A consistent description of kinetics and hydrodynamics of quantum Bose-systems
A consistent approach to the description of kinetics and hydrodynamics of many-Boson systems is proposed. The generalized transport equations for strongly and weakly nonequilibrium Bose systems are obtained. Here we use the method of nonequilibrium statistical operator by D.N. Zubarev. New equations...
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Інститут фізики конденсованих систем НАН України
2004
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| Цитувати: | A consistent description of kinetics and hydrodynamics of quantum Bose-systems / P.A. Hlushak, M.V. Tokarchuk // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 639–660. — Бібліогр.: 47 назв. — англ. |
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irk-123456789-1190332017-06-04T03:02:36Z A consistent description of kinetics and hydrodynamics of quantum Bose-systems Hlushak, P.A. Tokarchuk, M.V. A consistent approach to the description of kinetics and hydrodynamics of many-Boson systems is proposed. The generalized transport equations for strongly and weakly nonequilibrium Bose systems are obtained. Here we use the method of nonequilibrium statistical operator by D.N. Zubarev. New equations for the time distribution function of the quantum Bose system with a separate contribution from both the kinetic and potential energies of particle interactions are obtained. The generalized transport coefficients are determined accounting for the consistent description of kinetic and hydrodynamic processes. Запропоновано узгоджений підхід для опису кінетики та гідродинаміки багатобозонних систем. Отримано узагальнені рівняння переносу для сильно і слабо нерівноважних бозе-систем з використанням методу нерівноважного статистичного оператора Д.М. Зубарєва. Отримано нові рівняння для часових кореляційних функцій з виділеними внесками від кінетичної енергії і потенціальної енергій взаємодії частинок. Отримано узагальнені коефіцієнти переносу з урахуванням узгодженого опису кінетичних і гідродинамічних процесів. 2004 Article A consistent description of kinetics and hydrodynamics of quantum Bose-systems / P.A. Hlushak, M.V. Tokarchuk // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 639–660. — Бібліогр.: 47 назв. — англ. 1607-324X PACS: 67.40.-w, 47.37.+q DOI:10.5488/CMP.7.3.639 http://dspace.nbuv.gov.ua/handle/123456789/119033 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
A consistent approach to the description of kinetics and hydrodynamics of many-Boson systems is proposed. The generalized transport equations for strongly and weakly nonequilibrium Bose systems are obtained. Here we use the method of nonequilibrium statistical operator by D.N. Zubarev. New equations for the time distribution function of the quantum Bose system with a separate contribution from both the kinetic and potential energies of particle interactions are obtained. The generalized transport coefficients are determined accounting for the consistent description of kinetic and hydrodynamic processes. |
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Article |
| author |
Hlushak, P.A. Tokarchuk, M.V. |
| spellingShingle |
Hlushak, P.A. Tokarchuk, M.V. A consistent description of kinetics and hydrodynamics of quantum Bose-systems Condensed Matter Physics |
| author_facet |
Hlushak, P.A. Tokarchuk, M.V. |
| author_sort |
Hlushak, P.A. |
| title |
A consistent description of kinetics and hydrodynamics of quantum Bose-systems |
| title_short |
A consistent description of kinetics and hydrodynamics of quantum Bose-systems |
| title_full |
A consistent description of kinetics and hydrodynamics of quantum Bose-systems |
| title_fullStr |
A consistent description of kinetics and hydrodynamics of quantum Bose-systems |
| title_full_unstemmed |
A consistent description of kinetics and hydrodynamics of quantum Bose-systems |
| title_sort |
consistent description of kinetics and hydrodynamics of quantum bose-systems |
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Інститут фізики конденсованих систем НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/119033 |
| citation_txt |
A consistent description of kinetics and hydrodynamics of quantum Bose-systems / P.A. Hlushak, M.V. Tokarchuk // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 639–660. — Бібліогр.: 47 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT hlushakpa aconsistentdescriptionofkineticsandhydrodynamicsofquantumbosesystems AT tokarchukmv aconsistentdescriptionofkineticsandhydrodynamicsofquantumbosesystems AT hlushakpa consistentdescriptionofkineticsandhydrodynamicsofquantumbosesystems AT tokarchukmv consistentdescriptionofkineticsandhydrodynamicsofquantumbosesystems |
| first_indexed |
2025-07-08T15:07:24Z |
| last_indexed |
2025-07-08T15:07:24Z |
| _version_ |
1837091779529146368 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 3(39), pp. 639–660
A consistent description of kinetics and
hydrodynamics of quantum
Bose-systems
P.A.Hlushak, M.V.Tokarchuk
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received August 8, 2004
A consistent approach to the description of kinetics and hydrodynamics of
many-Boson systems is proposed. The generalized transport equations for
strongly and weakly nonequilibrium Bose systems are obtained. Here we
use the method of nonequilibrium statistical operator by D.N. Zubarev. New
equations for the time distribution function of the quantum Bose system
with a separate contribution from both the kinetic and potential energies
of particle interactions are obtained. The generalized transport coefficients
are determined accounting for the consistent description of kinetic and hy-
drodynamic processes.
Key words: Bose system, helium, kinetics, hydrodynamics, correlation
function, transport coefficients
PACS: 67.40.-w, 47.37.+q
1. Introduction
The theoretical investigation of nonequilibrium properties of vapour helium and
their change at transition with the decrease of the temperature lower than Tc = 4.2 K
in a fluid state HeI, and lower than Tλ = 2.17 K in a liquid state HeII that is charac-
terized by superfluidity, remains an urgent issue in the modern statistical theory of
nonequilibrium processes of quantum systems. To construct a nonequilibrium statis-
tical theory capable of consistensly describing vapour, liquid and superfluid helium in
view of phase transitions is a real problem for every theorist interested in the unique
physical properties of helium. The quantum system of Bose particles serves as a
physical model in theoretical descriptions of both the equilibrium and nonequilibri-
um properties of real helium. In particular, many articles [1–16] are devoted to the
hydrodynamic description of normal and superfluid states of such a system. A brief
review of the results of investigations within the framework of linear hydrodynamics
c© P.A.Hlushak, M.V.Tokarchuk 639
P.A.Hlushak, M.V.Tokarchuk
has been carried out in an article by Tserkovnikov [16]. In papers [17–19] theoretical
approaches are proposed to the description of nonlinear hydrodynamic fluctuations
connected with the problem of calculating the dispersion for the kinetic transport
coefficients and a spectrum of collective modes in the low-frequency area for a su-
perfluid Bose liquid. Problems of building the kinetic equation for Bose systems
based on the microscopic approach were considered in papers [20,21]. For normal
Bose systems, the calculations of the collective mode spectrum (without accounting
for a thermal mode), dynamic structure factor, kinetic transport coefficients [9, see
the reference] are carried out based on the hydrodynamic or kinetic approaches.
Nevertheless, these results are valid only in the hydrodynamic area (i.e., small val-
ues of wave vector k and frequency ω). For superfluid helium, some papers [22–24]
were devoted to the investigation of the dynamic structure factor and spectrum of
collective excitations.
In papers [25,26], a generalized scheme for theoretical description of dynam-
ic properties of semiquantum helium has been proposed based on the method of
nonequilibrium statistical operator. Here the set of equations of generalized hydro-
dynamics is obtained and the thermal viscous model with kinetic and hydrodynami-
cal collective modes is analyzed in detail. The closed system of the equations for time
correlation functions is obtained using the Markovian approximation for transport
kernels. Using these equations the analysis of dynamic properties of semiquantum
helium is carried out at two values of temperature above the transition to a super-
fluid state. Similar investigations were performed in papers [27–29] for helium above
the point of the phase transition.
