Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems
Starting from the first principles of statistical mechanics, the two-fluid hydrodynamics of superconductor in ideal approximation is constructed. The system of equations of motion for normal and anomalous correlation functions is used for hydrodynamic description. Transition to equations of hydro...
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Інститут фізики конденсованих систем НАН України
2005
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| Цитувати: | Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems / P. Shygorin, A. Svidzynsky // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 701–710. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1205152017-06-13T03:05:20Z Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems Shygorin, P. Svidzynsky, A. Starting from the first principles of statistical mechanics, the two-fluid hydrodynamics of superconductor in ideal approximation is constructed. The system of equations of motion for normal and anomalous correlation functions is used for hydrodynamic description. Transition to equations of hydrodynamics is performed using an expansion of equations of motion for correlation functions in terms of a small parameter. Виходячи з перших принципів статистичної механіки, побудовано дворідинну гідродинаміку надпровідника в ідеальному наближенні. Для побудови гідродинаміки використано систему рівнянь руху для нормальної та аномальної кореляційних функцій. Перехід до рівнянь гідродинаміки здійснюється через розклад рівнянь руху для кореляційних функцій за малим параметром. 2005 Article Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems / P. Shygorin, A. Svidzynsky // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 701–710. — Бібліогр.: 7 назв. — англ. 1607-324X PACS: 05.30.Fk, 47.37.+q DOI:10.5488/CMP.8.4.701 http://dspace.nbuv.gov.ua/handle/123456789/120515 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
Starting from the first principles of statistical mechanics, the two-fluid hydrodynamics
of superconductor in ideal approximation is constructed. The
system of equations of motion for normal and anomalous correlation functions
is used for hydrodynamic description. Transition to equations of hydrodynamics
is performed using an expansion of equations of motion for
correlation functions in terms of a small parameter. |
| format |
Article |
| author |
Shygorin, P. Svidzynsky, A. |
| spellingShingle |
Shygorin, P. Svidzynsky, A. Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems Condensed Matter Physics |
| author_facet |
Shygorin, P. Svidzynsky, A. |
| author_sort |
Shygorin, P. |
| title |
Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems |
| title_short |
Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems |
| title_full |
Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems |
| title_fullStr |
Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems |
| title_full_unstemmed |
Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems |
| title_sort |
microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120515 |
| citation_txt |
Microscopic derivation of the hydrodynamic equations for the superfluid fermi-systems / P. Shygorin, A. Svidzynsky // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 701–710. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT shygorinp microscopicderivationofthehydrodynamicequationsforthesuperfluidfermisystems AT svidzynskya microscopicderivationofthehydrodynamicequationsforthesuperfluidfermisystems |
| first_indexed |
2025-07-08T18:00:39Z |
| last_indexed |
2025-07-08T18:00:39Z |
| _version_ |
1837102678047457280 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 701–710
Microscopic derivation of the
hydrodynamic equations for the
superfluid fermi-systems
P.Shygorin, A.Svidzynsky
Lesya Ukrainka’s Volyn State University
Received August 3, 2005
Starting from the first principles of statistical mechanics, the two-fluid hy-
drodynamics of superconductor in ideal approximation is constructed. The
system of equations of motion for normal and anomalous correlation func-
tions is used for hydrodynamic description. Transition to equations of hy-
drodynamics is performed using an expansion of equations of motion for
correlation functions in terms of a small parameter.
Key words: two-fluid hydrodynamics, correlation function, superfluid
Fermi-systems
PACS: 05.30.Fk, 47.37.+q
1. Introduction
Due to specific properties the quantum liquids turn out to be a very interesting
object in physical studies. They include superfluids He-3 and He-4, a superconducti-
ng Fermi-systems, a trapped Bose gas, a core of neutron stars, etc [1,2]. In the
weakly-nonequilibrium states the equations of hydrodynamics are very important
tool of describing the afore-mentioned systems. The phenomena of superconduc-
tivity and superfluidity are known to be deeply related. The property of the non-
dissipative current states, that is caused by the phase transition to a more-organized
state, unites them [3]. It is not surprising that the equations of hydrodynamics of
these systems have a similar structure (the so-called two-fluid equations). The phe-
nomenological hydrodynamics of superfluid He-4 was constructed by Landau in 1941
[4]. At the microscopic level these equations were derived by Bogoliubov in 1963
[5]. The two-fluid model of superconductor was derived in 1965 by Svidzynsky and
Slusarev [6], and independently by Stephen [7]. The system of equations of moti-
on for the normal and the anomalous correlation functions is the starting point for
constructing the hydrodynamic equations in [5,6]. The transition to equations of hy-
drodynamics is performed using an expansion of equations of motion for correlation
functions in terms of a small parameter. The small parameter is introduced formally.
