o r

if we take

Y = exp ( - - Dr t - - D r t) F - - e-O* F, (A .7)

F ---~ (D - - D ---v - - D r ) - ' K], ix- (A .8)

Substitution of (A.7) and also of the cor responding Value for X a y into (A.4) leads to (2.10), while (A.8) is _ I equivalent to the express ion (D ~ T _ DT)F = KI,I~, i . e . , to the second equation in (2.10a).

3. S imi la r ly , solving Eq. (2.11), we obtain

k~ (t) ---- Z~: B ~ - - Y,r B ~ R~ v,r~ + perm., (A .9)

where Yot~ has the e a r l i e r meaning of (A.6), while t

Z , ~ = .[ d': (e-D(t--"),p (e-D'f--~))'~ "K~o, -- (e-D')~ . (A .10) 0

Changing to the modif ied m a t r i x notation and integrat ing, we obtain

Z ---- exp (-- _Dr t) O -- exp(-- D r - - Dr) O, (A.11)

where

G (D + D - - D r ) - ' ~x = -- Kl1,1 (A .12)

Expressions (2.13) and (2.14) follow f rom (A.9), (A.11), and (A.7).

1~

2. 3. 4. 5.

L I T E R A T U R E C I T E D

R. L. Stratonovich, Vestn. Mosk. Gos. Univ., Fiz . As t ren . , No. 4, 84 (1967). R. L. Stra tonovich, Vestn . Mosk. Gos. Univ., Fiz . As t ron . , No. 5, 479 (1970). R. L. Stratonovich, Izv. Vyssh . Uchebn. Zaved. , Radiofiz. , 13, No. 10, 1512 (1970). N. A. Krupennikov, Izv. Vyssh. Uchebn. Zaved. , Radiofiz. , 1__88, No. 3, 383 (1975). G. F. E f r e m o v , Zh. Eksp. Teor . F iz . , 51, 156 (1966).

P R O B A B I L I T Y C H A R A C T E R I S T I C S OF D Y N A M I C

S Y S T E M S S U B J E C T E D TO N O N - D E L T A - C O R R E L A T E D

R A N D O M F O R C E S

A. N . M a l a k h o v a n d O. V . M u z y c h u k UDC 538.56:519.25

The authors develop a method of analyzing dynamic s y s t e m s of fa i r ly genera[ fo rm acted on by intensive random fo rces with an a r b i t r a r y scale of cor re la t ion . Chains of kinetic equations a re se t up to de te rmine the one-point probabi l i ty c h a r a c t e r i s t i c s . Breaking of these chains at some step co r r e sponds to d i s regard ing the h i g h e r - o r d e r s ta t i s t i ca l re la t ions between the random action and some functional of it. A cor respondence is es tab l i shed between the proposed procedure of obtaining c losed kinet ic equations and another approach based on rep lacement of a Gaussian random force by a superposi t ion of s ta t i s t ica l ly independent te legraph p r o c e s s e s . Using some e x a m p l e s , the poss ib i l i ty of using the second approximat ion for de termining the probabi l i ty c h a r a c t e r i s t i c s of nonl inear s y s t e m s and of s y s t e m s with fluctuating p a r a m e t e r s is investigated.

1. In s ta t i s t ica l phys ics , in the theory of wave propagat ion in randomly inhomogeneous media , and in s ta t i s t i ca l radio engineer ing , extensive use is made of the kinetic equations of the Markov (or f i r s t diffusion) approximat ion to de te rmine the probabi l i ty c h a r a c t e r i s t i c s of p r o c e s s e s and fields [1, 2]. These equations

Gorki State Univers i ty . T rans l a t ed f rom Izves t iya Vysshikh Uchebnykh Zavedenii , Radiofizika, Vol. 23, No. 8, pp. 968-981, August , 1980. Original a r t i c le submit ted Apri l 4, 1979.

0033-8443/80/2308-0651507.50 �9 1981 Plenum Publishing Corpora t ion 651

are exact only in the case of de l ta -cor re la ted random actions. At the same t ime, it is c lear that the idealiza- tion of real physical p rocesses as being de l ta -cor re la ted itself involves certain assmm'ptions and may be un- acceptable in a number of cases . Problems of determining the f i rs t two moments of the output coordinate of stochastic l inear sys tems with nonwhite random actions that are Markov p rocesses have been fairly well in- vestigated at present [3-6]. A more general s tat is t ical description (nonlinear systems) has been achieved only for random forces that constitute Markov p rocesses with a finite number of s ta tes , p r imar i ly for "telegraph" actions with Poisson s ta t is t ics of jumps [4, 7, 8].

Such a probabil is t ic descript ion can theoret ical ly be achieved by employing the apparatus of mult ivariate Markovproces se s , b u t i t i s exceptionally difficult to obtain pract ical resul ts along these lines in view of the di.fficuity of solving the part ia l differential kinetic equations. In this paper we propose a method of setting up the kinetic equations for the probabili ty charac te r i s t i c s of dynamic sys tems of fairly general form with non- de l ta -cor re la ted random forces . Two models of such forces are considered: a Gaussian Markov p rocess and the square of a Gaussian p roces s . The procedure is essential ly a further general izat ion of the cumulant ap- proach [9] that was used in [6, 10] to determine the moments of the output coordinate of stochastic Linear sys - tems with nonwhite actions.

