On the hydrodynamics of polaron gas in the Bogolyubov reduced description method
On the basis of a generalization of the Chapman-Enskog method a new approach to derivation of hydrodynamic equations for weak density polaron gas has been elaborated taking into account the relaxation of temperature and velocity in the system. Non-locality of the collision integral of the used kinet...
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irk-123456789-1071132016-10-14T03:02:38Z On the hydrodynamics of polaron gas in the Bogolyubov reduced description method Sokolovsky, S.A. Section D. Theory of Irreversible Processes On the basis of a generalization of the Chapman-Enskog method a new approach to derivation of hydrodynamic equations for weak density polaron gas has been elaborated taking into account the relaxation of temperature and velocity in the system. Non-locality of the collision integral of the used kinetic equation was taken into account also. Both circumstances lead to some modification of the standard transport theory. На основе обобщения метода Чепмена-Энскога разработан новый подход к выводу уравнений гидродинамики для поляронного газа малой плотности с учетом релаксации скорости и температуры системы. Учтена также нелокальность интеграла столкновений используемого кинетического уравнения. Оба обстоятельства ведут к модификации стандартной теории переноса. На основі узагальнення метода Чепмена-Енскога розроблено новий підхід до виведення рівнянь гідродинаміки поляронного газу малої густини з урахуванням релаксації швидкості та температури. Також врахована нелокальність інтеграла зіткненнь кінетичного рівняння, що використовується. Обидві обставини ведуть до модифікації стандартної теорії переносу. 2012 Article On the hydrodynamics of polaron gas in the Bogolyubov reduced description method / S.A. Sokolovsky // Вопросы атомной науки и техники. — 2012. — № 1. — С. 217-220. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 72.10.Bg, 72.15.Eb, 72.15.Lh http://dspace.nbuv.gov.ua/handle/123456789/107113 en Вопросы атомной науки и техники Prydniprovs'k State Academy of Engineering and Architecture |
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Section D. Theory of Irreversible Processes Section D. Theory of Irreversible Processes |
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Section D. Theory of Irreversible Processes Section D. Theory of Irreversible Processes Sokolovsky, S.A. On the hydrodynamics of polaron gas in the Bogolyubov reduced description method Вопросы атомной науки и техники |
| description |
On the basis of a generalization of the Chapman-Enskog method a new approach to derivation of hydrodynamic equations for weak density polaron gas has been elaborated taking into account the relaxation of temperature and velocity in the system. Non-locality of the collision integral of the used kinetic equation was taken into account also. Both circumstances lead to some modification of the standard transport theory. |
| format |
Article |
| author |
Sokolovsky, S.A. |
| author_facet |
Sokolovsky, S.A. |
| author_sort |
Sokolovsky, S.A. |
| title |
On the hydrodynamics of polaron gas in the Bogolyubov reduced description method |
| title_short |
On the hydrodynamics of polaron gas in the Bogolyubov reduced description method |
| title_full |
On the hydrodynamics of polaron gas in the Bogolyubov reduced description method |
| title_fullStr |
On the hydrodynamics of polaron gas in the Bogolyubov reduced description method |
| title_full_unstemmed |
On the hydrodynamics of polaron gas in the Bogolyubov reduced description method |
| title_sort |
on the hydrodynamics of polaron gas in the bogolyubov reduced description method |
| publisher |
Prydniprovs'k State Academy of Engineering and Architecture |
| publishDate |
2012 |
| topic_facet |
Section D. Theory of Irreversible Processes |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/107113 |
| citation_txt |
On the hydrodynamics of polaron gas in the Bogolyubov reduced description method / S.A. Sokolovsky // Вопросы атомной науки и техники. — 2012. — № 1. — С. 217-220. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT sokolovskysa onthehydrodynamicsofpolarongasinthebogolyubovreduceddescriptionmethod |
| first_indexed |
2025-07-07T19:30:45Z |
| last_indexed |
2025-07-07T19:30:45Z |
| _version_ |
1837017749043281920 |
| fulltext |
ON THE HYDRODYNAMICS OF POLARON GAS IN THE
BOGOLYUBOV REDUCED DESCRIPTION METHOD
S.A. Sokolovsky ∗
Prydniprovs’k State Academy of Engineering and Architecture, Dnipropetrovs’k, Ukraine
(Received November 6, 2011)
On the basis of a generalization of the Chapman-Enskog method a new approach to derivation of hydrodynamic
equations for weak density polaron gas has been elaborated taking into account the relaxation of temperature and
velocity in the system. Non-locality of the collision integral of the used kinetic equation was taken into account also.