c© P.Shygorin, A.Svidzynsky 701
P.Shygorin, A.Svidzynsky
This is a noticeable drawback of these studies. In the [5] the so-called “parameter
of homogeneity” is chosen as small parameter, while in [6] it is the Plank constant,
which obviously was possible to put equal to unity.
In the present paper, where following the chart of work [6], starting from the
first principles of statistical mechanics, we derived the equations of two-fluid hydro-
dynamics of superconductor in an ideal approximation. An electronic liquid in the
superconductor is described by the four-fermion Hamiltonian, that reflects a direct
interaction between electrons. In terms of the Heisenberg equations we constructed
the equations of motion for correlation functions. The mean-field approach is used
for disclosure of higher correlation functions. By writing down equations of motion
in a dimensionless form it is possible to select a small parameter, which is equal to
the ratio of the length of coherence to a characteristic length of changes in macro-
scopic quantities (such as the mean number of particles, momentum, energy). The
expansion in terms of this small parameter coincides with the expansion in terms of
gradients of the macroscopic quantities.
2. Equations of motion for correlation functions
Let us consider a superconductor in the BCS model. In this model the Hamilto-
nian of the system of electrons in the presence of an external electromagnetic field
in the second quantization representation is as follows (we set h̄ = c = 1 throughout
this paper)
Ĥ =
∑
σ
∫
d~rΨ+
σ (~r)
{
1
2m
(~̂p − e ~A(~r, t))2 + eA0(~r, t)
}
Ψσ(~r)
+ g
∫
d~rΨ+
↑ (~r)Ψ+
↓ (~r)Ψ↓(~r)Ψ↑(~r). (1)
To construct the hydrodynamics of a system with spontaneous broken symmetry
we should proceed from the extended system of correlation functions [5], which is
formed both by a normal and by an anomalous correlation function. Therefore we
will start with a system of correlation functions in the next form
〈Ψ+
↑ (x1)Ψ↑(x2)〉, 〈Ψ↓(x1)Ψ↑(x2)〉. (2)
Here xi ≡ (~ri, t), and the dependence of the creation and annihilation operators on
time is given through a Heisenberg representation, for instance
Ψ+
↑ (x1) = eiĤtΨ+
↑ (~r1)e
−iĤt,
the angular brackets indicate an average at the local-equilibrium ensemble.
Using the Heisenberg’s equation of motion
i
∂Ψσ(x)
∂t
=
[
Ψσ(x), Ĥ
]
−
, (3)
702
Hydrodynamic equations for the superfluid fermi-systems
we obtain the equations of motion for correlation functions (2). These equations are
as follows:
{
i
∂
∂t
+ eA0(x1) − eA0(x2)
}
〈Ψ+
↑ (x1)Ψ↑(x2)〉
+
1
2m
[
(~̂p1 + e ~A(x1))
2 − (~̂p2 − e ~A(x2))
2
]
〈Ψ+
↑ (x1)Ψ↑(x2)〉
= −g〈Ψ+
↑ (x1)Ψ
+
↓ (x1)Ψ↓(x1)Ψ↑(x2)〉 + g〈Ψ+
↑ (x1)Ψ
+
↓ (x2)Ψ↓(x2)Ψ↑(x2)〉, (4)
{
i
∂
∂t
− eA0(x1) − eA0(x2)