In general, there exists a hard problem in describing the Bose systems going
out from the hydrodynamic area to the area of intermediate values of k and ω,
where the kinetic and hydrodynamic processes are interdependent and should be
considered simultaneously. This is one of the urgent problems in the statistical theory
of nonequilibrium processes of transport in a quantum liquid. It should be noted that
in the paper by J.O.Tserkovnikov [30], a problem of building the linearized kinetic
equation for the Bose-system above critical temperature was considered using the
method of two-time Green functions [31,32].
The main step in this direction has been made when investigating semiquantum
helium [25,26]. A considerable success was achieved in papers [33–36] in which the
approach to the consistent description of kinetics and hydrodynamics of classical
dense gases and fluids is proposed based on the method of nonequilibrium statistical
operator by D.N.Zubarev [37,38]. In the present study, we apply this approach to a
consistent description of kinetics and hydrodynamics of many-particle Bose systems.
In the second part of the paper we shall obtain a nonequilibrium statistical oper-
ator of the system at the consistent description of kinetics and hydrodynamics using
the method of the nonequilibrium statistical operator. The quantum nonequilibrium
one-particle distribution function and the average value of the potential energy of
interaction (for which the closed system of the transport equations is obtained) have
been selected as parameters of this consistent description of a nonequilibrium state.
In the third part, the kinetics and hydrodynamics of weakly nonequilibrium
640
Kinetics and hydrodynamics of quantum Bose-systems
Bose gas are considered. Here the self-consistent transport equations are obtained
for parameters of an abbreviated description based on their solutions, the kinetic
equation for the quantum one-partial distribution function is written in a form
where the transport kernels contain a renormalization of kinetical correlations by
hydrodynamical ones. Moreover, we also obtain the system of equations for time
correlation functions of parameters of a consistent description. From these equations,
the dynamic structure factor of the system as well as the time correlation functions
related to momentum and energy fluctuations are determined.
2. The nonequilibrium statistical operator for Bose system
Let us consider a normal Bose system with the Hamiltonian
Ĥ =
∑
k
∑
p
p2
2m
â+
p−k
2
âp+k
2
+
1
2V
∑
k
∑
q
∑
p
ν(q)â+
p+q−k
2
ρ̂q âp−q−k
2
, (2.1)
where âp and â+
p are the Bose operators of annihilation and creation of particles in
the state with momentum p, ν(q) =
∫
exp(−iqr)Φ(|r|)dr is the Fourier-component
of the interaction potential between particles, V is the volume,
ρ̂q =
1√
N
∑
p
â+
p−q
2
âp+q
2
(2.2)
is the Fourier-component of the operator of number particles density, N is the total
number of particles.
The nonequilibrium state of such a quantum system is completely described by
the nonequilibrium statistical operator ρ̂(t) which satisfies the quantum Liouville
equation:
∂
∂t
ρ̂(t) + iL̂N ρ̂(t) = 0, (2.3)
where Liouville operator iL̂N Â = i/h̄ · [Â, Ĥ]. To solve the equation it is necessary
to set initial conditions. We shall use the method of nonequilibrium statistical op-
erator [37,38]. From the very beginning we consider the problem of selecting such
solutions to the equation (2.3) which correspond to the ideas of abbreviated descrip-
tion [38]. These solutions depend on time only through average values of the set of
observable quantities
〈
P̂m
〉t
and do not depend on the initial moment of time t0:
ρ̂(t) = ρ̂(. . .
〈
P̂m
〉t
. . .). Such solutions can be obtained by including an infinitesimal
source in the right-hand side of Liouville equation (2.3) [37,38]:
∂
∂t
ρ̂(t) + iL̂N ρ̂(t) = −ε(ρ̂(t) − ρ̂q(t)), (2.4)
where ε → +0 after the limiting thermodynamic transition. The sourse breaks the
symmetry of the equation with respect to t → −t and selects retarded solutions
which correspond to an abbreviated description of the nonequilibrium system. The
641
P.A.Hlushak, M.V.Tokarchuk
quasiequilibrium statistical operator ρ̂q(t) is defined from the condition of an extreme
of the information entropy of the system at the conservation of the normalization
condition
Sp ρ̂q(t) = 1 (2.5)
and at fixed values of the quantities
〈
P̂m
〉t
(parameters of an abbreviated descrip-
tion).
When investigating a hydrodynamical nonequilibrium state of the normal Bose
liquid, which is characterized by transport of energy, momentum and mass, the
observable quantities such as the average values of energy density 〈ε̂q〉t, momentum〈
P̂q
〉t
, and particle number 〈ρ̂q〉t [10,14,16] are chosen as the parameters of an
abbreviated description. The averaging is defined as
〈(. . . )〉t = Sp[(. . . ) ρ̂(t)]. (2.6)
The Fourier-components of the energy density and momentum density
ε̂q =
1√
N
∑
p
(
p2
2m
− q2
8m
)
â+
p−q
2
âp+q
2
+
1
2V
∑
p
∑
k
ν(k)â+
p+q−k
2
ρ̂k âp−q−k
2
, (2.7)
P̂q =
1√
N
∑
p
p â+
p−q
2
âp+q
2
(2.8)
together with the number of particles density (2.2) satisfy local conservation laws.
Their averaged values satisfy the equations of the generalized hydrodynamics of
quantum systems (in the linear approximation of the equations of the molecular
hydrodynamics). The important feature of the densities of energy, momentum and
number of particles, which is locally conserved, is that they are defined through the
Klimontovich operator of the phase particles number density n̂q(p):
n̂q(p) = â+
p−q
2
âp+q
2
, (2.9)
ρ̂q =
1√
N
∑
p
n̂q(p), (2.10)
P̂q =
1√
N
∑
p
pn̂q(p), (2.11)
ε̂kin
q =
1√
N
∑
p
(
p2
2m
− q2
8m
)
n̂q(p), (2.12)
ε̂int
q =
1
2V
∑
p
∑
p′
∑
k
ν(k)â+
p+k−q
2
n̂k(p′) â
p−k−q
2
, (2.13)
where ε̂kin
q and ε̂int
q are the Fourier-components of operators of kinetic and potential
energy density. The averaged values for these quantities can be written as:
〈ρ̂q〉t =
1√
N
∑
p
f1(p,q, t), (2.14)
642
Kinetics and hydrodynamics of quantum Bose-systems
〈
P̂q
〉t
=
1√
N
∑
p
pf1(p,q, t), (2.15)
〈
ε̂kin
q
〉t
=
1√
N
∑
p
(
p2
2m
− q2
8m
)
f1(p,q, t). (2.16)
The one-particle distribution function f(p,q, t) = 〈n̂q(p)〉t satisfies the kinetic equa-
tion for the quantum Bose system. On the other hand (2.14)–(2.16), the average
value of the potential energy is defined through the quantum two-particle nonequi-
librium distribution function:
〈
ε̂int
q
〉t
=
1
2V
∑
p
∑
p′
∑
k
ν(k)f2
(
p +
k − q
2
;p′;p− k − q
2
, t
)
, (2.17)
where
f2
(
p +
k − q
2
;p′;p− k − q
2
; t
)
=
〈
â+
p+k−q
2
n̂k(p′) â
p−k−q
2
〉t
. (2.18)
At this stage there is a problem of a consistent description of kinetics and hy-
drodynamics of the quantum Bose system. For the hydrodynamical description of a
nonequilibrium state of the system it is enough to include in the set of parameters of
an abbreviated description the average values of particle number 〈ρ̂q〉t, momentum〈
P̂q
〉t
and full energy 〈ε̂q〉t. On the other hand, the quantum one-particle distribu-
tion function, which satisfies the kinetic equation, is a characteristic parameter for
the kinetic description of the nonequilibrium state of the system. The agreement
between kinetics and hydrodynamics for very dilute Bose gas does not cause any
problem because in this case the density is a small parameter. Therefore, only the
quantum one-particle distribution function f1(p,q, t) can be chosen for a parameter
of an abbreviated description.
