Radiative Transport in a Periodic Structure

Transcription

1 Journal of Statistical Physics, Vol. 95, Nos. 12, 1999 Radiative Transport in a Periodic Structure Guillaue Bal, 1 Albert Fanniang, 2 George Papanicolaou, 3 and Leonid Ryzhik 4 Received Septeber 16, 1998; final Deceber 1, 1998 We derive radiative transport equations for solutions of a Schro dinger equation in a periodic structure with sall rando inhoogeneities. We use systeatically the Wigner transfor and the Bloch wave expansion. The streaing part of the radiative transport equations is deterined entirely by the Bloch spectru, and the scattering part by the rando fluctuations. KEY WORDS Radiative transport; waves in rando edia, Wigner distribution; seiclassical liits; Bloch waves. 1. INTRODUCTION Radiative transport equations describe propagation of the phase space energy density of high frequency waves in a ediu with weak rando ipurities whose correlation length is coparable to the wave length *. (19) The background ediu ay vary but only on scales that are uch larger than *. The phase space energy density has also been studied for a periodic potential when the period is coparable to the wave length. (10, 15) It is then found that the liiting phase space energy density satisfies a syste of decoupled Liouville equations with Hailtonians given by the Bloch eigenvalues. Thus, we expect that the addition of sall rando fluctuations 1 Departent of Matheatics, Stanford University, Stanford, California 94305; e-ail balath.stanford.edu. 2 Departent of Matheatics, University of California Davis, Davis, California 95616; e-ail fannianath.ucdavis.edu. 3 Departent of Matheatics, Stanford University, Stanford, California 94305; e-ail papnicoath.stanford.edu. 4 Departent of Matheatics, University of Chicago, Chicago, Illinois 60637; e-ail ryzhikath.uchicago.edu Plenu Publishing Corporation

2 480 Bal et al. to the periodic structure will give rise to a syste of coupled radiative transport equations. This is what we show in this paper forally, using asyptotic expansions. The rigorous proof of our final result reains an open proble. Our ain result is the syste of the radiative transport equations (3.15) that describes the propagation of phase space energy densities _ (t, x, p) corresponding to various Bloch eigenvalues E (p). We derive here the transport equations only for the Schro dinger equation. However, the generalization to ore general types of waves in a periodic and rando ediu, like those described by hyperbolic systes considered in ref. 19, is straightforward. This results in the replaceent of the eigenvalues E (p) in (3.15) by the corresponding Bloch eigenvalues for the hyperbolic syste under consideration. The paper is organized is follows. First, we recall in Section 2 soe properties of the Bloch eigenfunctions and also give a foral derivation of the Liouville transport equations in the absence of rando inhoogeneities. These results were previously derived rigorously in refs. 15 and 10, and we show the connection between our foralis and their results in Section 2.4. Section 3 is the ain part of the paper, where we derive the radiative transport equations in the presence of a rando potential. 2. WAVES IN A PERIODIC STRUCTURE 2.1. The Schro dinger Equation We give here a derivation of the Liouville equation for the phase space energy density like the one in ref. 19. A different analysis is given in refs. 10 and 15. We then adapt the analysis of ref. 19 to the periodic-rando case. There is not a lot of atheatical work on the transport liit for the Schro dinger equation with rando potential. We cite here the work of Martin and Ech, (16) of Spohn, (20) of Dell'Antonio (3) and the recent extensive study of Ho, Landau and Wilkins. (11) These papers established the validity of the kinetic linear transport equation for a sall tie T>0. The global validity of this liit was proved recently by Erdo s and Yau in ref. 4. They treat only spatially hoogeneous probles but it is known how to extend the analysis to the spatially inhoogeneous case (slow x-dependent initial data and potential). (5) A really satisfactory atheatical treatent of radiative transport asyptotics fro rando wave equations is lacking at present. It is convenient for us to use the usual Wigner distribution and not the Wigner band-series as in refs. 10 and 15. The two forulations are,