}
〈Ψ↓(x1)Ψ↑(x2)〉
−
1
2m
[
(~̂p1 − e ~A(x1))
2 + (~̂p2 − e ~A(x2))
2
]
〈Ψ↓(x1)Ψ↑(x2)〉
= −g〈Ψ+
↑ (x1)Ψ↓(x1)Ψ↑(x1)Ψ↑(x2)〉 + g〈Ψ↓(x1)Ψ
+
↓ (x2)Ψ↓(x2)Ψ↑(x2)〉. (5)
The obtained system of equations of motion is not a closed one due to the
correlation functions of higher order in these equations. In order to make the closed
system of equations we use a disclosure of Hartree-Fock-Bogoliubov type (the mean-
field approximation). For instance
〈Ψ+
↑ (x1)Ψ
+
↓ (x1)Ψ↓(x1)Ψ↑(x2)〉 = 〈Ψ+
↑ (x1)Ψ
+
↓ (x1)〉〈Ψ↓(x1)Ψ↑(x2)〉
+ 〈Ψ+
↑ (x1)Ψ↑(x2)〉〈Ψ
+
↓ (x1)Ψ↓(x1)〉 − 〈Ψ+
↑ (x1)Ψ↓(x1)〉〈Ψ
+
↓ (x1)Ψ↑(x2)〉, (6)
and similarly the other averages. After using the disclosure of type (6) the equation
of motion (4), (5) will be as follows
{
i
∂
∂t
+ eA0(x1) − eA0(x2)
}
〈Ψ+
↑ (x1)Ψ↑(x2)〉
+
1
2m
[
(~̂p1 + e ~A(x1))
2 − (~̂p2 − e ~A(x2))
2
]
〈Ψ+
↑ (x1)Ψ↑(x2)〉 (7)
= ∆(x2)〈Ψ
+
↑ (x1)Ψ
+
↓ (x2)〉 − ∆∗(x1)〈Ψ↓(x1)Ψ↑(x2)〉,
{
i
∂
∂t
− eA0(x1) − eA0(x2)
}
〈Ψ↓(x1)Ψ↑(x2)〉
−
1
2m
[
(~̂p1 − e ~A(x1))
2 + (~̂p2 − e ~A(x2))
2
]
〈Ψ↓(x1)Ψ↑(x2)〉
= ∆(x1)δ(~r1 − ~r2) − ∆(x1)〈Ψ
+
↑ (x1)Ψ↑(x2)〉 − ∆(x2)〈Ψ
+
↓ (x2)Ψ↓(x1)〉. (8)
Here
∆(x) ≡ g〈Ψ↓(x)Ψ↑(x)〉 (9)
is the order parameter. We note that in the obtained equations (7), (8) the terms
not relevant to the effect of superconductivity (Hartree and Fock terms) have been
dropped. However, taking into account that the terms of Hartree type are trivial,
one may include them into potential A0(x).
703
P.Shygorin, A.Svidzynsky
The next step will be a separation of gauge-noninvariant multipliers (in fact we
will use a reference system in which the condensate is motionless). Such a separation
of phase has the following form [3]
〈Ψ+
σ (x1)Ψσ(x2)〉 = exp {im(χ(x2) − χ(x1))}Gσ(x1, x2),
〈Ψ↓(x)Ψ↑(x)〉 = exp {im(χ(x2) + χ(x1))}F (x1, x2),
∆(x) = exp {2imχ(x)} |∆(x)|. (10)
Then the equations of motion for G and F are as follows:
{
i
∂
∂t
+
1
2m
[
(~̂p1 − m~vs(x1))
2 − (~̂p2 + m~vs(x2))
2
]
}
G↑(x1, x2)
+ {mχ̇(x1) − mχ̇(x2) + eA0(x1) − eA0(x2)}G↑(x1, x2)
= |∆(x2)|F
∗(x2, x1) − |∆(x1)|F (x1, x2), (11)
{
i
∂
∂t
−
1
2m
[
(~̂p1 + m~vs(x1))
2 + (~̂p2 + m~vs(x2))
2
]
}
F (x1, x2)
− {mχ̇(x1) + mχ̇(x2) + eA0(x2) + eA0(x1)}F (x1, x2)
= |∆(x1)|δ(~r1 − ~r2) − |∆(x1)|G↑(x1, x2) − |∆(x2)|G↓(x2, x1), (12)
where
~vs = ∇χ −
e
m
~A
is superfluid velocity (velocity of the condensate). It is a gauge-invariant combinati-
on, because a change of calibration may be compensated by a gauge transformation
of the vector potential.
The transition to equations of hydrodynamics is performed using an expansion
of equations (11) and (12) in terms of space gradients. This expansion can be simply
performed by using the so-called mixed Wigner representation [3]. To this end we
introduce new variables
~R =
1
2
(~r1 + ~r2), ~r = ~r2 − ~r1 .
