Annalen der Physik. 7. Folge, Band 38, Heft 1, 1981, S. 7-23 J. A. Barth, Leipzig Effective Linear Response of Random Media to Stochastic Sources By G. DIENER Sektion Physik der Technischen Universitit Dresden A b s t r a c t . The notion of effective sourcesis introducedinorderto describe the mean field response of a random medium to stochastic sources. Effective sources and stochastic sources are connected by a linear functional which depends on the random material properties of the medium. A formal perturbation series for the functional is given. Further, a single-grain approximation is proposed to oalculate the effective sources in a strongly heterogeneous phase mixture. The approach is applied to the electrostatic field in a mixture consisting of approximately spherical grains. Detailed calculations are carried out for stochastic sources of a special type in a two-component mixture. Effektive lineare Response heterogener Medien auf stochastische Quellen I n h a l tsubersicht. Zur Beschreibung eines von stochastischen Quellen in einem heterogenen Material hervorgerufenen mittleren Feldes wird der Begriff der effektiven Quellen eingefiihrt. Die effektiven Quellen sind mit den stochastischen Quellen iiber ein lineares Fnnktional verkniipft, das die stochastischen Materialeigenschaften des Mediums enthiilt. Eine formale Stkirungsreihe fur dieses Funktional wird angegeben. Des weiteren wird eine ,,Ein-Korn"-Naherung z u r Berechnung der effektiven Quellen in stark heterogenen Phasengemischen vorgeschlagen. Zur Illustration wird die Methode auf das elektrostatische Feld in einem aus naherungsweise kugelformigen Kornern bestehenden Gemisch angewandt. Detaillierte Rechnungen werden fur stochastische Quellen eines speziellen Typs in einem zweikomponentigen Gemisch ausgefiihrt. 1. Introduction I n a previous work [l],in the following referred to as I, the effective linear response, i.e. the mean field response, and effective constitutive laws for mean fields in stochastically heterogeneous media have been considered under the assumpt,ion of non-randoni sources. This restriction has generally been adopted in studies on heterogeneous media (cf. [2--41). On the other hand, there is considerable work done on the problem of stochastic load or sources acting on homogeneous media [5-7]. This case is much easier t o treat than that of stochastic material properties, a t least if the system is linear. Wave emission from stochastic sources leads to the problem of coherence [8]. The purpose of the present paper is to join both these problenis and t o deal with the effective behaviour of a linear heterogeneous mediuiii under stochastic load or source terms. The considerations will be confined t o the study of the mean field response, as has been done in I. An almost trivial example of a stochastic load is the force of gravity acting on a heterogeneous elastic body because it contains the mass density as a randomly varying parameter. More interesting cases may occur if we consider heat conduction or diffusion phenomena with random sources due, e.g., t o chemical reactions, sinter processes, mechanical deformations and so on, radiation emission from random sources etc. s G . DIENER In order to develop the general notions which will be used t o treat the problem, we start from a linear equation for a certain field u(r, t ) L,tu(r, t ) = q d r , 4. (1.1) The subscript st indicates that the operator L,,, as well as the source density qat,show stochastic fluctuations. The inean field (u(r,t ) ) , averaged over an ensemble of samples, is governed by an equation of the saiiie type L(u(r. f ) : = q(r,t ) , (1.2) which contains an effective operator L and an effective source term q. The definitions of the effective quantities are somewhat arbitrary, but it seeins reasonable t o require L to be conipletely independent of the source teriii qst and to depend merely on the properties of the inaterial described by Lst.Consequently, the operator L in eq. (1.2) must be identified with the effective operator in the case of non-random sources which has been studied in I (and which there was denoted by Len).So we are left with the task of deducing a general expression for the effective source term q. To this end let u s forinally solve eq. (I .l)by ineans