At transition to quantum Bose liquids, the contribution of collective correla-
tions, which are described by average potential energy of interaction
〈
ε̂int
q
〉t
(2.17), is
more important than the one-particle correlations connected with f1(p,q, t). From
this fact it follows that for a consistent description of kinetics and hydrodynam-
ics of a Bose liquid, the quantum one-particle nonequilibrium distribution function
f1(p,q, t) and the average potential energy of interaction
〈
ε̂int
q
〉t
are indispensable in
order to be choosen as parameters of an abbreviated description for a nonequilibrium
state. Similar problems of the consistent description of kinetics and hydrodynamics
of classical dense gases and liquids, as was already noted above, were considered in
papers [33–35]. Therefore, using papers [33,35,36,44], we shall find the quasiequi-
librium statistical operator, which has been entered in (2.4), from the condition of
the extremum of information entropy at the conservation of normalization condition
(2.5) for fixed values of 〈n̂q(p)〉t = f1(p,q, t) and
〈
ε̂int
q
〉t
:
ρ̂q(t) = exp
{
− Φ(t) −
∑
q
β−q(t) ε̂
int
q −
∑
q
∑
p
γ−q(p; t) n̂q(p)
}
. (2.19)
643
P.A.Hlushak, M.V.Tokarchuk
The Lagrangian multipliers β−q(t), γ−q(p; t) are defined from the self-consistent
conditions:
〈
ε̂int
q
〉t
=
〈
ε̂int
q
〉t
q
, 〈n̂q(p)〉t = 〈n̂q(p)〉tq , (2.20)
〈(. . . )〉tq = Sp[(. . . ) ρq(t)].
The Massieu-Planck functional
Φ(t) = ln Sp exp
{
−
∑
q
β−q(t) ε̂
int
q −
∑
q
∑
p
γ−q(p; t) n̂q(p)
}
(2.21)
is determined from the normalization condition (2.5).
At the given quasiequilibrium operator ρ̂q(t) (2.19) we shall find the nonequi-
librium statistical operator ρ̂(t) that satisfies the quantum Liouville equation it the
presence of a source. For this purpose we shall write down the equation (2.4) as [38]:
[
∂
∂t
+ (1 − Pq(t)) iL̂N + ε
]
(ρ̂(t) − ρ̂q(t)) = − (1 − Pq(t)) iL̂N ρ̂q(t), (2.22)
where the generalized Kawasaki-Gunton projection operator acts only on statistical
operators
Pq(t)ρ̂
′ =
[
ρ̂q(t) −
∑
q
δρ̂q(t)
δ
〈
ε̂int
q
〉t
〈
ε̂int
q
〉t −
∑
q
∑
p
δρ̂q(t)
δ 〈n̂q(p)〉t
〈n̂q(p)〉t
]
Sp ρ̂′
+
∑
q
δρ̂q(t)
δ
〈
ε̂int
q
〉t Sp(ε̂int
q ρ̂′) +
∑
q
∑
p
δρ̂q(t)
δ 〈n̂q(p)〉t
Sp(n̂q(p)ρ̂′) (2.23)
and has the following properties:
Pq(t)ρ̂
′ = ρ̂q(t), Pq(t)ρ̂
′
q(t
′) = ρ̂q(t), Pq(t)Pq(t
′) = Pq(t).
The formal solution of the equation (2.22) is of the form:
ρ̂(t) = ρ̂q(t) −
t∫
−∞
dt′ exp{ε(t′ − t)} Tq(t, t
′)
(
1 − Pq(t
′)
)
iL̂N ρ̂q(t
′), (2.24)
where
Tq(t, t
′) = exp+
−
t∫
t′
dt′
(
1 − Pq(t
′)
)
iL̂N
(2.25)
is the generalized evolution operator which takes into account the projection. Fur-
ther, we shall act by operators (1 − Pq(t
′))iL̂N on ρ̂q(t
′) in the right-hand side of
(2.24). As a result, we obtain
(
1 − Pq(t
′)
)
iL̂N ρ̂q(t
′) = −
1∫
0
dτ(ρ̂q(t))
τ
{∑
q
β−q(t)I
int
ε (q, t)
+
∑
q
∑
q
γ−qp(t)In(p,q, t)
}
(ρ̂q(t))
1−τ , (2.26)
644
Kinetics and hydrodynamics of quantum Bose-systems
where the generalized flows are:
I int
ε (q, t) =
(
1 − P (t)
)
iL̂N ε̂
int
q , (2.27)
In(p,q, t) =
(
1 − P (t)
)
iL̂N n̂
int
q (p). (2.28)
Expressions (2.27) and (2.28) contain the generalized Mori projection operator:
P (t)b̂ = 〈b̂〉tq +
∑
q
δ〈b̂〉tq
δ〈ε̂int
q 〉t (ε̂
int
q −〈ε̂int
q 〉t)+
∑
q
∑
p
δ〈b̂〉tq
δ〈n̂q(p)〉t (n̂q(p)−〈n̂q(p)〉t), (2.29)
which acts only on operators of physical quantities and has the properties:
P (t)n̂q(p) = n̂q(p), P (t)ε̂int
q = ε̂int
q , P (t)P (t′) = P (t).