After the Fourier transformation with respect to relative coordinate ~r we obtain
f(x1, x2) → f(~R,~r, t) =
∫ d~p
(2π)3
f(~R, ~p, t)ei~p~r,
and
~r1 → ~R −
i
2
∇~p , ~r2 → ~R +
i
2
∇~p ,
~̂p1 → ~p −
i
2
∇~R, ~̂p2 → −~p −
i
2
∇~R . (13)
Any function of ~R + i/2 · ∇~p can be understood in terms of its power-series expansion
f
(
~R +
i
2
∇~p
)
= f
(
~R
)
+
i
2
∂f(~R)
∂ ~R
∂
∂~p
− · · · . (14)
704
Hydrodynamic equations for the superfluid fermi-systems
Using procedures (13) and (14) the equations for correlation functions can be written
as follows
i
{
∂
∂t
+ eEi
∂
∂pi
− mv̇si
∂
∂pi
−
∂
∂Rj
(
(pi + mvsi)
2
2m
)
∂
∂pj
+
(
pi
m
+ vsi
)
∂
∂Ri
}
G↑(~R, ~p, t) = |∆(~R, t)|
(
F ∗(~R, ~p, t) − F (~R, ~p, t)
)
−
i
2
∂|∆(~R, t)|
∂Ri
∂
∂pi
(
F ∗(~R, ~p, t) + F (~R, ~p, t)
)
, (15)
{
i
∂
∂t
− 2eA0(~R, t) − 2mχ̇(~R, t) − i
(
pi
∂vsi
∂Rj
∂
∂pj
− vsi
∂
∂Ri
)
−
(
p2
m
+ vs
2
)}
F (~R, ~p, t) = |∆(~R, t)|
(
1 − G↑(~R, ~p, t) − G↓(~R,−~p, t)
)
−
i
2
∂|∆(~R, t)|
∂Ri
∂
∂pi
(G↑(~R, ~p, t) − G↓(~R,−~p, t)). (16)
In the obtained equations the second order terms with respect to space gradient (the
terms proportional to ∇2
~R
) were neglected.
3. Two-fluid hydrodynamics
By solving equations (15), (16) we use the perturbation theory. Therefore, we
have to determine the order of other terms in these equations.
Let L be the length, on which the macroscopic quantities can change (it scales
with the size of the system). Then
|∇~R| ∼ L−1,
∂
∂t
∼ ūL−1 ∼ vF
Tc
EF
L−1, (17)
where Tc is critical temperature, EF = (mv2
F)/2 = (p2
F)/(2m) is Fermi energy. The
characteristic momenta are of the order of Fermi momentum, therefore we can put
p = pF +
ξ
vF
, ξ ∼ Tc .
Then
|∇~p| ∼
vF
Tc
∼ ξ0 ∼ 10−4cm,
where ξ0 is the coherence length. The gap ∆ is of the order Tc.
This theory has two small parameters. It is ξ0/L and a/ξ0 = Tc/EF, where a is
the interatomic distance. The first parameter is associated with the hydrodynamic
approach, while the second is semi-classical. Hereinafter the semi-classical motion
of electrons will be neglected. Therefore, we may put Tc ∼ EF.
We denote
α =
ξ0
L
� 1. (18)
705
P.Shygorin, A.Svidzynsky
Dividing equations (15) and (16) by Tc and using (18) we get
iα
{
∂
∂t
+ eEi
∂
∂pi
− mv̇si
∂
∂pi
−
∂
∂Rj
(
(pi + mvsi)
2
2m
)
∂
∂pj
+
(
pi
m
+ vsi
)
∂
∂Ri
}
G↑(~R, ~p, t) = |∆(~R, t)|
(
F ∗(~R, ~p, t) − F (~R, ~p, t)
)
− α
i
2
∂|∆(~R, t)|
∂Ri
∂
∂pi
(
F ∗(~R, ~p, t) + F (~R, ~p, t)
)
, (19)
{
iα
∂
∂t
− 2eA0(~R, t) − 2mχ̇(~R, t) − iα
(
pi
∂vsi
∂Rj
∂
∂pj
− vsi
∂
∂Ri
)
−
(
p2
m
+ vs
2
)}
F (~R, ~p, t) = |∆(~R, t)|
(
1 − G↑(~R, ~p, t) − G↓(~R,−~p, t)
)
− α
i
2
∂|∆(~R, t)|
∂Ri
∂
∂pi
(G↑(~R, ~p, t) − G↓(~R,−~p, t)). (20)
In order to solve the equations (19) and (20) we formally expand the functions
G, F and ∆ in powers of α
f = f (0) + αf (1). (21)
In the lowest order equation (19) gives
F (0)∗(~R, ~p, t) = F (0)(~R, ~p, t), (22)
that means a real function F (0). The equation (20) by using (22) in the lowest order
gives
(
eA0(~R, t) + mχ̇(~R, t) +
1
2
mv2
s(
~R, t)
)
F (0)(~R, ~p, t)
= −
p2
2m
F (0)(~R, ~p, t) −
1
2
|∆(0)(~R, t)|(1 − G
(0)
↑ (~R, ~p, t) − G
(0)
↓ (~R,−~p, t)). (23)
The separation of variables gives
eA0(~R, t) + mχ̇(~R, t) +
1
2
mv2
s(~R, t) + µ(~R, t) = 0, (24)
and
2ξpF
(0)
(
~R, ~p, t
)
+ |∆(0)
(
~R, t
)
|
(
1 − G
(0)
↑
(
~R, ~p, t
)
− G
(0)
↓
(
~R,−~p, t
))
, (25)
where ξp = p2/(2m) − µ and µ – is the order of separation of variables. In the local
equilibrium state µ – is chemical potential and equation (25) is reduced to the
equation for order parameter.
