Quasilinear theory of quantum Fermi liquid
Quasilinear theory of a weakly turbulent quantum Fermi liquid is presented. Landau's linear theory of Fermi liquids is generalized by consideration of weak nonlinear regime. A newly derived kinetic equation of the Fermi particles is used to derive a slowly varying distribution function f₀, whic...
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irk-123456789-1293322018-01-19T03:04:24Z Quasilinear theory of quantum Fermi liquid Tsintsadze, Nodar L. Alkhanishvili, Davit M. Квантовые жидкости и квантовые кpисталлы Quasilinear theory of a weakly turbulent quantum Fermi liquid is presented. Landau's linear theory of Fermi liquids is generalized by consideration of weak nonlinear regime. A newly derived kinetic equation of the Fermi particles is used to derive a slowly varying distribution function f₀, which satisfies the diffusion equation. It is shown that the magnitude of the diffusion coefficient D depends on weak interactions between atoms and the de Broglie waves diffraction. 2016 Article Quasilinear theory of quantum Fermi liquid / Nodar L. Tsintsadze Davit M. Alkhanishvili // Физика низких температур. — 2016. — Т. 42, № 12. — С. 1368-1371. — Бібліогр.: 7 назв. — англ. 0132-6414 PACS: 67.30.–n, 71.10.Ay, 74.20.Mn http://dspace.nbuv.gov.ua/handle/123456789/129332 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Tsintsadze, Nodar L. Alkhanishvili, Davit M. Quasilinear theory of quantum Fermi liquid Физика низких температур |
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Quasilinear theory of a weakly turbulent quantum Fermi liquid is presented. Landau's linear theory of Fermi liquids is generalized by consideration of weak nonlinear regime. A newly derived kinetic equation of the Fermi particles is used to derive a slowly varying distribution function f₀, which satisfies the diffusion equation. It is shown that the magnitude of the diffusion coefficient D depends on weak interactions between atoms and the de Broglie waves diffraction. |
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Article |
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Tsintsadze, Nodar L. Alkhanishvili, Davit M. |
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Tsintsadze, Nodar L. Alkhanishvili, Davit M. |
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Tsintsadze, Nodar L. |
| title |
Quasilinear theory of quantum Fermi liquid |
| title_short |
Quasilinear theory of quantum Fermi liquid |
| title_full |
Quasilinear theory of quantum Fermi liquid |
| title_fullStr |
Quasilinear theory of quantum Fermi liquid |
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Quasilinear theory of quantum Fermi liquid |
| title_sort |
quasilinear theory of quantum fermi liquid |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2016 |
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Квантовые жидкости и квантовые кpисталлы |
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http://dspace.nbuv.gov.ua/handle/123456789/129332 |
| citation_txt |
Quasilinear theory of quantum Fermi liquid / Nodar L. Tsintsadze Davit M. Alkhanishvili // Физика низких температур. — 2016. — Т. 42, № 12. — С. 1368-1371. — Бібліогр.: 7 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT tsintsadzenodarl quasilineartheoryofquantumfermiliquid AT alkhanishvilidavitm quasilineartheoryofquantumfermiliquid |
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2025-07-09T11:09:11Z |
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2025-07-09T11:09:11Z |
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1837167391485722624 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12, pp. 1368–1371
Quasilinear theory of quantum Fermi liquid
Nodar L. Tsintsadze and Davit M. Alkhanishvili
Faculty of Exact and Natural Sciences and E. Andronikashvili Institute of Physics, Javakhishvili Tbilisi State University
Tbilisi 0128, Georgia
E-mail: nltsin@yahoo.com
Received May 1, 2016, published online October 24, 2016
Quasilinear theory of a weakly turbulent quantum Fermi liquid is presented. Landau’s linear theory of Fermi
liquids is generalized by consideration of weak nonlinear regime. A newly derived kinetic equation of the Fermi
particles is used to derive a slowly varying distribution function f0, which satisfies the diffusion equation. It is
shown that the magnitude of the diffusion coefficient D depends on weak interactions between atoms and the de
Broglie waves diffraction.
PACS: 67.30.–n 3He;
71.10.Ay Fermi-liquid theory and other phenomenological models;
74.20.Mn Nonconventional mechanisms.
Keywords: Fermi liquid, quasiparticles, quasilinear theory.
