v 2 _ ~ ( 0 " - ! - s ) v + ~ O " s ] .-00

The sum on the right ise . t 1 3 ( ! i ~ ( 0 " - ! - s .

On account of the quasi-periodicity of the thetafunction, this is equal toe . t 1 3 ( ! i ~ ( 0 " - ! - s+ 1) )e-'("-s)

,- ,) ,-1= e " 8 t 1 3 ( ! i ~ ( 0 " - ! - ' Y I I e-( .- +k)

k==O

= t 1 3 0 i ~ ( 0 " - ! - ' Y e x p [ ! ~ s 2 + ! ~ s - ! ~ ' Y 2 + (O"-!)].Because of the factor e x p ( ! ~ s 2 ) , this asymptoticbehavior is irreconcilable with the normalizationcondition (5.1), unless the theta function is zero. Hence,! i ~ ( O " - ! - ' Y ) has to coincide with one of the zeros of t1 3,

! i ~ ( O " - !-'Y) = (m'+!)11"+ (n'+!) i ~ / 2 ) ,with any integers m', n'. The only solution within therestriction (4.2) is

Behavior for s ~ + 00I f T-O" is not an integer, the series does not break offfor v + 0 . Then, for s + 0 , the terms with largepositive v will predominate, so that the asymptoticexpression (3.14) may be used. Hence

c f > ( s ~ +00 ;O",T)+00'" L exp[ ! ~ V 2 _ ~ ( 0 " + T + ! - S ) V + ~ o " s ] .

This can be worked out like the previous example, withthe result that the normalization condition can only be17 S. Wigert, Arkiv Mat. Astron. Fysik 17, No. 18 (1923);G. Szego, Sitzber. Preuss. Akad. Wiss. Physik.-Math. Kl. (1926)242; L. Carlitz, Ann. Mat. Pura Appl. (4) 41,359 (1956). Szego'spolynomials K n ( ~ , q ) are related to our oI>n(1) by

oI>n(l)(s)=Kn(-e"+', e-).I t should be noted that his orthogonality relation involves anintegral, whereas our Eq. (7.5) involves a summation over .



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596 N. G. VAN KAMPENsatisfied if

liE(o+r+!- 'Y)= (m"+!)1r+ (n"+!) (iE/2).This yields a denumerable set of solutions, viz.,

o+r=n"+'Y+1ri/E, (n"=O, 1, 2, . . . ).6. VARIOUS TYPES OF EIGENFUNCTIONS

Summarizing the results, one finds four possible setsof values for u and r.(1) q=O, r=n (n=O, 1, 2, .);(2) q= 0, r= n+'Y+1ri/E

(n= , -2 , -1,0,1,2, . . . );(3) q='Y+1ri/E, r=n+'Y+1ri/E

(n= , -2 , -1 ,0 ,1 , 2, .) ;

(n= , -2 , -1 ,0 ,1 ,2 , . . . ).They give rise to the following four possible types ofeigenfunctions.

First type: polynomials in e", see (5.2).n (1) (s) ==(s; 0, n) = t (_1),[n] exp( - Ev2+EJIS);

,= 0 P

Second type :n(2) (s)==(s; 0, n+'Y+1ri/E)

An (2 ) = 1+e-(n+r) (n= ., - 2, -1 , 0, 1, 2, . . . ).Third type:

n(3) (s) ==(s; 'Y+1ri/E, n+'Y+1ri/E)n 1rl(n+'Y)

= exp[(E'Y+1ri)sJ L: (-1)"------00 1rlC'Y+p)1r(n-p)Xexp( - Ep2- 2E'YP+EPS); (6.2)

An (3) = 1+e-(n+r) (n= , - 2, -1 ,0 , 1, 2, . . . ).An alternative form isn (3) (s)= (-1)n exp( - En2- 2E'Yn)

Xexp[(en+E'Y+1ri)SJ f:. (_O'[n+'Y]p= Q p11

Xexp[ ~ v 2 + 2 E ( n + ' Y ) p - ~ v s ] . (6.3)

Fourth type :n(4) (s)==(s; 'Y+1ri/E, n)

