Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas
Nonequilibrium properties of an inhomogeneous electron gas are studied using the method of the nonequilibrium statistical operator by D.N. Zubarev. Generalized transport equations for the mean values of inhomogeneous operators of the electron number density, momentum density, and total energy dens...
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Kostrobii, P. Markovych, B. Vasylenko, A. Tokarchuk, M. Rudavskii, Yu. 2017-06-14T07:48:41Z 2017-06-14T07:48:41Z 2006 Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas / P. Kostrobii, B. Markovych, A. Vasylenko, M. Tokarchuk, Yu. Rudavskii // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 519–533. — Бібліогр.: 37 назв. — англ. 1607-324X PACS: : 05.60.Gg, 05.70.Np, 63.10.+a, 68.43, 82.20.Xr DOI:10.5488/CMP.9.3.519 https://nasplib.isofts.kiev.ua/handle/123456789/121353 Nonequilibrium properties of an inhomogeneous electron gas are studied using the method of the nonequilibrium statistical operator by D.N. Zubarev. Generalized transport equations for the mean values of inhomogeneous operators of the electron number density, momentum density, and total energy density for weakly and strongly nonequilibrium states are obtained. We derive a chain of equations for the Green’s functions, which connects commutative time-dependent Green’s functions “density-density”, “momentum-momentum”, “enthalpy-enthalpy” with reduced Green’s functions of the generalized transport coefficients and with Green’s functions for higher order memory kernels in the case of a weakly nonequilibrium spatially inhomogeneous electron gas. Дослiджуються нерiвноважнi властивостi просторово неоднорiдного електронного газу з використанням методу нерiвноважного статистичного оператора Зубарєва. Отримано узагальненi рiвняння переносу для середнiх значень густин операторiв числа електронiв, їх iмпульсiв та повної енергiї для опису як слабо, так i сильно нерiвноважних станiв системи. Отримано ланцюжок рiвнянь функцiй Грiна, який зв’язує комутаторнi часовi функцiї Грiна “густина-густина”, “iмпульс-iмпульс”, “ентальпiяентальпiя” iз приведеними функцiями Грiна узагальнених коефiцiєнтiв переносу в’язкостi, теплопровiдностi (включаючи перехреснi в’язко-тепловi коефiцiєнти переносу) та iз приведеними функцiями Грiна вищих функцiй пам’ятi для слабонерiвноважного просторово неоднорiдного електронного газу. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas Нерiвноважний статистичний оператор Зубарєва i функцiї Грiна для неоднорiдного електронного газу Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas |
| spellingShingle |
Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas Kostrobii, P. Markovych, B. Vasylenko, A. Tokarchuk, M. Rudavskii, Yu. |
| title_short |
Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas |
| title_full |
Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas |
| title_fullStr |
Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas |
| title_full_unstemmed |
Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas |
| title_sort |
nonequilibrium statistical zubarev’s operator and green’s functions for an inhomogeneous electron gas |
| author |
Kostrobii, P. Markovych, B. Vasylenko, A. Tokarchuk, M. Rudavskii, Yu. |
| author_facet |
Kostrobii, P. Markovych, B. Vasylenko, A. Tokarchuk, M. Rudavskii, Yu. |
| publishDate |
2006 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Нерiвноважний статистичний оператор Зубарєва i функцiї Грiна для неоднорiдного електронного газу |
| description |
Nonequilibrium properties of an inhomogeneous electron gas are studied using the method of the nonequilibrium
statistical operator by D.N. Zubarev. Generalized transport equations for the mean values of inhomogeneous
operators of the electron number density, momentum density, and total energy density for weakly
and strongly nonequilibrium states are obtained. We derive a chain of equations for the Green’s functions,
which connects commutative time-dependent Green’s functions “density-density”, “momentum-momentum”,
“enthalpy-enthalpy” with reduced Green’s functions of the generalized transport coefficients and with Green’s
functions for higher order memory kernels in the case of a weakly nonequilibrium spatially inhomogeneous
electron gas.
Дослiджуються нерiвноважнi властивостi просторово неоднорiдного електронного газу з використанням методу нерiвноважного статистичного оператора Зубарєва. Отримано узагальненi рiвняння переносу для середнiх значень густин операторiв числа електронiв, їх iмпульсiв та повної енергiї
для опису як слабо, так i сильно нерiвноважних станiв системи. Отримано ланцюжок рiвнянь функцiй
Грiна, який зв’язує комутаторнi часовi функцiї Грiна “густина-густина”, “iмпульс-iмпульс”, “ентальпiяентальпiя” iз приведеними функцiями Грiна узагальнених коефiцiєнтiв переносу в’язкостi, теплопровiдностi (включаючи перехреснi в’язко-тепловi коефiцiєнти переносу) та iз приведеними функцiями
Грiна вищих функцiй пам’ятi для слабонерiвноважного просторово неоднорiдного електронного газу.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121353 |
| citation_txt |
Nonequilibrium statistical Zubarev’s operator and Green’s functions for an inhomogeneous electron gas / P. Kostrobii, B. Markovych, A. Vasylenko, M. Tokarchuk, Yu. Rudavskii // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 519–533. — Бібліогр.: 37 назв. — англ. |
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2025-11-25T21:20:35Z |
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2025-11-25T21:20:35Z |
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| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 3(47), pp. 519–533
Nonequilibrium statistical Zubarev’s operator and
Green’s functions for an inhomogeneous electron gas
P.Kostrobii1, B.Markovych1, A.Vasylenko2, M.Tokarchuk2, Yu.Rudavskii1
1 National University “Lvivska Politekhnika” 12, Bandera Str., Lviv, UA-79013, Ukraine
2 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received June 21, 2006
Nonequilibrium properties of an inhomogeneous electron gas are studied using the method of the nonequi-
librium statistical operator by D.N. Zubarev. Generalized transport equations for the mean values of inhomo-
geneous operators of the electron number density, momentum density, and total energy density for weakly
and strongly nonequilibrium states are obtained. We derive a chain of equations for the Green’s functions,
which connects commutative time-dependent Green’s functions “density-density”, “momentum-momentum”,
“enthalpy-enthalpy” with reduced Green’s functions of the generalized transport coefficients and with Green’s
functions for higher order memory kernels in the case of a weakly nonequilibrium spatially inhomogeneous
electron gas.
Key words: nonequilibrium statistical operator, Green’s functions, generalized transport coefficients,
inhomogeneous electron gas
PACS: : 05.60.Gg, 05.70.Np, 63.10.+a, 68.43, 82.20.Xr
1. Introduction
Investigations of the nonequilibrium properties of many-electron systems are a topical prob-
lem in modern nanotechnology [1,2]. Nonequilibrium properties of spatially inhomogeneous many-
electron systems are of special importance both from theoretical and experimental points of view.
They play a determining role in the processes of absorption, desorption, in catalytic reactions be-
tween atoms and molecules at the surfaces of transition metals [3–5], in carbon nanostructures [6,7],
including the processes of excitation and ionization of gas atoms and molecules [8–10]. Nonequilib-
rium properties of many-electron systems were explored by means of various methods, in particular
by the Green’s functions method [11–17]. For further studies of the Green’s function and for specific
calculations, the time-dependent method of density functional [24–30] is developed [24–30] based
on the Kohn-Sham ideas [18–20], with formulation based on the hydrodynamic approach [21,22]
and Mori-like projection operators [23].