Let us substitute (2.26) in (2.24). Then for the nonequilibrium statistical operator
of Bose system we shall write:
ρ̂(t) = ρq(t) +
∑
q
t∫
−∞
dt′ exp{ε(t′ − t)} Tq(t, t
′)
×
1∫
0
dτ(ρq(t
′))τI int
ε (q, t′)(ρq(t
′))1−τβ−q(t
′)
+
∑
q
∑
p
t∫
−∞
dt′ exp{ε(t′ − t)} Tq(t, t
′)
×
1∫
0
dτ(ρq(t
′))τIn(p,q, t′)(ρq(t
′))1−τγ−q(p, t
′). (2.30)
The nonequilibrium statistical operator (2.30) is obtained at a consistent de-
scription of kinetics and hydrodynamics of the Bose system. Using it we shall find
the non-closed system of transport equations for the parameters of an abbreviated
description f1(p,q, t), 〈ε̂int
q 〉t. For this purpose we use the identities:
∂
∂t
f1(p,q, t) =
∂
∂t
〈n̂q(p)〉t = 〈 ˙̂nq(p)〉tq + 〈In(p,q)〉t,
∂
∂t
〈ε̂int
q 〉t = 〈 ˙̂εint
q 〉t = 〈 ˙̂εint
q 〉tq + 〈I int
ε (q)〉t, (2.31)
where
˙̂nq(p) = iL̂N n̂q(p), ˙̂ε
int
q = iL̂N ε̂
int
q . (2.32)
Now we shall perform the averaging in right parts (2.31) with the nonequilibri-
um statistical operator (2.30). As a result, we shall find the set of equations for
645
P.A.Hlushak, M.V.Tokarchuk
the nonequilibrium distribution function f1(p,q, t) and average value of interaction
energy density 〈ε̂int
q 〉t:
∂
∂t
〈n̂q(p)〉t = 〈 ˙̂nq(p)〉tq +
∑
q′
t∫
−∞
dt′ exp{ε(t′ − t)}ϕint
nε (q,p,q
′, t, t′)β−q(t
′)
+
∑
q′
∑
p′
t∫
−∞
dt′ exp{ε(t′ − t)}ϕnn(q,p,q
′,q′, t, t′)γ−q(p
′, t′),
∂
∂t
〈ε̂int
q 〉t = 〈 ˙̂εint
q 〉tq +
∑
q′
t∫
−∞
dt′ exp{ε(t′ − t)}ϕint
εε (q,q′, t, t′)β−q(t
′)
+
∑
q′
∑
p′
t∫
−∞
dt′ exp{ε(t′ − t)}ϕint
εn (q,q′,p′, t, t′)γ−q(p
′, t′). (2.33)
In the equations (2.33) the generalized transport kernels are introduced, which de-
scribe dissipative processes in the system:
ϕint
nε (q,p,q
′, t, t′) = Sp
[
In(q,p, t)Tq(t, t
′)
1∫
0
dτρτ
q(t
′)I int
ε (q′, t′)ρ1−τ
q (t′)
]
, (2.34)
ϕint
εn (q,q′,p, t, t′) = Sp
[
I int
ε (q, t′)Tq(t, t
′)
1∫
0
dτρτ
q(t
′)In(p,q′, t)ρ1−τ
q (t′)
]
, (2.35)
ϕnn(q,p,q
′,p′, t, t′) = Sp
[
In(q,p, t)Tq(t, t
′)
1∫
0
dτρτ
q(t
′)In(p′,q′, t)ρ1−τ
q (t′)
]
, (2.36)
ϕint
εε (q,q′, t, t′) = Sp
[
I int
ε (q, t′)Tq(t, t
′)
1∫
0
dτρτ
q(t
′)I int
ε (q′, t′)ρ1−τ
q (t′)
]
. (2.37)
The system of equations (2.33) for the one-particle distribution function and aver-
age density of potential energy presents a strongly nonlinear system and can be used
for the description of both strongly and weakly nonequilibrium states of the Bose
system at a consistent description of kinetics and hydrodynamics. These transport
equations for the Bose systems are new. Projecting them on the values of compo-
nents of the vector Ψ(p) =
(
1,p, p2/2m − q2/8m
)
, according to (2.14)–(2.16), we
shall receive the equations of nonlinear hydrodynamics in which the transport pro-
cesses of kinetic and potential parts of energy are described by two interdependent
equations.
It is obvious that such equations of hydrodynamics of nonlinear processes enable
us to describe more in detail the processes of mutual transformation of kinetic and
potential energy of particles at the transition from a vapour to a liquid state at the
change of density, pressure and temperatures.
The system of the transport equations becomes considerably simpler and is closed
for weakly nonequilibrium processes. We shall consider such a case in the next
section.
646
Kinetics and hydrodynamics of quantum Bose-systems
3. Kinetics and hydrodynamics of nonequilibrium state near
equilibrium
For kinetics and hydrodynamics of nonequilibrium processes which are close to an
equilibrium state, the parameters of an abbreviated description 〈n̂q(p)〉t, 〈ε̂int
q 〉t slow-
ly change in space and time and slightly differ from their equilibrium values. Then it
is sufficient to consider the deviations of parameters βq(t), γq(p, t) from equilibrium
values using the linear approximation. In this approximation, the generalized trans-
port equations (2.33) [39] transform into a transport equation for fk(p; t) = 〈n̂k(p)〉t,
hint
k (t) = 〈ĥint
k 〉t
∂
∂t
fk(p; t) +
ik · p
m
fk(p; t) =
= − ik · p
m
nf0(p)c2(k)
∑
p′
fk(p
′; t) + iΩnh(k,p)hint
k (t)
−
∑
p′
t∫
−∞
dt′ eε(t′−t)ϕnn(k,p,p′; t, t′)fk(p
′; t′)
−
t∫
−∞
dt′ eε(t′−t)ϕnh(k,p; t, t′)hint
k (t′), (3.1)
∂
∂t
hint
k (t) =
∑
p
iΩhn(k,p)fk(p; t) −
∑
p
t∫
−∞
dt′ eε(t′−t)ϕhn(k,p; t, t′)fk(p; t′)
−
t∫
−∞
dt′ eε(t′−t)ϕhh(k; t, t′)hint
k (t′), (3.2)
where iΩnh(k,p), iΩhn(k,p) are normalized static correlation functions:
iΩnh(k,p ) = 〈 ˙̂nk(p)ĥint
−k〉0Φ−1
hh (k), (3.3)
iΩhn(k,p) =
∑
p′
〈 ˙̂
h
int
k n̂−k(p
′)〉0Φ−1
k (p′,p), (3.4)
where
ĥint
k = ε̂k −
∫
dpdp′〈ε̂kn̂−k(p
′)〉0Φ−1
k (p′,p)n̂k(p) = ε̂int
k − 〈ε̂int
k n̂−k〉0S−1
2 (k)n̂k , (3.5)
ε̂int
k =
1
2
N∑
l 6=j=1
Φ(|rlj|)e−ik·rl, n̂k =
N∑
l=1
e−ik·rl (3.6)
are the Fourier-components of densities for the interaction energy and the number of
particles, respectively. Further, it is more convenient to use instead of the dynamical
variable of energy ε̂k the variable ĥint
k (3.5) which is orthogonal to n̂k(p) by means
of the equality:
〈ĥint
k n̂−k〉0 = 0. (3.7)
647
P.A.Hlushak, M.V.Tokarchuk
From the structure of the dynamical variable ĥint
k (3.3) it can be seen that it corre-
sponds to a potential part of the Fourier-component of the generalized enthalpy ĥk
which is introduced in the molecular hydrodynamics [25,26]:
ĥk = ε̂k − 〈ε̂kn̂−k〉0n̂k = ĥkin
k + ĥint
k , (3.8)
where
ĥkin
k = ε̂kin
k − 〈ε̂kin
k n̂−k〉0S−1
2 (k)n̂k (3.9)
is a kinetic part of the generalized enthalpy,
ε̂kin
k =
N∑
l=1
p2
l
2m
e−k·rl
is the Fourier-component of the kinetic energy density. S2 = 〈n̂kn̂−k〉0 is the static
structure factor of the system.
Φ−1
k (p′′,p′) =
δ(p′′ − p′)
nf0(p′′)
− c2(k), (3.10)
where c2(k) denotes a direct correlation function which is connected with the cor-
relation function h2(k) as: h2(k) = c2(k)[1 − nc2(k)]−1.
ϕnn(k,p,p′; t, t′) =
∑
p′′
〈In(k,p)T0(t, t
′)In(−k,p′′)〉0Φ−1
k (p′′,p′), (3.11)
ϕhn(k,p; t, t′) =
∑
p′
〈I int
h (k)T0(t, t
′)In(−k,p′)〉0Φ−1
k (p′,p), (3.12)
ϕnh(k,p; t, t′) = 〈In(k,p)T0(t, t
′)I int
h (−k)〉0Φ−1
hh (k), (3.13)
ϕhh(k; t, t′) = 〈I int
h (k)T0(t, t
′)I int
h (−k)〉0Φ−1
hh (k) (3.14)
are the generalized transport kernels (memory functions) which describe kinetic and
hydrodynamic processes.
