Fortschritte der Physik / Progress of Physics: Band 27, Heft 8 1979 [Reprint 2021 ed.] 9783112522745, 9783112522738

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN" IM AUFTRAGE D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

DEH DEUTSCH KN DKMOK.RATISC.11E N R E P U B L I K . VON F. RASCHI.I HN. V I.OSCHE. H. HITSCHE U M ) R. ROMPE

H E F T 8 • 1979 • BAJN D 27

A K A D E M I E

- V E R L A G EVP 1 0 , - M 31728

.

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an das Zeitungsvertriebsamt, an eine Buchhandlung oder an den AKADEMIE-VERLAG, D D R - 108 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung f ü r fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST U N D WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferungsstelle K U N S T UND WISSEN, Erich Bieber GmbH, CH - 8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ansland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, Postfach 160, oder an den A K A D E M I E - V E R L A G , D D R - 108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. D r . Artur Lasche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; F e m r u f : 2236221 und 2236229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Corki", D D R • 74 Altenburg, Carl-von-Ossietzky-Str&Oe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik 11 erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t I S , - M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses H e f t e s : 1027/27/8. (c) 1979 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. A N (EDV) 57 618

Fortschritte der Physik 27, 355-402 (1979)

Generating Functional» in Nonequilibrium Statistical Mechanics V . P . KALASHNIKOV a n d M . I . AUSLENDER

Institute for Metal Physics,

TJrals Scientific Sverdlovsk,

Centre Academy of Sciences of USSR

USSR,

Abstract The main ideas and methods of calculations within the framework of the generating functional technique are considered in a systematical way. The nonequilibrium generating functionals are defined as functional mappings of the nonequilibrium statistical operator and so appear to be dependent on a certain set of macroscopic variables describing the nonequilibrium state of the system. The boundary conditions and the differential equation of motion for the generating functionals are considered which result in an explicit expression for the nonequilibrium generating functionals in terms of the so-called coarse-grained generating functional being the functional mapping of the quasiequilibrium statistical operator. Various types of integral equations are derived for the generating functionals which are convenient to develop the perturbation theories with respect to either small interaction or small density of particles. The master equation for the coarse-grained generating functionals is obtained and its connection with the generalized kinetic equations for a set of macrovariables is shown. The derivation of the generalized kinetic equations for some physical systems (classical and quantum systems of interacting particles, the Kondo system) is treated in detail, with due regard for the polarization effects as well as the energy and momentum exchange between the colliding particles and the surrounding media. Contents Introduction

356

1. Formal Definition of Statistical Generating Functional for Arbitrary Nonequilibrium System. Examples 357 1.1. General definitions 357 1.2. Bogoljubov's classical and quantum generating functionals 358 1.3. Generating functionals for quantum statistics 362 2. Generating Functionals in Theory of Irreversible Processes 365 2.1. Boundary conditions and integral equations for generating functionals 365 2.2. Closed equations for coarse-grained generating functionals and generalized kinetic equations 368 2.3. Some generalizations 371 3. Derivation of Kinetic Equations for Systems of Particles with Pair Interactions 3.1. Formulation of the problem 3.2. Classical kinetic equations for the case of Coulomb interaction 3.3. Small density approximation for a classical system 3.4. Derivation of the quantum kinetic equations 29

Zeitschrift „Fortschritte der Physik", Heft 8

372 372 375 378 380

356

Y . P . KALASHNIKOV a n d M . I . ATJSLENDEB

4. Generating Junctionals in Kinetics of Mixed Fermion and Boson Systems 4.1. Generalized kinetic equations 4.2. Coupled equations for the correlation kernels of Markovian approximation

383 383 385

5. Application of Generating Functional Method of Kinetic Theory of Dilute Magnetic Alloys 388 5.1. GF for Kondo system 388 5.2. Generalized kinetic equations for Kondo system 389 5.3. Polarization approximation for kinetic equations 393 6. Conclusions

