Annalen der Physik. 7. Folge, Band 33, Heft 5 , 1976, S. 350-358 J.A. Barth, Leipzig Bound State Effects in Quantum Transport Theory') By W. EBELING Sektion Physik der Wilhelm-Pieck-Universitat Rostock A b s t r a c t . A quantum kinetical equation for theone-particle density operator for inhomogeneous multi component non-ideal gases is derived taking into account retardation effects and external fields. It is shown that the interactions give rise t o additional contributions to the quantities in the transport equation. I n some terms (e.g., internal energy, pressure, effective external force) the interaction effects are strong in the presence of bound states and yield contributions of the order exp (--E,,/kT) where El, is the ground state energy of bound pairs of particles. Oeher terms (e.g. friction forces) are rather insensitive with respect to the presence of bound states. An interpretation of these effects in terms of the mass action law is given using the principle of equivalence between bound states and particles. Effekte der Bindungszustande in der Quantentransporttheorie I n ha1 t s u b e r s i c h t. Unter Beriicksichtigung von Retardierungseffekten und PuBeren Feldern wird fur iuhomogene mehrkomponentige, nichtideale Gase eine quantenkinetische Gleichung fur den Einteilchen-Dichteoperator abgeleitet. Es wird gezeigt, daB die Wechselwirkungen zusatzliche Beitriige zu den GroBen in den Transportgleichungen erzeugen. In einigen Termen (z.B. innere Energie, Druck und effektive LuBere Kraft) sind die Wechselwirkungseffekte bei Anwesenheit gebundener Zustande sehr stark und Iiefern Beitrage der Ordnung exp (-E,,/kT), wobei El,den Grundzustand eines gebundenen Teilchenpaares darstellt. Andere Terme (z. B. die Reibungskriifte) sind gegeniiber dem Auftreten gebundener Zustfnde ziemlich unempfindlich. Unter Benutzung des Prinzips einer Aquivalenz zwischen gebundenen Zustanden und Teilchen wird eine Interpretation obiger Effekte mit Hilfe des Massenwirkungsgeset.zes gegeben. 1. Introduction The basic difficulty in applying statistical mechanics to many-body systems with bound states is the strength of the interaction between the particles. The main problem in the equilibrium theory o f these systems is the exponential growth of the coefficients in the density expansions of all thermodynamic properties, e.g., the second virial coefficient a t small temperatures behaves like [ 11 where El,is the ground state energy of a two particle system. The main idea in the statistical thermodynamics of equilibrium systems with bound states is a kind of refonnulation in such a way that bound states are treated as new composite particles [2, 3, 41. Formally this corresponds t o a regrouping of the expansions which corresponds to a summation of exponentially growing terms. Then the principle that bound states must l) 1975. This paper was presented a t the International Conference on Statistical Physics, Budapest Bound State Effects in Quantum Transport Theory 351 be treated on the same footing as particles immediately yields a representation which contains mass action laws for the formation of composite particles. Due to the existence of terms like exp (--E,,IkT) the treatment of bound states is a problem of the higher order terms in perturbation theory. Only after introducing the bound states as new physical entities (composite particles) the perturbation theory may be applied again. Therefore in the thermodynamic theory general ideas are available to treat systems with bound states. Up t o now the question is open whether these ideas may be applied to the statistical transport theory. I n most of the earlier papers on transport problems bound states are excluded from explicite consideration and there are only few investigations devoted to the effect of bound states on transport properties [5--81. The difficulties of the bound state problein are connected with the following circumstances : 1. The bound state problem is a proper quantum problem, i.e., the classical theory is not applicable. 2. Because the bound state contributions increase exponentially with the interaction, no perturbation theory can be applied. The present investigation is restricted as a n earlier one [5] to the region where hound states appear in the system in a very small concentration compared to the concentration of free particles. Our aim is the study of terms which are increasing in the limit of small inean kinetic energy (small temperature). 