Applying the operation ∇~R to equation (24) we obtain the equation of motion
for superfluid velocity
m
∂~vs
∂t
+ ∇~R
(
m~v2
s
2
+ µ
)
= e ~E. (26)
706
Hydrodynamic equations for the superfluid fermi-systems
By using the vector identity
1
2
∇~v2
s = ~vs × (∇× ~vs) + ~vs · ∇~vs = −
e
m
~vs × ~H + ~vs · ∇~vs .
Equation (15) can be written as follows
m
d~vs
dt
= e
[
~E + (~vs × ~H)
]
−∇µ. (27)
This is the first hydrodynamic equation and shows that the superfluid accelerates
freely under the applied fields. The remaining hydrodynamic equations are provided
by the conservation relations for the particle density ρ(~R, t), momentum density
~j(~R, t) and energy density E(~R, t).
Let us consider the first order of equations (19) and (20). These equations are
as follows:
∂G
(0)
↑ (~R, ~p, t)
∂t
+
(
pi
m
+ vsi
) ∂G
(0)
↑ (~R, ~p, t)
∂Ri
+
∂G
(0)
↑ (~R, ~p, t)
∂pi
×
(
eEi − mv̇si −
∂
∂Ri
(
(~p + m~vs)
2
2m
))
−
∂∆(0)(~R, t)
∂Ri
∂F (0)(~R, ~p, t)
∂pi
= 2|∆(0)(~R, t)|ImF (1)(~R, ~p, t), (28)
∂F (0)(~R, ~p, t)
∂t
+ 2i
(
eA0(~R, t) + mχ̇(~R, t) +
1
2
mv2
s(~R, t) +
p2
2m
)
F (1)(~R, ~p, t)
=
∂∆(0)(~R, t)
∂Rj
∂
∂pj
(
G
(0)
↑ (~R, ~p, t) − G
(0)
↓ (~R,−~p, t)
)
+
∂
∂pj
(
pi
∂vsi
∂Rj
F (0)(~R, ~p, t)
)
−
∂
∂Ri
(
vsiF
(0)(~R, ~p, t)
)
− i|∆(1)(~R, t)|
(
1 − G
(0)
↑ (~R, ~p, t) − G
(0)
↓ (~R,−~p, t)
)
. (29)
The relations for ρ, ~j and E follow simply from calculation of moments of (28)
and (29)(see [6]). By definition
ρ(~R, t) = 2m
∫ d~p
(2π)3
G
(0)
↑ (~R, ~p, t) (30)
and after integrating (28) over ~p we find
∂ρ
∂t
+ div~j = 0, (31)
where
~j(~R, t) = 2
∫ d~p
(2π)3
~pG
(0)
↑ (~R, ~p, t) + ρ~vs ≡ ~j0 + ρ~vs .