In(k,p) = (1 −P0) ˙̂nk(p), (3.15)
I int
h (k) = (1 −P0)
˙̂
hint
k (3.16)
are the generalized fluxes in the linear approximation, ˙̂nk(p) = iLN n̂k(p),
˙̂
hint
k =
iLN ĥ
int
k , T0(t, t
′) = e(t−t′)(1−P0)iLN is the time evolution operator with the projection
operator P0 which acts on the dynamical variables Âk
P0Âk′ =
∑
k′
〈Âk′ĥint
−k〉0Φ−1
hh (k)ĥint
k +
∑
k
∑
pp′
〈Âk′ n̂−k(p
′)〉0Φ−1
k (p′,p)n̂k(p). (3.17)
The system of transport equations (3.1), (3.2) is closed. We shall use the Laplace
transform with respect to time, assuming that at t > 0 the quantities fk(p; t = 0),
hint
k (t = 0) are known
A(z) = i
∫ ∞
0
dt eiztA(t), z = ω + iε, ε→ +0. (3.18)
648
Kinetics and hydrodynamics of quantum Bose-systems
Then, equations (3.1) and (3.2) are presented in the form:
zfk(p; z) +
ik · p
m
fk(p; z) = − ik · p
m
nf0(p)c2(k)
×
∑
p′
fk(p
′; z) + Σnh(k,p; z)hint
k (z)
−
∑
p′
ϕnn(k,p,p
′; z)fk(p
′; z) + fk(p; t = 0), (3.19)
zhint
k (z) =
∑
p′
Σhn(k,p′; z)fk(p
′; z) − ϕhh(k; z)hint
k (z) + hint
k (t = 0), (3.20)
Σnh(k,p; z) = iΩnh(k,p) − ϕnh(k,p; z), (3.21)
Σhn(k,p; z) = iΩhn(k,p) − ϕhn(k,p; z). (3.22)
In the next subsection, based on the system of transport equations for Fourier
components of the nonequilibrium one-particle distribution function and the poten-
tial part of enthalpy (3.21), (3.22), we shall obtain equations for time correlation
functions. We shall also investigate the spectrum of collective excitations and the
structure of generalized transport coefficients.
4. Time correlation functions, collective modes and genera l-
ized transport coefficients
Using combined equations (3.1), (3.2) one obtains a system for time correlation
functions:
Φnn(k,p,p′; t) =
∑
p′′
〈n̂k(p; t)n̂−k(p
′′; 0)〉0Φ−1
k (p′′,p′), (4.1)
Φhn(k,p; t) =
∑
p′
〈ĥint
k (t)n̂−k(p
′; 0)〉0Φ−1
k (p′,p), (4.2)
Φnh(k,p; t) = 〈n̂k(p; t)ĥint
−k(0)〉0Φ−1
hh (k), (4.3)
Φhh(k; t) = 〈ĥint
k (t)ĥint
−k(0)〉0Φ−1
hh (k), (4.4)
where n̂k(p; t) = e−iLN tn̂k(p; 0), ĥint
k (t) = e−iLN tĥint
k (0).
One uses the Fourier transform with respect to time
〈a〉ω =
∫ ∞
−∞
dt eiωt〈a〉t.
Then we write the system of equations (4.1), (4.2) in the form:
−iω〈n̂k(p)〉ω =
∑
p′
Σnn(k,p,p′;ω + iε)〈n̂k(p
′)〉ω + Σnh(k,p;ω + iε)〈ĥint
k 〉ω , (4.5)
−iω〈ĥint
k 〉ω =
∑
p′
Σhn(k,p′;ω + iε)〈n̂k(p
′)〉ω − ϕhh(k;ω + iε)〈ĥint
k 〉ω , (4.6)
649
P.A.Hlushak, M.V.Tokarchuk
where
Σnn(k,p,p′;ω + iε) = iΩnn(k,p,p′) − ϕnn(k,p,p′;ω + iε), (4.7)
Σnh(k,p;ω + iε) = iΩnh(k,p) − ϕnh(k,p;ω + iε), (4.8)
Σhn(k,p;ω + iε) = iΩhn(k,p) − ϕhn(k,p;ω + iε). (4.9)
It is more convenient to present the system of equations (4.5), (4.6) in a matrix
form:
−iω〈ãk〉ω = Σ̃(k;ω + iε)〈ãk〉ω , (4.10)
where ãk = col(n̂k(p), ĥint
k ) is a vector-column and
Σ̃(k;ω + iε) =
∑
p′ Σnn(k,p,p′;ω + iε) Σnh(k,p;ω + iε)
∑
p′ Σhn(k,p′;ω + iε) −ϕhh(k;ω + iε)
, (4.11)
Σ̃(k;ω + iε) =
∫ ∞
0
dt ei(ω+iε)tΣ̃(k; t).
Now, one uses the solution to the Liouville equation in approximation [40,41] without
introducing the projection operator Pq(t):
%
(
xN ; t
)
= %q
(
xN ; t
)
−
t∫
−∞
dt′ eε(t′−t)eiLN (t′−t)
(
∂
∂t
′
+ iLN
)
%q
(
xN ; t′
)
.
Then, from the self-consistency conditions 〈ãk〉t = 〈ãk〉tq we obtain a system of equa-
tions which connects the average values 〈n̂k(p)〉ω and 〈ĥint
k 〉ω with spectral functions
of time correlation functions:
iωΦ̃(k;ω + iε)〈ãk〉ω =
[
Φ̃(k) − i(ω + iε)Φ̃(k;ω + iε)
]
〈ãk〉ω , (4.12)
where
Φ̃(k;ω + iε) =
∑
p′ Φnn(k,p,p′;ω + iε) Φint
nh(k,p;ω + iε)
∑
p′ Φint
hn(k,p′;ω + iε) −Φint
hh(k;ω + iε)
.
Let us multiply equation (4.10) by the matrix Φ̃(k;ω + iε) and compare the result
with equation (4.12). So we find
zΦ̃(k; z) = Σ̃(k; z)Φ̃(k; z) − Φ̃(k), z = ω + iε,
Φ̃(k; z) = i
∫ ∞
0
dt eiztΦ̃(k; t), ε→ 0 (4.13)
or in an explicit form:
zΦnn(k,p,p′; z) =
∑
p′′
Σnn(k,p,p′′; z)Φnn(k,p′′,p′; z)
+ Σnh(k,p; z)Φint
hn(k,p′; z) − Φnn(k,p,p′), (4.14)
650
Kinetics and hydrodynamics of quantum Bose-systems
zΦint
nh(k,p; z) =
∑
p′′
Σnn(k,p,p′′; z)Φint
nh(k,p′′; z)
+ Σnh(k,p; z)Φint
hh(k; z), (4.15)
zΦint
hn(k,p′; z) =
∑
p′′
Σhn(k,p′′; z)Φint
nn(k,p′′,p′; z)
− ϕhh(k; z)Φint
hn(k,p′; z), (4.16)
zΦint
hh(k; z) =
∑
p′′
Σhn(k,p′′; z)Φint
nh(k,p′′; z)
− ϕhh(k; z)Φint
hh(k; z) − Φhh(k), (4.17)
where the condition Φhn(k,p′) = Φnh(k,p) = 0 is taken into account.
The system of equations (4.14)–(4.17) for time correlation functions of weakly
nonequilibrium Bose system is obtained based on a consistent description of ki-
netics and hydrodynamics. These are new equations for the quantum Bose-system.
Projecting them on eigenvalues (1,p) of quantum one-partial distribution function
f1(p,q, t) we shall obtain the corresponding system of equations for time correlation
functions which can be derived based on the equations of molecular hydrodynamics
or the method of Green functions [10] (in view of the connection of time correlation
functions with corresponding Green time functions).