In the present paper the method of nonequilibrium statistical operator (NSO) [15] by
D.N. Zubarev is applied to the description of nonequilibrium properties of an inhomogeneous
electron gas at a metal surface. This method is based on N.N. Bogolyubov’s ideas about the re-
duced description of nonequilibrium states of the system. In the second section the Hamiltonian
of a spatially inhomogeneous electron gas, based on the generalized “jellium” model, is presented.
Based on this the one- and two-electron distribution functions, free energy, surface energy, inho-
mogeneous electrical field of electrons of semi-infinity jellium are calculated in [31–34] using the
method of dynamical collective variables and the functional representation with taking screening
effects into account. These results of equilibrium theory are considered as basic ones for the devel-
opment of nonequilibrium statistical theory of transport processes for a spatially inhomogeneous
electron gas. In the third section, the NSO [15] for a nonequilibrium spatially inhomogeneous
electron gas in the generalized “jellium” model is obtained with taking into account electric and
magnetic fields, obeying averaged Maxwell equations. Mean values of the number density, mo-
c© P.Kostrobii, B.Markovych, A.Vasylenko, M.Tokarchuk, Yu.Rudavskii 519
P.Kostrobii et al.
mentum density, and total energy density operators, which satisfy the corresponding conservation
laws, are chosen as the parameters of a reduced description. In the same section, by means of NSO
with projection technique we obtain generalized transport equations for the reduced description
parameters, which are applicable to the consideration of both weakly and strongly nonequilib-
rium processes of charge transfer, electric current, and the energy transport in the case of spa-
tially inhomogeneous electron gas. In the fourth section the generalized transport equations for
a weakly nonequilibrium spatially inhomogeneous electron gas are obtained. The chain of equa-
tions for Green’s functions is constructed for the transport kernels (memory functions dealt with
generalized coefficients of viscosity, heat conductivity etc.) within the framework of Tserkovnikov
approach [35–37]. This chain of equations interrelates the commutative time-dependent Green’s
functions “density-density”, “momentum-momentum”, “enthalpy-enthalpy” with reduced (time
evolution operator includes projection) Green’s functions of the generalized transport coefficients
and reduced Green’s functions of higher-order memory functions for weakly nonequilibrium spa-
tially inhomogeneous electron gas.
2. Hamiltonian of the equilibrium inhomogeneous electron g as within the
“jellium” model
We consider a semi-restricted metal surface with an unlimited ionic subsystem in the OXY
plane with “metal-vacuum” transitions in the direction, perpendicular to this plane. The Hamil-
tonian of this system is chosen to be the following:
H = T + Vee + Vei + Vii, (2.1)
where
T = −
~
2
2m
N
∑
i=1
∆i (2.2)
is an operator of the kinetic energy of conductivity electrons, m denotes the electron mass, N
stands for the number of electrons;
Vee =
1
2
N
∑
i6=j
e2
|ri − rj |
(2.3)
is the potential energy of interaction between electrons, ri denotes the radius-vector of i-th electron
position;
Vii =
1
2
∫
dR1
∫
dR2
ρ̂ion (R1) ρ̂ion (R2)
|R1 − R2|
(2.4)
is the potential energy of an ionic subsystem, ρ̂ion (R) = Ze
Nion
∑
i=1
δ (R − Ri) denotes a microscopic
density of the ion charge distribution, Ri means the radius-vector of the ion position, Ze stands
for the ion charge and Nion is the number of ions.
Vie = −e
N
∑
i=1
∫
dR
ρ̂ion(R)
|ri − R|
(2.5)
denotes the energy of interaction between the conductivity electrons and the ionic subsystem. The
system obeys electro-neutrality condition:
∫
dRρ̂ion (R) = eN .
To specify the “jellium” subsystem, let us present the distribution function of the ion charge
in the following form: ρ̂ion (R) = ρ̂jell (R) + δρ̂ion (R), where ρ̂jell (R) stands for the distribution
function of ion charge in the “jellium” model, δρ̂ion (R) = ρ̂ion (R) − ρ̂jell (R) is the difference
520
Short title: Nonequilibrium statistical Zubarev operator and Green’s functions
between the charge distribution of the ionic subsystem and of the “jellium” model. The Hamiltonian
of the “jellium” model could be presented as
H = Hjell − e
N
∑
i=1
∫
dR
ρ̂ion(R)
|ri − R|
, (2.6)
where
Hjell = −
~
2
2m
N
∑
i=1
∆i +
1
2
N
∑
i6=j=1
e2
|ri − rj |
+
N
∑
i=1
V (ri), (2.7)
V (ri) =
∫
dR1dR2
[
1
2N
δρ̂(R1)δρ̂(R2)
|R1 − R2|
+
1
N
δρ̂(R1)ρ̂jell(R2)
|R1 − R2|
− e
δρ̂(R1)δρ̂(R2 − ri)
|R1 − R2|
]
(2.8)
is the surface potential. In the second quantization representation, based on the wave functions
formalism [31,32], Hamiltonian (2.6) can be written down as:
H = Hjell + ∆H −
H
2S
∑
q
′υ(q |0) , (2.9)
where
Hjell = H0 +
1
2SL
∑
k,q
′υk(q)ρ̂k(q)ρ̂k(−q) (2.10)
is the Hamiltonian of a “jellium” model for the inhomogeneous electron gas [31,32] in the collective
variables representation.
H0 =
∑
p,α
Eα (p) a+
α (p) aα (p). (2.11)
S is the OXY surface area; L determines the range of values of a coordinate Z{−L
2 ,
L
2 } in the
surface, perpendicular to OXY ; νk(q) = 4π e2
(q2+k2) denotes the Fourier transform of Coulomb inter-
action, k = 2π n
L
, n = 0,±1,±2, . . . ., q = (qx, qy) = 2πmx,y, mx,y = 0,±1,±2, . . . , ν(q |z) =
2πe2 exp{−q |x|}/q is the two-dimensional Fourier transform of Coulomb interaction. Eα(p) is the
electron energy, while a+
α (p), aα(p) stand for the electron creation and annihilation operators
with momentum p in the α-state, respectively.
ρ̂k(q) =
∑
p,α,α′
〈α| exp(ikz) |α′ 〉a+
α (p)aα′(p − q) (2.12)
is a mixed Fourier transform of the local density of electrons, 〈α| . . . . |α′ 〉 =
∫
dzϕ∗
α(z) . . . ϕα′(z),
ϕα(z) is a wave function of an electron in the α-state in the direction, perpendicular to the “metal-
vacuum” surface. The second term of (2.9) describes interaction between electronic and ionic
subsystems:
∆H =
Nion
2SL
∑
k,q
′Sk(q)ωk(q)ρ̂−k(−q), (2.13)
where Nion is the number of ions, Sk(q) = 1
Nion
Nion
∑
n=1
exp(iqRn) means an equilibrium structure
factor of ions, Rn denotes the radius-vector of the ion in OXY plane, ωk(q) stands for a form-
factor of the ion-electron interaction.
