quantum distribution-function transport equations systems...

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 265-273Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

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Publishers imprint.Printed in India.

Quantum Distribution-function Transport Equationsin Non-normal Systems and in Ultra-fast Dynamics

of Optically-excited Semiconductors

F. A. BUOT

Naval Research Laboratory, Washington, D.C. 20375-5320

The derivation of the quantum distribution-function transport equations combines theLiouvillian super-Green’s function technique and the lattice Weyl-Wigner formulationof the quantum theory of solids. A generating super-functional is constructed whichallows an algebraic and straightforward application of quantum field-theoreticaltechniques in real time to derive coupled quantum-transport, condensate, and pair-wavefunction equations. In optically-excited semiconductors, quantum distribution-function transport equations are given for phonons, plasmons, photons, and electron-hole pairs and excitons by transforming the Bethe-Salpeter equation into a multi-timeevolution equation. The virtue of quantum distribution function is that it allows easyapplication of ’device-inflow’ subsidiary boundary conditions for simulating femtose-cond device-switching phenomena.

Keywords: Quantum distribution function, quantum transport, ultra-fast optics, excitons, non-equilibrium superconductivity, nonequilibrium superfluidity, lattice Weyl transform, Liouvillian,super Green’s function

1. INTRODUCTION

There is a need for generalized quantum distribu-tion-function transport equations, valid for non-normal, non-uniform, and ultra-fast systems, asbases for large-scale computer simulations. Thisbecomes urgent with advances in material science,ultra-fast laser probes, nanofabrication, and thedevelopment of more powerful energy beams. Thedrive to produce systems which are functionallymore dense and have wider bandwidths will leadnanostructure devices to atomic-scale dimensions

with different materials: insulators, semiconduc-tors, metals, and superconductors.The nonequilibrium quantum transport theory

including pairing dynamics is formulated in termsof the Liouville-space (L-space) quantum-fieldtheory [1- 2] and lattice Weyl transform technique[3, 4]. For normal systems, this reduces to thenonequilibrium Green’s function technique ofSchwinger [5], Kadanoffand Baym [6], and Keldysh[7], coupled with the lattice Weyl-Wigner formula-tion of the quantum theory of solids [3, 4]. Severalnew results are derived with the present approach.

265

266 F.A. BUOT

This L-space approach has provided the actionprinciple for a multi-variable functional theory ofnonequilibrium condensed-matter systems [8-10].Thus, the method set forth here may opendoors for the investigation of ultra-fast dynamicsin quantum nanostructures. So far, only thedistribution-function approach has characterized,in time-domain, a highly-nonlinear and highly-nonequilibrium quantum behavior [1 4].

2. QUANTUM DYNAMICS IN LIOUVILLESPACE

The density-matrix equation of quantum statisticaldynamics in Hilbert space (H-space) becomes a

super-Schrodinger equation for the super-statevector in L-space as

0ih -t IP(t))) lP(t)))- (1)

p(t) is the density-matrix operator for the wholemany-body system in H-space, and Ip(t))) is thecorresponding super-state vector in L-space. Thesuper-operator corresponds to the commutator

[oCt, p], and is referred to in this paper as theLiouvillian. Thus, we may write the Liouvillianas 5e- 07g_ f, which define and 3f. Thesehave the property that lp(t)))= IWp(t))), andlp(t))) Ip(t)Ygt)). These relations are valid forfermions and bosons. For number-conservingfermion operator o, 07f 3gt. The quantum fieldsuper-operators, ()and t(t), are definedthrough their commutation relations in H-Space.

where 1)) is a unit super-vector. We have

{-if }Ip(t)>> Zexp T o(t’)dt’ (t, -x)lPeq)o

o

{-i/’O(t’) )IA(t)))- Zac exp -- dt’ N(t,

.(3)where Tac denotes anti-chronological time oraer-ing and

{ifo’ }(t, -) -Texp .’o(t’)dt’

T exp -ff ’(t’)dt’

{ -ifo’ }T exp - ’o(t’)dt’

The "transition probability" obeys the equality

((llS(c, -)lPeq)) exp W (4)

where W is identified as the effective action. It canbe shown that

0W- ((A(t)lih-- [p(t)))dt

This relation forms the basis of a time-dependentfunctional theory of condensed matter discussed byRajagopal and Buot in a series of papers [8-1 0].