3 Radiative Transport in a Periodic Structure 481 however, equivalent. (15) Let, = (t, x) be the solution of the initial value proble i=, = t +=2 2 2, =&V =+ \x =0 (2.1), = (0, x)=, 0 =(x) The initial data, 0 (x) is uniforly bounded in = L2 (R d ) &, 0 & = L2C. We also assue that it is =-oscillatory, that is, for any test function # C c (R d ) 0 li 2 = dk 0 as R goes to + (2.2) = 0 k R= Finally we assue that the faily, 0 = is copact at infinity li sup = 0 x R, 0 = (x) 2 dx 0 as R goes to + (2.3) A sufficient condition for (2.2) is &{, 0 = &C=. The case of =-independent initial data was studied in ref. 2, where the Liouville equation for the wave aplitudes (not energies) was derived. The potential V(z) is periodic V(z+&)=V(z) where vector & belongs to the period lattice L L= { d =1 n e } n # Z = (2.4) and e 1,..., e d for basis of R d with the dual basis e defined by (e } e k )=2?$ k and the dual lattice L* defined by (2.4) with e replaced by e. We denote by C the basic period cell of L and by B the Brillouin zone B=[k # R d k is closer to +=0 than any other point ] We define the Wigner distribution by W = (t, x, k)= R d dy (2?) d eik } y, = (t, x&=y), =(t, x) (2.5)

4 482 Bal et al. This definition is equivalent to its syetric version W =(t, x, k)= R d dy (2?) d eik } y, =\ t, x&=y 2+ =\, t, x+=y 2+ in the sense that W = and W =(x, k) have the sae weak liit as = 0. (9) We also have that E = (t, x)=, = (t, x) = 2 dk W = (t, x, R d k)= dk W =(t, x, k) R d The basic properties of the Wigner distributions are reviewed in detail in refs. 10 and 18. In particular the weak liit W(t, x, k) ofw = (t, x, k) exists in S$(R d _R d ) and under assuptions (2.2) and (2.3) it captures correctly the behavior of the energy E = li dx E =(t, x)= dx dk W(t, x, k) = 0 We deduce fro (2.1) and (2.5) the following evolution equation for W = (t, x, k) W = t +k } { xw = + i= 2 2 xw = = 1 i= Here V (+) are the periodic Fourier coefficients of V(y) so that if V(y) is sooth we have and e i+ } x= V (+)[W = (x, k&+)&w = (x, k)] (2.6) V (+)= 1 C C dy e &i+ } y V(y) (2.7) 1 C V(y)= e i+ } z = e i+ } y V (+) & # L We introduce a ultiple scales expansion for W = $(z&&) (2.8) W = (t, x, k)=w 0\ t, x, x =, k + +=W 1\ t, x, x =, k + + } } } (2.9)

5 Radiative Transport in a Periodic Structure 483 and assue that the leading ter W 0 (t, x, z, k) is periodic in the fast variable z=x=. As usual, we replace then { x { x + 1 = { z in (2.6) and rewrite it as W = t +k } _ { x+ 1 = { z& W =+ i= 2 \ { x+ 1 = { z+ } \ { x+ 1 = { z+ W = = 1 i= e i+ } z V (+)[W = (k&+)&w = (k)] (2.10) We insert the perturbation expansion (2.9) into (2.10) and get at the order = &1 where the skew syetric operator L is given by LW 0 =0 (2.11) Lf(z, k)=k } { z f + i 2 2 z f& 1 i e i+ } z V (+)[ f(z, k&+)&f(z, k)] 2.2. The Bloch Functions The eigenfunctions of L are constructed as follows. Given p # R d consider the eigenvalue proble & z9(z, p)+v(z) 9(z, p)=e(p) 9(z, p) 9(z+&, p)=e ip } & 9(z, p), for all & # L (2.12) 9 9 ip } & (z+&, p)=e (z), for all & # L z z This proble has a coplete orthonoral basis of eigenfunctions 9 (z, p) in L 2 (C) (9, 9 ; )= C dz C 9 (z, p) 9 ; (z, p)=$ $ ; (2.13) They are called the Bloch eigenfunctions, corresponding to the real eigenvalues E (p) of ultiplicity r. Here =1,..., r labels eigenfunctions

