Quasi-simple secondary waves in a gas of quasi-particles
The equations describing simple secondary waves are obtained in gas dynamics with conserved and non-conserved number of quasi-particles at interactions. The non-linearity parameter in phonon and magnon gas dynamics is found, which appears to be in its value of an order of unity. The generalized Burg...
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| citation_txt | Quasi-simple secondary waves in a gas of quasi-particles / V.D. Khodusov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 315-317. — Бібліогр.: 11 назв. — англ. |
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| description | The equations describing simple secondary waves are obtained in gas dynamics with conserved and non-conserved number of quasi-particles at interactions. The non-linearity parameter in phonon and magnon gas dynamics is found, which appears to be in its value of an order of unity. The generalized Burgers equation is obtained describing the quasi-simple secondary waves.
|
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QUASI-SIMPLE SECONDARY WAVES
IN A GAS OF QUASI-PARTICLES
V.D. Khodusov
Departmen of Physics and Technology, Kharkov National University by V.N. Karazin
Kharkov, Ukraine
e-mail: khodusov@pem.kharkov.ua
The equations describing simple secondary waves are obtained in gas dynamics with conserved and non-con-
served number of quasi-particles at interactions. The non-linearity parameter in phonon and magnon gas dynamics is
found, which appears to be in its value of an order of unity. The generalized Burgers equation is obtained describing
the quasi-simple secondary waves.
PACS: 05.40+j, 63.20.-e
INTRODUCTION
A system of equations of gas dynamics of bose
quasi-particles has been obtained in the work [1]. This
system of equations is non-linear relative to the inde-
pendent variables: the drift velocity u , the relative tem-
perature ( ) 00 TTT −=θ ( 0T is the equilibrium temper-
ature) and chemical potential µ , if the number of quasi-
particles at interactions is kept constant. In the linear ap-
proximation, this system of equations describes, in cer-
tain circumstances, the weakly attenuating secondary
waves [1, 2] being similar to the second sound waves in
He II [3]. These waves relate to a hyperbolic type, i.e.
their dispersion is absent. The paper deals with the non-
linear secondary waves in a gas of quasi-particles,
which are described by a system of non-linear equations
of gas dynamics of quasi-particles in the second approx-
imation. This system of equations is similar to the sys-
tem of non-linear equations of gas dynamics of parti-
cles, describing the propagation of non-linear acoustic
waves in a medium without dispersion [4-6]. Using this
system of equations for the one-dimensional case, when
there is no dissipation, we can derive an equation de-
scribing simple Riemann waves. Taking into account
the dissipative processes one can find the generalized
Burgers equation describing quasi-simple waves. These
waves are featured by the distortion of their profiles,
like, for example, in the case of a periodic wave - a saw-
tooth profile is formed.
SIMPLE SECONDARY WAVE EQUATION
The simplest way is to obtain the simple secondary
wave equation in isotropic gas of quasi-particles with
the non-conserved number of quasi-particles. In the case
of a plane along the x-axis, which can be realized in, for
instance, an infinite plate of finite thickness, ( ),, txuu =
( )txTT ,= . A system of gas dynamics equations in the
second approximation, in terms of independent vari-
ables and kinetic equation will be written in the follow-
ing form
ru
x
u
x
TS
x
uP
x
Pu
t
P −
∂
∂
+=
∂
∂+
∂
∂+
∂
∂+
∂
∂
2
2
0
~~
3
42 ζη ,
2
2~
x
T
Tx
uS
x
uu
t
u
T
P
x
Tu
t
T
T
C
∂
∂=
∂
∂+
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂ κ
,
(1)
where S , C are the densities local equilibrium entropy
and heat capacity of a gas of quasi-particles, uP
ρ~= is
the pulse density, ρ~ is the density of a gas of quasi-par-
ticles, ζη ~,~ are the first and second viscosity kinetic co-
efficients, κ~ is the hydrodynamic thermal conductivity
at account of normal processes of quasi-particle interac-
tions. r is the coefficient of external friction, as a result
of the quasi-particle interaction effects leading to non-
conservation of their momentum, e.g. the umklapp pro-
cesses (U-processes). In the absence of dissipation the
system of equations (1) describes simple secondary
waves similar to Riemann waves [4-6]. In a simple
wave, all quantities are the functions of one quantity.