The Hamiltonian (2.9) is a generalization of the “jellium” model because it takes into account
the interaction between electronic and ionic subsystems via the form-factor of the ion-electron in-
teraction in ∆H. The equilibrium state of the system can be completely described by the statistical
operator
ρ0 =
1
Q
e−β(H−Nµ), (2.14)
521
P.Kostrobii et al.
where
Q = Sp e−β(H−Nµ) (2.15)
is a partition function of the equilibrium statistical operator in the generalized “jellium” model. µ
denotes a chemical potential of the electronic subsystem, β = 1
kBT
, kB is the Boltzmann constant.
T stands for an equilibrium value of the system temperature. The equilibrium mean value 〈ρ̂k (q)〉0
defines local electric field of the electronic subsystem at the metal surface,
ikqEk (q) = 4πe 〈ρ̂k (q)〉0 . (2.16)
In [31–34] for the presented model there has been proposed a new approach to the calculation
of thermodynamic potential, which consists in the reduction of the corresponding non-Gaussian
integral to the Gaussian form with a renormalized correlation function “density-density”. It was
shown that an effective potential of the electron interaction completely defines the thermodynamic
potential. By means of functional integration there has been evaluated the one-particle distribu-
tion function of the electrons for the “jellium” model of semi-infinite domain of the metal with
consideration of exchange and correlation effects. One has also calculated an effective potential
of the electron interaction and a two-particle correlation function “density-density” for a semi-
restricted metal in the “jellium” model; their asymptotic behaviour has been studied at large
distances between electrons in the plane of dividing surface. Such investigations of the equilibrium
structure distribution functions and thermodynamic values form a good basis for the creation of a
noneqiulibrium statistical theory of the spatially inhomogeneous electron gas.
3. Nonequilibrium statistical operator of an inhomogeneou s electron gas
A Hamiltonian of the nonequilibrium inhomogeneous electron gas of the metal surface with
taking into account the electromagnetic processes in the “jellium” model can be written as:
H̃ (t) = T̃ (t) +Hph + Vee + Vei + Vii, (3.1)
where
T̃ (t) =
~
2
2m
N
∑
i=1
(
~∇i −
e
c
a (r; t)
)2
+
N
∑
i=1
eϕ (ri; t), (3.2)
c denotes the light velocity, a (r; t), ϕ (ri; t) are, correspondingly, the vector and scalar potentials
of the quantized electromagnetic field.
Hph =
∞
∫
0
dw~w
∫
drf̂
+
(r;w) f̂ (r;w), (3.3)
~w means the energy of a photon, f̂+ (r;w) , f̂ (r;w) are the boson field creation and annihilation
operators of the electromagnetic field quantum in the space-frequency representation:
[
f̂i (r;w) f̂+
j (r′;w)
]
= δijδ (r − rj) δ (w − w′) ,
[
f̂i (r;w) f̂j (r′;w)
]
=
[
f̂+
i (r;w) f̂+
j (r′;w)
]
= 0. (3.4)
a (r; t) corresponds to the microscopic value of the vector potential operator, expressed via the
creation and annihilation operators of the photons of electromagnetic field [13]:
â (r) =
∑
λk
(
2πe2
V ωk
)
1
2 [
ekλf̂kλeikr + e∗kλf̂
+
kλe−ikr
]
, (3.5)
where ekλ is the photon polarization vectors, which satisfy orthogonality conditions:
ekλk = 0,
2
∑
λ=1
ekλi
e∗kλj
= δij −
kikj
k2
,
522
Short title: Nonequilibrium statistical Zubarev operator and Green’s functions
where λ = 1, 2 is the photon polarization, V stands for a volume. â (r) defines the quantized
electrical and magnetic field operators
Ĥ (r) =
−→
∇ × â (r) , Ê (r) = Êt (r) + Êl (r) ,
Êt (r) = −
1
c
∂
∂t
â (r) , Êl (r) = −
−→
∇ ·
(∫
dr′
ρ̂ (r′)
|r − r′|
+
∫
dR
ρ̂ion (R)
|r − R|
)
. (3.6)
Êt (r) is the quantized transverse electrical field operator; Êl (r) means the longitudinal Coulomb
field, created by the electron subsystem in positively charged ion field with a fixed distribution
Sk (q) at the surface. Ĥ (r) and Ê (r) satisfy the corresponding microscopic Maxwell-Lorentz equa-
tions:
−→
∇ × Ê (r) = −
1
c
∂
∂t
Ĥ (r) ,
−→
∇ × Ĥ (r) =
1
c
∂
∂t
Ê (r) +
4π
c
ĵe (r) . (3.7)
−→
∇ · Ĥ (r) = 0,
−→
∇ · Ê (r) = 4π (eρ̂ (r) + Zeρ̂ion (R)) , (3.8)
where
ĵe (r) =
ie
2m
{(
~∇ +
ie
c
a (r; t)
)
ψ+ (r)ψ (r) − her.con.
}
(3.9)
denotes a current of the electron density operators without account of the spin degrees of freedom,
which are taken into account, respectively, by the term
−→
∇ × ψ+ (r)σψ (r) ; σ is a matrix of the
intrinsic magnetic momentum of electron; ψ+ (r) , ψ (r) stand for the field creation and annihilation
operators of the electrons: ρ̂ (r) = ψ+ (r)ψ (r) . The vector and scalar potentials in the Hamiltonian
(3.1) are determined from the averaged Maxwell-Lorentz equations:
−→
∇ ×
〈
Ê (r)
〉t
= −
1
c
∂
∂t
〈
Ĥ (r)
〉t
,
−→
∇ ×
〈
Ĥ (r)
〉t
=
1
c
∂
∂t
〈
Ê (r)
〉t
+
4π
c
〈
ˆ̂
je (r)
〉t
, (3.10)
−→
∇ ·
〈
Ĥ (r)
〉t
= 0,
−→
∇ ·
〈
Ê (r)
〉t
= 4π
(
〈eρ̂ (r)〉
t
+ 〈Zeρ̂ion (R)〉
t
)
, (3.11)
〈
Ĥ (r)
〉t
=
−→
∇ × 〈â (r)〉
t
=
−→
∇ × a (r; t) ,
〈
Ê (r)
〉t
=
〈
Êt (r)
〉t
+
〈
Êl (r)
〉t
,
〈
Êt (r)
〉t
= −
1
c
∂
∂t
〈â (r)〉
t
= −
1
c
∂
∂t
a (r; t) ,
〈
Êl (r)
〉t
= −
−→
∇ ·
(
∫
dr′
〈eρ̂ (r′)〉
t
|r − r′|
+
∫
dR
〈Zeρ̂ion (R)〉0
|r − R|
)
, (3.12)
where the mean values 〈. . .〉
t
= Sp (. . . ρ (t)) are calculated with the NSO ρ (t) , obeying the Liouville
equation:
∂
∂t
ρ (t) + iL (t) ρ (t) = 0, (3.13)
with the normalization condition Sp ρ (t) = 1. iL(t) is a Liouville operator, which corresponds to
the Hamiltonian (3.1), so that iL(t)ρ (t) = i/~
[
H̃ (t) , ρ (t)
]
.