2. Let us f i rs t consider a one-dimensional dynamic sys tem descr ibed by the equation

dx =~(x)+g(x) ~(t), x(O)=xo, dt

where f(x) and g(x) are a rb i t r a ry functions, while random force a(t) is a Gaussian Markov process . it by the auxil iary s tochast ic equation

d~ a--i- + rt ~ = ~ (t), (2)

where r/(t) is Gaussian de l t a - co r r e l a t ed noise , for which

< ~ > = 0 , <~( t )~ ( t - -~ ) )=D~(~) , D ~ = 2 [ I ( ~ >

[the Last equation s tems f r o m the fact that p rocess a(t) is assumed to be stationary]. System (1), (2) defines a Markov population {a (t), x(t)} which can be descr ibed stat is t ical ly on the bas is of the two-dimensional proba- bility density Wax(a , x; t) Since it is virtually never possible to solve the corresponding "two-dimensional" F o k k e r - Pianck equation, we will consider another approach.

Assume that ~(x) is an a rb i t r a ry physically "good" function. On the basis of (1) we write the Liouvitle equation for the evolution of the mean value < r

The last t e rm of this equation is not closed relative to the one-dimensional probabili ty density of the output coordinate. Putting

~,(t) g(x) oq,(x____~) = ~(x, O, Ox

we can write the kinetie equation for {qJ,(x, t)) on the bas is of (1), (2):

It makes sense to r epresen t the last s tat is t ical mean in the following form:

where (~ , a , Z) is a cumulant bracket (joint t h i rd -o rde r cumuLant); Z = [g(O/0x)12Lb is some function of ran- dom force a (t). Introducing the notation*

< ( x\ < ~ ( x , O > = ~rS~(O, gox ~ ( ) / ' s = l , 2 . . . . . (4)

* Without the stat is t ical averaging opera tor , function ~ (x, t) is the so-calLed "incomplete cumulant bracket" (see the Appendix and also [9]).

(1)

We specify

652

and using the proper t ies of cumuiant b racke t s [9], we a r r ive at the following chain of kinetic equations:

ot - \ r Ox / + ( ~ ) '

( O + n ) <q''>='/fOq~'\+<~'>/{gO'~'N Ox / \~, Ox ] *~-- + ( "> ' (5)

. . . . . . . o ~ . . . . . . . . . . . . . * �9 �9 , �9 �9

�9 �9 * . . . . . . . . . . * . . . . . . . . . . . . . . . . .

In view of the s tat is t ical independence of random force a (t) and any functional of the form Z [a] = Z(x[a]) at the initial instant t = 0 the auxil iary var iables ( S s ) posses s zero initial conditions:

(~s(x, 0)) = 0 , s = 1, 2 .....

white the unknown function ($(x)) can have an a rb i t r a ry initial condition.

System (5) is exact; as s ~ oc it contains all the stat ist ical information on the output coordinate x(t) which is available in the two-dimensional F o k k e r - Planck equation. Chain (5) can be closed at some step in two ways: a) by omitting the function ( ~ ) in the equation for ($n-1); b) by closing the equation for (g?n-1), formal ly assuming the random force to be de l ta -cor re la ted only in the equation for (~b n) . Following the t e rmin - nology of [6,10], we will cat[ the solution for ($(x)) corresponding to method of closure a) the resul t of the n- th approximation, white that corresponding to case b) will be catted the n- th diffusion approximation. Method of c losure a) cor responds to d is regarding the h ighe r -o rde r stat ist ical relationships between the random force and some functional of it.

Making the passage to the limit to de l t a -cor re la ted noise

11 -+ oo, (a ~) ~ o% 2 (~2)/[[ = D = const (6)

in the equation for (~n), we obtain

"D (g(x) O ~ t ) )

Using this last resul t for n = 1, we a r r ive at the f i rs t diffusion (Markov) approximation for the unknown func- tion (~b(x)):

ot = \ ox / ~ \ \ ox } ~>. (7) Let us also give the sys tem of equations of the second diffusion approximation:

o(q~) = / f oq~ \ + (0,) Or' \ Ox /

(8)

\ o x /

Dropping the last t e rm in the second equation of (8), we obtain the second approximation for (r which is widely known in the case of a l inear s tochast ic sys tem as the Bourre t approximation.

Let us call attention to the following c i rcumstance : the condition for obtaining the second approximation

< involves d is regarding the joint cumulant (~2) = a, ~, g ~ which corresponds to the following "dis-

honest" opening of the mixed moment :

(~z(t) z, [~l ) ~ <~) ( z~ [~l >, z, = g q, (x, t).

It is known, however, that this last formula is exact for any functional Zt[a] if random force a(t) is a tele- graph (dichotomous) p rocess with a Poisson stat is t ic of jumps [7, 8]. This process possesses the same c o r - relation function Bo~(T)= ( a 2 ) e x p ( - l q l T I ) a s the Gaussian process under consideration. Thus, limiting

653

o u r s e l v e s to the second approx ima t ion c o r r e s p o n d s to r ep l ac ing a Gaussian r a n d o m act ion by a t e l eg raph act ion with the same c o r r e l a t i o n funct ion. Compar ing with the r e s u l t s of [11], we can a s s e r t that conf ining o u r s e l v e s to the n - t h approx ima t ion in this r e s p e c t is adequate to a r e p l a c e m e n t of a (t) by a supe rpos i t ion of n - 1 s t a - t i s t i ca l ly independent t e l e g r a p h p r o c e s s e s .