4. GENERALIZED QUANTUMDISTRIBUTION FUNCTIONS

3. SUPER S-MATRIX THEORY IN L-SPACE

The "transition probability" in L-space is given bythe following relation

((A(t)lp(t)))- ((I[S(,-)lPeq)) (2)

We introduce a 4-component second quantizedquantum field super-operators given by

ff(1)T--((1) t(1)*(1)(1)) (6)

This is a generalization of the multi-componentquantum field operator first introduced by Nambu[15], and by De Dominicis and Martin [16]. In

QUANTUM TRANSPORT EQUATIONS 267

terms of the fields, a system Liouvillian 5 can ingeneral be expressed as

M

Z v(1,2, 3,... ,N) (1)(2)(3)N=I

(N) / ex.(7)

where the Schwinger source term is given byext =ext- e,ext- U(1)(1)+ U(12)(1)(2).The field super-operator averages can be written interms of the S-matrix, e.g.,

if(1 2)(ih) 2 5((1(’5(2)5()<<11(, -)]Peq))

(8)

We obtained the following (e= for bosons,e- for fermions),

F(1,2) G(1,2))(1,2) ih eG T(1,2) F(1,2) (9)

where the superscript T indicates the taking of thetranspose, and,

expanded about the condensate for Wick’s theo-rem to be applicable) to evaluate the aJ’s in termsof diagrams or graphs, (1,2,... n) represents thetopologically distinct ’connected’ subset of graphs.We have

6" ln((11S(oc,0C(1,2,..., n) (ih)n 6u(n) 5u(n 1) 5u(1)(11)

Similar functional derivative relations can beobtained between GQDF with even number ofindices by using the variation with respect to theexternal Schwinger source term u(1,2). Sinceu (1,2) is an ordinary c-number, the order of theu(i, j)’s is not critical in taking the functionalderivatives.

4.1. Self-Consistent Equations for GeneralizedQuantum Distribution Functions

We have

gCF(1,2)

gh

fF(1 2)- ( gceeeg>ee

G(1,2)G>

eGr(1, 2) ( eGcrG< r

gh ) (1 2),gege<e )gea (1,2)

eG< )eaac (1,2),

eG>r

)Ga (1,2)

(10)

G(1,2) corresponds to the Keldysh nonequilibriumGreen’s function. We refer to the aj(1,2) simply asmoment quantum distribution function. Momentsare defined for time-ordered quantum field super-operators. We also define quantum correlationfunctions or quantum cumulants, g(, analogous tothe classical statistical theory. This distinction isimportant in treating the quantum transport ofsuperfluids and quanta of real classical fields. Wewill also refer to both as generalized quantumdistribution functions (GQDF).

In the application of Wick’s theorem (forsuperfluid Bose system it is assumed that 0 is

r(12) -1 (12)ih

w h e r e o(,,, 2)-1_ (7.g)-1/(,,, 2)O/Ot.-((’, 2), where the (T)-matrix arise from the

"’s ((’commutation relation of the (i, j) ,2) is aone-body potential matrix, and E(12) is theparticle super self-energy matrix. For fermions,(1,2)-5U(1,2). E(12) is expressed in terms ofYg’s (up to second-order cumulants) and vertexfunctions. These vertex functions obey equationssimilar to the Dyson equation, involving functionalderivative of the self-energy with respect to second-order GQDF hence decoupling the BBKGYhierarchy. The self-energy due to e-e interactionincludes the electron-plasmon vertex function.

5. QUANTUM TRANSPORT EQUATIONS

The time-evolution equation for Yg(i, j) isobtained. We write the resulting equations for

268 F.A. BUOT

the 2x2 matrix elements of (i, j) as

(13)

(14)

(15)

(16)

where A is the pair potential or gap function, andG-1 is a diagonal matrix with elements propor-tional to ih6(12)O/Ot2-,(12) with re(12) propor-tional to a one-body external potential. Theseequations were also given by Aronov, et al. [17a]and formally resemble the well-known Gorkovequations [17b] for superconductors at thermalequilibrium.