Let us assume that ρ~ and T are the functions of the
drift velocity u
and ( )( ) ;,~~ uTuρρ = ( )uTT = ,
u
T
dT
d
udu
d
∂
∂+
∂
∂= ρρρ ~~~
. By substituting these values into
(1), we obtain:
0~2
~~~ =
∂
∂
++
∂
∂+
∂
∂
∂
∂+
∂
∂+
x
u
du
dTSu
x
uu
t
u
du
dT
T
u
u
u ρρρρ
0
~
=
∂
∂+
∂
∂+
∂
∂
∂
∂+
x
uS
x
uu
t
u
T
u
du
dT
T
C ρ . (2)
If 0≠∂∂ xu , then from the conditions of existence of a
solution it follows that:
∂
∂+
∂
∂−+±−=
u
u
TC
SMM
C
T
du
dT ρ
ρ
ρρ ~
~ln
~ln211
~
2 ,
(3)
where
IIW
uM = is the number analogue to the Mach
number in acoustics ( )1< <M ( ) 212
0II
~ρCTSW = is the
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 315-317. 59
quasi-equilibrium velocity of secondary waves. By sub-
stituting (3) into any of the equations (2), we obtain the
following simple wave equation in the second approxi-
mation by the Mach number M
( )
0
ln
~ln
2 II
0II =
∂
∂
+
∂
∂+
∂
∂±
∂
∂
T
W
C
S
x
uu
x
uW
t
u ρ
, (4)
where
C
STW
0
2
02
0II ~ρ
= is the equilibrium velocity of sec-
ondary waves. The line signifies the equilibrium values
of the quantity are taken.
It is conveniently to deal with the variables xx ′=
and ( )0IIWxt −=τ , when solving the boundary-value
problems. Then ;
τ∂
∂=
∂
∂
t
xWx ′∂
∂+
∂
∂−=
∂
∂
τ0II
1
and
from (4) it follows (omitting the prime x ):
0
2
0II
=
∂
∂−
∂
∂
τ
ε uu
Wx
u
, (5)
where the non-linearity parameter equals
( )
T
W
C
S
ln
~ln
2 II
∂
∂
+=
ρ
ε .
If the number of quasi-particles is conserved, then a
simple secondary wave equation can be derived simil-
arly. Assuming the quantities T , µ and ρ~ to be the
functions of the drift velocity u
: ( )uTT = , ( )uµµ = ,
( ) ( )( )uuTu µρρ ,,~~ = and du
d
du
dT
Tdu
d µ
µ
ρρρ
∂
∂+
∂
∂=
~~~
, in the
absence of dissipation, one can obtain
+
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂+
x
uu
t
u
du
du
du
dT
T
u
u
u µ
µ
ρρρρ
~~~~
0~2 =
∂
∂
µ++ρ+
x
u
du
dn
du
dTSu ,
0=
∂
∂+
∂
∂+
∂
∂
+
∂
∂+
x
un
x
uu
t
u
du
d
Tu
n
du
dT
T
µβα
, (6)
0
~
=
∂
∂+
∂
∂+
∂
∂
µα+
∂
ρ∂+
x
uS
x
uu
t
u
du
d
TT
u
du
dT
T
C
.
From the solvability conditions of the system (6), we
can find the relation between dudT and dudµ and
the following expression for dudT in zero approxima-
tion by the Mach number M
**
II
*
CW
TS
du
dT ±= (7)
where ( ) ;* βαnSS −= ( ) ;2* βα−= CC
+=
βρ *
2*
*
*
*
2*
I ~ S
nCS
C
TSWI , the hydrodynamic quantities
n , α , β are determined in [1].
Proper substitution of variables gives the following
simple secondary wave equation in gas dynamics with
the conserved number of quasi-particles
( ) 02*
0II
=
∂
∂−
∂
∂ ∗
τ
ε uu
Wx
u
, (8)
where the non-linearity parameter equals
( ) ( )
µ
ραρε
ln
~ln
ln
~ln2
*
II
*
***
II
*
*
∂
∂−+
∂
∂+=∗ W
C
SnC
T
W
C
S . (9)
Because secondary waves are essentially the thermal
waves, it is more convenient to deal with the equations
with more natural variable θ rather than with for the
drift velocity. In the linear approximation, the drift ve-
locity u and θ , as it follows from (3), (7) are related
to each other in gas dynamics with the non-conserved
number of quasi-particles by the relationship
SCWu II0θ= , and by the relationship ***
II0 SCWu θ=
in the case of the conserved number. By substituting
these values into (5) and (8), we obtain the simple sec-
ondary wave equation in terms of θ :
0=
∂
∂−
∂
∂
τ
θν θθ
x
, (10)
where
0IISW
Cεν = ( *
0II
*
*
*
WS
Cε=ν ).
Let us present the values of the non-linearity param-
eters ε and *ε for some specific gases of quasi-parti-
cles. In gas dynamics of phonons, using the values of
the necessary thermodynamic quantities given in [1] ,
we obtain 32=ε [7]. In gas dynamics of magnons in
ferromagnetics with magnetic anisotropy of the "light
axis" type in the case when the number of magnons is
not conserved and energy of magnon activation is small
( Ta < <ε ) and using the given in [1] values of thermo-
dynamic quantities characteristic of magnon gas, we
have 34=ε . If an interchange magnon scattering ap-
pears to be decisive at a number of magnons being con-
stant, one can easily verify that in this case the non-lin-
earity parameter equals as well 34* =ε .