Set t ing $(x) = x n in (4) and (5), we obtain a s y s t e m of coupled equat ions fo r de t e rmin ing the m o m e n t ( x n ) . In the case o f an addi t ive ac t ion (g(x) = 1), this chain has f ini te o r d e r n . We should no te , however , that fo r any non l inea r function f(x) the s y s t e m is unsolvable without addi t ional a s s u m p t i o n s , s ince it includes h igher m o m e n t s of the output coord ina te . F o r the l i nea r p r o b l e m (f(x) = - a x ) the exac t value of the m o m e n t can be r ead i ly de t e rmined . In the case of a Linear s t ochas t i c equat ion if(x) = - a x , g(x) = - x ) s y s t e m (5) r educ e s to an infinite chain r e l a t ive to the v a r i a b l e s

(x"), O, x"> .... , Oral, x">, .:.,

whose solution can be represented by a continued fraction [6].

3. Since for the nonlinear problem system (8), like any closed system of equations of the n-th or n-th* diffusion approximation, does not enable us to determine the moments, let us consider the construction of a system of kinetic equations to obtain the one-dimensional probability distributions of the output coordinate W(x; t). Set t ing ~b(x) = f i x - x(t)] in (4) and (5), and p e r f o r m i n g some manipu la t ions that a re d e s c r i b e d in the Appendix , we can Obtain

OW O O o-7- + ~ (f w) = ~ (g w,),

. . . . . . . . . . . . . . . . . . . . . . . . . . (9)

s-----l, 2 ....

Here W(x; t) ~-W0(x , t ) , while the aux i l i a ry v a r i a b l e s Ws(x , t) a r e r e l a t ed to the condi t ional cumulan t s of the p r o c e s s ~ (t) (see the Appendix):

w,(x, 0 = ( - - 1 p W(x; t) O~sh>x. (10)

They p o s s e s s ze ro ini t ial condi t ions and "ze ro weight" :

W,(x, O),= O, ~ Ws(x, t)dx = O, s = l, 2 .... - - c o

Chain of equat ions (9) is c l o sed at some s tep in a m a n n e r s i m i l a r to (5). Thus , the s y s t e m of equat ions of the n - t h diffusion approx ima t ion fo r the probabi l i ty dens i ty W(X; t) can be obta ined by p a s s a g e to the l imi t to white noise in the equat ion fo r Wn(x, t) , where

W,~(x, t ) = D 0 2 o---; g(x) w~_, (x, O.

Using the las t e x p r e s s i o n fo r n = 1 and subs t i tu t ing into the f i r s t equat ion in (9), we read i ly obtain the F o k k e r - PLanck equat ion. The c o r r e s p o n d i n g p robab i l i ty dens i ty will be ca l led the r e su l t of the diffusion approx imat ion and denoted by Wd(X; t) in what fo l lows. Omit t ing the va r i ab le W 2 in the second equat ion of chain (9), we ob- ta in the c losed s y s t e m of equat ions of the second approx ima t ion :

OW 0 Tx 0--7 + ~ (fW)= (gW,), (11) (0 ) 0

J7 + ~ ~' + -~x (/w,) = o'> (gW).

The corresponding probability density willbe denotedby W(2 ) in what follows. Assuming that there exists a stationary probability density W(2)(x) = W(2)(X; oo), on the basks of (II) we can readily obtain

C l g l exp ( I I . [ f a x ) /,(x) < O'> g~(x) (12) Wr = <~2) g, _/2 _ <~2> g, __ f2

O, f=(x) > <~'> g~(x)

*As in R u s s i a n o r ig ina l -- P u b l i s h e r .

654

This impl ies that funct ion W(2)(x) is f ini te . A s i m i l a r r e su l t was f i r s t obtained in [8], where this p robabi l i ty dens i ty was the exac t solut ion of the in t eg rod i f f e ren t i a l k inet ic equat ion c o r r e s p o n d i n g to t e l eg raph p r o c e s s a ( t ) . Making the p a s s a g e to the l imi t (6) in (12), we obtain the wel l -known s t a t i ona ry solut ion of the F o k k e r - P lanek equat ion:

W~(x) - - I g ( x ) I g ' ( x ) " (13)

4 . Le t us ana lyze some p a r t i c u l a r c a s e s in g r e a t deta i l . Cons ide r a dynamic s y s t e m with an addit ive ex te rna l f o r ce (g(x) = 1). In this ca se Eq. (1) d e s c r i b e s the mot ion of a pa r t i c l e in a potent ia l well ac ted upon by a r a n d o m fo rce with finite c o r r e l a t i o n t ime . The s e c o n d - a p p r o x i m a t i o n s y s t e m (11) y ie lds the fol lowing equat ion f o r the s t a t i ona ry p robab i l i ty dens i ty :

d "dx [((d-') - - [2 (x)) W(2)(x) ] = 11 [ (x) W(2i (x). (14)

Mult ip lying (14) th rough by x and in tegra t ing by p a r t s , we obtain

&-> + r~ <xf (x) > = < p (x) >, (15)

where the second and th i rd t e r m s a re a v e r a g e d with p robab i l i ty dens i ty W(2)(x). Note that if we were to use the s t a t i ona ry f o r m of the exac t chain of equat ions (9), the t e r m dW2(x)/dx would be added to the r igh t side of (14). As fol lows f r o m (9), the s t a t i ona ry value of the aux i l i a ry var iab le W2(x) has the f o r m of the der iva t ive with r e s p e c t to x of some e n s e m b l e of func t ions , and hence the p r e s e n c e of th is t e r m does not affect r e su l t (15). In o t h e r w o r d s , e x p r e s s i o n (15), obta ined on the bas i s of the s e c o n d - a p p r o x i m a t i o n probabi l i ty dens i ty , is exac t . This can a lso be e s t ab l i shed d i r e c t l y f r o m the ini t ial equat ions (1) and (2) (see the Appendix).