Both circumstances lead to some modification of the standard transport theory.
PACS: 72.10.Bg, 72.15.Eb, 72.15.Lh
1. INTRODUCTION
In our paper [1] relaxation phenomena in spatially
uniform low density polaron gas have been investi-
gated. The consideration was conducted in standard
model in which phonons form an ideal equilibrium gas
with the temperature T0. Electron spin, phonon po-
larization and zone structure of electron spectrum are
neglected; energy of electron is chosen as εp = p2/2m.
In [1] kinetic equation for strong non-uniform
states of polaron (low density polaron gas) in weak
electric field has been obtained too and has the form
∂fp(x, t)
∂t
= −pn
m
∂fp(x, t)
∂xn
+ eEn(x)
∂fp(x, t)
∂pn
+
+Ip(x, f(t)). (1)
Here collision integral Ip(x, f) is given by the formula
Ip(x, f) =
1
4π3h̄2
0∫
−∞
dτ
∫
d3kg2
k
{
nkfp−kh̄
(
x + k
h̄τ
2m
)
− (1 + nk)fp
(
x − k
h̄τ
2m
)}
×
× cos
τ
h̄
(εp−kh̄ + h̄ωk − εp)+
+
1
4π3h̄2
0∫
−∞
dτ
∫
d3kg2
k
{
(1 + nk)fp+kh̄
(
x − k
h̄τ
2m
)
− nkfp
(
x + k
h̄τ
2m
)}
×
× cos
τ
h̄
(εp + h̄ωk − εp+kh̄) , (2)
where nk = (e
h̄ωk
T0 − 1)−1 is the Planck distribution
for phonons, e is module of electron charge, h̄ωk is
energy of a phonon. This result does not amply that
gradients of the polaron distribution function fp(x, t)
are small. In present paper on the basis of this ki-
netic equation and with the help of a generalization of
the Chapman-Enskog method transport phenomena
in polaron gas are discussed. In contrast to the stan-
dard theory we use kinetic equation with the collision
integral which is expanded in a series in gradients of
the polaron distribution function
Ip(x, f) ≡
∫
d3p′M(p, p′)fp′(x)+
+
∫
d3p′Mn(p, p′)
∂fp′(x)
∂xn
+ .... (3)
Also we do not assume that the distribution function
fp(x, t) in hydrodynamic states in zero order in gradi-
ents approximation coincides with the local Maxwell
distribution.
2. BASIC EQUATIONS
Transport phenomena in the polaron gas are dis-
cussed in the framework of its hydrodynamics. Ac-
cording to the kinetic equation laws of conservation
(variation) of the gas mass, energy and momentum
have the form
∂σ
∂t
= −∂πn
∂xn
,
∗E-mail address: sersokol@list.ru
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 217-220.
217
∂ε
∂t
= −∂qn(f)
∂xn
− e
m
πnEn + R0(f),
∂πl
∂t
= −∂tln(f)
∂xn
− e
m
σEl + Rl(f) (4)
where mass, energy and momentum densities are de-
fined by the formulas
σ = m
∫
d3p fp, πl =
∫
d3p plfp,
ε =
∫
d3p εpfp. (5)
Flux densities of energy qn(x, f), momentum tnl(x, f)
and sources R0(x, f), Rl(x, f) are given by expres-
sions:
tln(x, f) =
∫
d3p pl
∂εp
∂pn
fp(x),
qn(x, f) =
∫
d3p εp
∂εp
∂pn
fp(x);
R0(x, f) =
∫
d3p εpIp(x, f),
Rl(x, f) =
∫
d3p plIp(x, f). (6)
Basic hydrodynamic parameters, temperature
T (x, t), velocity ul(x, t) and number of particles den-
sity n(x, t) of the system, are defined by usual formu-
las
ε =
3
2m
σT +
1
2
σu2, πl = σul, σ = nm. (7)
Equations (4) give hydrodynamic equations if their
right sides are expressed through these variables
ξμ(x, t) : ξ0(x, t) = T (x, t), ξl(x, t) = ul(x, t),
ξ4(x, t) = n(x, t). This is possible if after some time
τ0 the polaron distribution function has the structure
fp(x, t)−−−−−→
t�τ0
fp(x, ξ(t)), (8)
where fp(x, ξ) is a functional of the variables ξμ(x).