2. Quantum Kinetic Equation The kinetic equation will be formulated using the density operator method, in contrast to a n earlier paper of KLIMONTOVICH and EBELING [5] where the Wigner function iiiethod was used. Consider a quantum system consisting of N , particles of species a. h’b particles of species 6 , etc. Assume that the external forces acting on particles of species c(, b, . . are given by K,, K,, ... The von Neuniann equation for the density operator Q is given by . The last term of the r.h.s. of eq. (2) describes a source of small stochastic perturbation, e.g., a radiation field. The operator Di is a dissipator in the sense of INGARDEN and KOSSAKOWSKI [9]. The most simple “ansatz” for the dissipator is given by I 1 (4) I V (p) where Ezq is the one-particle density operator in equilibrium. If the particles 1 and ‘1 belong to the species a and b the one-particle and two-particle density operators are defined after BOGOLIGBOV [lo] by F u ( l ) = v T r e ( l , 2 , ...)N ) , (5) Di == - ~ i1 - -3’:q(i) Tr , .. (2. N ) Fab(12) = v2( 3T. . .rN@ ( l2,, 3, ..., 3). ) Performing in eq. ( 2 ) the trace over the particles 2 , 3, (6 1 ... we obtain W.EBELING 352 ... An equation for Fa, is found by performing the trace over the particles 3, 4 , ,N in eq. ( 2 ) . Neglecting here terms of order ( N I B ) and introducing a correlation operator gab(12) we obtain = a - at gab = Labgab f (9) Fa(1) Fb(2) 1 LBab) (10) ) Introducing eq. ( 1 2 ) into eq. (8) we get I, = D,Fa + .Tio' + IL", Bab, F a F b Now we perform the limit to -+ - 00 will be destroyed lim Ii')(t, to) = 0. to-+- " 1 t-to afi 0 f7 J dt&b(t) 1 [ V a b , F a ( t - Z) F b ( t - T ) ] (161 and assume that the effect of initial correlations (17) Furtheron we assume that the integral in Iiv)reaches a plateau value. These assumptions correspond to BOGOLIUBOV'S condition of total destruction of initial correlations [ l o ] . We note that the resulting collision operator contains retardation in time because in eq. ( 1 6 ) appear arguments (t - t ) .Hxpanding with respect to T and neglecting second order terins we find This collision operator corresponds to the Wigner representation of the collision integral for nonideal gases derived in a n earlier paper [ 5 ] . Another form is obtained using the equation of motion Bound State Effects in Quantum Transport Theory Eliminating 353 [v&F,Fi] by iiieans of eq. (19) we obtain the suin representation + 1,l + + I,, + I,, + 1, = Iuo I,, = D,F,, Ia5 = Ia2 z-Tr [ -7 zfi b fa5, W vab, (2) $ dr &(r) 0 (0, f n,)F a F b ] * I n this representation IaOdescribes the usual external dissipation and I,, its niodification by interaction effects. The term I,, is an operator form of the quantum Boltzniann collision integral, Ia2 and I,, describe effects connected with the inhomogeneity in time effects of the external force. In general the expressions and space respectively and la3 given in (21) may be simplified by neglection of several cross effects, e.g., the propagator Aaomay be approxiniated in the following way (f - arbitrary operator) f(12) uab(z) = uab(T) [ exp - f(l2) i (H!b u&(T) + (22) v a b ) 7)* Pnrtheron in general the coordinates ri may be treated as c-numbers with the exception of Ia4where spatial derivatives of first order are to be taken into account. For the probleni of bound state effects especially the contributions Ia2, I,,. and Iu4 are of importance as will be shown in the next sections. 3. Quantum Transport Equations Transport equations represent a closed causal description of transport phenomena by partial differential equations for local mean values. We define the local mean value of an operator A,(r,, p,) by Ill] (23) a -?z,(r. t ) at % , a a +V J 3 r , t) Ja(r,t ) = 0, +v - (12, \?,(' (24) <I*,)) +v - uef'(r,t ) + v .jef'(r,t ) = &(r, t ) . %t I>,(?-, t ) = kEff(r,t ) + f;"(r, t ) , (25) (26) 354 W. EBELING The transport quantities are defined in the following way + T' Tr (6 ( r h'a (1) r1) r dur)7 (2) v .