of the stochastic Green’s function introduced in I u ( r ,t ) = Jdr’ Jdt‘ gst(r,r’. t , t‘) qsl(r’,t’) = : gstqst(r,t ) . (1.3) I n order to avoid boundary effects, we choose infinite samples. The sources are imagined to be localized in a finite region, and t o produce fields which are vanishing a t infinity. This boundary condition can be satisfied by a n appropriate choice of the Green’s function which has to tend to zero at great distances. Splitting up qsl into the average value (qst) and the fluctuation q’ + 4x1 = (qxi) q’ and averaging eq. (1.3), we obtain (1.4) + (u> = (981) (qxt? (gstq’?. (1.5) The brackets always denote ensemble averages. Now we take into account the general expression for the effective operator found in I L = 9-1. g : = (g,,). Applying this operator to eq. (1.5) gives (1.6) L ( u ) = (q.Qr> L(g,tq’>* A comparison with eq. (1.2) yields a n expression for the effective source density (1.7) + + q = (qsr) L(gsiq’) = L(gd7,t) = : QSt. (1.8) The operator P is independent of the sources, and only containst he properties of the inaterial which are characterized by the stochastic Green’s operator gat.Therefore. according to (1.8). the relation between q and qst is a linear one. If there is no correlation between the fluctuations of the inaterial parameters on the one hand and the sources on the other hand, the last term in eq. (1.7) vanishes, and the effective source density reduces to the mean value q = ‘%gat) (981) = <qSt>’ (1.9) The stochastic Green’s function being unknown, an exact evaluation of q is, of course, impossible. I n the following section, a perturbation series for q is given which can be applied to weakly heterogeneous materials. I n Section 3 the method developed in I is iiiodified in order to evaluate q in strongly heterogeneous bodies. To illustrate the general ideas, Section -1 is devoted to the special case of electrostatics, which involves only one stochastic material parameter and a scalar potential field. Finally Section 5 gives detailed calculations for a siniple type of stochastic source distributions. Linear Response of Random Media to Stochastic Sources 9 2. Perturbation Treatment Analogously to ( 1 . 4 ) , we split up the stochastic operator L8,into + L,, = Lo L'. (2.1) Lo represents a homogeneous operator (which must not necessarily be identical with the mean value of L8,).Its inversion go = L o 1 is supposed t o be known. The inhomogeneous deviations L' are considered as small perturbations in coniparison with Lo. Thus the linear operators L and P may be expanded in terms of L'. If we insert the decomposition ( 2 . 1 ) into eq. ( l . l ) ,we find Lou = - L u q8t. (2.2) Averaging leads to + + (2.3) Lo<iLi = -(L'u) (%>* The difference between the last two equations may be written as (2.4) Lou = Lo<u) + (1- M ) (q8t- L'u). Here, the operator M ... = (...) prescribes a n ensemble average of the whole expression on its right, whereas the projection operator 1 - M singles out the fluctuating part of this expression. Application of go = L r 1 yields + (2.5) = <u> go(1 - M , (q8l - L'u) * According to the boundary condition discussed above, the Green's function in go has to vanish a t large distances. Equation (2.5) can be solved by iteration which leads to a n infinite series m = 2' n=O - L')n [<u> f go(' - M , (2.6) NOW. inserting this result into eq. (2.3), we write all terms acting on ( u ) on the left hand side and all the other terms acting on qSton the right hand side of the eqaution, and obtain m + M c (-L'go(l - M))nL'l(u> = M c [-L'go(l - Wlnqat- [Lo n=O (2-7) m n=O A coniparison with eqs. ( 1 . 2 ) and (1.8) yields the following perturbation series for the effective quantities L and q CQ L =Lo+ M 03 2 [-L'go(l - M)lnL'= M n=O 2 [--L'go(l n=O - M)]"L,t. (2.8) The result for L is well known [9]. I f we use the notation of eq. (1.4) and specialize Lo t o the average operator <L,t), the first terms of (2.9) are (2.10) G. DIENER 10 From eqs. (2.8), (2.9) we can deduce a relation between the operators L and P L = PL,,. (2.11) The perturbation series (2.9) clearly shows P t o be an integral operator (2.12) Ppst(r,t ) = J dr' J dt' P ( r , r', t - t') qst(r',t') because it contains the integral operator go. Moreover, from (2.11), the range of the integral kernels of L and P can be expected t o be of the same order of magnitude. Thus, recalling the results of I for the effective operator L, we can assume the integrand in P t o decay rapidly for distances 1 r - r' j beyond the correlation length .2, Along range behaviour may occur only for extreme heterogeneities. 