In 1961 quasilinear theory for plasma particles was cre-
ated by A.A. Vedenov et al. [1] and W. Dramond and
D. Pines [2]. The quasilinear theory describes the dynam-
ics of the interaction between the resonance particles and
the waves. The theory is able to treat such processes when
the energy of the medium oscillations is appreciably less
than the total internal energy of the particles, but is, at the
same time, much greater than the noise energy of the sound
waves.
The quasilinear theory lies in the division of the particle
distribution function into two parts: a rapidly oscillating
part and a slowly varying part, and also in calculating the
influence of the mean square of the oscillating part of the
number density of particles on the slowly varying part. It is
found that the behavior of the slow part of the distribution
function can be described by a diffusion equation in the
momentum space, but the rate of damping or growth of the
rapid oscillations is given by the linear theory equations in
which the slowly varying part of the distribution function
varies slowly with time.
We have applied quasilinear theory to the Fermi liquid.
It is well known that at temperatures 1–2 K only two quan-
tum liquids exist in nature, the isotopes of helium 3He and
4 He, and all other substances solidify. The peculiarly
weak interaction between the helium atoms is the reason
for helium to remain liquid. Based on this fact, namely that
in 3He the weak interactions take place between atoms at
sufficiently low temperatures, Landau has created the theo-
ry of Fermi liquid [3]. In which he took into account only
the weakly excited energy levels of the liquid, lying fairly
close to the ground state. Landau assumed that any weakly
excited state of a macroscopic body can be represented as
an assembly of separate elementary excitations (quasipart-
icles). Moreover, the elementary excitations are represent-
ed as the collective motion of atoms in liquid and it can not
be identified with individual atoms. Therefore, an import-
ant characteristic of the energy spectrum is the establish-
ment of the dispersion relation ( )pε for elementary excita-
tions. Landau has then shown that the undamped zero
sound can exist in an almost ideal Fermi gas, which was
confirmed in experiment by W.R. Abel et al. [4].
Pomeranchuk has shown that Landau’s liquid can be un-
stable [5]. Landau’s theory of Fermi liquids was general-
ized by incorporating the de Broglie waves diffraction [6].
In this article, we describe the quasilinear theory of Fer-
mi liquids by taking into account the de Broglie wave dif-
fraction and derive the corresponding diffusion equation in
momentum space written for equilibrium distribution func-
tion. To achieve this we employ a novel quantum kinetic
equation derived in Ref. 6, where use was made of the
quasiclassical function = exp ( / )A iSψ . The difference
between the Landau kinetic equation and ours is that in our
equation an additional term, namely the Madelung term is
incorporated due to the diffraction of de Broglie waves.
Nonequilibrium states of a Fermi quantum liquid are
described by the one particle distribution function ( , , ),f r p t
which satisfies the quantum Boltzmann equation [6]:
2 1( ) = ( )
2
f f ff n C f
t m n
∂ ∂ ∂
+ ⋅∇ −∇ε ⋅ + ∇ ∆
∂ ∂ ∂
v
p p
, (1)
© Nodar L. Tsintsadze and Davit M. Alkhanishvili, 2016
Quasilinear theory of quantum Fermi liquid
ε is the energy of the quasiparticle, m and n are mass and
density of particles, respectively, and ( )C f is the collision
integral, which describes the variation of the distribution
function due to particle collisions.
Note that when the spin of the particles is taken into ac-
count, the distribution function is an operator with respect
to the spin variables σ . The quasiparticles in a Fermi liquid
have spin 1/2. However, there is a wide range of problems
in which it is sufficient to consider a distribution independ-
ent of spin variables, so that f becomes the ordinary quasi-
classical distribution function ( , , )f tr p . We recall here that
the condition for quasiclassical motion is that the de Brog-
lie wavelength = /d Fpλ ( Fp is Fermi momentum) of
the particle must be very small compared with the charac-
teristic length L , over which ( , , )f tr p varies considerably.
Following the Landau’s theory, hereafter, we consider
Fermi liquid as a spinless, and the energy ε of quasiparticle
is a functional of the distribution function; a variation of
distribution function
0( , , ) = ( ) ( , , )f t f f t+ δr p p r p (2)
produces a variation of energy given by
3
3
2= ( , ) ( , , )
(2 )
df t
′
′ ′δε φ δ
π∫
pp p r p
(3)
where the factor 2 appears due to spin, 0 ( )f p and ( , )′φ p p
are the equilibrium distribution function and the quasi-
particle interaction function, respectively; in a Fermi gas
= 0φ . Thus the distribution function (2) refers to the ener-
gy of quasiparticle
0= ( ) ( , , )tε ε + δεp r p (4)
where 0 ( )ε p is the energy corresponding to the equilibrium
state.