This relation (called the functional hypothesis) is
the basis of the Chapman-Enskog method which is
a special case of the Bogolyubov reduced descrip-
tion method (see, for example, [2]). Time τ0 in
our consideration is assumed satisfying the condition
τ0 � τT , τu where τT , τu are relaxation times of the
temperature and velocity of the polaron gas. As a
result, the hydrodynamic equations can be written
as
∂ξμ(x, t)
∂t
= Lμ(x, f(ξ(t))), (9)
where Lμ(x, f) are some functionals of fp(x). For the
distribution function fp(x, ξ) from the kinetic equa-
tion (1) with (8) we obtain the equation
∑
μ
∫
d3x′ δfp(x, ξ)
δξμ(x′)
Lμ(x′, f(ξ)) = −pn
m
∂fp(x, ξ)
∂xn
+
+eEn(x)
∂fp(x, ξ)
∂pn
+ Ip(x, f(ξ)). (10)
Definitions (6) of the parameters ξμ(x) which describe
state of the system give additional conditions to this
equation∫
d3p fp(x, ξ)pl =mn(x)ul(x),
∫
d3p fp(x, ξ) = n(x),
∫
d3p fp(x, ξ)εp =
3
2
n(x)T (x) +
1
2
mn(x)u(x)2.
(11)
The solution of the equation (10) taking into ac-
count conditions (11) we found in the form of a double
series in the gradients of the parameters ξμ(x) (g is
their small parameter) and small parameter ε defined
by the estimations
υl(x, t) ∼ ε, T (x, t) − T0 ∼ ε, El(x) ∼ ε;
∂υn(x, t)
∂xl
∼ gε,
∂T (x, t)
∂xl
∼ gε,
∂n
∂xl
∼ g, (12)
which describes the proximity of the system to equi-
librium (A(m) is contribution of the order gm to A,
A(m,n) is contribution of the order gmεn to A).
Zero order in gradients approximation f
(0)
p has
been investigated in our paper [1]. It was established
that the main contributions to this distribution func-
tion have the form
f(0)p = wo
p + wo
p{A(p)pnυn + C(p)pnEn+
+B(p)(T − T0)} + O(g0ε2);
wo
p(ξ) ≡
n
(2πmT0)3/2
e−
p2
2mT0 . (13)
Functions A(p), B(p), C(p) satisfy the equations
λuA(p)pn =
∫
d3p′ K(p, p′)A(p′)p′n,
〈εpA(p)〉 = 3/2; (14)
λT B(p) =
∫
d3p′ K(p, p′)B(p′),
〈B(p)〉 = 0, 〈εpB(p)〉 = 3/2; (15)
{μλuA(p) − e/mT0}pn =
∫
d3p′ K(p, p′)C(p′)p′n,
〈εpC(p)〉 = 0, (16)
where the notation
〈g(p)〉 ≡
∫
d3pwo
pg(p)
is introduced. In Eqs. (14)–(16) instead of the kernel
M(p, p′) defined by the formula (3) the kernel K(p, p′)
is used
M(p, p′)wo
p′ ≡ −wo
pK(p, p′). (17)
In standard approach (see, for example, [3, 4]) the
distribution function f
(0)
p (x, ξ) is not calculated but
assumed to be equal to the local Maxwell distribution
wp(ξ(x)) where
wp(ξ) ≡ n
(2πmT )3/2
e−
(p−mu)2
2mT . (18)
218
The main in ε contributions to the distribution func-
tion of the first order in gradients f
(1)
p have the struc-
ture
f (1)
p = wo
p {D(p)pn + Qnl(p)ul + R(p)pn(T − T0)+
+ Snl(p)El} ∂n
∂xn
+ wo
p
{
Fnl(p)
∂un
∂xl
+
+ G(p)pl
∂T
∂xl
+ Hnl(p)
∂El
∂xn
}
+ O(gε2). (19)
This expression describes dissipative processes in the
system. Function D(p) gives the main contribution
to f
(1)
p and functions
G(p), Fnl(p) ≡ F1(p)δnl + F2(p)Δnl(p) (20)
(Δnl(p) ≡ pnpl − 1
3p2δnl) allow to calculate viscosity
and heat conductivity of the system. They satisfy the
integral equations with additional conditions:
{a A(p) + α(p) − 1
m
}pn =
∫
d3p′K(p, p′)D(p′)p′n,
〈εpD(p)〉 = 0; (21)
{λT G(p) − b1A(p) − 1
m
B(p) + B1(p)}pn =
=
∫
d3p′K(p, p′)G(p′)p′n,
〈εpG(p)〉 = 0; (22)
A1(p) − a1B(p) − 1
3
A(p)p2 + 1 =
=
∫
d3p′K(p, p′)F1(p′),
〈F1(p)〉 = 0, 〈εpF1(p)〉 = 0; (23)
− 1
m
A(p)Δnl(p) =
=
∫
d3p′K(p, p′)F2(p′)Δnl(p′). (24)
Here functions α(p), A1(p), B1(p) are defined by the
formulas ∫
d3p′Mn(p, p′)wo
p′ ≡ wo
pα(p)pn,
∫
d3p′Mn(p, p′)wo
p′A(p′) ≡ wo
pA1(p)pn,
∫
d3p′Mn(p, p′)wo
p′B(p′) ≡ wo
pB1(p)pn (25)
and are absent in the standard theory because they
take into account non-locality of our collision inte-
gral (see (1), (2)). Taking into account additional
conditions for functions A(p), B(p) (see (14), (15),
scalar values μ (mobility of the polaron in a steady
state), a, a1, b1 in equations (16), (21)–(23) are ex-
pressed through solutions of these equations. There-
fore, these expressions are not needed for solution of
integral equations (16), (21)–(23).