3. Single-grain approach to q The perturbation treatment fails in the case of strongly heterogeneous bodies. Therefore, a self-consistent single-grain approximation has been used in I to calculate L. This method presupposed a body consisting of a statistical arrangement of homogeneous regions (grains), which differ from one another by their material properties ci. Hence, the stochastic material parameters involved in L,, could he written as c(r) = z c i O i ( r ) , (3.1) i 1inside Oi(r) = 0 outside the i-th grain, C Oi(r)= 1 . z Further, L,, had been decomposed into a homogeneous part Lo and deviations Li due t o the i-th regions, respectively L, = L, 3- 2 Li, i (3.2) where Lo was chosen in such a way that, for a special harmonic mean field, the relation (u> N 6p.r9 L,(u) = L(u) (3.2') held. Now, the method applied in I will be carried over t o the calculation of q. To this end we have to assume that, analogously to (3.2)) the fluctuating part q' of the source term can be split up into a sum of contributions qi owing to the i-th regions, respectively q' = c qi + 9". (3.3) 2 The additional term q" contains all the fluctuations which are not correlated with the structure of the material and which, therefore, are due t o exterior stochastic parameters. According t o eq. (1.9)) this term does not contribute t o q. Thus we obtain froni (1 3) and (3.3) = (qd + L<F 77i), qi:= gstqi. (3.1) The functions qi obey t o the equations These equations being practically unsolvable, we introduce a single-grain approximation, in the sense of I, by dropping the mixed terms j =+ i in eq. (3.3) (Lo + Li) qi = Qi* (3.6) Linear Responss of Rmdom Media to Stochastic Sources 11 This equation is of the same type as eq. (2.10) of I, and may be solved in simple cases. Let us denote the corresponding Green's operator by gi l;li = (Lo + L p q i = : g iq i. (3.7) It depends only on the properties of the i-th grain and the effective properties involved in Lo. I n the case of statistical homogeneity, it is convenient t o transform q into the Fourierspace. Then, the integro-differential operator L becomes a number L ( k ) ,as has been seen in I q(k)= J dr e-*" + q(r) = (qat(k)) L(k)J dr e-ikr (2 qi). i (3-8) 'Here, a n eventual time dependence is omitted because it does not influence further considerations. I n case of need, the time can be handled analogously to the space coordinates. Moreover, it is useful t o refer the coordinates in the Green's function gi t o the center rf of the i-th grain 7; = giqi = J dr' gi(r - ri, r' - ri) qi(r'). (3.9) The Fourier-transformation then yields Jdr e-ik'r l;li = J d r dr' e-*" gi(r - ri, r' - ri) qi(r') = J dr dr' e-*'r gi(r, r') e-ik'ri qi(r' + ri). (3.10) This expression contains the following statistical parameters : the kind of the i-th grain, which will be denoted by (x, and which comprehends its shape, size and material properties, and the position ri of this grain. Furthermore, qi may depend on the random surroundings of the i-th grain. The ensemble average of (3.4) will be performed in two steps. First, we average, for a fixed kind a,with respect to the position ri and, if necessary, t o the surroundings, and define + 1 (3.11) k), N where N means the total number of grains. Then we introduce the mean number N , of grains of kind cx andthe probability pa = N,/N of a grain t o beof kind (x. With theaid of these notions the integral in (3.8) can be written (3.12) J d r e - * . r ( 2 l;li(r))= 2 pa J dr dr' e-ik'rga(r, r') qa(r', k). (ecik'"iqi(r z =: -qa(r, a 4. Application to Electrostatics Further calculations are specified t o the case of electrostatics, which has been chosen for the seek of simplicity. Then, the various operators take the forms Lo a- a = - E~ A , Li = - - ( E ~ E ~ Oi(r) ) -, - ar 2r L(k) = k ~ ( kk) (cf. I),where E , ~ , e0, E~ and E denote the corresponding dielectric parameters. The field u and the sources qat correspond t o the electrostatic potential and the charge density, respectively. Let