Solving quantum transport problems [11] cen-ters on the evolution of p<(12)-eihG <(12),which happens to be one of the matrix elementsof the nonequilibrium matrix Green’s function ofEq. (9). (>,< contains all information about thestatistical aspects of the field intensity. This iscoupled to the ’advanced’ and ’retarded’ propaga-tors often directly related to the experiment andcontain all the energetics and dynamical informa-tion of the system. We obtain the transportequation for G>,< given by the following expres-sions.

0 0 )Gih + (12) [(9 + 2r), Gr]

>+ [hhgee ghhee] + {h<ge + ghhee}

< gr < > g< g>{mhhgee nt- hhmee} --{-- [ghh hh]mee hh

gh<h] [Ae> Ae<e]

(8)

Equations for the pair wavefunctions, g>’<ee geer,ghh>’<, grhh, etc, are also obtained. Using latticeWeyl transformation, we can transform the abovetotal time evolution equations, Eqs. (17) (18)into quantum transport equations in (p, q, E, t)phase space for superconductive, systems.

5.2. Nonequilibrium Bose Superfluids

The expressions for ih(O/Otl + O/Ot2) (>,<(12),ih(O/Otl +O/Otz)Gr(12), and pair-wavefunctionequations, are similarly obtained by taking e

in Eqs. (13)- (16) and applying the self-consis-tency condition for the boson-particle self-ener-gies. We can also transform the total timeevolution equations into quantum transport equa-tions in (p, q, E, t) phase space for Bose superfluidsystems [18]. This will not be given in this paper.

6. ULTRA-FAST DYNAMICS OF EXCITEDSEMICONDUCTORS

5.1. Nonequilibrium Superconductivity

We have the following expressions for super-conductors,

0 0 ) G>,< G>,<+ 22 (12) [ + RePr,

[Re Gr, >’<]

.i{P’ G>,}+<i{A, p>,<}>,<q- {Arhh ge>{< --}- ghh A

ee }-->,< a a >,<-’}- { /hh gee + ghhAee } (17)

The physics of highly-excited semiconductor het-erostructures has been of continuing researchinterest [19- 23]. The electron- hole (e-h) pairtheory is formally identical to the theory ofsuperconductivity for extremely high-density ofe-h pairs. However, in the low density limit realelectron-hole pair bound states do occur, theseare localized composite e-h elementary excita-tions called ’excitons’ [23]. The device physics [24]of interacting matter and radiation requires theconsideration of all ranges of e-h densities andshort-time dynamics.The multi-component (in the "hat" and "tilde"

indices) quantum field super-operators for electro-


magnetic field (transverse and longitudinal) andlattice vibrations are given, respectively, by

7.1. Schr6dinger- Wannier Equationfor the Exciton Wavefunction

A() 4 i() 0()and U() (v/-() )"

(19)

It is helpful to derive from the Bethe-Salpeterequation, the Schr6dinger-Wannier equation forthe electron-hole pair wavefunction. Neglecting theself-energy terms and retaining only the Coulombinteraction terms, we obtain after setting tl t2

We used a "composite field operator" as thefourth field, (1,2), since coupling to fermionsonly occur through bilinear product of fermionfields. This is made up of (12), (12), er(12),and t-(12), consisting of different combination ofthe "hat" and "tilde" indices. We have,

ffs(1,2) r- (-(1,2) (1,2) er(1,2) *-(1,2))(20)

The time-ordered averages of the components givethe familiar quantities, namely,

F(1,2)(TtP(1,2))-(1,2)-- eGr(1,2)G(1,2)*F(1,2))

(21)

The scalar potential field super-operator O() isnot an independent field [25].

7. ELECTRON-HOLE AND EXCITON TIMEEVOLUTION EQUATION

The super propagator for electron-hole andexciton is defined by

g3((23)(22)(23; 12) fext(12

fext(12) includes the matrix element of the dipolemoment containing the selfconsistent transverseelectromagnetic field in the microcavity, forexample. In the matrix definition of 3((12; 34),the nonequilibrium pair propagator is identified asone of the matrix elements, namely, the ’(1,4th)’-component.