Near the surface of the Fermi sphere the variation of
distribution function ( , , )f t′δ r p is appreciably different
from zero, i.e., the magnitude = = Fp p′p . The same is
true for the function ( , )′φ p p . So that both depend only on
directions of the vectors p and ′p . Hence, the quasiparticle
interaction function φ and fδ can be expressed at the Fer-
mi surface as
2 3
*
( , ) = ( )
F
Q
m p
π′φ θp p , (5)
= ( ) ( , , )Ff F t′δ δ ε − ε n r (6)
where * = /F Fm p v is the effective mass of quasiparticle,
′n is the unit vector in the direction of ′p , and ( )Q θ is the
function of the angle θ between p and ′p .
We now employ Eq. (1) to study the propagation of
small perturbations in the Fermi quantum liquid. We sub-
stitute the Eqs. (2) and (4) into Eq. (1) and linearize it with
respect to the perturbation fδ to obtain
2
0 0
0
( ) = ( )
4
f ff nf C f
t m n
∂ ∂∂δ δ
+ ⋅∇ δ −∇δε ⋅ + + ∇ ⋅∆ δ
∂ ∂ ∂
v
p p
. (7)
We look for wave solutions in space and time for
( , , )F t′n r , assuming that it is proportional to exp [ ( )]i t⋅ −ωk r .
Taking into account Eqs. (3)–(6), ∇ε can be written as
= ( ) ( , , ) = ( ) ( )
2 2
d dQ F t i Q F
′ ′Ω Ω′ ′ ′ ′∇δε θ ∇ θ∫ ∫n r k n (8)
where = sind d′ ′ ′Ω θ θ .
If we assume that ( )Q ′θ is constant, i.e. 0( ) =Q Q′θ ,
then
( )
0= e ( )
2
i t di Q F⋅ −ω ′Ω′∇δε ∫k rk n . (9)
For density perturbation nδ we get
3 *
3 3
42= ( , , ) = ( )
(2 ) (2 )
Fm pdn f t F d
′ π′ ′ ′δ δ Ω
π π∫ ∫
pr p n
. (10)
Since 3
0 = 8 / 3(2 )Fn pπ π ,
* ( )
2
0
3= e ( )
2
i t
F
n m dF
n p
⋅ −ω ′δ Ω′∫k r n (11)
and
2
0 * 0
=
3
Fp ni Q
nm
δ
∇δε k . (12)
To see what impact does quasiparticle collisions have
on Fermi liquid, for a rough estimate of collision integral,
we can put
0( ) = = =
f f fC f f
− δ
δ − −νδ
τ τ
(13)
where 1= −τ ν is the mean free time.
We substitute Eqs. (12) and (13) into Eq. (7) and finally
for perturbation fδ we get
2 2 2
0 0
* *0 0
1 1=
3 4
Fp Q fk nf
n n im m
∂δ
δ − + ⋅ ω+ ν − ⋅ ∂
k
k v p
. (14)
The main criteria for using linear approximation is
0| |f fδ . (15)
In other words this means, that oscillation energy due to
perturbation must be less than Fermi liquid internal energy.
For degenerate Fermi liquid this condition has form
1
F
W
Nε
(16)
where FNε is an internal energy of the Fermi liquid and
W is the oscillation energy due to perturbation.
Condition (16) holds true for the most mass of the liq-
uid, but there exists some number of particles for which
this condition is not maintained. Those are the particles,
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1369
Nodar L. Tsintsadze and Davit M. Alkhanishvili
which move in phase with the sound waves and due to this
they become involved in Landau damping. In this process
the distribution function changes. This change has a non-
linear character and because of this, its behavior greatly
depends on spectral composition of the sound wave. In
linear theory influence of oscillations on distribution func-
tion is neglected. Contrary to this, in quasilinear theory
waves are still considered linear, but distribution function
change due to oscillations is not neglected.
To study Fermi liquid in quasilinear theory we consider
perturbations, which are ensemble of waves and their wave
vectors are located inside narrow k∆ interval around some
arbitrary 0k wave vector. Because wave vector interval is
narrow, small amount of quasiparticles will be involved in
the Landau damping and as a result their distribution func-
tion will change significantly.