In the p a r t i c u l a r ca se of a l i nea r s y s t e m (f(x) = - a x ) we can obtain the exac t value of the va r i ance of the output coord ina te f r o m (15):

<x~> = a 2 (I + 2~) -x , (16)

where 2 = < a )a-2, v = II (2a) -i are, respectively, the dimensionless variance and the spectral width of random force ~ (t). The stationary second-approximation probability density has the following form here:

,V(2)(x) = a '-2~ B - ' (v; I ) ( a ' - - x ' ) ~-', x ..< ~, (17)

where ]3 (v; 1/2) is a beta function. Of course, direct integration of (17) also yields result (16). We should note, however, that in determining the higher-order moments [which have the form <x 2n ) = (2n - l)l!<x2> n in view of the Gaussian nature of probability distribution W(x)], expression (17) leads to incorrect results. Figure 1 shows the form of distribution (17) as a function of the width v of the fluctuation spectrum (for fixed value of A 2 = Cr2/2V = 1). It Can be seen that f o r v < 1 function W(2)(x) d i f fe rs e n t i r e l y f r o m a Gauss ian c u r v e ,

t,~'c~) / 0.6 ~ / .

0,4 ~ 5

�9 E

1 2 x

F ig . 1. P robab i l i ty dens i ty (17) as a function of r e l a - t i r e s p e c t r a l width v: cu rve 1) diffusion approx imat ion (v =~,) ; cu rve 2) v = 2 ; curve 3) v = l ; cu rve 4) v = 0.5; cu rve 5) exac t p robab i l i ty d is t r ibu t ion for v = 1.

655

0,8 [

t

a4 I

~2 I

i

Fig . 2. S e c o n d - a p p r o x i m a t i o n p r o b a b i l i t y d e n s i t y fo r a s y s t e m w i t h p u r e l y cub ic n o n l i n e a r i t y fo r A = 1: c u r v e 1) d i f fus ion a p p r o x i m a t i o n (v = oo); do t t ed c u r v e i s f o r v = 5 v * ; c u r v e 2) v = v * ; c u r v e 3) v = 0 . 2 v * .

a l though , a s n o t e d a b o v e , i t l e a d s to an e x a c t va lue of the v a r i a n c e . A s v i n c r e a s e s , W(2)(x) , l ike the so lu t ion of the F o k k e r - P t a n c k e q u a t i o n , r a p i d l y " c o n v e r g e s " to the e x a c t r e s u l t .

F o r a s y s t e m wi th p u r e l y cub ic n o n l i n e a r i t y if(x) = - a x a ) , e x p r e s s i o n (15) l e a d s to an equa t ion tha t l inks the fou r th and s ix th m o m e n t s :

( x 6 ) + 2 ~ ( x 4 ) = & (18)

In the c a s e of a d e l t a - c o r r e l a t e d f o r c e (0-2 = 2vA2, v --* 0o), we ob ta in f r o m th i s the fou r th m o m e n t (X4)d = A 2, whi le in the c a s e of a q u a s i s t a t i c a c t i on (0 -2 = c o n s t , v ~ 0) we ob ta in the e x a c t va lue of the s ix th m o m e n t (x6)lv=o = 0-2. H e r e the s e c 0 n d - a p p r o x i m a t i o n p r o b a b i l i t y d e n s i t y has the f o r m

C (p _ x 2) ,/6p exp arctg p W(2)(x) = P a-x6 P~'q-PX2+ x4 pV3- p V 3 ' (19)

1 x l < g p - , p = ~/3.

It can be s een f r o m t h i s tha t t h e r e e x i s t s s o m e c r i t i c a l va lue of the f luc tua t ion band v* = 3p (which depends on the n o i s e in t ens i ty ) such tha t f o r v > u* p r o b a b i l i t y d e n s i t y (19) has a f o r m tha t i s q u a l i t a t i v e l y s i m i l a r to d i f - fus ion d e n s i t y (13), w h e r e g(x) = 1, f(x) = - a x 3. T h e s e c u r v e s a r e e n t i r e l y d i f f e r e n t in shape fo r v < v*, s i n c e d i s t r i b u t i o n ( 1 9 ) b e c o m e s unbounded a t the ends of the r e g i o n of e x i s t e n c e Ixl = pq-pT. F i g u r e 2 shows the t r a n s - f o r m a t i o n of the shape of p r o b a b i l i t y d i s t r i b u t i o n (19) a s a funct ion of the r e l a t i v e s p e c t r a l width v. It should b e n o t e d t h a t , even f o r v = 5v*, W(2)(x) v i r t u a l l y c o i n c i d e s wi th Wd(X).

F o r the c a s e of a q u a s i s t a t i c ac t ion (v --* 0) the s e c o n d - a p p r o x i m a t i o n m o m e n t s can be d e t e r m i n e d a n a - l y t i c a l l y . In p a r t i c u l a r ,

( x~)~2~ = p, ( x~}(~, -=- p~.

The c o r r e s p o n d i n g e x a c t v a l u e s can b e o b t a i n e d b y a v e r a g i n g the q u a s i s t a t i c so lu t ion of i n i t i a l equa t ion (1) wi th r e s p e c t to the GaussiarL d i s t r i b u t i o n a :

(x2)=2~/a~-~laF(5)p~O.8Op,

(x4> = 22/a~_l12r pZ~. 0.83p2.