Neglecting of the dissipative processes, time equa-
tions for temperature and velocity can be written in
the form
∂T
∂t
= −λT (T − T0) + a1
∂ul
∂xl
+ a2
∂El
∂xl
+
+(a3ul + a4El)
∂n
∂xl
+ O(g0ε2, g1ε2),
∂ul
∂t
= −λu(ul − μEl) + b1
∂T
∂xl
+ b2(T − T0)
∂n
∂xl
+
+O(g0ε2, g1ε2), (26)
where a1, ..., a4, b1, b2, μ are scalar values some of
which can be calculated from equations (16), (21)–
(23). The kernel K(p, p′) of the integral equations
(14)–(16), (21)–(24) has important properties which
can be expressed in the terms of the bilinear form
{g(p), h(p)} =
∫
d3pd3p′wo
pg(p)K(p, p′)h(p′), (27)
that gives
{g(p), h(p)} = {h(p), g(p)}, {g(p), g(p)} ≥ 0. (28)
These formulas show that eigenvalues λT , λu of op-
erator with the kernel K(p, p′) are positive. Ac-
cording to (26) they describe relaxation phenomena
in the system in the spatially homogeneous states
(the above mentioned relaxation times τT = λ−1
T ,
τu = λ−1
u ).
Expressions for energy and momentum fluxes tak-
ing into account dissipative contributions have the
form
ql = n(c1ul + c2El) + c3
∂n
∂xl
− κ
∂T
∂xl
+ O(g0ε2, g1ε2),
tnl = nTδnl − η
(
∂un
∂xl
)s
− ζδnl
∂um
∂xm
+
+d1
(
∂En
∂xl
)s
+ d2δnl
∂Em
∂xm
+
+d3
(
En
∂n
∂xl
)s
+ d4δnlEm
∂n
∂xm
+ O(g0ε2, g1ε2),
(29)
where η, ζ are viscosities, κ is heat conductivity;
c1, c2, c3, d1, d2, d3, d4 are some coefficients;
(Anl)
s ≡ Anl + Aln − 2
3
δnlAmm.
3. CONCLUSIONS
On the basis of a generalization of the Chapman-
Enskog method a new approach to derivation of hy-
drodynamic equations for weak density polaron gas
has been elaborated taking into account the relax-
ation of temperature and velocity in the system. Non-
locality of the collision integral of the used kinetic
equation was taken into account also. Both circum-
stances lead to some modification of the standard the-
ory [3, 4]. Solution of the obtained integral equation
219
with the help of expansion in the Sonine polynomial
series will be discussed in another paper (in spatially
uniform states this was done in [1]).
This work was supported by the State Foundation
for Fundamental Research of Ukraine under project
No. 25.2/102.
References
1. S.A. Sokolovsky. Toward polaron kinetics in the
Bogoliubov reduced description method // Theo-
retical and Mathematical Physics. 2011, v. 168 (2),
p. 1150-1164.
2. A.I. Akhiezer, S.V. Peletminsky. Methods of Sta-
tistical Physics. Oxford: Pergamon Press, 1981,
368 p.
3. A.I. Akhiezer, V.F. Aleksin, and V.D. Khodusov.
Gas dynamics of quasi-parlicles. I. General the-
ory // Low Temperature Physics. 1994, v. 20 (12),
p. 1199-1238.
4. E.M. Lifshitz, L.P. Pitaevskii. Physical Kinetics.
Oxford: Pergamon Press, 1981, 452 p.
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