us notice that electrostatics is mathematically equivalent t o other stationary problems as electric or heat conduction, diffusion and so on. We have only t o change the interpretation of the notations. It must be emphasized that, in general, G. DIENER 12 ~ ( kdefined ) by eq. (4.1) in connection with eq. (1.6) doesnot describe the relationbetween the mean electric displacement ( D ) and the mean electric field ( E ) as it does in the case of non-stochastic sources. Prom Maxwell's equations it follows for the mean displacement (D) a (4.2%) ar ( D W >= (qst(r)) -* or in the Fourier-space (4.2b) ik ' ( D ) k == (Pbt)kOn the other hand, the mean electric field ( E ) is governed by equation (1.2))where the operator L is given by the last expression of eq. (4.1) k - &(k)k ( U ) k = ik . & ( k )( E ) k = q ( k ) . Comparison of both these equations yields (4.3) (4.4) Therefore, the relation between (D)and ( E ) depends on the stochastic properties of the source term. A combination of eqs. (4.2) and (4.4) shows i. t o connect also the mean field with the mean charge ik * i ( k )( E ) k = <qat>k(4.5) If the stochastic fluctuations of ESt(r) are small, the perturbation series (2.9)) (2.10) can be used to evaluate the effective sources q. We obtain In the case of strong heterogeneity, we use the single-grain approximation (3.8)) (3.12), and start with the evaluation of the integral t a ( r ' ,k) := Jdr e--ik'r ga(r, r') = J d r e--ik'r gd(r',r) . (4.7) Taking into account the definition of the Green's function (3.7)) (3.9) and its symmetry with respect t o a permutation of the coordinates r and r') we can see that 6, satisfies the equation (Lo La)t a ( r ,k ) = e--ik.r. (4.8) The operator Labelongs t o a grain of kind oc centered a t the origin r = 0. Equation (4.8) resembles very much t o the equation obtained in I for the field fluctuations ui.Therefore, it can be solved in a similar way. For this aim, we have to restrict ourselves t o a special grain shape. We assume that all grains may be approximated by spheres of radii R,. Then, the solution of (4.8) is found analogously t o I. Outside a grain a we have + --E At, = e--ik*r, 1 r > R,, 00 + 2 A;(?) (4.9) I+ 1 Pl([), 5 = cos ( + k , r) z=o where the rotational symmetry with respect to the axis k is taken into account. Terms diverging at infinity are omitted. The Pl are Legendre's polynomials. Inside the grain similar equations hold Ea = - e--ik.r k2& Linear Response of Random Media to Stochastic Sources 13 The coefficients A,"and B," are determined by the boundary conditions a t the grain face r = R, SLW- (1.11) For this end, we have to expand the exponential term exp (--ik. r ) into a series of spherical harmonics - ik.r - c (21 + 1)(-ill 1 j,W P l ( 0 , I- (1.12) where the j l are spherical Bessel functions. Inserting eqs. (4.9), (4.10), (4.12) into the boundary conditions (4.11) we obtain &(I + 1)A? + E,IB~= 0 , (4.13) (4.14) where the abbreviation (4.1.5) is introduced. The first term in eq. (4.14) vanishes because of 2 p a J d r e-ik'r qa(r, k) = (2J d r e-ik'r qi(r)) i a - J d r e--ik." (q' - q") = 0. (4.16) Thus, the effective charge density is finally given by (4.17) This representation of q(k) using the moments Y r ( k )proves t o be convenient for an expansion in a power series of kR,. Restricting ourselves to a la-th order approxiniation G . DIENER 14 in kR,, we can neglect all nionients Yr with I > n on account of the factor jl(kRa)in eq. (4.17), the leading term of which is of I-th order. For k --f 0, eq. (4.17) simplifies t o (4.18) I n the ordinary space this result.takes the form J d r d r ) = s d r <Qst(r)). (4.19) The total effective charge is equal to the total mean charge. Thus the operator P only effects a redistribution of the mean charge. The same result can be obtained by integrating the perturbation series (2.9), because all ternis except the zeroth-order term are divergences of vector fields which are vanishing a t large distances froin the sources. Consequently, their integrals are equal to zero. The zerot,h-order contribution leads back t o eq. (4.19), which, therefore, is valid independently of the single-grain approximation. It follows that the behaviour of the mean field at large distances froni the sources agrees with the field which