400

7. References

401

Introduction The theoretical description of macroscopic systems is based usually on the concept of statistical distribution function or density matrix which determine the mean values of dynamic variables. There exists an alternative approach to the problem which consists in utilizing linear mapping of the distribution function (density matrix) — the so called generating functional (GF). Statistical distributions for complexes of particles, reduced density matrices and averages m a y be obtained from the G F by variational differentiation with respect to its functional arguments. Generally the G F in statistical mechanics plays the same role as the generating function for the moments of a stochastic variable in probability theory. Within the framework of nonequilibrium statistical mechanics, the G F was considered first by B O G O L J T T B O V [ 1 ] for the case of a classical system of particles interacting by pair forces. Further development and generalization of this approach by a number of authors enabled formulation of the whole statistical mechanics of classical and quantum nonequilibrium systems in terms of generating functionals. The explicit expressions and equations of motion were obtained for the GF's which correspond to the description of a nonequilibrium system in terms of a set of macroscopic variables [2—15]. This development of the G F technique is closely connected with the nonequilibrium statistical operator (NSO) method in the form proposed b y D. N. Z T J B A B E V and one of the authors [16—18]. I t is worth mentioning t h a t the concept of the G F implies not just a new language in the theory of irreversible phenomena, b u t leads to essential technical and methodical advantages in solving concrete problems. These advantages are specified hereinafter. 1. I t is well known t h a t the theory of nonequilibrium processes involves two noncommutative limit, viz. thermodynamic limit and the limit s —> ± 0 which selects retarded or advanced solutions of equations of motion. I t is essential that the latter limit should be taken only after calculating the averages and after proceeding to the thermodynamic limit. By definition, the G F appears to be an average value of a certain dynamic variable where the thermodynamic limit has already been taken; all other manipulations over GF, involving, in particular, the limit e -»- ± 0 , could be made only later. This ensures a correct order of sequence of these limits [17, 19]. 2. Evaluation of the G F to a certain approximation corresponds to calculating, in the same approximation, the average values of the whole set of dynamic variables for the system considered. This compactness is especially convenient if the problem involves a large number of correlation functions simultaneously. 3. I n many cases an explicit form of the GF describing the nonequilibrium state of the system in terms of a limited set of macrovariables is much simpler than the corresponding NSO.

Generating Functionals in Nonequilibrium Statistical Mechanics

357

4. For quantum systems, the ordinary Liouville operators can be represented as matrices with four groups of indices (tetradics) which are too complicated to deal with. In the GF picture we obtain, instead, fairly simple functional differential operators of low order in terms of which the equations of motion simplify considerably. 5. The concept of GF leads to some new and convenient ways of evaluating the collision integrals and correlation functions which will be demonstrated below with reference to particular examples. A mathematical technique, similar to that of the GF method, has been developed in recent years in the theory of coherent states and in critical dynamics \20~\. In the present paper we shall consider in a systematic way the main ideas and methods of calculations in the generating functional technique. Since this technique appears to be unusual to many of the physicists concerned with statistical mechanics and solid state physics, it seems appropriate to show some of the calculations in detail. We hope the examples considered will suffice to fully understand the GF method. 1. Formal Definition of Statistical Generating Functional for Arbitrary Nonequilibrium System. Examples

1.1. General definitions Let us consider a nonequilibrium system with a Hamiltonian H. Statistical description of the system is generally given by the total distribution function (classical statistics) or statistical operator (quantum statistics) g(t) which satisfy the Liouville equation ( ¿

+ iz;)e(i) = o

(1.1.1)

where L is the Liouville operator defined as (1.1.1a)

LP = i{H,P} for the classical case, {,} being the Poisson brackets, or (h = 1)

(1.1.1b)

LP = [H,P]

for the quantum case, where the brackets [,] denote the commutator. The average value of the dynamic variable P is evaluated as follows (Py = Lim Sp cAt) P

(1.1.2)

where the symbol Lim denotes the thermodynamic limit, and the operation Sp stands for integration over phase space in the classical case or the trace in the Hilbert space of states in the quantum case. Proceeding from Q(t) one can obtain in a well-known way a sequence of reduced distributions [2] or reduced density matrices for complexes of particles [22], There exists another general method of calculating these quantities, which is based on the concept of a generating functional defined hereinafter. Let U — {u} be an algebra of functions of one-particle dynamic variables. The linear mapping of the statistical operator (distribution function) Q(t)^>

29*

^{t-,u),

ue U

(1.1.3)

358

V. P. Kalashnikov and M. I. Atjslbndee

onto the space of functionals over U will be called a generating functional (GF) if the coefficients of the expansion of ; u) in Taylor series with respect to its functional argument u coincide with the reduced density matrices (distribution functions), and (J

M

))

0

= g p e (t) = 1.

(1.1.3a)

The subscript "0" implies that in this expression the functional argument should be set equal to zero. By analogy with the generating functions, any G F can be written in the following general form u) = (A(u))< = Lim Sp g(t) A(u) (1.1.3b) where A(u) is some dynamic variable depending on u so that (A(w))0 = 1. Two corollaries follow from the linearity of the functional mapping (1.1.3): a) the evaluation of the average value (1.1.2) must be equivalent to a certain linear operation on the G F (FY = t; »))« (1.1.4) where the structure of the linear functional operator $ is determined unambiguously by the dynamic variable P ; (b) there exists a functional mapping of the Liouville equation (1.1.1)

+ if^J

u) = 0

(1.1.5)

where, in accordance with eq. (1.1.3b) 2 3?{t\u)

=

(1.1.5a)

-{LA(u\)*.