Moreover, designing the system of equations (4.14)–(4.17) for higher moments
of the quantum one-partial distribution function results in the corresponding sys-
tems of hydrodynamical equations for the time correlation functions related to the
densities of operators of particle number, momentum, enthalpy, generalized tensor
of viscous tension and generalized flow of enthalpy. Similar equations for the Green
time functions have been obtained in papers by Tserkovnikov [10,13,15]. A princi-
pal interest in such an approach is the study of collective modes and generalized
transport coefficients such as viscosity and thermal conductivity of quantum Bose
systems.
In order to solve the system of equations (4.14)–(4.17) we also apply the projec-
tion procedure [42,43]. Let us introduce the dimensionless momentum ξ = k(mv0)
−1,
v2
0 = (mβ)−1. Then the system of equations (4.13)–(4.17) can be rewritten in the
matrix form [43]:
zΦ̃(k; ξ, ξ′; z) − Σ̃(k; ξ, ξ′′; z)Φ̃(k; ξ′′, ξ′; z) = −Φ̃(k; ξ, ξ′), (4.18)
where it is clear that the integration must be performed with respect to the repeating
indices ξ′′. Further, let us introduce the scalar product of two functions, φ(ξ) and
ψ(ξ), as
〈φ|ψ〉 =
∑
ξ
φ∗(ξ)f0(ξ)ψ(ξ). (4.19)
Then, the matrix element for some “operator” M can be determined as
〈φ|M |ψ〉 =
∑
ξ
dξ′ φ∗(ξ)M(ξ, ξ′)f0(ξ
′)ψ(ξ′). (4.20)
651
P.A.Hlushak, M.V.Tokarchuk
Let φ(ξ) = {φµ(ξ)} be the orthogonalized basis of functions with the weight f0(ξ),
so that the following condition is satisfied:
〈φν |φµ〉 = δνµ,
∑
ν
|φν〉〈φν| = 1, (4.21)
where
φµ(ξ) = φlmn(ξ) = (l!m!n!)−1/2H̄l(ξx)H̄m(ξy)H̄n(ξz), (4.22)
H̄l(ξ) = 2−l/2Hl(ξ/2), Hl(ξ) is a Hermite polynomial. Then, each function in the
system of equations (4.18), which depends on momentum variables ξ, ξ′, can be
expanded over functions φµ(ξ) in the series:
Φ̃(k; ξ, ξ′; z) =
∑
ν,µ
φ∗
ν(ξ)Φ̃νµ(k; z)φµ(ξ′)f0(ξ
′), (4.23)
Σ̃(k; ξ, ξ′; z) =
∑
ν,µ
φ∗
ν(ξ)Σ̃νµ(k; z)φµ(ξ′)f0(ξ
′), (4.24)
where
Φ̃νµ(k; z) = 〈φν |Φ̃(k; ξ, ξ′; z)|φµ〉 =
∑
ξ′
φ∗
ν(ξ)f0(ξ)Φ̃(k; ξ, ξ′; z)φµ(ξ′), (4.25)
Σ̃νµ(k; z) = 〈φν |Σ̃(k; ξ, ξ′; z)|φµ〉 =
∑
ξ′
φ∗
ν(ξ)f0(ξ)Σ̃(k; ξ, ξ′; z)φµ(ξ′). (4.26)
Let us substitute expansions (4.23)–(4.26) into equation (4.18). As a result, one
obtains:
zΦ̃νµ(k; z) −
∑
γ
Σ̃νγ(k; z)Φ̃γµ(k; z) = −Φ̃νµ(k). (4.27)
In actual calculations, a finite number of functions from the set φν(ξ) is used. Taking
into account this fact let us introduce the projection operator P which projects
arbitrary functions ψ(ξ) onto a finite set of functions φµ(ξ):
P =
n∑
ν=1
|φν〉〈φν | = 1 −Q, P 〈ψ| =
n∑
ν=1
〈ψ|φν〉〈φν|. (4.28)
Here n denotes a finite number of functions. Then, from (4.27) we obtain a system
of equations for a finite set of functions φµ(ξ),
n∑
γ=1
[
zδ̄νγ − iΩ̃νγ(k) + D̃νγ(k; z)
]
Φ̃γµ(k; z) = −Φ̃νµ(k), (4.29)
where
D̃νµ(k; z) = 〈φν|ϕ̃(k; z) + Σ̃(k; z)Q
[
z −QΣ̃(k; z)Q
]−1
QΣ̃(k; z)|φµ〉 (4.30)
are generalized hydrodynamic transport kernels and
iΩ̃νµ(k) = 〈φν|iΩ̃(k)|φµ〉 (4.31)
652
Kinetics and hydrodynamics of quantum Bose-systems
is a frequency matrix. Note that matrices iΩ̃(k) and ϕ̃(k; z) are defined according
to (4.3), (4.4) and (4.11)–(4.14).
Let us find solutions to the system of equations (4.19) in the hydrodynamic region
when a set of functions φµ(ξ) presents five moments of a one-particle distribution
function:
φ1(ξ) = 1, φ2(ξ) = ξz , φ3(ξ) =
1√
6
(ξ2 − 3),
φ4(ξ) = ξx , φ5(ξ) = ξy . (4.32)
Then, the following relations are fulfilled:
〈1|n̂k(ξ) =
∑
ξ
n̂k(ξ) = n̂k ,
〈ξγ|n̂k(ξ) =
∑
ξ
n̂k(ξ) ξγ = p̂γ
k ,
〈6−1/2(ξ2 − 3)|n̂k(ξ) =
∑
ξ
n̂k(ξ) 6−1/2(ξ2 − 3) = Êkin
k − 3n̂kβ
−1 = ĥkin
k , (4.33)
for the Fourier components of densities for the number of particles, momentum
and the kinetic part of generalized enthalpy (for the Bose system). Besides that, the
microscopic conservation laws for densities of the number of particles and momentum
can be written in the form:
〈1| ˙̂nk(ξ) = −ikγ p̂
γ
km
−1 ,
〈ξα| ˙̂nk(ξ) = −ikγT̂k
γα ,
(4.34)
where T̂k
γα is a Fourier component of the stress tensor.
If we choose the direction of wavevector k along oz-axis, then φν(ξ), ν = 1, 2, 3
will correspond to longitudinal modes, whereas φν(ξ) at ν = 4, 5 they will be related
to transverse modes.
From the system of equations (4.29), at ν = 4, 5, φ4(ξ) = ξx, φ5(ξ) = ξy, one
obtains an equation for the Fourier component of the time correlation function
connected with the transverse component of the momentum density Φ44(k; z). From
this equation one finds:
Φ44(k; z) = Φ⊥
pp(k; z) = − 1
z +D⊥
pp(k; z)
, (4.35)
where
Φ⊥
pp(k; z) = 〈ξx|Φnn(k, ξ, ξ′; z)|ξ′x〉, (4.36)
D⊥
pp(k; z) = D⊥(kin)
pp (k; z) +D⊥(int)
pp (k; z), (4.37)
D⊥(kin)
pp (k; z) = 〈ξx|ϕnn(k, ξ, ξ
′; z)|ξ′x〉, (4.38)
D⊥(int)
pp (k; z) =
〈
ξx|
[
Σ̃(k, ξ, ξ′; z)Q
[
z −QΣ̃(k, ξ, ξ′; z)Q
]−1
QΣ̃(k, ξ, ξ′; z)
]
nn
|ξ′x
〉
,
(4.39)
D⊥
pp(k; z) = ik2η(k; z)(mn)−1, (4.40)
653
P.A.Hlushak, M.V.Tokarchuk
where η(k; z) denotes a generalized coefficient of shear viscosity (for the Bose
system). Such a coefficient consists of two main contributions. The first one is
D⊥(kin)
pp (k; z), whereas the second contribution D⊥(int)
pp (k; z) describes an interplay
of kinetic and hydrodynamic processes. The investigation of the contributions
D⊥(kin)
pp (k; z) and D⊥(int)
pp (k; z) in the shear viscosity of the quantum Bose system
at the decrease of temperature to 2.17 K presents a significant interest in connec-
tion with the beginning of the superfluid component.