would be created by a mean charge distribution. Here, distances can be considered as large, if they considerably exceed the correlation length of the niaterial. This results from the fact stated at the end of Section 2 that the operatorP perfornis displacements of the mean charges which are of the order of magnitude of the correlation length. Only in the case of extrenie heterogeneities, we can expect larger displacements in accordance with the non-local properties of P [11. Finally, it inay be noticed that, for slowly varying fields kR,--f 0, the relation (4.4) between (D)and ( E ) simplifies t o i ( k = 0) = &(k = O ) , (4.20) ( D ( k = 0)) = &(k = 0) ( E ( k = 0)) as can be seen froni eq. (4.18). Therefore, a t sufficiently large distances froin the charge distribution, a local, charge-independent dielectric constant can be used. 5. A Simple Example Now, the method outlined above will be illustrated by an special, relatively simple example. We assume a charge density (5.1) (4edr)) = qo(r), + <2&(rD = a 0. The factor (ai 1)varies stochastically froni grain to grain. Its value is uniquely determined by the kind of grain 01 (i.e. ai= a&).Equation (5.1) suggests the following definition of c/i qdr) = %@iW QO(') * Averaging over the grain positions, according t o eq. (:4.11),yields (5.2) (5.3) where @,(r)describes a grain of the kind grains this function is O,(r) = @ ( R ,- r ) = N located at the origin. For spherically shaped (3.4) Linear Response of Random Media to Stochastic Sources 16 lnserting eq. ( 3 . 3 ) into the definition (4.15) of the nioinents Yf, we obtain Y; N + 1),<Ha J dr = -uJZ V N . 4nR’ qO(k)(!%@ - 1 jl(x)} . 1)(5.5) V kR, dz L x=kR, Tlic calculation is given in the Appendix. With the use of this result, the effective source distribution (3.1 7 ) beconies - - -ua,az (1 + (5.6) xvherc the abbreviations S(.c. y) : = 2 41 2 T(.L..Y) : = + 1)(22 + 1) (jl(z))2, (5.7) 1+1+lY F 1Z(2l1+ 1)Ey + + (iLC4)’ * are introduced. The quantity v, = paN 4nR213V characterizes the volume fraction of the grains of kind a.The funtion S had already been defined in I, where a good approxinmtion for the infinite sun1 had been found S ( x , Y) w + Y Y ( 2 + Y) + Y(3& 52 (1 - 3p +y) + + cos 2x + 2 2 Si(2x) - 2 Y2(X - y)l, (5.8) [ B(x) := - -x2 Si(2x) 22 The suiii T is related t o S b y T ( z ,y) 1 = -[(l Y + y) S(n,y) - x2a(x)] (5.9) (cf. Appendix). The function W ( x ,y) vanishes in the limiting case of homogeneousinean fields LR, + 0 in accordance with (4.18). In the opposite case of extremely inhomogeneous fields k R + 00, W reduces t o 1 (5.m) W(.r+m,y)=-1. Y Consequently, in this liinit, q ( k ) becomes (5.11) G. DIENER 16 The last expression holds because of eq. (4.17) of I. This result is exact, as will be discussed below. We apply the result (5.6) t o a two-phase mixture whose grainsare ofequalsize R,= R. Thus, there are only two kinds of grains a = 1, 2 characterized by the dielectric constants + E, and occupying the volume fractions vl, v2, respectively. I n order t o satisfy the third equation of (5.1), we write a U a, = -, a,=--. (5.12) Vl v2 -Then, the effective charge density (5.6) takes the form =: 1 (5.13) +UAW. + Depending on the choice of a, the effective charge for k 0 can inore or less differ froin the average charge. The expression A W comprised in the braces is plotted in Figs. 1 and 2 for different parameter values. It increases with the heterogeneity of the material, but remains finite if the heterogeneity tends t o extreme values, El/&2 --f 0,oo. Now we consider the quantity i defined in eq. (4.4), which connects the inem field with the mean displacement and the mean charge, respectively and, therefore, represents a (charge-dependent) dielectric parameter. For k --f 0, itends to E , as has been quoted in eq. (4.20). I n the opposite limit k + 00, eq. (4.4) together with (5.11) and with (4.17) of I yield (5.14) I 5 . I/ 7 * 5 O I 10 kR Fig. 1. Effective source distribution for a two-component mixture with volume fractions 0.8:0.2 and different heterogenities EJE, according to eq. (5.13) Linear Response of Random Media to Stochastic Sources 17 Fig. 2. Effective source distribution for a two-component mixture with volume fractions 0,5: 0,5 according to eq. (5.13) This result coincides with the exact one which can be found by the following reasoning (cf. [2]). The mean potential is given by 1 (u(k))= -J dr e-ik*rJ dr’ <g,,(r, r’) qSt(r’)). (W3 (5.15) I n the limit k + 00, only small distances dr = I r - r’ 1 < k-1 contribute t o the integral. But for small distances Ar Q R the stochastic Green’s function may be replaced by Thus, we obtain for kR + 00 (5.16) Therefore, (5.17) and for the special charge density given in (5.1) + -cv,--- 1 1 a, (5.18) S(k=0O) , &a in accordance with (5.14). The dielectric parameter ~ ( = k 0 0 ) follows for nonstochastic sources a, = 0 1 = 2v, (5.19) , &, .S(k=00) 2 Ann. Pliysik. 