For all physical systems the quantity Ï , hereinafter referred to as the functional Liouville operator, appears to be a functional differential operator; its structure is determined by the Hamiltonian of the system. The relations (1.1.3)—(1.1.5) will be considered further in more detail for some model systems. 1.2. Bogoljubov's classical and quantum generating functionals Let us consider a classical system of N particles without internal degrees of freedom. Then the state of any particle, say, of the sort a, is determined by its position and momentum vectors ( r a , p a ) = xa. The reduced distribution functions /^...^(t, x l t . x s ) can be defined as average values of the dynamic variables P0,

x,)

=

E

n à ( x

{lSifcSJV^} k = 1

k

- x

o A

)

(1.2.1)

(oii,)+ — #(o»tt)

where Na is the number of particles of the type a, £ Na = N. a

I t is evident that /8j..,0).This agrees fully with the general definition of the GF given in the foregoing. Now define the correlation functions gax...a,{t\ xlt •••, xs) by means of the recursion relations /o,(i; — gai(t; X!) (1.2.4) s r fay-a.it; Xi, ..., Xs) =27 E n^alq{-alipjt-, ^„.J where {r} denotes the various permutations of the type

Substituting equations (1.2.4) into the functional series (1.2.3), we obtain the following exponential representation of Bogoljubov's GF 3 u ) oo

= exp

«)]

r

(1.2.5)

a

;u) = E («!)_1 E / àsxgay..at{t-, xu ...,x,) 77u 0i (Xj). 8=1 ay-a,J )=1 Thé functional &(t;u) is similar to the generating function of semiinvariants (cumulants) in probability theory. If there are no correlations in the system, i.e. gai -a, = 0 for s ^ 2, equation (1.2.5) yields / dxjai(t; xt) m ^ J .

¿F(t; u) = exp p

(1.2.6)

Let us give a representation for the classical functional Liouville operator in the case of interaction of particles by pairs, when the Hamiltonian of the system is Na

H =E where

1

Na

Nb

E Ha(xai) + — E E E QaMra; - rbi |) a ¿= 1 ^ ab ¡ = 1 j = 1 Ha(xa) = pa2/2ma + Va(xa)

(at) H= (bj)

(1.2.7) (1.2.7a)

V. P . KALASHNIKOV and M. I . ATTSLENDER

360

is the Hamiltonian of one sort a particle, and & + jf(i) (1.2.12) 2

) In [2—4] a one-component system was considered.

Generating Functionals in Nonequilibrium Statistical Mechanics *«»

-5? (1) = 4 - / * ab J

a J

fd

X l

¿»2 «¿^'[«nfe.

+ ua(xu *,')

dx,' ua(xu

«i')

*')]

x,') La(xu

«,') t oua(xj,

à{x2 — x2') + ub{x2,

361

^ a;, )

(1.2.12a)

a; 2 ') ¿(a^ — a ; / )

*2l 1 ' , * / ) ^

^

(1.2.12b)

where, by definition, La(xu Lab{xlt

x 2;

x/) =

-

ar2') = ^ ( l ^ -

r2|)

(1.2.13) -

0ai)(|r/-

r2'\)

and integration over x denotes conventionally integration over r and summation over a. In the paper by A l e x a n d r o v , K u k h a r e n k o and Niukkanen [7], the equation of motion for Bogoljubov's quantum G F was studied to derive a hierarchy of equations of motion of reduced density matrices. The general properties of Bogoljubov's quantum G F and the equation of its evolution in time were studied in detail by Ziesche [5, 6]. A grand ensemble representation of Bogoljubov's quantum G F was obtained in these papers 3 ). Ziesche's representation follows directly from relation [21] fai...a,(f,

X s I x")

=

TO*/)...

Wt(x/)

• •. vai(x,)>'

¥a.(xs)

(1.2.14)

where and ipa(x) are relevant creation and destruction operators complying with well-known commutation relations i f a i x ) , ipaix')],„ [y>a{x), fa +{x')]r,a

=

= d(x — x');

[y>a +(x), %+(«')],„

= 0

[ f a ( x ) , fh(x')]

=

0,

a #= b

where [A, E\n = AB — rjBA and rja is equal either to + 1 for bosons or to —1 for fermions. Taking (1.2.11) and (1.2.14) into account, one obtains the desired representation ; u) = (Nw

exp J J dx dx' £

(1.2.15)

y>+(x) ua(x, x') y>a(x')^y

with Nv being a normal ordering symbol. Such a representation of Bogoljubov's quantum G F gives permits a convenient use of a second quantization scheme. Bogoljubov's quantum G F rfl&y be introduced for any, not necessarily continuous system when the number of particles of every sort is conserved. Indeed, let {|ra)} be a complete set of one-particle states of the a-th sort. Then one ought to put in eq. (1.1.3b): (t) can be seen to be a projection operator 0>(t) 8?(t') = 3P(t). Equation (2.2.1) becomes closed with respect to u) if the explicit expression (2.1.6) for the G F is substituted into its right-hand side. This result is the most general form of the functional master equation. Now let the representation (2.1.9) have the property &>(t) XoA&Q; u) = [1 - 0>{t)] Jf«F(