In order to derive the equations in the quasilinear ap-
proximation, we split up the distribution function ( , )f t p
into a large slowly and a small fast developing part:
( )
0 0= ( , ) ( , ) = ( , ) ei t
k
k
f f t f t f t f ⋅ −ω+ δ + δ∑ k rp p p , (17)
2
( )0
0
= e
3
i tF k
k
p Q n
i
m n
⋅ −ωδ
∇δε ∑ k rk , (18)
where perturbation kfδ is defined in Eq. (14).
Second term in the Eq. (17) is a rapidly oscillating term
and so it vanishes after a time averaging over fast oscilla-
tions. First term 0 ( , )f t p represents slowly varying aver-
aged part of the distribution function.
Now we substitute Eqs. (17) and (18) into Eq. (1) repre-
senting quantum-kinetic equation and take a time average
over the fast oscillations. The time interval 0τ must satisfy
the conditixon
0
2π
τ
ω
. (19)
Here 0τ is the time during which the oscillations influence
the equilibrium quantum liquid state. Then we have
0
0
0 0
1( , , ) = ( , ) =f t f t fdt
τ
〈 〉
τ ∫r p p . (20)
Due to averaging Eq. (1), we obtain for the slow part of
the distribution function:
2
0
0
= 0
4
f f n f
t m n
∂ ∂δ δ ∂δ
− ∇δε ⋅ + ∇ ⋅∆
∂ ∂ ∂p p
. (21)
As a result, after taking an average of Eq. (21), we ob-
tained quasilinear equation for the slow part of the distribu-
tion function in momentum space, which has the form
0 0= ij
i j
f f
D
t p p
∂ ∂∂
∂ ∂ ∂
(22)
where ijD is the tensor of diffusion coefficients
2 22 2 2
0
2 2
0
2=
3 4 ( )
F k
ij i j
k
p Q nkD k k
m m n
δ ν
+ ω− ⋅ + ν
∑
k v
. (23)
Theory that is established on this equation is called
quasilinear theory. The process absorption can be describ-
ed with quasilinear theory, which, as we said above, takes
into account changing of equilibrium distribution function
due to oscillations caused by sound waves. It is apparent,
that oscillations must affect distribution function in such
way to decrease the absorption of waves (decrease interac-
tion with particles). According to this, by increasing wave
amplitude it absorption must decrease and in Fermi liquid,
where we neglect collisions, can be established such distri-
bution, where absorption does not happen. From linear theo-
ry we know that wave absorption is determined with the
value of 0 /f p∂ ∂ (due to Landau damping). Hence, describ-
ed situation is possible, if due to impact of the wave on
the Fermi liquid plateau (area where 0 / = 0f p∂ ∂ ) is created
on the equilibrium distribution function and at this moment
wave is still not absorbed.
When we consider the collision processes, plateau can
not be created on the distribution function, because particle
collisions tend to establish equilibrium state in the system.
We can clearly observe this from the Eq. (14), when
=ω ⋅k v (particles which move in phase with the waves
are involved in the plateau formation). We get
2 2 2
0 0
0
=
3 4
Fp Q fk ni f
m m n
∂ δ
− νδ + ⋅ ∂
k
p
. (24)
It is clear that 0 / 0f p∂ ∂ ≠ , so plateau can not formed on the
distribution function.
In the absence of interaction between quasiparticles,
i.e., 0 = 0Q , we see that
2 22 2
0 0
2 2
0
2=
4 ( )
k
i j
i jk
f n fkk k
t p m n p
∂ δ ∂∂ ν
∂ ∂ ∂ω− ⋅ + ν
∑
k v
. (25)
From this we can deduce, that de Broglie wave diffraction
causes diffusion of distribution function in the momentum
space, when interaction between quasiparticles is absent.
To illustrate the solution of Eq. (22) for simple cases,
we assume that diffusion coefficient (23) does not depend
on velocity of the quasiparticles and propagating waves
wave vectors are parallel to each other. This is possible if
ω ≈ ⋅k v or 2 2( )ω− ⋅ νk v . In this case
2
0 0
2=
f f
D
t
∂ ∂
∂ ∂p
, (26)
2 22 2 2
2 0
0
2=
3 4
F k
k
p Q nkD k
m m n
δ
+ ν
∑ . (27)
1370 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12
Quasilinear theory of quantum Fermi liquid
The solution of Eq. (26) has the form
2
3
0 3/2
1 ( )( , ) = ( ) exp
48( )
f t F d p
DtDt
′−′ ′−
π
∫
p pp p (28)
where ( )F ′p is the initial distribution function, i.e.,
0( ) = (0, )F fp p .