L e t us a l so g ive the r e s u l t s of G a u s s i a n a p p r o x i m a t i o n , which we ob t a in b y s e t t i n g v = 0 in (18), and a l so (X 2n) = (211 - 1) I ! (x2)n:

<x~>r ~ 0,40 p, <x 4 )r ~ 0.49 pL

656

Thi s e x a m p l e shows tha t the a g r e e m e n t wi th the e x a c t v a l u e s of the m o m e n t s o b t a i n e d by i n t e g r a t i o n of the s e c o n d - a p p r o x i m a t i o n p r o b a b i l i t y d e n s i t y i s s a t i s f a c t o r y even in the c a s e of a m a x i m a l l y s low ac t ion (and the a g r e e m e n t i s b e t t e r than tha t y i e l d e d b y the G a u s s i a n a p p r o x i m a t i o n ) . S ince in the o t h e r l i m i t i n g c a s e of a d e l t a - c o r r e l a t e d ac t i on the s e c o n d a p p r o x i m a t i o n goes o v e r into a M a r k o v a p p r o x i m a t i o n and b e c o m e s e x a c t , t h e r e i s r e a s o n to a s s u m e tha t i t can be s u c c e s s f u l l y u s e d a t l e a s t to d e t e r m i n e the f i r s t few m o m e n t s of the output c o o r d i n a t e o f n o n l i n e a r s t o c h a s t i c s y s t e m s .

H i g h e r a p p r o x i m a t i o n s shou ld be e m p l o y e d when i t i s n e c e s s a r y to a t t a in g r e a t e r a c c u r a c y ; in s o m e c a s e s the "d i f fu s ion" c l o s u r e in the e q u a t i o n s f o r <$s> (or Ws) , s = 2, 3 . . . . . can y i e l d b e t t e r r e s u l t s than s i m p l y d i s c a r d i n g v a r i a b l e s wi th a g r e a t e r n u m b e r [6 ,101. S y s t e m s (5), (9) s p e c i f y a r e l a t i v e l y s i m p l e and e o m p u t e r - i m p l e m e n t a b l e a l g o r i t h m f o r c o n s t r u c t i n g the h i g h e r a p p r o x i m a t i o n s .

5. L e t us c o n s i d e r one n o n - G a u s s i a n m o d e l of a r a n d o m ac t ion a (t) tha t a d m i t s a s i m i l a r s t a t i s t i c a l d e s c r i p t i o n of a s y s t e m . A s s u m e tha t a (t) = ~ 2(0, whi le ~ (t) i s G a u s s i a n n o i s e s p e c i f i e d b y the a u x i l i a r y s t o - c h a s t i c equa t ion

.d~ + H ~ = ~ ( t ) , (2a) dt 2

w h e r e 77(t), a s b e f o r e , i s whi te G a u s s i a n n o i s e . A f t e r some m a n i p u l a t i o n s s i m i l a r to t h o s e in Sec . 2, we can ob t a in the fo l lowing cha in s y s t e m of k i n e t i c e q u a t i o n s * :

= a x / + ( ~ ) ' at \ ax / �9 �9 . . . . * * �9 , , �9 . , �9 �9 �9 . . �9 ; , . �9 �9 ,

(20)

<,(o), , +2s(2s--l)<~2>2 g-ox ~z,-~ + < ~ + 2 > , s - - - - l , 2 . . . . .

H e r e we have i n t r o d u c e d the a u x i l i a r y v a r i a b l e

< ~ , ( x , t) > \~[2sl

which p r o c e s s e s z e r o i n i t i a l c o n d i t i o n s ( l ike t h o s e u s e d above) . F o r the c a s e of a l i n e a r s t o c h a s t i c s y s t e m , E q s . (20) e n a b l e us to f ind an e x a c t so lu t ion fo r the m o m e n t s of the output c o o r d i n a t e s in the f o r m of a c o n - t i n u e d f r a c t i o n [10] a s i n t h e e a s e of a G a u s s i a n a c t i on . To a n a l y z e a n o n l i n e a r s y s t e m , we should change o v e r f r o m (20) to the c o r r e s p o n d i n g cha in of e q u a t i o n s fo r the p r o b a b i l i t y d e n s i t y . P e r f o r m i n g m a n i p u l a t i o n s a n a l o - gous to t h o s e in the A p p e n d i x , we ob ta in the fo l lowing s y s t e m of k ine t i c e q u a t i o n s :

a w + 0__ a e ~ , at ox (f + < ~> g) w = Ox

. . . . . . . . . . . . . . . . . . . . . . . �9 (21)

(o ) o o ~ + sn ~/~, + "~x ( /+ (4s + l) < ~,> g) ~'2s - - -~x e ( ~ + 2 + 2 s ( 2 s - ~) < ~ > - ~ _ ~ ) , s = 1, 2 . . . . .

w h e r e , a s b e f o r e , the v a r i a b l e s W2s(X, t) a r e de f ined by (I0) fo r even v a l u e s of the n u m b e r s in (.10). When the i n i t i a l d y n a m i c equa t ion c o n t a i n s no t the s q u a r e of the G a u s s i a n p r o c e s s o~ = g2(t) bu t only f l uc tua t i ons of t h i s quan t i t y , ~ = ~ 2 _ <~2>, t e r m s con ta in ing the v a r i a n c e <~2) in the f i r s t d e g r e e in (21) shoutd be s e t equa l to z e r o . In t h i s e a s e , in any equa t ion of (21) b e g i n n i n g wi th the s e c o n d , we can f o r m a l l y change o v e r to "whi te" n o i s e ~ (t) b y m a k i n g the p a s s a g e to the l i m i t

H -+ oo, 2 < "~2 >/1I = 4 < ~ >2/II = D = const

and can construct higher diffusion approximations for the probability density W(x; t). If the initial equation contains ~ 2(I;) (as is the case in a number of applications, e .g. , in analyzing noise self-compensation systems with correlational feedback [12]), this passage to the limit makes no physical sense. In this case, in order to obtain the n-th approximation for the probability density we should omit the function W2n in the equation for

* S i m i l a r k ine t i c e q u a t i o n s fo r a s t o c h a s t i c l i n e a r s y s t e m w e r e d e r i v e d in [10].