If we put ν = 1, 2, 3, φ1(ξ) = 1, φ2(ξ) = ξz, φ3(ξ) − 6−1/2(ξ2 − 3) in the system
of equation (4.29), then we obtain:
zΦna(k; z) − iΩnp(k)Φpa(k; z) = −Φna(k), (4.41)
zΦpa(k; z) − iΩpn(k)Φna(k; z) +D||
pp(k; z)Φpa(k; z) − Σphkin(k; z)Φhkina(k; z)
−Σphint(k; z)Φhinta(k; z) = −Φpa(k), (4.42)
zΦhkina(k; z) − Σhkinp(k; z)Φpa(k; z) +Dhkinhkin(k; z)Φhkina(k; z)
+Dhkinhint(k; z)Φhinta(k; z) = −Φhkina(k), (4.43)
zΦhinta(k; z) − Σhintp(k; z)Φpa(k; z) +Dhinthkin(k; z)Φhkina(k; z)
+Dhinthint(k; z)Φhinta(k; z) = −Φhinta(k), (4.44)
where a = {n̂k, p̂k, ĥ
kin
k , ĥint
k } and
Σphkin(k; z) = iΩphkin(k) − Dphkin(k; z),
Σphint(k; z) = iΩphint(k) − Dphint(k; z),
Σhkinp(k; z) = iΩhkinp(k) − Dhkinp(k; z),
Σhintp(k; z) = iΩhintp(k) − Dhintp(k; z),
(4.45)
iΩab(k) and Dab(k; z) are determined according to (4.31), (4.30) and describe the
correlation between the viscous, kinetic and potential parts of the generalized en-
thalpy of the Bose system.
From the system of equations (4.41)–(4.44) one can define the Fourier compo-
nents of the particle number density correlation functions
Φnn(k; z) = Φ11(k; z) = 〈1|Φnn(k, ξ, ξ′; z)|1′〉,
as well as of the longitudinal component of the momentum density
Φ||
pp(k; z) = Φ22(k; z) = 〈ξz|Φnn(k, ξ, ξ′; z)|ξ′z〉,
for the kinetic part of generalized enthalpy
Φhkinhkin(k; z) = Φ33(k; z) = 〈6− 1
2 (ξ2 − 3)|Φnn(k, ξ, ξ
′; z)|6− 1
2 ((ξ′)2 − 3)〉
as well as for the potential part of generalized enthalpy Φhinthint(k; z) and cross
correlation functions, especially Φhinthkin(k; z), Φnhkin(k; z), Φnhint(k; z), Φphkin(k; z),
654
Kinetics and hydrodynamics of quantum Bose-systems
Φphint(k; z). Solving the system of equation (4.41)–(4.44) at a = n, one obtains an
expression for the correlation function “density-density” Φnn(k; z)
Φnn(k; z) = −S2(k)
[
z − iΩnp(k)iΩpn(k)
z + D̄
||
pp(k; z)
]−1
, (4.46)
where
D̄||
pp(k; z) = D||
pp(k; z) − Σ̄phkin(k; z)
[
z + D̄hkinhkin(k; z)
]−1
Σ̄hkinp(k; z)
− Σphint(k; z) [z + Dhinthint(k; z)]−1 Σhintp(k; z), (4.47)
Σ̄phkin(k; z) = Σphkin(k; z)
− Σphint(k; z) [z +Dhinthint(k; z)]−1Dhinthkin(k; z), (4.48)
Σ̄hkinp(k; z) = Σhkinp(k; z) −
−Dhkinhint(k; z) [z +Dhinthint(k; z)]−1 Σhintp(k; z), (4.49)
D̄hkinhkin(k; z) = Dhkinhkin(k; z)
−Dhkinhint(k; z) [z +Dhinthint(k; z)]−1Dhinthkin(k; z). (4.50)
In expressions (4.47)–(4.50) we can observe an interesting renormalization of the
functions Σab and Dab via the generalized transport kernels for fluctuations of flows
of the potential part of enthalpy density. However, D̄||
pp(k; z) is connected only with
the generalized longitudinal viscosity η||(k; z), since the densities of the number of
particles n̂k and momentum p̂k are included in the set of variables of an abbreviated
description. In our case D̄||
pp(k; z) takes into account both thermal and viscous dy-
namical correlation processes. Excluding from (4.46) the imaginary part Φ||
nn(k;ω)
of the correlation function Φnn(k; z), one obtains an expression for the dynamical
structure factor S(k;ω) in which the contributions of transport kernels correspond-
ing to the kinetic and potential parts of the enthalpy density ĥk are separated. It
is evident that the main contribution of the transport kernel Dhinthint(k; z) to the
S(k;ω) for liquids was in the hydrodynamical region (the region of small values of
wavevector k and frequency ω), whereas Dhkinhkin(k; z) will contribute to the kinetic
region (of the order of interatomic distances and small correlation times).
It is natural, that their contributions will significantly differ in both the imperfect
Bose gas and Bose liquid. We can expect then that the difference of contributions
from Dhkinhkin(k; z) and Dhinthint(k; z) will show itself in the dynamic structure factor
of semiquantum Bose system in the region of intermediate values of k, ω [25,26]. In
the region of intermediate values of wavevector k and frequency ω, it is necessary
to take into account all the transport kernels Σphkin(k; z), Σphint(k; z), Dhkinhkin(k; z),
Dhinthkin(k; z), Dhinthint(k; z). Since it is impossible to perform exact calculations of
the above described functions, it is necessary in each separate region to accept
models corresponding to the physical processes. Obviously, it is necessary to pro-
vide the modelling on the level of generalized transport kernels ϕnn(k,p,p′; t, t′),
ϕnh(k,p; t, t′), ϕhn(k,p
′; t, t′), ϕhh(k; t, t′) (4.11)–(4.14). In the limit k → 0, ω → 0,
the cross correlations between the kinetic and potential flows of energy and shear
655
P.A.Hlushak, M.V.Tokarchuk
flows become not so important and the system of equations (4.46)–(4.50) gives a
spectrum of the collective modes inherent in molecular hydrodynamics [25]. For in-
termediate values of k and ω, the spectrum of collective modes can be found from
the condition
z iΩnp(k) 0 0
iΩpn(k) z +D||
pp(k; z) Σphkin(k; z) Σphint(k; z)
0 Σhkinp(k; z) z +Dhkinhkin(k; z) Dhkinhint(k; z)
0 Σhintp(k; z) Dhinthkin(k; z) z +Dhinthint(k; z)
= 0, (4.51)
in which the contributions of kinetic and potential parts of generalized enthalpy
are separated. This will be reflected in the structure of a heat mode at concrete
model calculations of the wavevector- and frequency-dependent transport kernels
Dhkinhkin(k; z), Dhkinhint(k; z), Dhinthkin(k; z), Dhinthint(k; z) depending on k and ω.