-,;f'= v . (T $ ; {d(r - r l ) (A) 2ma Fa}) (2(j - I,, 1 , q ~ f f = ~ J a . K a + C - Z ' r{ 6 ( r - r l ) (2); ) v 2ma --z-+~r{d(r-r,) ft a (33) (1) a (1) (34) \Ye see that all effective transport quantities such as local energy density (28), particle flow density (B), external force density (30), friction force density (31), local pressure (32), energy flow density (33), energy source density (34) consist of ,an ideal part and a Ia3, I,,, and Ia5. nonideal contribution which is given bj- the collision operators la2, Especially these ternis may be affected by thc appearance of bound states. I n the usual transport theory the contributions Ia2,I,,, Ia4,and I,, are neglected. Obviously the transport theory af nonideal systems is nmch iiioi'e complicated as the usual transport theory. We note that in the case of qnantnni systeiiis the nonidealitp influences also the one-particle deilsity operator [5]. As a rule for the calculation of the nonideal terms we note that in all cases r1 can be treated as a c-number with the exception of I,, where linear spatial derivatives must be taken into account. Here we are especially interested in the contributions of the terms Irl, Ia3,Iu4,We note that thenon-ideal parts of the collision integral Ia2,Ia3,I,, respectively yield nonideal contributions to the tinie derivatives, the force effects, and the spatial derivatives of the transport quantities respectively. These terms include also the bound state effects which are of interest here. 4. Calculation of Interaction Effects The explicite calciilation of the nonideal contributions is a coinplicated quantuni statistical problein which can be solved up to now only in part. Bound State Effects in Quantum Transport Theory 365 Consider as a n example the potential energy; after some manipulations we find where - Fkk) = uub(m) Fu(19 t ) F b ( 2 , t ) u&(Oo) (36) We shall use the assumption of a Maxwellian distribution which is for quantum systems correct only a t sinall densities [ 5 ] A, = h(2nmukT)-112. Taking into account the following properties of the evolution operator [ll] f ( A )u&(r) =f ( U a b ( r ) A u&(T)) Uub(Oo)H , " b u & ( a )= Uab(Oo)Habu&(Oo) we find the well known formula uab(r) (38) > = Hab (39) Introducing this into eq. (35) we get the usual representation for the potential energy of nonideal gases in the approximation of the second virial coefficient. I n order to calculate the full local energy we shall use the following relations Finally we get the expression For the pressure we find in the given approximation 1 PEff= nukT 1 - -2(1:A2723 T r (exp (-H,b/kT) -exp (-Hj$/kT)) 2 b (12) I n the limit of low temperature the trace can be restricted to the contribution of the ground state. As an example we consider a system with two species e and i with bound states of pairs (e, i ) where the ground state of the energy of relative motion is Elo; for illustration we can imagine an electron-ion plasma without long-range forces. Then we 23* W. EBELING 356 get from eqb. (42)-(43) ,eff - 3 kT { (ne + nil + Elonemi exp (--Eio/LT) - n,niAb exp (--E,,/kT) + A,i = h [2nm,i kT]-1'2, Pzff=nekT{l - niA:iexp (--E,,/kT) (44) a}, + ...>. (45) For the effective force density we find in the given approximation I n the low temperature limit we can restrict ourselves to the ground state contribution which is for the model disussed above given by 6. Discussion of the Bound State Contributions The analysis of the formulae (44)) (45), and (47) for a inodel system with two species e and i forming bound pairs (e, i) shows that in the local energy, in the local pressure, and in the effective external force the hound state effects are strong and yield contributions of order exp (--E,/kT) where El, is the ground state energy of the relative niotion of pairs. An interpretation of these terms can be given in terms of the mass action law. I n a system with bound states we have to differentiate between free and bound particles, i.e., n, = nX + no*, ni = nf + no*, (48) where nz, n f are the densities of free electrons or ions respectively and n,* is the density of bound pairs (atoms). I n the ground state approximation the relation of the new densities is given by the following mass action law (n,*/n,*nf) = A:i exp (--E',,/kT), (49) where the r.h.s. represents the mass action constant. A t low densities we find using eqs. (48)-(49) approximately n,* = n,ni exp (--E,,/kT) A$ +- n,* = n, - neniA:iexp (-Elo/kT) nt = ni - n,n&$ exp (-&/kT) * ) + + ---. (50) Introducing