6 484 Bal et al. inside the eigenspace. The eigenvalues E (p) are L*-periodic in p and have constant finite ultiplicity outside a closed subset F of p # R d of easure zero. They ay be arranged E 1 (p)<e 2 (p)< }}} <E (p)< } } } with E (p) as, uniforly in p. (2, 13, 22) We consider oenta p outside the set F. The proble (2.12) ay be rewritten in ters of periodic functions 8(z, p)=e &ip } z 9(z, p) & z8 +V(z) 8 &ip } { z8 + p 2 8 =E (p) 8 (z, p) (2.14) We differentiate (2.14) with respect to p and take the scalar product of the resulting equation and 8 ; to get E p which ay be rewritten as $ ; = p $ ; &i \8 z, 8 ; + E p $ ; =i \9, 9 ; (2.15) z + There is no suation over in (2.15). Recall the Bloch transfor of a function,(x)#l 2 (R d ), (p)= R d It has the following properties dz,(z) 9 (z, p) (i) (ii),(x)=(1 B ) =1 r =1 B dp, (p) 9 (x, p), x # Rd. Let,(x), '(x)#l 2 (R d ), the Plancherel forula holds dx,(x) ' (x)= 1 R d B, B dp, (p) '~ (p) (iii) The apping,, is one-to-one and onto, fro L 2 (R d ) L 2 (B). We deduce fro these properties the orthogonality relations $(y&x)= 1 B, B dp 9 (x, p) 9 (y, p)

7 Radiative Transport in a Periodic Structure 485 and $ $ ; $ per (p&q)= 1 B R d dx 9 (x, p) 9 ; (x, q) (2.16) The periodic delta function $ per in (2.16) is understood as follows for any function,(p)#c (B),(p)= B dq,(q) $ per (p&q) Given any vector k # R d we ay decopose it uniquely as k=p k ++ k (2.17) with p k # B and + k # L*. We then define the z-periodic functions (z, +, p),, p # B by Q ; n Q ; n (z, +, p)= C Then a direct coputation shows that LQ ; dy C ei(p++)}y 9 (z&y, p) 9 ; n (z, p) (2.18) (z, +, p)=i(e n (p)&e n (p)) Q ; n (z, +, p) (2.19) with +=+ k, p=p k,soq ; n are eigenfunctions of L The Liouville Equations Now (2.19) iplies that, for any p, ker L is spanned by the functions Q ;, which we denote by Q; (to indicate that there is no suation over ). Then (2.11) iplies that W 0 (t, x, z, k) ay be written as W 0 (t, x, z, k)=w 0 (t, x, z, p++) =,, ; _ ; (t, x, p) Q ; (z, +, p), p # B, (2.20) with +=+ k, p=p k. This defines _, which is scalar if the eigenvalue E (p) is siple, and is a atrix of size r _r if E (p) has ultiplicity r >1. We call _ the coherence atrices in analogy to the non-periodic case. (19) They are defined inside the Brillouin zone p # B but it is convenient to extend the as functions in R d, L*-periodic in p.

8 486 Bal et al. Next we look at = 0 ters in (2.10). We get W 0 t +k } { xw 0 +i { x } { z W 0 =&LW 1 (2.21) ; We now integrate both sides of (2.21) against Q (z, +, p) over z in C and su over +. To evaluate the left side we note that we have after suing over + using (2.8), and using (2.13) and a change of variables y z&y C dz C Q$;$ (z, +, p) Q ; (z, +, p) dy dz = C_C C 9 $ ;$ (z, p) 9 (y, p) 9 2 (z, p) 9 ; (y, p) =$ $ $ $ ;;$ (2.22) Further, using the change of variables and equations above and also (2.15), we get ((p l ++ l ) Q $;$, Q ; ) dz dy 1 dy 2 = e i(p++)}(y C 3 C 3 1 &y 2 ) ( p l ++ l ) _9 $ (z&y ;$ 1, p) 9 (z, p) 9 (z&y 2, p) 9 ; (z, p) = 1 dz dy 9 i $ (y, p) 9 C_C C 2 y (y, p) 9 ; (z, p) 9 ;$ (z, p) l = 1 $ i $ 9 $ ;;$\, 9 y l + =$ E $ $ $ ;;$ (2.23) p l A siilar calculation shows that the third ter on the left vanishes \ Q $;$, Q ; z l dz dy 9 = $ (z&y, p) 9 C_C C 2 z (z&y, p) 9 ;(x, p) 9 ;$ (z, p) l + dz dy + C_C C 9 $ (z&y, p) 9 (z&y, p) 2 9; (x, p) 9 ;$ (z, p) z l =& 1 i $ ; $ ;;$ $ E p l + 1 i $ ; $ ;;$ $ E p l =0