+00 1r(n)=exp[(E'Y+1ri)sJL: ( -1) , - - - - - --00 1rl(V+'Y)1rl(n-p-'Y)X e x p ( - ~ v 2 - 2 E ' Y P + E P S ) . (6.4)

An(4)=1-e- n (n=O, 1,2, . . . ).For negative n this expression is meaningless because1r(n) is infinite. We therefore define for n


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FL U C T U A T IO N S P E e T RU M FOR A NO N L IN EAR M OD E L S Y S T E M 597..yo and 'l>n in the following way:

-.IfO=Z-l'1ro,'l>n ( 1 )= 1 I " ( n ) e . n J - } ~ n (1),

or, with the aid of (3.1), (3.6), and (7.10),2[ 00 e-n1l"(n-1)SN(W)=- :E - - - -11" n=l (1 - e-w )2+w2

Z' e-2.'Y +0 0 e-n1l"1(n+'Y-1) ]+--:E .Z 11"1(1') -0 0 (1+e-(n+y2+ w2

The general orthonormality relation isSWon(i)'l>nW = ~ i J - o " m .

Accordingly, we also put-.If"U)=-.Ifo'l>,,(i).

In order to compare this with the result of reference 1, one hasto replace w by ow and multiply the whole expression with(7.5) because of the difference in time scale. The first term then agreeswith Eq. (44) of reference 1. I t involves the relaxation times

A special example of (7.5) isS-.Ifn (i ) = i 1 ~ " O . (7.6)

We here mention a result, which is obtained bysimilar methods in Appendix D :

Ss'lrn(l) = -1I"(n-1)Z, 1) (7.7)11"1 (n+'Y-1)Ss'lrn (2) Z"11"1(1'-1)

Here Z" is a third constant defined byZ"= s'lr0 (2)

I t is shown in Appendix F thatG8 Z'

(7.8)

(7.9)

Z"= -exp( -!E'Y2+!ey)__ =---. (7.10)11"1(-1') l-e''YI t should be noted that, on account of (7.6), the relations(7.7) and (7.8) remain valid if the factor s on the leftis replaced with N=s+g.

8. FLUCTUATION SPECTRUMThe spectral density of the fluctuations in the numberN of electrons is given byB

(8.1)the summation extending over all normalized eigenfunctions -.Ir with their eigenvalues Inserting theresults of the previous section, one gets

2 00 1-e-n {1I"(n-1)}2SN(W)=- :E- - - ----1" n=l (1-e-n)2+w2 1I"(n)ew

2+00 l+e-(n+y) {1I"1(n+'Y-1)}2+-:E- - - - - - - - -11" - 0 0 (1 +e-(n+y2+w2 11"1(1'-1)

Z"2 11"1(-y)X- e-en,ZZ'1I"1(n+'Y)

1- -RoC=-RoC+O( . )1-e-n n 'where RoC is the RC time of the circuit in the linear region. Thesecond term is new; it involves relaxation times of order

E ffJRo1+e .(n+yloC=kT"+. ..The physical meaning has been discussed in the introduction,and in reference 1. Of course, these relaxation times loose theirphysical meaning when they become comparable with the time offlight of the electrons between both electrodes.

The expansion of SN(W) for large W s

(8.2)I t has been shown before1,4 that this should be independent of the nonlinearity, which implies that thequantity [ ] should be unity. I t can readily be verified(by comparing powers of e-E on both sides) that

00:E e-En1l"(n-l)=1-G.n= l

Moreover, it is shown in Appendix G that+0 0:E e- En1l"1(n+'Y- 1)=GeE'Y. (8.3)-Substituting this in (8.2) and using (2.4), (7.4), and(3.5), one finds the desired result.This check is important for the following reason.Equation (8.1) has been derived for a complete set ofnormalized eigenfunctions. Unfortunately, we have notbeen able to prove that the set consisting of the functions

-.Irn(l) and ~ n ( 2 ) is complete. However, it is clear thatcompleteness is not actually necessary for (8.1); it issufficient that no eigenfunction has been skipped forwhich~ { S N - . I r F > O .