0t(12)- -mV2+mzV22[rlr2----l (12)

(23)

This is the familiar Schr6dinger equation for a

two-particle system consisting of an electron and ahole. Using a dielectric function to screen thepotential, the resulting bound states correspond toexciton states [23].

7.2. Time Evolution Equations for C (12; 34)

The total time derivative of 3((12; 34) is obtainedfrom the Bethe-Salpeter equation, the details willbe published elsewhere. The ’(1,4th)’ elementdescribing the electron-hole pair propagator isdenoted by (ffexciton (12; 34), which in turn containsin its ’(1, 4)th’ element the e-h pair density matrix<CNexciton (12; 34).

7.3. Transport Equation for the NonequilibriumPair Propagator

What needs to be done is to single out the equationfor the electron-hole pair density matrix, which we

<denote by CNexciton(12; 34). The result is very longand will be published elsewhere. Interested readersare encouraged to contact the authors for thisdetail. The final transport equation is obtained bysetting l---t2, and t3 t4 and taking the latticeWeyl transform of the above transport equation[in general double lattice Weyl transform in spatialvariables]. To demonstrate this, let us take thesimplest approximation of this equation, which we

270 F.A. BUOT

write us

0 0)ih -+-3 Cffexcitn(12’ tl; 34, t3)

_{ [9(12")]aJe<xciton(2"2, tl;34, t3) )< tt--c-ffexciton(12 tl, 34, t3)[gT(2"2)]

(ffexciton tl; 3"4, t3)<--Cffexciton(12, tl; t3)[Vr(4"4)] (24)

Upon taking the lattice Weyl transform and takingthe gradient expansion the result is

(25)

The last equation is the appropriate Wignerdistribution transport equation of a two-particledistribution corresponding to the Schr6dinger-Wannier equation of Eq. (23)

8. TRANSPORT EQUATIONSFOR PHONONS

The "reduced density matrix" for phonons isdefined by

) lj )) (26)

The lattice Weyl transform can be cast in the form

-iSj(k, k’; p", lc; ca, tc)Aij(k, k’; p", lc; ca, tc) n(p", lc; ca, tc)

(27)

Similarly, we can write the phonon "hole"distribution as

-iS (k, k’; p", lc; ca, to)Aij(k, k’; p", lc; ca, tc) [1 + n(p", lc; ca, tc)]

(28)

8.1. Phonon Transport Equations

We obtain the following expressions

0 0Ot2 Ot t2

+i{Im l-lr, x>’<} -i{ II>’<, ImS}--[Rel-[r_d(2), S>,< + [I-I>’<,ReS r]

H>’<,s<,> H<,>,s

02Ot2

(29)

Ot,2Sr’a d (2) Sr.a + Hr’a,sr,a

(30)

where we have used the definition for the forceconstants

d2)(cc’) -J (31)V/mkmk

and the following identities

S> S< l-I> l-I<ilmSr

2ilmII (32)

2

What then take the lattice Weyl transform of Eqs.(29) and (30) to obtain the transport equations in

(, lc, ca, t)- (/, , E, t) phase-space.


8.2. The Phonon Boltzmann Equation

More revealing equations can be seen by neglect-ing off-diagonal or "inter branch" terms, andexpanding the equations in terms of the gradients.We defined a renormalized kinematic frequencyby f (p, q, E, t) ,2 Rei-i (p, q, E, t). TheBoltzmann equation for the distribution functionof phonons, n (p, q, E, t), from vibration branchA readily follows by neglecting leading quantumcorrections. We obtain the familiar interpretation[22] relating (i/2){hII(p,q,E,t)/E} as the scat-tering-out rate and (i/2){hII (p, q, E, t)/E} as thescattering-in rate. By taking the renormalizedfrequency to be given by the solution of theequality f (p, q, a; , with this solutiondenoted by a;(p, t), we finally obtain

0O---t n (p, q, 9, t) fl- phCV (19, t) Vqn (p, q, co, t)

I-I (p, q, a;, t) n,q,,t)

+ + q’ t)

(33)

mann equation is obtained by means of gradientexpansion. We have

2

--tDo (p, q, E, t) +

_Re i_Iro (p, q, E, t)

VqD’> (p, q, E, t)

i(I-I>o’<(p,q,E,t) <,> )=qz- hoRei_iro(p,q,E,t) Do (p,q,E,t)

i{ I-i<o’>(p,q,E,t) >,< }where we have left out terms involving diffusionin momentum and energy space. We would liketo point out the notable use of the renormali-zed plasmon group velocity given by

(p- VpRel-Iro(p,q,E,t)) /oReI-[o(p,q,E,t). The

use of the renormalized diffusion velocity arisesfrom the same reason as that given in thederivation of the Boltzmann equation for pho-nons. The right-hand side are the familiar collisionterms. The omitted terms describe the kinematicsof the dynamical motion of plasmons.