The nonequilibrium statistical operator ρ (t) as a solution of the Liouville equation has to be
found using the method of NSO by D.N. Zubarev [15]. In this method the general solution for
(3.13) with taking into account the Kawasaki-Gunton projection can be presented in the following
form:
ρ (t) = ρq (t) −
t
∫
−∞
eε(t′−t)Tq (t, t′) (1 − Pq (t′)) ρq (t′) dt′, (3.14)
523
P.Kostrobii et al.
where transition ε→ +0 has to be performed after a thermodynamic limit N → ∞, V → ∞ and
selects retarded solutions of the Liouville equation (3.13) [15].
Tq (t, t′) = exp+
t′
∫
t
(1 − Pq (τ)) iL (τ)dτ
is the evolution operator with taking into account the Kawasaki-Gunton projection operator Pq (t) .
Its structure depends on the auxiliary statistical operator ρq (t) in the formulation of the Cauchy
problem for the Liouville equation with t = t0, ρ (t0) = ρq (t0) [15].
In the NSO method [15] ρq (t), according to the Gibbs’ hypothesis, can be found from the ex-
tremum of the informational entropy with fixed parameters of the reduced description and obeying
the normalization condition Spρ (t) = 1. To describe nonequilibrium properties of spatially inhomo-
geneous electron gas at the metal surface with taking into account the electromagnetic processes,
the mean nonequilibrium values of the number density of the electrons 〈ρ̂ (r)〉
t
, momentum den-
sity 〈p̂ (r)〉
t
, and total energy density 〈ε̂ (r)〉
t
, which satisfy the corresponding conservation laws,
can be chosen as the parameters of the reduced description. 〈ρ̂ (r)〉
t
determines a nonequilibrium
value of density of the electron charge 〈ρ̂e (r)〉
t
= e 〈ρ̂ (r)〉
t
,
〈
ĵe (r)
〉t
is a mean nonequilibrium
value of the electric current of the electrons,
〈
ĵe (r)
〉t
= e/m 〈p̂ (r)〉
t
. At the same time,
〈
ĵe (r)
〉t
,
〈ρ̂e (r)〉
t
, according to (3.10)–(3.12), define the mean nonequilibrium values of the magnetic and
electric fields and their potentials. With this choice of the parameters of reduced description, one
can write down the following expression (in the nonrelativistic approximation) for ρq (t) [15]:
ρq (t) = exp
{
−Φ(t) −
∫
drβ (r; t) (ε̂′ (r; t) − µel (r; t) ρ̂ (r))
}
, (3.15)
where Φ (t) is the Massieu-Planck functional:
Φ (t) = ln Sp exp
{
−
∫
drβ (r; t)
(
ε̂′ (r) − µel (r; t) ρ̂ (r)
)
}
, (3.16)
β (r; t) and µel (r; t) denote the Lagrange multipliers, determined, respectively, from the self-
consistency conditions
〈
ε̂′ (r)
〉t
=
〈
ε̂′ (r)
〉t
q
, (3.17)
〈ρ̂ (r)〉
t
= 〈ρ̂ (r)〉
t
q (3.18)
and thermodynamic relations:
δΦ(t)
δβ (r; t)
= −
〈
ε̂′ (r)
〉t
q
= −
〈
ε̂′ (r)
〉t
, (3.19)
δΦ(t)
δβ (r; t)µel (r; t)
= −〈ρ̂ (r)〉
t
q = −〈ρ̂ (r)〉
t
, (3.20)
δS (t)
δ
〈
ε̂′ (r; t)
〉t = β (r; t) , (3.21)
δS (t)
δ 〈ρ̂ (r; t)〉
t = −β (r; t)µel (r; t) , (3.22)
and define the inverse nonequilibrium temperature β−1 (r; t) = kBT (r; t) and the electrochemi-
cal potential of the electronic subsystem: µel (r; t) = µ (r; t) + eϕ (r; t). Here µ (r; t) is the local
nonequilibrium chemical potential of the electrons, 〈. . .〉
t
q = Sp (. . . ρq (t)) . ε̂′ (r) is the total en-
ergy density in the moving frame: ε̂′ (r) = ε̂ (r)− (v (r; t) + e/c · a (r; t)) p̂ (r) +m/2 · v2 (r; t) ρ̂ (r),
524
Short title: Nonequilibrium statistical Zubarev operator and Green’s functions
v (r; t) denotes a vector of mean velocity of the electrons. S (t) is the nonequilibrium Gibbs’ entropy
defined as follows:
S (t) = −〈ln ρq (t)〉
t
q
= Φ(t) +
∫
drβ (r; t)
(
〈
ε̂′ (r)
〉t
q
− µel (r; t) 〈ρ̂ (r)〉
t
q
)
,
or, taking into account the self-consistency conditions (3.17)–(3.18), S (t) could be rewritten as:
S (t) = Φ (t) +
∫
drβ (r; t)(〈ε̂ (r)〉
t
−
(
v (r; t) +
e
c
a (r; t)
)
〈p̂ (r)〉
t
− ν (r; t) 〈ρ̂ (r)〉
t
− eϕ (r; t) 〈ρ̂ (r)〉
t
), (3.23)
where ν (r; t) = µ (r; t)−m/2 ·v2 (r; t) . The nonequilibrium entropy is a functional of the nonequi-
librium average densities of the total energy, momentum and number of the electrons, conjugate
thermodynamic parameters β (r; t) , v (r; t) , µ (r; t), and the field potentials a (r; t) , ϕ (r; t) .
Taking into account the definition of ε̂′ (r), one can present the statistical operator ρq (t) in the
following form:
ρq (t) = exp
{
−Φ(t) −
∫
drβ (r; t)(ε̂ (r) −
(
v (r; t) +
e
c
a (r; t)
)
p̂ (r)
−ν (r; t) ρ̂ (r) − eϕ (r; t) ρ̂ (r))
}
, (3.24)
where contributions of the quantized electromagnetic field are selected. According to the structure
of the statistical operator (3.24), the Kawasaki-Gunton projection operators have the following
form:
Pq (t) ρ′ = ρq (t) −
∫
dr(
δρq (t)
δ 〈ε̂ (r)〉
t 〈ε̂ (r)〉
t
+
δρq (t)
δ 〈p̂ (r)〉
t 〈p̂ (r)〉
t
+
δρq (t)
δ 〈ρ̂ (r)〉
t 〈ρ̂ (r)〉
t
)Spρ′)
+
∫
dr(
δρq (t)
δ 〈ε̂ (r)〉
t Sp (ε̂ (r) ρ′) +
δρq (t)
δ 〈p̂ (r)〉
t Sp (p̂ (r) ρ′) +
δρq (t)
δ 〈ρ̂ (r)〉
t Sp (ρ̂ (r) ρ′)
with the operator properties: Pq (t)Pq (t′) = Pq (t), Pq (t) ρ′ = ρq (t), Pq (t) ρq (t′) = ρq (t). Speci-
fying the action of the Liouville operator iL (t) and (1 − Pq (t)) on the quasi-equilibrium statistical
operator ρq (t) (3.24) according to (3.14), one can obtain, for the nonequilibrium statistical operator
of the inhomogeneous electron gas, the following result:
ρ (t) = ρq (t) +
∫
dr
t
∫
−∞
dt′eε(t′−t)Tq (t, t′)
1
∫
0
dτρτ
q (t′)Iε (r; t′) ρ1−τ
q (t′)β (r; t′)
−
1
∫
0
dτρτ
q (t′)Ip (r; t′) ρ1−τ
q (t′)β (r; t′)
(
v (r; t′) +
e
c
a (r; t′)
)
, (3.25)
where Iε (r; t′), Ip (r; t′) are the generalized fluxes of energy and momentum:
Iε (r; t′) = (1 − P (t)) iL (t) ε̂ (r) ,
Ip (r; t′) = (1 − P (t)) iL (t) p̂ (r) ,
(3.26)
P (t) is a Mori-like projection operator. Its structure is related to the structure of the Kawasaki-
Gunton operator and could be presented as
P (t) Â =
〈
Â
〉t
q
+
∫
dr
δ
〈
Â
〉t
q
δ 〈ε̂ (r)〉
t
(
ε̂ (r) − 〈ε̂ (r)〉
t
)
+
∫
dr
δ
〈
Â
〉t
q
δ 〈p̂ (r)〉
t
(
p̂ (r) − 〈p̂ (r)〉
t
)
+
∫
dr
δ
〈
Â
〉t
q
δ 〈ρ̂ (r)〉
t
(
ρ̂ (r) − 〈ρ̂ (r)〉
t
)
.