7. Folge, Bd. 38 18 G. DIENEB (cf. eq. (4.17) of I).Finally, according to eq. (4.4), the effective sources q are connected with (5.18) and (5.19) by (5.20) which, for k -+ 00, leads t o the result already given in (5.11). For a strongly heterogeneous medium .sg % E ~ expression , (5.14) niay be of the order of magnitude of .sl (like E(k +00))but it may also attain the order of .s2, if 1 a, tends to zero. This latter case signifies that there are no charges situated in the first component with small E ~ Then. . the k-dependence of ;(k) turns out to be completely different from the behaviour of E discussed in I. Above the percolation threshold v2 > v, = 113, e.g., the dielectric parameter ;(k) remains of the order of .s2 for all values of k, whereas ~ ( k ) decays rapidly to values of the order of .sl. Accordingly, the sharp peak at k = 0 observed in I for ~ ( k )does , not occur in i ( k ) .Therefore, after transition to the ordinary space, the kernel function i ( r )does not necessarily show the long range behaviour which had been obtained for E(r).This example illustrates the great influence that statistical properties of the source distribution exert on the effective dielectricity i(k). + Fig. 3. Effective charge densities q(7) for a point charge situated in the second component of two-componcent mixture Linear Response of Random Media to Stoohastic Sources 19 To illustrate more in detail the results given in (5.13), we consider the mean field response to a point charge 1 qo(r) = Q d(r), qo(k) = - Q . (5.21) (2nI3 The effective charge density in the physical space q(r) is given by the three-dimensional Fourier-transform of eq. (5.13) 4n dkk sin kr[q(k)- ~(oo)]. +I q ( r ) = ( 2 ~ d(r) ) ~q(k = oo) O0 (5.22) 0 It consists of a point charge (5.23) at the center and a surrounding cloud a& 1 O0 T dt t sin-t [ A W ( t )- A W ( o o ) ] 2n2R2 r R -.- (5.24) c 0.8 0 1 2 f IR Fig. 4. Effective charge densities p(r) for a point charge situated in the second component of a two-component mixture 2' G.DIENER 20 where A W ( k R ) : = W ( k R ,E ~ / E ( ~ ) ) W(LR,E ~ / E ( ~ ) ) . This charge distribution is drawn in Figs. 3 and 4 for several two-component phase mixtures and for a = -vl (no charges in component 1).The extension of the cloud is about the grain size R except for very strong heterogeneities E & + C G above the percolation threshold v2 > 1/3 (cf. Fig. 4).The corresponding mean potential can be obtained froni (4.3) by Fourier transformation 4n Q ( u ( r ) ) = -- " d k 1+ a A W ( k R ) sinkr r (2n)3$ - T E(kR) dt 2 ~ ( 0 ) ---(1 Q 4na(k = 0 ) r t 7c ~ ( t ) + a A W ( t ) )sin(% 1). (5.25) The first factor in the last line describes the potential of a point charge Q in a medium with a n effective dielectric constant E (k = 0). The integral gives the correction due t o the dispersion of the effective dielectric parameter e(E) and due t o the effective sources. This correct,ion factor is plotted in Figs. 5 and 6 for the special choice a = -z+ (no 0 0.5 1.0 r IR 1.5 Fig. 5. Mean potentials ( u ( r ) ) for a point charge situated in the second component of a two.component mixture (full lines) and for a non-random point charge (dashed lines) 21 Linear Response of Random Media to Stochastir Sourres charges in component 1-solid lines). For comparison, the mean potential of a non-stochastic point charge (a = 0) is also represented (dashed lines). It shows a completely different behaviour. This clearly demonstrates the great influence of stochastic properties of the source distribution on the mean field response a t small distances. I n general the correction factor tends rapidly t o unity for increasing distances from the