9 Radiative Transport in a Periodic Structure 487 ; The right side of (2.21) integrated against Q vanishes since L is skew syetric, and Q ; # ker L. Putting together (2.21), (2.22) and (2.23) we obtain the Liouville equations for the coherence atrices t +{ pe } { x _ =0 (2.24) The initial data for equations (2.24) is constructed as follows. Let W 0 = (x, k) be the Wigner transfor of the initial data, 0 (x). Then _ = (0, x, p) is given by _ ; (0, x, p)= li = 0 W 0 = (x, p++) Q ; \ with the liit understood in the weak sense. x =, +, p The Wigner Band Series The Liouville equations (2.24) were previously derived in refs. 15 and 10 in ters of the Wigner series defined by w = (t, x, k)= C e ik } &, (2?) d = (x&=&), =(x)= W = (x, k++) & # L It was shown that the weak liit w(t, x, k) ofw = (t, x, k) has the for w(t, x, k)= w (t, x, k) Here w is the liit Wigner series of the proection, = of the solution, = of the Schro dinger equation on the Bloch spaces S = S = = { f # 1 L2 (R d ) f(x)= B B dp = f (p) 9 32 \ x =, p += Each proection w evolves according to the Liouville equation (2.24). This result ay be related to our approach as follows. The Wigner distribution W = (t, x, k) has the asyptotics W = (t, x, p++)r,, ; _ ; (t, x, p) Q; \ x =, +, p (2.25) +

10 488 Bal et al. Then the weak liit of w = is given by dz w(t, x, p)= _ ; (t, x, p),, ; C C Q; (z, +, p)= Tr _ (t, x, p) If we take the trace of (2.24) we get the transport equations for w (t, x, p) =Tr _ (t, x, p) obtained in refs. 15 and 10. However, the representation (2.25) captures not only the weak liit of w = but also the oscillations of W = on the fine scale. Moreover, the cross-polarization of various odes corresponding to the sae eigenvalue is also taken into account by the offdiagonal ters in the coherence atrices. The oscillations of the spatial energy density, which is the physically interesting quantity, are given by E = (t, x, p)r dp _ ; (t, x, p) 9,, ; B \ x =, p + 9 ; \ x =, p + ( dp Tr _ (t, x, p) as = 0 B This inforation on the energy oscillations ay be useful in nuerical siulations. We see that _ are phase space resolved energy densities of different odes inside the Brillouin zone. 3. THE RANDOM PERTURBATION 3.1. Siple Eigenvalues We assue first that the Bloch eigenvalues E (p) have ultiplicity one for all p # B. This assuption is known to be true for the leading eigenvalue when the Fourier transfor (2.7) of the periodic potential V(y) is negative. (1) In any physical probles the absence of level crossings restricts our results to the lower part of the spectru. We consider now sall rando perturbations of the periodic proble (2.1) with randoness being on the sae scale as the periodic potential but weak i=, = t +=2 2 2, =&V \x =+ &- =N \ x =+ =0, = (0, x)=, 0 = (x) Here N(y) is a tie independent ean zero spatially hoogeneous rando process with covariance tensor R(x) defined by (N(y) N(y+x)) =R(x), (N (p) N (q)) =(2?) d R (q) $(p+q) (3.1)