525
P.Kostrobii et al.
P (t) (unlike Pq (t)) acts on the dynamical values and possesses the properties of the projection on
the space of reduced description parameters ε̂ (r) , p̂ (r) , ρ̂ (r): P (t)P (t′) = P (t), P (t) ε̂ (r) = ε̂ (r),
P (t) p̂ (r) = p̂ (r), P (t) ρ̂ (r) = ρ̂ (r). At that the equation (1 − P (t)) iL (t) ρ̂ (r) = 0 is satisfied.
Thus, we obtained the NSO of an inhomogeneous electron gas in the generalized “jellium” model
based on the idea of a reduced description. The nonequilibrium statistical operator is a functional
of dynamical quantities, whose mean values are observable quantities. It is also a functional of the
generalized fluxes (3.26) that describe dissipative processes, related both to the motion of electrons
and to the quantum electromagnetic field. Taking into account the identities,
∂
∂t
〈ρ̂ (r)〉
t
= 〈iL (t) ρ̂ (r)〉
t
, ∂
∂t
〈p̂ (r)〉
t
= 〈iL (t) p̂ (r)〉
t
q +
〈
Îp (r; t)
〉t
,
∂
∂t
〈ε̂ (r)〉
t
= 〈iL (t) ε̂ (r)〉
t
q +
〈
Îε (r; t)
〉t
,
one can obtain generalized transport equations of the inhomogeneous electron gas at the metal
surface in the following form:
∂
∂t
〈ρ̂ (r)〉
t
= −
1
m
∂
∂r
· 〈p̂ (r)〉
t
, (3.27)
∂
∂t
〈p̂ (r)〉
t
= 〈iL (t) p̂ (r)〉
t
q +
∫
dr′
t
∫
−∞
dt′eε(t′−t)ϕIpIε
(r, r′; t, t′)β (r′; t′)
−
∫
dr′
t
∫
−∞
dt′eε(t′−t)ϕIpIp
(r, r′; t, t′)β (r′; t′)
(
v (r′; t′) +
e
c
a (r′; t′)
)
, (3.28)
∂
∂t
〈ε̂ (r)〉
t
= 〈iL (t) ε̂ (r)〉
t
q +
∫
dr
t
∫
−∞
dt′eε(t′−t)ϕIεIε
(r, r′; t, t′)β (r′; t′)
−
∫
dr′
t
∫
−∞
dt′eε(t′−t)ϕIεIp
(r, r′; t, t′)β (r′; t′)
(
v (r′; t′) +
e
c
a (r′; t′)
)
, (3.29)
where ϕIpIp
(r, r′; t, t′) , ϕIεIε
(r, r′; t, t′) , ϕIpIε
(r, r′; t, t′) , ϕIεIp
(r, r′; t, t′) denote spatially inhomo-
geneous generalized memory kernels related to the generalized viscosity and heat conductivity
as well as to the coefficients that describe cross-correlations of viscous and heat processes of the
electron gas. They have a structure of Kubo functions:
ϕInIm
(r, r′; t, t′) = Sp
In (r; t)Tq (t, t′)
1
∫
0
dτρτ
q (t′) Im (r′; t′)ρ1−τ
q (t′)
. (3.30)
The system of equations (3.27)–(3.29) describes strongly nonlinear transport processes of the
generalized “jellium” model. This system of equations is unclosed, because it contains (functionally)
thermodynamic parameters β (r; t) ,v (r; t) , µ (r; t), determined by the self-consistency conditions
(3.17), (3.18) as well as the vector and scalar potentials a (r; t), determining magnetic and electric
fields, which obey the averaged Maxwell equations (3.10),(3.11). Thus, the generalized transport
equations (3.27)–(3.29) should be considered self-consistently with the averaged Maxwell equations
(3.10)–(3.11) for electromagnetic field. It is clear from the proposed approach that in the case of
interaction of inhomogeneous electron gas of the metal surface with unbound atoms or molecules
of gas (for instance, in catalysis processes), first of all the electromagnetic polarization of atoms
and molecules takes place, and then the processes of their absorption, desorption and chemical
reactions are possible.
The generalized transition kernels (3.30), dealing with the corresponding viscosity and heat
conductivity, which describe many-electron dissipative processes, are a special problem in the
526
Short title: Nonequilibrium statistical Zubarev operator and Green’s functions
transport equations (3.27)–(3.29). Calculation of their time-spatial dependences is one of the main
tasks in the modern theory of nonequilibrium processes. To solve these problems for classical and
quantum systems of interacting particles, the method of the Mori-like projection operators and
the method of time-dependent Green’s functions are used. For weakly nonequilibrium systems the
chain of transport equations (3.27)–(3.29) is considerably simplified, and the NSO method makes
it possible [36,37] to obtain the system of equations for time correlation functions or the cor-
responding “density-density”, “momentum-momentum”, “energy-energy” Green’s functions. The
same approach allowed us [36] to obtain the chain of equations for the time-dependent Green’s
functions related to generalized transport coefficients. A similar chain of equations for Green’s
functions was proposed by Tserkovnikov [35].