The fact that the quantity [ ] in (8.2) is actually



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598 N. G. VAN KAMPENunity guarantees that this condition is met. In particular, it follows that eigenfunctions belonging to A= 1, ifany, cannot contribute.

APPENDIX A. PROOF THAT 4n(4) ISPROPORTIONAL TO 4n( l)According to (6.S) and (3.7),

exp[ (e'Y+1I'i)sJ-1 (4) (s)11'1 ( -1-'Y)1I'1('Y)

+0 0XL (-1)" exp[ -!ep2+e(s-'Y+!)P].-0 0

The infinite sum ist?-4( -! ie(s- 'Y+!=O.

Thus, _1(4) = 0. Moreover, the relation (4.5) whenapplied to _m (4) with m ~ akes the form(1-e.mE)_m(4) = _m-l(4).Substituting successively m= 1, 2, "', one finds thatall _m(4) are zero. Substituting m=O, one finds

(1 - E)O(4) = 0; hence, 0(4) = constant.Thus 0(4) is proportional to 0 (1). I t then followsimmediately from (4.6) that each n (4) for sproportional to n (1) (with the same proportionalityconstant).

APPENDIX B. PROOF THAT 4n(3) ISPROPORTIONAL TO 4n( l)First one obtains from (6.2) directly

" (3) (2n+2'Y-s) = (-l)nn (2) (s)Xexp[-e(n+y+1I'i/e)s+en2+2e'Yn+2e'Y2+211'i'YJ.

Substitute n=O, S='Y,0(3) ('Y)=O(2) ('Y) exp(e'Y2+1I'i'Y)' (B.1)

Substitute s='Y+1,0(3) ('Y-1) = -0(2) ('Y+ 1) exp(e'Y2+1I'i'Y- E'Y).

On the other hand, from the difference equation for0(2) ,

Hence,0(3) ('Y-1)=O(2)('Y-1) exp(E'Y2+1I'i'Y). (B.2)

From (B.1) and (B.2) and the fact that 0(2)(S) and0(3)(S) satisfy the same difference equation (4.1), itfollows that they must be proportional to each other.Finally, using the relations (4.5) and (4.6), one findsgenerally

" (3) (s) =" (2) (s) exp(E'Y2+II"i'Y). (B.3)

APPENDIX C. PROOF OF THE NORMALIZATIONFORMULAS (7.1) AND (7.2)I f u and 7' have such values that the norm exists, onefinds using the result (2.5)

SWO(s)[ (s ;U,7')]2= X-lSWO(S)[ (1-E-l)(S; U,7')J2.From (4.7), this may be written

( l - e - OT)Swo(s)e2e(&-Il[E-2(s; u, 7'-1)]2,which according to (2.3) is equal to(1-e-OT)e'S[E-2wo(s)J[E-2(s; u, 7'-1)]2

= (1-e-'T)e'Swo(s)[(s; u, 7'-1)]2.Thus, we have obtained a recursion formula for thenorm. When applied to the n(1), it yieldsSWO[n (1)J2= (1-e- w ) (1-e-(n-l . . .

X (1-e-)enSwo[o(l)J2,which is just (7.1). When applied to n(2), it yields in asimilar way (7.2).

APPENDIX D. PROOF OF (7.7) AND (7.8)We first derive a recursion formula with the aid of(4.6).

Ssw(s; U,7') =Ss(1-e-'(T-l)E-l)w(s; u, 1'-1)=S'l'(s; u, 1'-1) (1-e-(T-!)E)s= (1-e-(T-lSsw(s; u, 7'-1)

-e-(T-l)Sw(s; u, 7'-1).According to (7.6), the last term vanishes unless 7'= 1and u=O. By iterating this recursion formula,Ssw" 1) (s)= -1I'(n-1)Z,

SSW n (2) (s)/1I'1(n+'Y-1) = independent of n.These are the relations (7.7) and (7.8), respectively.