11 Radiative Transport in a Periodic Structure 489 Here ( } ) denotes the enseble average and the Fourier transfor N (q) is N (q)= R d dx e &iq } x N(x) (3.2) The Wigner distribution W = (t, x, k) satisfies the evolution equation W = t +k } { xw = + i= 2 2 xw = = 1 i= + 1 i - = R d e i+ } x= V (+)[W = (k&+)&w = (k)] dq (2?) d eiq } x= N (q)[w = (k&q)&w = (k)] Here V (+) is the periodic Fourier transfor (2.7) and N (q) is the Fourier transfor (3.2) over R d. We consider the asyptotic expansion W = (t, x, k)=w 0\ t, x, x =, k + +- =W 1\ t, x, x =, k + +=W 2\ t, x, x =, k + +}}} with the leading order ter W 0 being deterinistic. We introduce as before the fast variable z=x=, replace { x { x +(1=) { z and collect the powers of =. The order = &1 gives as before LW 0 =0 Thus we still have the decoposition (2.20) W 0 (t, x, z, p++)= _ (t, x, p) Q (z, +, p) (3.3) with _ being scalar because the spectru is siple. Recall that the functions _ (t, x, p) are L*-periodic in p and the z-periodic functions Q are given by (2.18) with =n. The order = &12 ters give LW 1 = 1 i R d dq (2?) d eiq } z N (q)[w 0 (z, k&q)&w 0 (z, k)] (3.4) The distribution W 1 need not be periodic in the fast variable z. Therefore it ay not be expanded in Q n and we use the basis functions P n (z, +, p, q)= C dy C ei(p++)}y 9 (z&y, p) 9 (z, p+q) (3.5)

12 490 Bal et al. defined for z # R d and p, q # B, in place of the periodic functions Q n (z, +, p). The functions P n are quasi-periodic in z P n (z+&, +, p, q)=p n (z, +, p, q) e &i& } q We use (2.16) and (2.8) to obtain the orthogonality relation for the functions P n R d dz B P n(z, +, p, q) P l(z, +, p, q 0 )=$ $ n $ per (q&q 0 ) (3.6) The operator L acts on these functions as LP n =i(e (p)&e n (p+q)) P n (z, +, p, q) which is the analog of (2.19) for the functions Q n. Note that the integrand in (3.5) is periodic in y so no boundary ters are produced by integration by parts. We decopose W 1 in this basis as dq W 1 (t, x, z, p++)=, n B B ' n(t, x, p, q) P n (z, +, p, q) (3.7) with z # R d, p # B and. We insert (3.7) into (3.4), ultiply (3.4) by P l(z, +, p, q 0 ), su over and integrate over z # R d. Then we get using (3.6) on the left side dz dq e ' l (t, x, p, q 0 )= iq } z N (q) R 2d (2?) d _ [W 0(z, p++&q)&w 0 (z, p++)] E l (p+q 0 )&E (p)+i% P l(z, +, p, q 0 ) (3.8) where % is the regularization paraeter. We let % 0 at the end. We insert expression (3.3) for W 0 and the definition (2.18) of Q into (3.8). The resulting expression ay be siplified using (2.16) and (2.8) ' l (t, x, p, q 0 )= 1 (2?) d & R d _R d N (&+&q 0 ) _ l (p+q 0 ) E l (p+q 0 )&E (p)+i% A l(p+q 0 ++, p) dz dq N (q) _ (p) 9 (2?) d B eiq } z l (z, p+q 0 ) 9 (z, p) E l (p+q 0 )&E (p)+i% (3.9)

13 Radiative Transport in a Periodic Structure 491 Here the aplitude A l (q, p) is given by A l (q, p)= C The next order equation is dy C e&i(q&p)}y 9 l (y, q) 9 (y, p) (3.10) W 0 t +k } { xw 0 +i { z } { x W 0 +LW 2 = 1 i R d dq (2?) d eiq } z N (q)[w 1 (z, k&q)&w 1 (z, k)] We ultiply this equation by Q (z, +, p), integrate over z and su over, and take average. Then as before the left side is The right side is where and I 1 = 1 i LHS= _ t +{ pe } { x _ (3.11) RHS=I 1 +I 2 (3.12) dz dq e C C iq } z (N (q) W R d (2?) d 1 (p++&q) Q (z, +, p)) (3.13) I 2 =& 1 i C dz C R d dq e iq } z (2?) d (N (q) W 1 (p++) Q (z, +, p)) We insert expression (3.7) for W 1 into (3.13) to get I 1 = 1 i dz C C R d _, n dq dq (2?) 0 (N (q) Q (z, +, p) d B B ' n (p++&q, q 0 ) P n (z, +, p&q, q 0 )) We ay split I 1 =I 11 &I 12 according to the two ters in (3.9). We insert the first ter in (3.9) into the expression for I 11 and average using spatial