point charge. For distances large compared to the grain size the mean potential is that of the mean charge produced in an effective material without dispersion. The = 50, 100) indicates a breakdown of strange behaviour of some curves in Fig. 6 the approximation scheme for extremely strong heterogeneities. Especially the change in the sign of the potential is in conflict with physical expectation. This feature is niathematically connected with the maxima in the curves of Fig. 2, which, therefore, also have t o be attributed to the inaccuracy of the approximation used. 50 0 / 0.5 1.0 rf R 15 Fig. G. Mean potentials (u(r)>for a point charge situated in the second component of a two-component mixture (full lines) and for a non-random point charge (dashed lines) 6. Conclusion I n the present paper, the notion of effective sources has been introduced and procedures for calculating this quantity have been proposed. I n spite of the basic character of this notion, practical evaluations of effective sources and the corresponding mean fields G. DIENEB 22 near the sources will be, in general, very cumbersome. Moreover, it is to expect that in most cases a precise knowledge of the stochastic source distribution will not be available. Fortunately, the problem simplifies rigourously as long as we are only interested in static fields at large distances from the sources, because in this case the effective sources may be approximated by the mean sources and the nonlocality of the effective constitutive laws may be disregarded. On the contrary, in the dynamical case of wave emission from stochastic sources the stochastic source effects probably remain existent even at large distances from the sources. Finally, let us notice that stochastic initial conditions in non-stationary problems may be treated in the same manner as random sources. I n fact, initial conditions may always be transformed into supplementary source terms. For time-independent material parameters, for example, this can easely be done by performing a Laplace-transformation with respect t o the time. A c k n o w l d g e m e n t . The author is indebted t o W. HASSLERfor numerical calculations. Appendix I n the appendix the calculations leading to the expressions for Yf (5.5) and T (5.9) will be sketched. The integral in ( 5 . 5 ) may be transformed into Twofold integration by parts yields The last contribution vanishes because of [($3P1(5)]= 0 . (A 3) To evaluate the surface integrals, we apply the expansion (4.12), and take into account the orthogonality of Legrendre's polynomials 1 2 SU*. j- d5 5 ( 5 ) P245) = -1 I n this way, we obtain the equation which has been used in (5.5). A relation between the sums S and T appearing in (5.7) may be established with the aid of the identity 1 1 (1 +Y)W+ 1) 21 1 -- 2 (A 6) J+l+lY Y I+l+lY Inserting this equation into the definition of T ( x ,y) (5.7) and taking into account the exact relations [l, 101 do 1 sin (2s) cos 2m 2 s Si (2s) = : 1 2x%(z), + =-[ 1=0 do + ++I. + 1 + Linear Response of Random Nedia to Stochastic Sources 23 yields 1 T ( z ,y) = - [(l y) S ( x , y) - x2ar(x)]. Y + If we make use of the approximation (5.8) for S(.u,y) derived in I, eq. (A 8) becomes References [l] G. DIENER,F. KASEBERC, Int. J. Solids Struct. 12, 173 (1976). [2] If. J. BERAN, J. J. McCoy, Int. J. Solids Struct. 6, 1035 (1970). [3] 11. J. BERAN, Phys. Status Solidi (a) 6, 365 (1971). [A] E. KROXER,H. KOCH,SM Archives 1, 183 (1976). [5] s. CRANDALL(Ed.), Random Vibrations. M.T.T. Press, Cambridge Mass., Vol. I, 1958; Vol. I1 1963. [6] V. A. LOMAKIN, Statistical Problems in the Mechanics of Deformable Solids. Moscow, 1970 (in Russian). [7] L. ARNOLD, Stochastic Differential Equations. Wiley-Interscience,New York, 1974. [8] J. PEBm-4, Coherence of Light. van Nostrand-Reinhold, London, 1973. [9] U. FRISCH, Wave Propagation in Random Nedia, in: Probabilistic Methods in Applied Mathematics, Vol. I, Ed. by A. T.BACHURA-REID, Academic Press, New York, 1968. [lo] G . N. WATSON, A Treatise on the Theory of BesseI Functions. Cambridge, 1958. Bei der Redaktion eingegangen am 3. Juli 1980. Anschrift. d. Verf.: Dr. G. DIENER Sektion Physik der Technischen Universitiit Dresden DDR-8027 Dresden Mommsenstr. 13

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