4. Time correlation functions and Green’s functions of a wea kly inhomoge-
neous electron gas
Let us consider the case of a nonequilibrium state of an inhomogeneous electron gas in the
“jellium” model, when the parameters β (r; t) , µ (r; t) weakly deviate from their equilibrium val-
ues β, µ, and the corresponding fields are small. Then expanding the quasi-equilibrium statistical
operator (3.15) in a series in deviations of thermodynamic parameters and excluding them from
ρq (t) by means of self-consistency conditions (3.17), (3.18), one can obtain the following form for
ρq (t) in the linear approximation:
ρ̃q (t) = ρ0 +
∑
q
∫
dz
∫
dz′ 〈δρ̂ (q; z)〉
t
Φ−1
ρρ (q; z, z′) ρ̂ (q; z′, τ)
+
∑
q
∫
dz
∫
dz′ 〈δp̂ (q; z)〉
t
Φ−1
pp (q; z, z′) p̂ (q; z′, τ)
+
∑
q
∫
dz
∫
dz′
〈
δĥ (q; z)
〉t
Φ−1
hh (q; z, z′) ĥ (q; z′, τ). (4.1)
We have used the (q, z) representation of the functions; z denotes a coordinate in a normal direction
q to the dividing surface, where δρ̂ (q; z) = ρ̂ (q; z) − 〈ρ̂ (q; z)〉0 , δp̂ (q; z) = p̂ (q; z) − 〈p̂ (q; z)〉0
〈p̂ (q; z)〉0 = 0, δĥ (q; z) = ĥ (q; z) −
〈
ĥ (q; z)
〉
0
, and 〈. . .〉0 = Sp (. . . ρ0) stands for the equi-
librium averaging with operator (2.14). ĥ (q; z) denotes the generalized enthalpy operator of the
inhomogeneous electron gas:
ĥ (q; z) = ε̂ (q; z) −
∫
dz′
∫
dz′′ 〈ε̂ (q; z) ρ̂ (q; z′)〉
0
Φ−1
ρρ (q; z′, z′′) ρ̂ (q; z′′) , (4.2)
where Φ−1
ρρ (q; z′, z′′) is an equilibrium correlation function, defined from the condition
∑
qq′
∫
dz′Φρρ (q; z, z′) Φ−1
ρρ (q′; z′, z′′) = δ (z − z′′) δqq′ , (4.3)
Φρρ (q; z, z′) = 〈ρ̂ (q; z) ρ̂ (q; z′, τ)〉0 = Sp
ρ̂ (q; z)
1
∫
0
dτρτ
0 ρ̂ (q; z′) ρ1−τ
0
(4.4)
is the Kubo equilibrium “density-density” correlation function for the inhomogeneous electron gas.
Functions Φ−1
pp (q; z, z′) and Φ−1
hh (q; z, z′) are determined from the following integral relations:
∑
qq′
∫
dz′Φ−1
pp (q; z, z′) Φ−1
pp (q′; z′, z′′) = δ (z − z′′) δqq′ ,
∑
qq′
∫
dz′Φ−1
hh (q; z, z′) Φ−1
hh (q′; z′, z′′) = δ (z − z′′) δqq′ .
527
P.Kostrobii et al.
In the quasi-equilibrium statistical operator (4.1) the quantities ρ̂ (q; z) , p̂ (q; z) , ĥ (q; z)
(ρ̂ (q; z, τ) , p̂ (q; z, τ) , ĥ (q; z, τ) have the following structure: â (q; z, τ) =
1
∫
0
dτρτ
0 â (q; z) ρ1−τ
0 )
and are orthogonal in the average values sense: 〈ρ̂ (q, z) p̂ (q; z′, τ)〉0 = 0,
〈
p̂ (q, z) ĥ (q; z′, τ)
〉
0
= 0,
〈
ρ̂ (q, z) ĥ (q; z′, τ)
〉
0
= 0.
In the approximation (4.1), the nonequilibrium statistical operator of the inhomogeneous elec-
tron gas has the form
ρ (t) = ρ̃q (t) −
∑
q
∫
dz
∫
dz′
t
∫
−∞
dt′eε(t−t′)T 0
q (t, t′)
{
Ĩp (q; z′, τ) Φ−1
pp (q; z, z′) 〈δp̂ (q; z)〉
t′
+Ĩh (q; z′, τ) Φ−1
hh (q; z, z′)
〈
δĥ (q; z)
〉t
}
. (4.5)
where T 0
q (t, t′) is an evolution operator with the Kawasaki-Gunton projectors in the linear approx-
imation (4.1),
Ĩp (q; z, τ) =
1
∫
0
dτρτ
0 Ĩp (q; z) ρ1−τ
0 , Ĩh (q; z, τ) =
1
∫
0
dτρτ
0 Ĩh (q; z) ρ1−τ
0 ,
Ĩp (q; z) = (1 − P ) iL (t) p̂ (q; z) , Ĩh (q; z) = (1 − P ) iL (t) ĥ (q; z) (4.6)
are the generalized fluxes in the linear approximation; P is the Mori-like operator, which has the
following structure:
PÂ =
〈
Â
〉
0
+
∑
q
∫
dz
∫
dz′
{〈
Âρ̂ (q; z)
〉
0
Φ−1
ρρ (q; z, z′) ρ̂ (q; z′)
+
〈
Âp̂ (q; z)
〉
0
Φ−1
pp (q; z, z′) p̂ (q; z′) +
〈
Âĥ (q; z)
〉
0
Φ−1
hh (q; z, z′) ĥ (q; z′)
}
(4.7)
with properties P ρ̂ (q; z) = ρ̂ (q; z), P p̂ (q; z) = p̂ (q; z), P ĥ (q; z)= ĥ (q; z), PP = P , P (1 − P )= 0.
In the approximation (4.2), the generalized transport equations (3.27)–(3.29) have a closed
structure:
∂
∂t
〈δρ̂ (q; z)〉
t
+
∫
dz′iΩρp (q; z, z′) 〈δp̂ (q; z′)〉
t
= 0, (4.8)
∂
∂t
〈δp̂ (q; z)〉
t
+
∫
dz′iΩph (q; z, z′)
〈
δĥ (q; z′)
〉t
−
∫
dz′
t
∫
−∞
dt′eε(t′−t)ϕ̃ph (q; z, z′; t, t′)
〈
δĥ (q; z′)
〉t′
−
∫
dz′
t
∫
−∞
dt′eε(t′−t)ϕ̃pp (q; z, z′; t, t′) 〈δp̂ (q; z′)〉
t′
= 0, (4.9)
∂
∂t
〈
δĥ (q; z′)
〉t
+
∫
dz′iΩhp (q; z, z′) 〈δp̂ (q; z′)〉
t
−
∫
dz′
t
∫
−∞
dt′eε(t′−t)ϕ̃hp (q; z, z′; t, t′) 〈δp̂ (q; z′)〉
t′
−
∫
dz′
t
∫
−∞
dt′eε(t′−t)ϕ̃hh (q; z, z′; t, t′)
〈
δĥ (q; z′)
〉t′
= 0, (4.10)
528
Short title: Nonequilibrium statistical Zubarev operator and Green’s functions
where iΩρp (q; z, z′) , iΩph (q; z, z′) , iΩhp (q; z, z′) are normalized correlation functions:
iΩρp (q; z, z′) =
∫
dz′′
〈
˙̂ρ (q; z) p̂ (q; z′′; τ)
〉
0
Φ−1
pp (q; z′′, z′),
iΩph (q; z, z′) =
∫
dz′′
〈
˙̂p (q; z) ĥ (q; z′′; τ)
〉
0
Φ−1
hh (q; z′′, z′),
iΩhp (q; z, z′) =
∫
dz′′
〈
˙̂
h (q; z) p̂ (q; z′′; τ)
〉
0
Φ−1
pp (q; z′′, z′), (4.11)
are the elements of the frequency matrix in the Mori theory of the projection operators, which de-
scribe non-dissipative processes. ϕ̃hp (q; z, z′; t, t′) , ϕ̃ph (q; z, z′; t, t′) mean the generalized trans-
port kernels that describe dynamical correlations between viscous and heat conduction processes for
a weakly nonequilibrium inhomogeneous electron gas in the generalized “jellium” model.