I t should be noted that these equations do no t giveSs'l'o(1), but this quantity does not occur in the fluctuation spectrum either. Actually, it is easy to computeSswo(l) (s) = 'YZ+ (2i)-1t?-3'(!iE('Y-! exp( _!e-y2+!E'Y),where the prime denotes differentiation with respect tothe argument of the theta function.

APPENDIX E. COMPUTATION OF Z'In the definition (7.3) of Z', we substitute one factor0(2) from (6.1) and the other from (6.3), using (B.3).

Z'=S exp( -!eS2+tES)X { E-1)"[1 exp( -EP2+EPS)}X {exp( - E'Y2-1I'i'Y) exp[ (E'Y+1I'i)SJ

X f. (_1)1'['Y] eXP(-EJ.L2+2E'YJ.L-EJ.LS)}.1'=0 J.L 1Downloaded 05 Feb 2013 to 129.97.58.73. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions


9/11

FLU C T U A T ION S PE e T RUM FOR A NON L IN EAR MOD E L S Y S T E M 599Pu t s='Y+N, so that tke summation variable N takesintegral values,

+N lX lim L: exp( t ~ N 2 + t ~ N +7riN)Nl-+oo -N l

Since we have cut off the range of N, the triple sum isabsolutely convergent, and is therefore equal to

+N lX L: (_1)N exp[ t ~ N 2 + E ( V - JL+t)N]. (E.1)-N l

I t is readily checked that for integral V-IJ.,+0 0L: (_1)N exp[-tEN2+E(V-IJ.+t)N]=O. (E.2)- 0 0

Hence, the summation over N in (E.1) may be replaced,+N l 00 -N lL: - t- L: - L: . (E.3)-N l Nl

We consider the contribution of the first of these twoterms to the sum (E.1).First put N - v+IJ.= N' in (E.1);

00X L: (_1)N' exp[ t ~ N ' 2 + t E N ' ] .Nl- . . t j l

The last sum clearly tends to zero as N 1 -t 00 for anyfinite value of v (regardless of JL). Hence, one makes noerror by using the asymptotic value (3.14),

[ 'YJ,....., 1 exp[tEv2-E('Y+t)V],v 1 7rl(-'Y-1)

Moreover, pu t v=N1+IJ.+X; the result is

00

X L: exp( - EJ.lX)X ~ N l - j l00X L: (_1)N' exp( -tEN'2+tEN'). (E.4)

N'-X

Consider separately the terms with J.I) 1. Summationby parts yields for the repeated sum over X and N00 e-'j lXL: --( 1)X exp( -tEX2_hX)

-N l - j l 1 - e-p.

The second term, when inserted in the sum over J.I, givesrise to a sum which in absolute value is less than 1'] exp[ t E J . l 2 + ~ ( ' Y _ t ) J . I ]j l - l J.I 1 1-e- '

Obviously, this vanishes in the limit Nl - t 00 . Thefirst term in (E.5) is majorized by

When inserted in the sum (E.4), the result is less than

Hence, only the term with J.I=O survives in the limitNl -t 00 . It s value is found from (E.4)-1 00 00---- lim L: L: (_1)N' exp(-tEN'2+tEN').7rl( -1 ' -1) Nl-+OO X -N l -x

Again, summing by parts, one finds for the sum00- L: (X+N1)(-1)-(Hl) exp[-tE(X+1)2_tE(X+1)]-N l

00+ L: (_1)N' exp( -tEN'2+hN').Nl



10/11

600 N. G. VAN KAMPENThe last term vanishes in the limit and so does

00- (N 1-1 ) L (-I'exp(-!X2-tE:X)- N I + l

-N I= (N 1-1 ) L (-I)}. exp( _!X2_!fX)[using the identi ty (E.2) for 1'- Jl. = -1] . The remainingterm is

+0 0- L X(-I)}.exp(-tfX2-hX)= - (2i)-II1/(lif)= -GJ.

The second term in (E.3) may be shown, in a verysimilar way, to vanish. Thus, collecting results, onefinds (7.4).APPENDIX F. COMPUTATION OF Z"

According to the definition (7.9) and (7.6),+0 0Z"= L C'Y+N)'l10(2)('Y+N)-00

+0 0= exp( - f'Y2-'Il"i'Y) L N'l10f>0(3).