The renormalized group velocity [26] emerges sincephonons do not diffuse freely but interact with theenvironment as well as collide with each other.Similar situation arises in deriving the plasmon andphoton Boltzmann equations. We expect theleading term of Re I-I (P, q, E, t) to be independentof time. An example of the RHS of Eq. (33) forphonon-phonon interaction can be found in thework of the author [27] and in the study of phononhydrodynamics and second sound [28].

9. TRANSPORT EQUATIONSFOR PLASMONS

10. TRANSPORT EQUATIONSFOR PHOTONS

We will also neglect the off-diagonal terms inpolarization indices. The result to leading order is

given by

(35)

9.1. The Plasmon Boltzmann Equation

The plasmon super-propagator is given byDo((,(’) 6(O(’))/6pext("). The plasmon Boltz-

where a2_x,x (p, q, E, t) c2p2 ReP ,,x (P, q, E, t) andwe have neglected the leading quantum correctionson the right-hand side. We note that p is relatedto the transverse dielectric tensor [29]. We may

272 F.A. BUOT

consider fa(p, q, E, t) > 0 to constitute the valid-ity of the Boltzmann equation for photons withdiffusive term. The use of renormalized groupvelocity corresponds to the use of refractive indexwell-known in optics.A self-consistent Boltzmann transport equ-

ation follows by setting, fZ2(p,q,03, t)- c2p-RePr(p, q, 03, t) 032 > 0. Upon substituting the

solution for 03, which we will denote by 03 (p, ),we obtain

0Ot Fa< (p’ q’ 03’ t) + Vph03a (p, t) VqFA< (p, q, 03, t)

2i { P (P’ q’ 03’ t) }F(p’ q’ E’

{ Pa,q,a,t) }Fa,q,E,t) (36)+

11. CONCLUDING REMARKS

A major challenge in analyzing the .ultra-fastdynamics of semiconductor gain material forinvestigating microcavity lasers is the fact thatthe electron-hole system properties are stronglyaffected by many-particle Coulomb interactions,and strong coupling to the light and crystal lattice-phonon fields [24]. In device physics of highlyexcited semiconductor systems, involving allranges of e-h pair densities, the exciton, e-hCooper pairs, and e-h plasma energetics and theiraccompanying distributions greatly affect thepolaritons, phonons, biexcitons, and higher-order’pairing’ (excitonic molecules) distributions, theirwavefunctions and energies. For the bosons of realfields, the diffusion velocity is not equal to thegroup velocity of bare excitations but is definedonly by its renormalized value. Quanta of theseclassical fields interact with the environment or

collide with each Other as it diffuse in space. Thecondensate and normal excitation energetics andtheir accompanying distribution also influence thegap function and energy gap. The full transportequations will be published elsewhere.

Acknowledgement

The author is grateful to Dr. A. K. Rajagopal forhelpful discussions. This work is supported in partby the Office of Naval Research.

References

[1] Schmutz, M. (1978). "Real-time Green’s functions inmany-body problems", Z. Physik, B30, 97-106; Buot, F.A. and Rajagopal, A. K. (1994). "Quantum transportusing Liouvillian quantum-field dynamics and functionalapproach to self-consistent many-body and scatteringeffects", in Proc. Third Int. Workshop on Comp. Electro-nics, Corvallis: Oregon State University, pp. 183-186.

[2] Arimatsu, T. and Umezawa, H. (1987). "Generalstructure of non-equilibrium thermo field dynamics",Prog. Theor. Phys., 77, 53--66; Suzuki, M. (1985)."Thermo field dynamics in equilibrium and non-equili-brium quantum systems", J. Phys. Soc. Jpn., 54, 4483-4485.