The system of equations (4.41)–(4.44) for time correlation functions allows us an
extended description of collective modes in the quantum Bose system, which includes
both hydrodynamic and kinetic processes. Including some additional functions based
on the functions φν(ξ) (4.32),
ψl
Q(ξ) =
1
5
(ξ2 − 5)ξl, ψlj
Π(ξ) =
√
2
2
(ξlξj −
1
3
ξ2δlj), (4.52)
which correspond to a 13-moment approximation of Grad, one obtains a system of
equations for time correlation functions of an extended set of hydrodynamic variables
{n̂k, p̂k, ĥkin
k , Π̂k, Q̂k, ĥ
int
k } ( Π̂k =
∑
ξ ϕΠ(ξ)n̂k(ξ), Q̂k =
∑
ξ ϕQ(ξ)n̂k(ξ)):
iΩ̃H(k) =
0 iΩnp 0 0 0 0
iΩpn 0 iΩphkin iΩpΠ 0 iΩphint
0 iΩhkinp 0 0 iΩhkinQ 0
0 iΩΠp 0 0 iΩΠQ 0
0 0 iΩQhkin iΩQΠ 0 iΩQhint
0 iΩhintp 0 0 iΩhintQ 0
(4.53)
is an extended hydrodynamic frequency matrix,
ϕ̃H(k; z) =
0 0 0 0 0 0
0 DH
pp DH
phkin DH
pΠ DH
pQ DH
phint
0 Dhkinp Dhkinhkin DhkinΠ DhkinQ Dhkinhint
0 DΠp DΠhkin DΠΠ DΠQ DΠhint
0 DQp DQhkin DQΠ DQQ DQhint
0 Dhintp Dhinthkin DhintΠ DhintQ Dhinthint
(4.54)
is a matrix of generalized memory functions, elements of which are transport kernels
(4.30) projected based on the functions φν(ξ) (4.32), (4.52). For such a description,
the spectrum of generalized collective modes of the system is determined for inter-
mediate k and ω by the relation det
∣∣∣zĨ − iΩ̃H(k) + ϕ̃H(k)
∣∣∣ = 0 which takes into
account kinetic and hydrodynamic processes. In the hydrodynamic limit k → 0,
656
Kinetics and hydrodynamics of quantum Bose-systems
ω → 0, when the contribution of cross dissipative correlations between the kinetic
and hydrodynamic processes practically vanishes, the system of equations for the
time correlation function, after some transformations, can be reduced to a system of
equations for time correlation functions of densities for the number of particles n̂k,
momentum p̂k, total enthalpy ĥk, the generalized stress tensor π̂k = (1−PH)iLN p̂k
and the generalized enthalpy flow q̂k = (1−PH)iLN ĥk, where PH is the Mori oper-
ator constructed on the dynamical variables {n̂k, p̂k, ĥk}.
In this case, in the hydrodynamic limit k → 0, ω → 0 the spectrum of collective
excitations coincide with the spectrum [25,26] for semiquantum helium.
However, at fixed values of k and ω, the transport kernels ϕππ, ϕπQ, ϕQπ, ϕQQ
are expressed via the generalized transport kernels Dνµ(k; z) of matrix (4.54), i.e. via
ϕnn(k,p,p
′; t, t′), ϕnh(k,p; t, t′), ϕhn(k,p
′; t, t′), ϕhh(k; t, t′) (4.11)–(4.14), according
to the definition Dνµ(k; z) (4.30). Here, it is important to point out that passing
from the system of transport equations of a self-consistent description of kinetics and
hydrodynamics to the equations of generalized hydrodynamics, we can connect the
generalized transport kernels (4.11)–(4.14) with the hydrodynamic transport kernels
Dνµ(k; z) in (4.51) or (4.54).
5. Conclusion
In this work we have introduced the statistical approach of a consistent descrip-
tion of kinetic and hydrodynamic processes for quantum Bose system far from a
point of phase transition. For this purpose we used a method of nonequilibrium
statistical operator by D.Zubarev. For parameters of a consistent description, the
nonequilibrium one-partial distribution function and average potential energy of in-
teraction Bose particles have been chosen. As a result, the generalized equations of
transport (2.33) have been obtained, which in a case of weakly nonequilibrium pro-
cesses are closed (3.1), (3.2). By means of the latter we have obtained a new system
of consistent equations for time correlation functions (4.14)–(4.17) from which the
projection method can define a dynamic structure factor, the functions ”flow-flow”,
etc. based on the eigenvectors of nonequilibrium one-partial function. In the hydro-
dynamical limit this system can be transformed to a system of equations for time
correlation functions constructed on the dynamic variables related to the density of
particle number, momentum, enthalpy, generalized stress tensor and enthalpy flow
for which the spectrum of collective excitations is known [25,26]. However, a special
feature here is that the generalized memory functions are constructed on memo-
ry functions (2.34)–(2.37) initially obtained in the coupled system of equations for
nonequilibrium one-partial distribution functions and average potential energy of
interaction. In this way the relation between one-particle and many-particle corre-
lations is traced. It can be actual in a relationship with the studies of the kinetics
of one-particle and pair-particle Bose condensate [45,46].
In our approach, a part of the average potential energy of interaction connected
with the nonequilibrium distribution function of a pair condensate can be extracted
at once. Further we can obtain the system of equations for time correlation func-
657
P.A.Hlushak, M.V.Tokarchuk
tions with separation of an one-particle and pair-particle Bose condensate both in
Hamiltonian [46] and in expressions. Moreover, in our approach we can start with
the relevant statistical operator (2.19) to construct a chain of BBGKY equations
for nonequilibrium distribution functions of particles with the modified boundary
conditions (taking into account multipartial correlations), which is similar both for
classical [43] and Fermi systems [47]. This research can be carried out above and the
below the points of phase transition with the allocation of condensation distribu-
tion functions. In the area of phase transition to a superfluid state of the quantum
system, the account of kinetic and hydrodynamical nonlinear fluctuations is impor-
tant. Here our approach can be modified by choosing as parameters a consistent
description of the Wigner nonequilibrium distribution function and nonequilibrium
distribution function of collective variables: density of particle number, momentum,
energy. In this case the relevant statistical operator can be presented as:
%q(t) = exp
{
−Φ(t) −
∑
qp
γ−q(p; t)n̂q(p) −
∫
daF (a; t)f̂(a)
}
,
where f(a; t) = Spf̂(a)%(t) is the nonequilibrium distribution function of collective
variables amk: f̂(a) = δ(â − a) =
∏3
m=1
∏
k δ(âmk − amk), and â1k = n̂k, â2k = P̂k,
â3k = ε̂k. f(a; t) satisfies the generalized Fokker-Plank-type equation [38]. These
problems will be considered in our subsequent papers.
Acknowledgement
This study was partially supported by State Foundation for Fundamental Re-
searches, project No. 02.07/418.
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Узгоджений опис кінетики та гідродинаміки
квантових бозе-систем
П.А.Глушак, М.В.Токарчук
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 8 серпня 2004 р.
Запропоновано узгоджений підхід для опису кінетики та гідродина-
міки багатобозонних систем. Отримано узагальнені рівняння пере-
носу для сильно і слабо нерівноважних бозе-систем з використан-
ням методу нерівноважного статистичного оператора Д.М. Зубарє-
ва. Отримано нові рівняння для часових кореляційних функцій з ви-
діленими внесками від кінетичної енергії і потенціальної енергій вза-
ємодії частинок. Отримано узагальнені коефіцієнти переносу з ура-
хуванням узгодженого опису кінетичних і гідродинамічних процесів.
Ключові слова: бозе-система, гелій, кінетика, гідродинаміка,
кореляційна функція, коефіцієнти переносу
PACS: 67.40.-w, 47.37.+q
660
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