ϕ̃pp (q; z, z′; t, t′) , ϕ̃hh (q; z, z′; t, t′) are the corresponding transport kernels related to the gener-
alized viscosity and heat conductivity. According to (3.30), they have the following structure:
ϕ̃ĨnĨm
(q; z, z′; t, t′) =
∫
dz′′Sp(Ĩn (q; z)T 0
q (t, t′)
1
∫
0
dτρτ
0 Ĩm (q; z′′) ρ1−τ
0 )Φmm (q; z′′, z′) ,
Ĩm (q; z′′) =
(
Ĩp (q; z′′) , Ĩh (q; z′′)
)
. (4.12)
In the NSO method [37], the transport equations (4.8)–(4.10) make it possible to obtain a closed
system for the time correlation functions of dynamical variables ρ̂ (q; z) , p̂ (q; z) , ĥ (q; z) . In a
matrix form it can be presented as:
∂
∂t
Φ̄ (q; z, z′; t) +
∫
dz′′iΩ(q; z, z′′) Φ̄ (q; z′′, z′; t)
−
∫
dz′′
t
∫
−∞
dt′eε(t′−t)ϕ̄ (q; z, z′′; t, t′) Φ̄ (q; z′′, z′; t′) = 0, (4.13)
where
Φ̄ (q; z, z′; t) =
Φρρ Φρp Φρh
Φpρ Φpp Φph
Φhρ Φhp Φhh
(q;z,z′;t)
(4.14)
is a matrix of time correlation functions of the dynamical variables ρ̂ (q; z) , p̂ (q; z) , ĥ (q; z) for the
inhomogeneous electron gas. They have the following structure:
Φnm (q; z, z′; t) = 〈â (q; z; t) â (q; z′; τ)〉0 ,
where â (q; z; t) = eiLtâ (q; z) .
iΩ̄ (q; z, z′) =
0 iΩρp 0
iΩpρ 0 iΩph
0 iΩhp 0
(q;z,z′)
(4.15)
is a frequency matrix, which describes non-dissipative correlations in the system,
ϕ̄ (q; z, z′; t, t′) =
0 0 0
0 ϕ̃pp ϕ̃ph
0 ϕ̃hp ϕ̃hh
(q;z,z′;t,t′)
(4.16)
denotes a matrix of the memory functions (transport kernels) that describe dissipative processes
in a spatially inhomogeneous electron gas. In the case of spatial homogeneity in the hydrody-
namic limit |k| → 0, ω → 0 (here k stands for a three-dimensional wave vector, ω is the fre-
quency)} ϕ̃pp ∼ k2η, ϕ̃hh ∼ k2λ, ϕ̃ph = ϕ̃hp ∼ 0, where η is viscosity, λ means heat con-
ductivity of the spatially homogeneous electron gas. It is important to note that omitting the
529
P.Kostrobii et al.
energy fluctuations (or generalized enthalpy fluctuations) in the system of equations (4.10), i.e.
with iΩph = iΩhp = 0, ϕ̃ph = ϕ̃hp = 0, ϕ̃hh = 0, we actually obtain the system of equations
of [23], which has been obtained by means of projection operators and the method of dynami-
cal variables ρ̂ (q; z) , p̂ (q; z) for the inhomogeneous electron gas in TDDFT formulation. Our
approach generalizes the results of [23] by taking into account the enthalpy fluctuations. In the
spatially inhomogeneous case, using Fourier transformation for the space coordinate z, Fourier and
Laplace transformations for the time coordinate, defined by the relations,
f (ω) =
∞
∫
−∞
dteiωtf (t), f (t) =
1
2π
∞
∫
−∞
dωe−iωtf (ω),
f (s) =
∞
∫
0
dteistf (t), Ims > 0, f (s) =
0
∫
−∞
dte−istf (t), Ims < 0, s = ω ± iε
one can present the system of equations (4.8)–(4.10) in the following form:
sΦ̄ (q; k, k′; s) +
∑
k′
iΩ̄ (q; k, k′′; s) Φ̄ (q; k′′, k′; s)
∑
k′′
ϕ̄ (q; k, k′′; s) Φ̄ (q; k′′, k′; s) = Φ̄ (q; k, k′; 0) ,
(4.17)
where ϕ̃pp (q; k, k′′; s) = q2kη (q; k, k′′; s) k′′, ϕ̃hh (q; k, k′′; s) = q2kλ (q; k, k′′; s) k′. Here
η (q; k, k′′; s) , λ (q; k, k′′; s) are the generalized viscosity and heat conductivity for the inhomoge-
neous electron gas in the generalized “jellium” model. To evaluate these quantities, one can apply
the method of Green’s function, developed by Tserkovnikov [35], or, alternatively, the method of
NSO [36,37]. For this purpose we introduce the matrix of the reduced Green’s functions, related
to the matrix of memory kernels
iϕ̄ (q; z, z′; s) = Ḡ(r) (q; z, z′; s) = −i
∞
∫
0
dteistϕ̄ (q; z, z′; t), Ims > 0, (4.18)
Ḡ(r) (q; z, z′; t) = −
i
~
Θ(t) ϕ̄ (q; z, z′; t) . (4.19)
In the presented equations Ḡ(r) (q; z, z′; t) are the retarded reduced Green’s functions, Θ (t) denotes
the Heviside’s step function.
It is natural to introduce the piecewise-analytic functions, determined in a complex plane s
with a cut along the real axis (Im s 6= 0). Then
Ḡ (q; z, z′; s) =
∞
∫
−∞
dω(e~βω − 1)
ϕ̄ (q; z, z′;ω)
s− ω
(4.20)
with Im s > 0, coincides with the analytic continuation of the Fourier transforms of the retarded
reduced Green’s functions Ḡ(r) (q; z, z′; s), and with Im s < 0, coincides with the analytic continu-
ation of the Fourier transformation of the advanced Green’s functions Ḡ(a) (q; z, z′; s),
Ḡ(a) (q; z, z′; t) =
i
~
Θ(−t) ϕ̄ (q; z, z′; t) , (4.21)
and
Ḡ(a) (q; z, z′; s) = i
∞
∫
−∞
dteistϕ̄ (q; z, z′; t), Im s < 0.