657

W2n_ 2 [in complete a g r e e m e n t with the construct ion of higher approximat ions in the case of a Gaussian action a(t)] . We should note , however , that this b reak ing of chains (20) and (21) does not co r r e spond to r ep l acemen t of a rea l Gaussian action ~ (t) [or ~2(t)] by a superposi t ion of t e legraph p r o c e s s e s , as was the case for Gaus- sian p r o c e s s c~(t).

As in the case of a Gaussian act ion ~ (t), it is poss ib le to obtain the s ta t ionary probabi l i ty densi ty of the second approximat ion W(2)(x) (if it exis ts ) f rom (21). Setting 0 /a t = 0 in (21) and confining ou r se lves to the f i r s t two equat ions, we a r r i v e at the express ion

= S h (x) h (x) (22)

h (x) = p (x) + 6 < }~ > ~ (x) g (x) + 3 < ~2 >~ g"(x),

which somewhat r e ca l l s (12).*

As an example , let us cons ider the s tochas t ic equation

dx + bx + ~" (t) x = a ~2 (t), (23)

dt

which d e s c r i b e s , under ce r ta in condit ions, the re laxat ion of the control voltage (x) in a noise s e l f - c o m p e n s a - tion s y s t e m with co r re la t iona l feedback [12]. Gaussian noise ~ (t) (the envelope of the in te r fe rence) is spec i - fied by auxi l ia ry equation (2a). On the b a s i s of (22) we can de te rmine the s ta t ionary probabi l i ty density of f luctuations of the contro l vol tage, but we will cons ider only the second momen t <x 2> = <x2(t)> I t . - . oo. Setting r = x 2 in (20) and confining ou r se lves to the second-approx imat ion s y s t e m , we can readi ly obtain the follow- ing equations for the s ta t ionary value <x2):

(1 + 8) < x2 ) + ~tx 2= a~ (<x) + :q), (24)

4~ <x~> + 2(1 + �9 + 5~) x2 = 2a~(3 < x > -- a) + a(1 + 2~ + 15,~)x,,

where n l = <f, ~, x><~2>-1, n2 = (~, ~, x2) <}2)-t a re aux i l i a ry va r i ab l e s , r ep re sen t ing joint cumulants ; ~ = <~ 2>b-1, t, = II(2b) - t a r e d imens ion less p a r a m e t e r s that c h a r a c t e r i z e the power and spec t r a l width of the noise

(t). The values of <x) and ~1 can be readi ly de te rmined in advance f rom the s i m i l a r s y s t e m of equations [for which we need to se t r = x in (20)1:

a~ I + ~ < X > = l + ~ + 2 ~ ( 1 + 3~ + 2~)_x, x , = a + ~ < x > . (25)

It is e x t r e m e l y difficult to find the exact value of <x 2) fo r a r b i t r a r y v and/3, but Eq. (23) yie lds the quasis ta t ic probabi l i ty density

W(x)['=~ = (2~x)-'/~(a " x)-3/~ exp ( 2~(aX x) ' (26)

0 < x < a ,

which enables us to obtain <x2) by d i rec t integrat ion.

Figure 3 shows the quas is ta t ic values of the second momen t <x2>lt,=0 as a function of the noise power/~. Curve 1 is the exact solution, obtained by numer i ca l integrat ion of probabi l i ty density (26). Curve 2 is the r e su l t of the second approximat ion , obtained f rom (24) and (25) for v = 0. Curve 3 co r re sponds to the "f i rs t approximat ion ," i . e . , to d i rec t opening of the mixed momen t s <xk~ 2> ~ <xk><~ 2>, k = 1, 2. Thus, even for a quas is ta t ic act ion, the second approximat ion vi r tual ly coincides with the exact r e su l t in the region of modera te noise /~ ~ 1.

6. The above approach is fully appticabte to the s ta t i s t ica l descr ip t ion of mul t iva r i a t e dynamic s y s t e m s with one random force :

dxk fk(x) + a (t) gk (x), k = 1, N, (27) dt

* The probabi l i ty density (22) of the second approximat ion , like the cor responding density for a Gaussian action, is finite; in the region h(x) -< 0 we have W(2)(x) - 0.

658

0.4

o21 0,1

,hZ"

J 0,4 4

F i g . 3

w h e r e x = (xi, x2, . . . . x N) i s the N - d i m e n s i o n a l s t a t e v e c t o r of the s y s t e m , whi le p r o c e s s ~(t) i s s p e c i f i e d by a u x i l i a r y s t o c h a s t i c equa t ion (2). The chain of k i n e t i c equa t ions fo r the a v e r a g e va lue of a r b i t r a r y funct ion $(x) now has the f o r m

o <q~> ot (~ f ~ \ + ( ~k>,

Ox /

s ~ I, 2 ..... (28)

w h e r e fo r b r e v i t y we have put

f O ~ f ~ ( x ) 0 ( 0 )2 0 g](x) c)

S u m m a t i o n i s p e r f o r m e d o v e r r e p e a t i n g i n d i c e s f r o m 1 to N. The a u x i l i a r y func t ions e m p l o y e d h e r e a r e d e - f ined by c u m u l a n t b r a c k e t s of the f o r m

~k . . . . n (x , t ) = ( / ~ m ( t ) , gk(x) 0 O~ n ~.,~ \ ~ ... g.(x) ~(x) , s = 1, 2 . . . . .

