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Bound State Effects in Quantum Transport Theory.

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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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