+0 0L (- I)NNexp(-!fN2+tfN)N ~ o o

I f N is cut off at the lower end, the summations may beinterchanged so that one getslim i:, (_1)'[1'] exp(-V2+f'YV)

Nl---+-OO v=0 V 100X L (- I)NN exp(-tN2+!E:N-mV). (F.1)

-NI

I t can be shown, by methods similar to those inAppendix E, that

00 -NI-llim L .. . L =0.Nl-OO p=o N ~ o o

Hence, in (F.l) one may simply replace N 1 with 00, sothat the summation over N becomes+0 0L ( - I )NNexp[-hNLf(v- t )N]

= (2i)-lt14 '(tif(V-!= (2i)-I(_1)'-1 exp(tfv2-!fV)I1/ (lif)= (_1)'-1 exp(!eJl2-!ev)GJ.

I t remains to evaluate the sum

We shall first show

Define for p=O, 1, 2, . . .

By use of (3.12), one finds the recurrence relationQp= ( l-e- P)Qp-l.

From this follows by iteration

Hence, it suffices to find Qp for p 00. For fixed VI ,Vl P 00Qp= L + L + L (F.3)p=o '1+1 P+l

For the second term, one has

00const. L exp( -tfV2).PI+l

The third term in (F.3) approaches, for large p,00 1L exp[-!V2+f('Y-!)V]P+l 'Il"1('Y+P-V)

exp[-!p2+f('Y-!)P] 00=G L 'Il"1(-'Y- 1+ v)'Il"1 C'Y)'Il"1(-1 ' -1) v=1 Xexp(-fPV-fV),which obviously vanishes as P 00 . The first term,for P ~ 00 , is

PI 1L -- exp[-!fV2+('Y-t)v].v=O 'Il"(v)

Now let VI also go to infinity; the result is (F.2).In order to evaluate the series (F.2), define

00 1Rp== L -- exp[-!fV2+f('Y-!)V- epv],v=0 11"(1')



11/11

F L U C T U A T I O N S P E C T R U M FO R A N O N L I N E A R M O D E L S Y S T E M 601so that Ro=GQo. With the aid of (3.1), one finds therecurrence relation

Rp = exp ( - ey+ ep) (Rp--1- Rp).From this follows by iteration

R O= 7r1(P-,),)/7r1(-')').As R",= 1, one finally has RO=G/7r1( -') '). On collectingresults, one finds (7.10).

APPENDIX G. COMPUTATION OF+00

P(z) = ~ e-n?t(z+n-l)Define

+00Pp = L: e- pn7r(z+n-p).

Inserting for 7r(z+n-p) the identity (3.1), one finds therecurrence relation

P p=e-'PPp-e-'(z-p)P ooH.From this follows by iteration

7r(p-1)Pp= (-l)r----7r(p+r-1)Xexp[!er2+e(p-z-!)r]Pp+r. (G.1)

Hence, it suffices to find the asymptotic value of P p forlarge p.For this purpose, write

+00P p= exp( - ep) L: 7r(z+n) exp( - epn)00 -n l=exp( -ep){ L: +L: }.-n1+1 - 00

The first term is in absolute value not greater than00exp(-ep2+ m1p) L: [7r(z+n) [ exp(-en)

= o exp[-ep]) for fixed n1.In the second term, the asymptotic value (3.4) for7r(z+n) may be used because n1 may be chosen arbitrarily large.

-"1exp( - ep2) L: 7r(z+n) exp( - mp)7r(z)7r(- z- l )

~ e x p ( - ep)-- - G

-ntX L: (- l)n exp[- !m 2-e (p+!+z)nJ

On account of (3.5),P ( - l ) p G 2 exp[- !ep+ep(z+!)] .

By inserting this result in (G.1) for P p+r, one obtains

In particular for p= 1,P(z)= -GeEZ

On substituting z="y+7ri/e, one obtains (8.3).

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Exact Calculation of the Fluctuation Spectrum for a Nonlinear Model - Van Kampen