[3] Buot, F. A. (1974). "Method for calculating Tr gon insolid state theory", Phys. Rev., B10, 3700-3705; "Weyltransform and the magnetic susceptibility of relativisticDirac electron gas", Phys. Rev., A8, 1570-1581, 1973 &Phys. Rev, A9, 211, 1974; "Formalism of distribution-function method in impurity screening", Phys. Rev, B14,977-989, 1976; "Magnetic susceptibility of interactingfree and Bloch electrons", Phys. Rev., B14, 3310-3328,1976; "Real-space tight-binding and discrete phase-spacemany-body quantum transport", Superlattices and Mi-crostructures 11, 103-111, 1992. See also P. Kasperko-vitz,"Wigner-Weyl formalisms for toroidal geometries",Ann. Phys., 230, 21, 1994 (Note that for crystals withinversion symmetry, there is an odd number of latticepoints obeying the Born-von Karman periodic boundarycondition).

[4] Buot, F. A. and Jensen, K. L. (1990). "Lattice Weyl-Wigner formulation of exact many-body quantum trans-port theory and applications to novel quantum-baseddevices", Phys. Rev., B42, 9429-9457; Buot, F. A."Exact integral operator form of the Wigner distributionfunction equation in many-body quantum transporttheory", J. Stat. Phys., 61, 1223-1256, 1990.

[5] Schwinger, J. (1961). "Brownian motion of a quantumoscillator", J. Math Phys., 2, 407-432.

[6] Kadanoff, L. P. and Baym, G. (1962). Quantum statisticalMechanics. New York: Benjamin.

[7] Keldysh, L. V. (1964). "Diagram technique for none-quilibrium processes", Zh. Eksp. Theor. Phys., 47, 1515,[1965, Sov. Phys.-JETP 20, 1018]. See also V. Korenman,"Nonequilibrium quantum statistics: application tolasers", Annals Phys., 39, 72-126, 1966; P. Danielewicz,"Quantum theory of nonequlibrium processes, I", AnnalsPhys., 152, 239- 304, 1995.

[8] Rajagopal, A. K. and Buot, F. A. (1965). "A none-quilibrium time-dependent functional theory based onLiouvillian quantum field dynamics", Int. J. QuantumChem., 56, 389-397.

[9] Rajagopal, A. K. and Buot, F. A. (1995). "None-quilibrium time-dependent functional theory for coupled


interacting fields", Phys. Rev., BS1, 1883-1897; "Time-dependent functional theory of superconductors", Phys.Rev., B52, 6769-6774, 1995.

[10] Rajagopal, A. E. and Buot, F. A. (1996). "Generalizedfunctional theory of interacting coupled Liouvillianquantum fields of condensed matter", Topics in CurrentChemistry 181 Density functional theory II. New York:Springer Verlag.

[11] Buot, F. A. (1993). "Mesoscopic physics and nanoelec-tronics: nanoscience and nanotechnology", Phys. Re-ports, 234, 73-174.

[12] Jensen, K. L. and Buot, F. A. (1991). "Numericalsimulation of intrinsic bistability and high-frequencycurrent oscillations in resonant tunneling structures",Phys. Rev. Lett., 66, 1078-1081; Biegel B. A. andPlummer, J. D. "Comparison of self-consistency itera-tion options for the Wigner function method ofquantum device simulation", Phys. Rev. B, 54, 8070-8082, 1996.

[13] Buot, F. A. and Jensen, K. L. (1991). "Intrinsic high-frequency oscillations and equivalent circuit model in thenegative differential resistance region of resonant tunnel-ing devices", COMPEL, 10, 241.

[14] Buot, F. A. and Rajagopal, A. K. (1993). "High-frequency behavior of quantum-based devices: equivalentcircuit, nonperturbative response, and phase-space ana-lyses", Phys. Rev., B48, 17217-17232. See also, "Theoryof novel nonlinear quantum transport effects in resonanttunneling structures", Materials Sc. and Engng., B35,303-317, 1995.

[15] Nambu, Y. "Quasi-particles and gauge invariance in thetheory of superconductivity", Phys. Rev., 117, 648-663,1960 see also P.W. Anderson, "Random-phase approx-imation in the theory of superconductivity", Phys. Rev.,112,1900, 1958 A. K. Rajagopal, "Spin waves in aninteracting electron gas", Phys. Rev., 142, 152, 1966.