The s y s t e m of equa t i ons c o r r e s p o n d i n g to (9) f o r d e t e r m i n i n g the N - d i m e n s i o n a l o n e - p o i n t p r o b a b i l i t y d e n s i t y W{x; t) i s

OW (x; t) 0 0 Ot -t- ~ f~(x) W(x; t) = ~ g~(x) Wx(x, t),

�9 o . . . . ~ �9 o ~ ~ o , . . . . . . . . . . . . .

--~- + s n ~s(x, 0 + /k(x)~/Ax, 0 = g,(x)(W~+l(x, O + s ( ~ l~'~_~(x, 0),

w h e r e the func t ions Ws(x , t) p o s s e s s z e r o i n i t i a l cond i t ions and a r e de f ined by e x p r e s s i o n (10) to wi th in the s u b s t i t u t i o n x ~ x. C l o s u r e of s y s t e m s of k ine t i c equa t i ons (28) and (29) can be e f f e c t e d as in the o n e - d i m e n - s i o n a l e a s e c o n s i d e r e d above . In p a r t i c u l a r , m a k i n g the p a s s a g e in the l i m i t to whi te n o i s e ~ (t) in t h e equa t ion f o r the funct ion Wl(x, t ) , we ob ta in an e x p r e s s i o n fo r the l a t t e r in t e r m s of the unknown p r o b a b i l i t y d e n s i t y :

D 0 lV,Cx, t) - - g . ( x ) tV,(x; t). (30)

2 Ox~

Subs t i t u t i ng (30) into the f i r s t equa t ion in (29), we a r r i v e at an N - d i m e n s i o n a l F o k k e r - P i a n c k equa t ion :

(0 0 ) o 0 --g/- + ~ f~(x) v~J~(x; 0 = 2 o,,, g,(x) g.(x) tV~(x; t).

If the r a n d o m f o r c e in (27) i s the s q u a r e of a G a u s s i a n p r o c e s s ~ (t), then it i s a l so no t d i f f i cu l t to g e n e r - a l i z e the c o r r e s p o n d i n g r e s u l t s of See . 5. We shou ld n o t e , h o w e v e r , tha t i t is not p o s s i b l e to ob ta in in the g e n e r a l c a s e an a n a l y t i c so lu t ion fo r the s e c o n d - a p p r o x i m a t i o n p r o b a b i l i t y d e n s i t y s i m i l a r to (12) o r (22) f o r the m u l t i d i m e n s i o n a l problem.

659

A P P E N D I X

1. Taking r in (4) and (5) to be a del ta function

9(x) = ~ (x' - x) - - ~ (x)~, x = x ( t ) ,

we t r a n s f o r m the s t a t i s t i ca l a v e r a g e (4). We in t roduce the s o - c a l l e d incomple te cumulan t b r a c k e t s (paren- theses ) [9]. By def ini t ion, b r a c k e t (x, y . . . . . z) c is some function of r andom va r i ab l e s and the i r s t a t i s t i ca l a v e r a g e s , w h i e h b e e o m e s a e u m u l a n t b r a e k e t a f t e r a v e r a g i n g , e . g . ,

(x, y), = xy - <x> y,

<(x, Y),> = <xy> - <x> <y> - - <x, y>.

Using this no ta t ion , we wr i t e the s t a t i s t i ca l mean < Ss(X)6 > ill the f o r m

rs <~(x , t)~ > = -~ w..(~', x'; 0 ct~l(O, g ~ x ' ~(x'--x) ~ acax',

where W~x (o~', x ' ; t) is the joknt p robab i l i ty dens i ty of p r o c e s s e s c~ (t) and x(t). In tegra t ing by pa r t s with r e - speet to x ' , we have

�9 <%(x, t)~ > = ( ' 1 ) ~ 0 g (r x; t) de .

- -OO

Since the t w o - d i m e n s i o n a l p robabi l i ty dens i ty is r e l a t ed to the condit ional densi ty (~V) and to the o n e - d i m e n - s ional densi ty by the e x p r e s s i o n

w ~ . (~, x; t ) = w(~, /x; t) W ( x ; t);

we finally obtain

where

)s < 9s (x, t)~ > = ( - - 1). -bTx g(x) Ws(x, t) , CA .1)

W~(x, t) = W(x; t) < r (t), >x,

< ~[~] (t), >. = ~ (~ '%)c W (~,'/x;t) d e . - - o o

CA .2)

Thus , the aux i l i a ry va r i ab l e s Ws(X, t) a re p roduc t s of the o n e - d i m e n s i o n a l probabi l i ty densi ty and the condi t ional cumulan t s of p r o c e s s ~ (t) [the incomple te cumulan t b r a c k e t in CA.2) is a v e r a g e d with the condit ional probabi l i ty dens i ty] .

S i m i l a r l y , it is not diff icult to e s t ab l i sh that the fol lowing re la t ions a re val id:

/ = - -~-x g(X) -~x f (X) w~(x, O, CA .3)

",\((g (x)-~-x 2 q,,_,(x,, t)~ >= / o~...~_x g(X)),~+~ tv~_,(x, t).