The quantities ϕ̄ (q; z, z′;ω) can be calculated using discontinuous jump of Green’s functions on
the real axis, see equation (4.20), and can be presented as
Ḡ (q; z, z′;ω + iε) − Ḡ (q; z, z′;ω − iε) =
2π
i~
(e~βω − 1)ϕ̄ (q; z, z′;ω) . (4.22)
530
Short title: Nonequilibrium statistical Zubarev operator and Green’s functions
Now, according to the Green’s functions method, we obtain the equation for Ḡ (q; z, z′; s) in the
following form:
sḠ (q; z, z′; s) +
∫
dz′′iΩ(1) (q; z, z′′) Ḡ (q; z′′, z′; s)
−
∫
dz′′ϕ̄(1) (q; z, z′′; s) Ḡ (q; z′′, z′; s) = ϕ̄ (q; z, z′; 0) , (4.23)
where iΩ(1) (q; z, z′′) denotes the frequency matrix and ϕ(1) (q; z, z′′; s) is the matrix of memory
functions, constructed on the operators of the generalized dissipative fluxes
Ĩ(1)
a (q; z) =
{
Ĩ(1)
ρ (q; z) = 0, Ĩ(1)
p (q; z) =
(
1 − P (1)
)
iL(1)Ĩ(1)
p (q; z) ,
Ĩ
(1)
h (q; z) =
(
1 − P (1)
)
iL(1)Ĩ
(1)
h (q; z)
}
,
iL(1) =
(
1 − P (1)
)
iL,
where Mori-like projection operator P (1) is constructed on the generalized fluxes (4.6) Ĩρ (q; z),
Ĩp (q; z) , Ĩh (q; z) and has the following structure (in the matrix form):
P (1) · · · = 〈· · ·〉0 +
∫
dz
∫
dz′
〈
· · · Ĩ(+) (q; z, τ)
〉
0
〈
Ĩ (q; z) · Ĩ(+) (q; z′)
〉−1
0
Ĩ (q; z′) (4.24)
with the corresponding properties of the projection operator P. Here
Ĩ (q; z) = col
(
Ĩρ (q; z) , Ĩp (q; z) , Ĩh (q; z)
)
denotes a column vector,
Ĩ(+) (q; z) =
(
Ĩρ (q; z) , Ĩp (q; z) , Ĩh (q; z)
)
is a line vector. In the accepted notation, the matrix iΩ̄(1) (q; z, z′) has the structure
iΩ̄(1) (q; z, z′) =
∫
dz′′
〈
iL(1)Ĩ (q; z) · Ĩ(+) (q; z′′, τ)
〉
0
〈
Ĩ (q; z′′) · Ĩ(+) (q; z′, τ)
〉−1
0
, (4.25)
and the matrix of higher memory functions has the following form:
ϕ̄(1) (q; z, z′; t) =
∫
dz′′〈
(
1 − P (1)
)
iL(1)Ĩ (q; z)T (1)
q (t)
1
∫
0
dτρ−τ
0
(
1 − P (1)
)
iL(1)Ĩ(+) (q; z′′) ρτ
0〉0
×
〈
Ĩ (q; z′′) · Ĩ(+) (q; z′, τ)
〉−1
0
, (4.26)
where T
(1)
q (t) = exp
{(
1 − P (1)
)
iL(1)t
}
is a reduced time evolution operator. At that, the following
relation between the matrix of Green’s functions Ḡ(1) (q; z, z′; s) and the matrix of Green’s functions
Ḡ (q; z, z′; s) is valid:
Ḡ(1) (q; z, z′; s) = Ḡ (q; z, z′; s)−
∫
dz′′
∫
dz′′′ḠIa (q; z, z′′; s)Ḡ(0) (q; z′′, z′′′; s)
−1
ḠaI (q; z′′′, z′; s) ,
(4.27)
where Ḡ(0) (q; z′′, z′′′; s) stands for the matrix of commutative Green’s functions, constructed on the
parameters âm (q; z) =
{
ρ̂ (q; z) , p̂ (q; z) , ĥ (q; z)
}
of the reduced description for inhomogeneous
electron gas, and is defined by
G(0)
nm (q; z′, z′′; s) =
1
i~
∞
∫
0
dteist 〈[ân (q; z′; t) , âm (q; z′′)]〉0 . (4.28)
531
P.Kostrobii et al.
Green’s functions ḠIa (q; z, z′′; s), ḠaI (q; z′′′, z′; s) have a structure, similar to (4.28)
ḠInm (q; z′, z′′; s) =
1
i~
∞
∫
0
dteist
〈[
Ĩn (q; z′; t) , âm (q; z′′)
]〉
0
, (4.29)
where Ĩn (q; z′; t) = exp{(1−P )iLt}Ĩn (q; z′). The matrices of Green’s functions (4.29), (4.28) obey
the equations, similar to (4.23) with the corresponding matrices of memory functions. Then, taking
into account the relations like (4.27) for higher memory functions, we obtain a chain of equations
for the time correlation functions.
We have proposed one of the possible approaches to the description of nonequilibrium properties
of spatially inhomogeneous electron gas in the generalized “jellium” model. To this end the method
of nonequilibrium statistical operator by D.N. Zubarev has been used, based on the Bogolubov’s
ideas of the abbreviated description of the nonequilibrium states of the system. Our choice of the
mean densities of particles number, momentum and total energy as the parameters of a reduced
description of nonequilibrium properties of the spatially inhomogeneous electron gas provides re-
alization of the conservation laws. In this approach we obtained the nonequilibrium statistical
operator and the generalized transport equations for mean values of the number density, momen-
tum density and total energy density, which are coupled with the averaged Maxwell equations for
the electromagnetic field via equations for the mean vector and scalar potentials. The obtained
generalized transport equations take into account the memory effects and many-electron viscous
and heat processes. These equations can be applied to the description of both weakly and strongly
nonequilibrium states and to the calculation of spatially inhomogeneous mean values 〈ρ̂ (q; z)〉
t
,
〈p̂ (q; z)〉
t
and 〈ε̂ (q; z)〉
t
.
The obtained chain of equations interrelates the commutative time-dependent “density-density”,
“momentum-momentum”, “enthalpy-enthalpy” Green’s function and the reduced Green’s functi-
ons, constructed on the generalized transport kernels (dealing with viscosity, heat conductivity and
cross-correlation coefficient) as well as reduced Green’s function of the higher order memory func-
tions for weakly nonequilibrium spatially inhomogeneous electron gas. The chain of equations for
the Green’s functions contains the elements of the frequency matrix (4.15) and equilibrium correla-
tion functions (4.4), which could be expressed via interaction potentials and one- and two-electron
distribution functions, calculated in [31–34].
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Нерiвноважний статистичний оператор Зубарєва i функцiї
Грiна для неоднорiдного електронного газу
П.П.Костробiй1, Б.М.Маркович 1, А.I.Василенко2, М.В.Токарчук2,
Ю.К.Рудавський 1
1 Нацiональний унiверситет “Львiвська полiтехнiка”, Львiв 79013, вул. С.Бандери 12, Україна
2 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1
Отримано 21 червня 2006 р.
Дослiджуються нерiвноважнi властивостi просторово неоднорiдного електронного газу з викори-
станням методу нерiвноважного статистичного оператора Зубарєва. Отримано узагальненi рiвнян-
ня переносу для середнiх значень густин операторiв числа електронiв, їх iмпульсiв та повної енергiї
для опису як слабо, так i сильно нерiвноважних станiв системи. Отримано ланцюжок рiвнянь функцiй
Грiна, який зв’язує комутаторнi часовi функцiї Грiна “густина-густина”, “iмпульс-iмпульс”, “ентальпiя-
ентальпiя” iз приведеними функцiями Грiна узагальнених коефiцiєнтiв переносу в’язкостi, теплопро-
вiдностi (включаючи перехреснi в’язко-тепловi коефiцiєнти переносу) та iз приведеними функцiями
Грiна вищих функцiй пам’ятi для слабонерiвноважного просторово неоднорiдного електронного га-
зу.
Ключовi слова: нерiвноважний статистичний оператор, функцiї Грiна, узагальненi коефiцiєнти
переносу, неоднорiдний електронний газ.
PACS: : 05.60.Gg, 05.70.Np, 63.10.+a, 68.43, 82.20.Xr
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