14 492 Bal et al. hoogeneity (3.1) of the rando process N(z), orthogonality (2.13) of the Bloch functions 9 (z, p) and also su over using (2.8). Then we get I 11 = 1 i dq R (q++) A (p, p&q&+) 2 _ (p) B B (2?) d E (p)&e (p&q)+i% Here we have replaced integration over q # R d by integration over B and su over. The second ter I 12 is evaluated siilarly I 12 = 1 i B dq R (q++) A (p, p&q&+) 2 _ (p&q) B (2?) d E (p)&e (p&q)+i% Thus we have I 1 = 1 i dq R (q++) A (p, p&q&+) 2 [_ (p)&_ (p&q)] B B (2?) d E (p)&e (p&q)+i% (3.14) One ay verify that I 2 =I 1 in (3.12). We insert then (3.14) into (3.12), take the liit % 0, ake a change of variables q p&q, and cobine it with (3.11) to get the syste of radiative transport equations _ t +{ dq pe } { x _ = B B Q (p, q)[_ (q)&_ (p)] $(E (p)&e (q)) (3.15) The differential scattering cross-sections Q (p, q) are given by Q (p, q)= 1 (2?) d&1 R (p&q++) A (p++, q) 2 This is the ain result of this paper we have derived a syste of coupled radiative transport equations for the phase space energy densities of the Bloch odes. The transition probabilities Q (p, q) are real and syetric Q (p, q)=q (q, p) as seen fro the definition (3.10) of A i (p, q). Therefore the total energy is conserved E(t)= R d dx B dp _ (t, x, p)=const Transport equations like (3.15) are well known in the theory of resistance of etals and alloys. (17) Their systeatic derivation fro the Schro dinger equation with a periodic and rando potential (3.1) is new.

15 Radiative Transport in a Periodic Structure 493 APPENDIX. MULTIPLE EIGENVALUES When eigenvalues E (p) are not siple, but their ultiplicity is independent of p, so that there are still no level crossings, the analysis in the previous section can be extended to this case. This case is probably very rare for the Schro dinger equation but is iportant for other types of waves, for instance, in syetric hyperbolic systes. We present it here for the sake of copleteness. The result is as follows. Let the atrices T ; (p, q), =1,..., r, ;=1,..., r, where r and r are the ultiplicities of E and E be defined by T ; (p, q)= C dz (2?) (d&1)2 C ei(p&q)}z 9 ; (z, q) 9 (z, p) Then the coherence atrices _ (p) satisfy the syste of radiative transport equations _ t +{ pe } { x _ = ; # N B dq B R (p&q++) T (p, q&+) _ (q) _T* (p, q&+) $(E (p)&e (q))& i dq R (p&q++) 2? B B (p, q&+) T* (p, q&+) _ (p) T E (p)&e (q)+i0 & _ (p) T (p, q&+) T* (p, q&+) E (p)&e (q)&i0 & These equations have the sae structure as the radiative transport equations for polarized waves derived in ref. 19. The expression of the total scattering cross-section as a principal value integral and not as a failiar integral against $[E (q)&e (p)] is known in transport theory for (12, 6, 19) polarized waves. They reduce typically to the for coon in scalar transport equations under additional syetries, like rotational invariance of the original wave equations and the power spectru tensor. ACKNOWLEDGMENTS The work of G. Bal and G. Papanicolaou was supported by AFOSR Grant F and by NSF Grants DMS The work of A. Fanniang was supported by NSF Grant DMS The work of L. Ryzhik was supported in part by the MRSEC Progra of the National Science Foundation under Award Nuber DMR

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