these relations into eqs. (44)) (45), (47) one derives 3 + nr + no*)+ Elon,* + Ueff = ,k,.T(n,* p e f f - n,*k~ a - (61) + . .. (52) - (63) kzff == n,*K, + * *. Bound State Effects in Quantum Transport Theory 357 Evidently the bound states behave as new composite particles. We see that the rule of equivalence of bound states and particles which was worked out for the thermodynamic equilibrium is applicable to certain transport situations too. According to this rule a strong interacting system with bound states behaves like a gas of weakly interacting free and composite particles. A reasonable transport theory of systems with bound states must take into account this fundamental physical observation, i ,e., bound states must enter the theory similar to composite particles. Formally this can be achieved by collecting terms which show an exponential divergence a t low temperatures like exp (--El0/kT). It was the aim of this paper to give an illustration of this general idea. We underline that not all transport quantities contain exponentially growing terms. An example for this type of quantities is the effective friction force densityfiff(r,t ) which does not contain terms of order exp (-Elo/kT) as far as we see. I n contrast to this, the effective external force density kzff(r,t ) contains bound state contributions as shown above (cf. eqs. (471, (53)). We note that this kind of effects is responsible for the drastic decrease of conductivity of systems with charged particles in the region where neutral bound states are formed. Similar ideas were used in earlier papers for the development of a theory of electrolytic conductance in the region of ion asociation [ 121. Obviously the problem of the conductance of partially ionized quantum plasmas is much more complicated and refines the solution of some principal quantum statistical problems some of which are discussed here. Symmetry effects not discussed here can be taken into account following a paper of KLIMONTOVICH and KRAEFT[13]. Effects of higher orders with respect to the density are discussed by OLMSTED and CURTISS[14]. I n conclusion we want to underline the main result of this paper that the bound state effects contribute terms of order exp (-E,,/kT) not only to the thermodynamic quantities but also to certain proper transport quantities, e.g., to the force density. Therefore the bound state contribution may be of importance a t lower temperatures and needs careful analysis. The author would like to thank Prof. Yu. L. KLIMONTOVICET, Moscow, Dr. A. KOSTorun, Dr. W. D. KRAEFT,Rostock, and Prof. D. KREMP,Gustrow, for helpful discussions. SAKOWSKI, References [l] D. KREMP,W. EBELING and W. D. K R ~ E F T Physica , 61, 146 (1971). S. MA and H. J. BERNSTEIN, Phys. Rev. 187, 345 (1969). [2] R. DASHEN, [3] W. EBELINQ. Physica 73, 573 (1974). [4] W. EBELING, W. D. KRAEFT and D. KREMP, Theory of Bound States and Ionization Equilibrium in Plasmas and Solids, Akademie-Verlag Berlin (in press). [5] Yu. L. KLIMONTOVICH and W. EBELING, Zh. Eksp. Teor. Fiz. (USSR) 63. 917 (1972). [GI J. T. LOWRY and R. F. SNIDER, J. Chem. Phys. 61, 2320 (1974). Kinetic Theory of theNonidea1 Gas and the Nonideal Plasma (in Russian) [7] Yu. L. KLIMONTOVICH, Moscow 1975. [8] S. W. PELETM~NSKI, Teor. Mat. Fiz. (USSR) 6, 123 (1971). E. S. YAKUB,Tepl. Vyss. Temp. (USSR) 10, 507 (1972). and A. KOSSAKOWSKI, Ann. Phys. (N. Y.) (in press). [9] R. S. INQARDEN [lo] N. N. BOGOLIUBOV, Selected Work, Vol. 2, (in Russian) Kiev 1970. [ll] D. N. SUBAREW, Statistische Thermodynamik des Nichtgleichgewichtes. Berlin 1975. 358 W. EBELING [le] H. FALKENHAGEN, W. EBELING and W. D. KRAEFT,Mass Transport Properties of Ionized Dilute Electrolytes. In: Ionic Interactions Vol. 1 (Ed. S. Petrueci) New York 1971; W. D. KRAEFT and W. EBELINO,Ann. Physik, Leipzig 18, 246 (1966); D. KREMP, W. D. KRAEFTand W. EBELING,Z. phys. Chem. 8.10, 141 (1969); W. EBELING,Z. phys. Chem. 249, 140 (1973). and W. D. KRAEFT,Tcor. Mat. Fiz. (USSR) 19, 364 (1974). [13] Yu. L. KLIMONTOVICH [14] R. D. OLMSTED and C. F. CURTISS,J. Chem. Phys. 62, 3979 (1975). Bei der Redakt,ion eingegangen am 8. Oktober 1975. Anschr. d. Verf.: Prof. Dr. W. EBELIKG,Sektion Physik d. Univ., DDR-25 Rostock, Universitatsplatz 3

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