707
P.Shygorin, A.Svidzynsky
By analogy, a simple calculation gives
∂jk
∂t
+
∂Πik
∂Ri
=
e
m
ρEk +
e
mc
Hkiji , (32)
where Hki is magnetic field intensity tensor and stress tensor Πik is given by
Πik =
2
m
∫ d~p
(2π)3
(pi + mvsi)(pk + mvsk)G
(0)
↑ (~R, ~p, t) + δik
∆(0)(~R, t)
g
. (33)
The energy density (without mean-field energy) is
E(~R, t) =
1
m
∫ d~p
(2π)3
(~p + m~vs)
2G
(0)
↑ (~R, ~p, t) +
|∆(0)(~R, t)|2
g
, (34)
or
E = E0 +~j0~vs +
1
2
ρv2
s ,
where
E0 =
1
m
∫ d~p
(2π)3
p2G
(0)
↑ (~R, ~p, t) +
|∆(0)(~R, t)|2
g
.
Using (28), (29) and (24) we obtain
∂E
∂t
+ div ~Q =
e
m
~E~j, (35)
the energy current is given by
~Q =
∫ d~p
(2π)3 (~p + m~vs)
(
~p
m
+ ~vs
)2
G
(0)
↑
(
~R, ~p, t
)
+ 2
∆(0)
(
~R, t
)
g
~vs . (36)
The flows of hydrodynamic quantities may be calculated if we assume that in
zeroth approximation the thermodynamic local equilibrium is provided by the gra-
dients. In local equilibrium
G
(0)
↑ = υ2
~p + u2
~pf
(
ε~p − ~u~p
T
)
− υ2
~pf
(
ε~p + ~u~p
T
)
, (37)
where
f(x) = (ex + 1)−1, u2
~p =
1
2
(
1 +
ξ~p
ε~p
)
, υ2
~p =
1
2
(
1 −
ξ~p
ε~p
)
, ε~p =
√
ξ2
~p + ∆2.
Using expression (37) we obtain
~j = ρs~vs + ρn~vn, ~vn ≡ ~u + ~vs ,
Πik = ρnvnivnk + ρsvsivsk + δikP,
~Q =
(
v2
s
2
+
µ
m
)
~j + TS~vn + ρn~vn(~vn · (~vn − ~vs)). (38)
708
Hydrodynamic equations for the superfluid fermi-systems
Here
ρn =
1
u2
∫ d~p
(2π)3
~p~uf
(
ε~p − ~u~p
T
)
is normal density,
ρs = ρ − ρn
is superfluid density,
P = TS − E0 + u2ρn + ρµ/m
is pressure, and entropy is represented as follows:
S = 2
∫ d~p
(2π)3
ln
(
1 + exp
(
ε~p − ~u~p
T
))
+
2
T
∫ d~p
(2π)3
f
(
ε~p − ~u~p
T
)
.
The equations (27), (31), (32), (35) and (38) form a complete system of magneto-
hydrodynamic equations for superconductor.
References
1. Nozières P., Pines D. The Theory of Quantum Liquids. Addison-Wesley, Redwood City,
California, 1990.
2. Yakovlev D.G., Levenfish K.P., Shibanov Yu.A., Usp. Fiz. Nauk, 1999, 169, 825 (in
Russian).
3. Svidzynsky A.V. Microscopic Theory of Superconductivity. Lutsk, Veza, 2001 (In
Ukrainian).
4. Landau L.D., JETF, 1941, 11, 592 (in Russian).
5. Bogolubov N.N. To the question about hydrodynamics of superfluid liquid. Preprint
UINR, Dubna, 1963 (in Russian).
6. Svidzynsky A.V. The Space Inhomogenous Problems Theory of Superconductivity.
Moscow, Nauka, 1982 (in Russian).
7. Stephen M.J. Phys. Rev. A, 1965, 139, 197.
709
P.Shygorin, A.Svidzynsky
Мікроскопічне виведення рівнянь гідродинаміки
для надплинних фермі систем
А.Свідзинський, П.Шигорін
Волинський державний університет ім. Лесі Українки
Отримано 3 серпня 2005 р.
Виходячи з перших принципів статистичної механіки, побудовано
дворідинну гідродинаміку надпровідника в ідеальному наближенні.
Для побудови гідродинаміки використано систему рівнянь руху для
нормальної та аномальної кореляційних функцій. Перехід до рівнянь
гідродинаміки здійснюється через розклад рівнянь руху для кореля-
ційних функцій за малим параметром.
Ключові слова: дворідинна гідродинаміка, кореляційна функція,
надплинні Фермі-системи
PACS: 05.30.Fk, 47.37.+q
710
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