[16] De Dominicis, C. and Martin, P. C. (1964). "Stationaryentropy principle and renormalization in normal andsuperfluid systems. I. Algebraic formulation", J. Math.Phys., 5, 14- 30.

[17a] Aronov, A. G., Gal’perin, Yu M., Gurevich, V. L. andKozub, V. I. in Langenberg, D. N. and Lakin, A. I."Kinetic approach", Eds., Nonequilibrium Superconduc-tivity, New York: North Holland, 1986, pp. 325-376.

[17b] For a discussion on Gorkov equations, see for exampleA. L. Fetter and J. D. Walecka, Quantum Theory ofMany-Particle Systems, New York: McGraw Hill, 1971,p. 183.

[18] Kirkpatrick, T. R. and Dorfman, J. R. (1985). "Trans-port in dilute but condensed nonideal Bose gas: kineticequations", J. Low Temp. Phys., 58, 30!-331.

[19] Yokoyama, H. (1992). "Physics and device applicationsof optical microcavities", Science, 256, 66.

[20] Keldysh, L. V. (1992). "Coherent excitonic molecules",Solid St. Comm., 84, 37-43; L. V. Keldysh, "Macro-scopic coherent states of excitons in semiconductors", inBose-Einstein Condensation, eds. A. Griffin, D.W. Stoke,and S. Stringari, Cambridge: Univ. Press, 1995

[21] Schmitt-Rink, S., Chemla, D. S. and Haug, H., (1988)."Nonequilibrium theory of the optical Stark effect and

spectral hole burning in semiconductors", Phys. Rev.,B37, 941.

[22] Koinov, Z. G. and Glinskii, G. F. (1988). "A newapproach to the theory of polaritons in semiconductors atfinite temperatures: local-field effects and crystal opticsapproximation", J. Phys. A." Math. Gen., 21, 3431-3450;see also Z. G. Koinov (1990). "Self-consistent approachto the theory of Wannier excitons in polar semiconduc-tors", J. Phys. Condens. Matter., 2, 6507-6518.

[23] Knox, R. (1963). "Excitons", Solid State Phys. Suppl., 5,2.

[24] Jahnke, F. and Koch, S. W. (1995) "Many-body theoryfor semiconductor microcavity lasers", Phys. Rev., A52,1712-1727.

[25] DuBois, D. F. (1967). "Nonequilibrium statistical me-chanics of plasmon and radiation", in Lectures inTheoretical Physics, New York: Gordon & Breach, pp.469-620.

[26] Horie, C. and Krumhansl, J. A. (1964). "Boltzmannequation in a phonon system", Phys. Rev., A136, 1397-1407.

[27] Buot, F. A. (1972). "On the relaxation rate spectrum ofphonons", J. Phys. C: Solid State Phys., 5, 5-14.

[28] Benin, D. (1975). "Phonon viscosity and wide-anglephonon scattering in superfluid helium", Phys. Rev., Bll,145-149; H. Beck, "On the temperature behavior ofsecond sound and Poiseuille flow", Z. Physik, B 20, 313-322, 1975 B. Perrin, "Sound propagation and vibra-tional relaxation in molecular crystals", J. Chemie Phys.,82, 191-197, 1985

[29] Agronovich, V. M. and Konobeev, Yu. V. (1964)."Theory of the dielectric permittivity of crystals", Soy.

Phys.-Solid state, 5, 1858.

Authors’ Biography

Dr. Felix A. Buot has served on the research staffof the University of London, ICTP (Trieste, Italy),McGill, St. Francis Xavier, Stanford, and CornellUniversity. He is a Fellow of the WashingtonAcademy of Sciences, and President of thePhilippine-American Academy of Science andEngineering. He is a member of the EditorialBoard, Transport Theory and Statistical Physics.He was a UNDP Consultant, 1993 and 1996. Hisinterests include device performance & reliability,optoelectronics, nonequilibrium quantum theory,multiband dynamics, and physics of computation.He is a Research Physicist at the U.S. NavalResearch Laboratory.

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