In our case ( )F ′p is distribution function which de-
scribes degenerate Fermi liquid at = 0 KT temperature.
Mathematically it is Heaviside step function. So
,
.
1
( ) =
0 >
F
F
p p
F p
p p
≤
(29)
Solution of Eq. (26) in 1D is
0
1( , ) = erf erf
2 2 2
x F F x
x
p p p p
f t p
Dt Dt
+ −
+
(30)
where erf ( )x is an error function, which is defined as
2
0
2erf ( ) = e
x
tx dt−
π ∫ . (31)
Corresponding plot of Eq. (30) for various moments of
time illustrate Fig. 1.
From this plot we can deduce that diffusion of distribu-
tion function in momentum space causes violation of the
Fermi surface.
Since the diffusion is happening in the momentum
space, density of the particles must be constant and must
equal to density of the degenerate Fermi liquid. We can
check this by integrating Eq. (28).
2
3 3
3 3/2
2 1 ( )= ( ) exp ,
4(2 ) 8( )
n F d p d p
DtDt
′−′ ′−
π π
∫∫
p pp
(32)
3
2 3=
3
Fpn
π
. (33)
As we expected solution (28) gives physical results and
thus our theory is correct.
In the similar way we calculated mean energy using the
distribution function (28)
2 2
3 3
3 3/2
2 1 ( )= ( ) exp ,
2 4(2 ) 8( )
pE F d p d p
m DtDt
′−′ ′−
π π
∫ ∫
p pp
(34)
23 6=
10 2
Fp n DtnE
m m
+ . (35)
First term of this expression is the mean energy in absence
of diffusion and corresponds to energy of degenerate Fermi
liquid. Second term arises due to diffusion and increases
linearly with time.
To summarize, we have developed quasilinear theory of
Fermi liquids by taking into account the diffraction of the
de Broglie waves. To this end we used the quantum kinetic
equation derived by N.L. Tsintsadze and L.N. Tsintsadze
in recent paper [6]. It should be noted that our kinetic
equation is considerably richer than the Landau’s kinetic
equation. There is an additional physical feature included
here, namely the Madelung term is incorporated due to the
diffraction of the de Broglie waves. This term is responsi-
ble for the diffusion in the momentum space even in an
ideal Fermi gas. Quasilinear theory can be also applied to
electrons in most metals. The theory has found further ap-
plication in nuclear an neutron star matter, superfluid 3He
and contemporary problems in superconductivity.
Acknowledgments
N.L.T. and D.M. Alkhanishvili would like to acknowledge
the partial support of GNSF Grant Project N FR /101/6-140/13.
1. A.A. Vedenov, E.P. Velikov, and R.Z. Sagdeev, Usp. Fiz.
Nauk 23, 701 (1961).
2. W. Drammond and D. Pines, Proceeding of the Int. Confe-
rence on Plasma Physics and Controlled Thermonuclear
Fusion, Salzburg (1961), Paper No. 134.
3. L.D. Landau, Zh. Eksp. Teor. Fiz. 30, 1058 (1956); ibid. 32,
59 (1957); ibid. 35, 97 (1958).
4. W.R. Abel, A.C. Anderson, and J.C. Wheatley, Phys. Rev.
Lett. 17, 74 (1966).
5. I.Ia. Pomeranchuk, Sov. Phys. JETP 35, 524 (1958).
6. N.L. Tsintsadze and L.N. Tsintsadze, Europhys. Lett. 88,
35001 (2009); From Leonardo to ITER: Nonlinear and
Coherence Aspects, Jan Weiland (ed.), AIP Proc. No.
CP1177 AIP, New York (2009).
7. N.L. Tsintsadze and L.N. Tsintsadze, Fiz. Nizk. Temp. 37,
982 (2011) [Low Temp. Phys. 37, 782 (2011)].
Fig. 1. (Color online) Plot of the distribution function, represent-
ed by Eq. (30) for various moments of time.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1371
http://dx.doi.org/10.1063/1.3665869
Acknowledgments
|