Subst i tut ing the r e s u l t s into s y s t e m (5) and "eanee l ing" by the o p e r a t o r g we a r r i v e at (9).

2. Cons ide r initial s y s t e m (1) with an addit ive ac t ion , se t t ing g(x) = 1. We mul t ip ly (1) th rough by x s and a v e r a g e . If f(x) is such that the re ex i s t s a s t a t i ona ry probabi l i ty d i s t r ibu t ion , then, let t ing t - - oo, we obtain

<x s f ( x ) > = - < r s>, s~-~0, 1 .... (A.4)

It a lso fol lows f r o m (1) and (2) that as t - - 0o we have

O= d <~x>=<~f(x)>+<~>_ii<~x>. dt

CA .5)

We wri te f(x) as a power s e r i e s , f(x) = ak xk (summat ion being p e r f o r m e d o v e r r e c e i v i n g indices) . Then, on the b a s i s of CA.4),

660

- <~x'> = <xsf(x)> =a~<x~+'>.

We mult iply this last equation through by a s and sum o v e r the index s:

- - a s < ~ x s ) = a k a s < x k + s ) ,

o r

- < ~ / ( x ) > = < p ( x ) ) .

On the basis o f ( A . 4 ) - ( A . 6 ) w e finally obtain

< ~ > + • < x f ( x ) > = < f ( x ) >.

Thus, express ion (15) is an exact co ro l l a ry of (1) and (2).

(A .6)

L I T E R A T U R E C I T E D

1. R . L . Stratonovich, Selected Topics in the Theory of Fluctuat ions in Radio Engineer ing [in Russian], Soy. Radio, Moscow (1961).

2. V. I. Klyatskin , Stat is t ical Descr ip t ion of Dynamic Sys tems with Fluctuating P a r a m e t e r s [in Russian] , Nauka, Moscow (1975).

3. N . G . Van Kampen, Phys. Rep. , 24, No. 3, 171 (1974). 4. A. B r i s s a u d and V. F r i s h , J . Math. Phys . , 15, No. 5, 524 (1974). 5. Yu. E . K u z o v l e v a n d G. N. Bochkov, Izv. Vyssh . Uchebn. Zaved. , Radiofiz. , 20, No. 10, 1505 (1977). 6. A . N . Malakhov, 0 . V. Muzychuk, and I. E. Pozumentov, Izv. Vyssh. Uchebn. Zaved. , Radiofiz. ,

2_!1, No. 9, 1279 (1978). R. B o u r r e t , V. F r i s h , and A. Pouquet, Phys ica , 6._~5, 303 (1973). 7.

8. V . I . Klyatskin , Izv. Vyssh. Uchebn. Zaved. , Radiofiz. , 20, No. 4, 562 (1977). 9. A . N . Malakhov, Cumutant Analys is of Non-Gauss ian Random P r o c e s s e s and Thei r T rans fo rma t ions

[in Russian] , Soy. Radio, Moscow (1978). 10. O . V . Muzychuk, Izv. Vyssh. Uchebn. Zaved. , Radiofiz. , 2_~2, No. 10, 1246 (1979). 11. V . I . Klyatskin, Izv. Vyssh. Uchebn. Zaved. , Radiofiz. , 2__22, No. 6, 716 (1979). 12. A . A . Ma l ' t s ev , 0 . V. Muzychuk, and I. E. Pozumentov , Radiotekh. Elek t ron . , 23, No. 7, 1401 (1978). 13. A . A . Mal ' t sev and A. I. Saichev, Radiotekh. Elekt ron. , 23, No. 12, 2543 (1978).

RADIATIG'N OF A CHARGE UPON CONVERSION OF AN

ISOTROPIC MEDIUM INTO A UNIAXIAL CRYSTAL

V. A. Davydov UDC539.12

The a r t i c le cons ide r s radiat ion of a charge that moves uniformly in a nonsta t ionary medium with i so t ropy that va r i e s in t ime. The fields and the radiat ion energy of the ord inary and e x t r a o r d i - n a r y waves a re calcula ted for the case of instantaneous convers ion of the isotropic medium into a uniaxial c ry s t a l .

The appearance of paper s [1, 2] has s t imulated fur ther in te res t in radiation of sources in nonsta t ionary med ia . P a p e r s [3, 4] calcula ted the radiat ion c h a r a c t e r i s t i c s of a charge in media with step and smooth non- s ta t ionary p rope r t i e s . In many c a s e s , var ia t ion of the e lec t romagne t ic p rope r t i e s of the medimn in t ime is also accompanied by violation of i so t ropy . It was shown in [5] that in media with varying anisot ropy it is pos - sible to have radiat ion of even a fixed charge ; the c h a r a c t e r i s t i c s were computed for the case of instantaneous convers ion of an i so t ropic med ium into a uniaxial c rys t a l . In what follows we will investigate the radiation of a uniformly moving charge upon convers ion of an i so t ropic medium into a uniaxial c rys ta l for the ease of a r b i t r a r y mutual or ientat ion of the optical axis and of the direct ion of motion of the charge .

Moscow State Univers i ty . Trans la ted f rom Izves t iya Vysshikh Uchebnykh Zavedenii , Radiofizika, Vol. 23, No. 8, pp. 982-987, August , 1980. Original a r t ic le submit ted June 28, 1979; revis ion submit ted F e b r u a r y 1, 1980.

0033-8443/80/2308-0661507.50 �9 1981 Plenum Publishing Corpora t ion 661