In gas dynamics of magnons in anti-ferromagnetics
with magnetic anisotropy of the "light plane" type, in
the low temperature range, a three-magnon interaction
[1] appears to be decisive; in this case the number of
phonons is not conserved. In this case the value of the
non-linearity parameter will be of order of unity 1~ε .
Solutions of the equation (10) are well known [4 - 6].
As the simple wave front spins, the importance of dissi-
pation coefficients is increasing. Further evolution of
the non-linear secondary waves will be then described
by the system (1).
GENERALIZED BURGERS EQUATION
FOR QUASI-SIMPLE SECONDARY WAVES
At small (but finite) amplitudes and small dissipa-
tion coefficients, there is a solution of the system (1),
which can be considered as an analogue of the simple
waves propagating along one-way direction [5, 6]. Such
waves are called quasi-simple. To obtain the equations
describing these waves, we shall assume - in much the
same way as it was made in [8] - that all quantities are
the functions of one of them with an accuracy of some
arbitrary small function, i.e.
( ) ( ) ( )txuTTTu ,;,~~ ψρρ +== (11)
We are seeking such a form of this function when
the corresponding solution is the most close to a simple
60
wave. Let us consider the function ( )tx,ψ to be a quan-
tity of the second order smallness. Obviously, it will sat-
isfy the secondary wave equation up to the second order
terms, when the waves are propagating along the posi-
tive direction of the x axis
00II =
∂
∂+
∂
∂
x
W
t
ψψ
. (12)
Substituting (11) and (12) in (1), we received in the
second approximation a following system:
+
∂
∂+
∂
∂
∂
∂+
∂
∂+
x
uu
t
u
du
dT
T
u
u
u ρρρ
~~~
ru
x
u
x
S
x
u
du
dTSu −
∂
∂
ζ+η=
∂
ψ∂+
∂
∂
+ρ+ 2
2~~
3
4~2 ,
2
2
~~
x
u
du
dT
tT
C
x
uS
x
uu
t
u
T
u
du
dT
T
C
∂
∂κ=
∂
ψ∂+
∂
∂+
∂
∂+
∂
∂
∂
ρ∂+
(13)
Using expression (3) to within the linear members on u
and expressing from equation (12)
t∂
∂ ψ
through
x∂
∂ ψ
,
from (13) we received the value
x∂
∂ ψ
:
∂
∂κ−
−
∂
∂
ζ+η
ρ
ρ=
∂
ψ∂
2
2
2
2 ~~~
3
4
~
1
2
~
x
u
C
ru
x
u
Sx
. (14)
If we substitute (14) into the first equation in (1), in
which the function ψ is taken into account, we find the
following generalized Burgers equation for quasi-simple
waves propagating along the positive directions of the x
axis
( ) =
∂
ρ∂+
∂
∂+
∂
∂+
∂
∂
T
W
C
S
x
uu
x
uW
t
u
ln
~ln2 II
0II
−
∂
∂
++=
ρ
ζη
ρ
κ
~
~~
3
4
~
1~
2
1
2
2 ru
x
u
C
. (15)
Proceeding to the variables τ and x ′ and assuming
that the wave profile is changed slowly, we can neglect
the derivatives with respect to x assuming that they in-
crease the order of smallness. With account of the
above, we shall obtain the following generalized Burg-
ers equation.
−
∂
∂
++=
∂
∂−
∂
∂
ρτ
ζη
ρ
κ
τ
ε
~
~~
3
4
~
1~
2
1
2
2
0II
ruu
C
uu
Wx
u
(16)
Using the relationships between the drift velocity u
and θ , we shall write the Burgers equation (16) in the
more convenient form in order to carry out further in-
vestigations in terms of θ :
γ θ−
τ∂
θ∂δ=
τ∂
θ∂ν θ−
∂
θ∂
2
2
x
, (17)
where
ζ+η
ρ
+κ=δ
~~
3
4
~
1~
2
1
2
II0 CW
, 2
II00
~ W
r
ρ
γ = . If
one employs the condition of existence ("window") of
secondary waves [1], then γδ > > . In this case, the last
term in (17) could be neglected and we come to the con-
ventional Burgers equation [5,6,8,9], being one of the
most comprehensively studied evolution equations in
the non-linear wave theory
2
2
τ∂
θ∂δ=
τ∂
θ∂ν θ−
∂
θ∂
x
. (18)
As it was shown by Hopf [10] and Cole [11], by substi-
tuting the variables
τ
ϕ
ν
δθ
∂
∂= ln2
and performing the
one-fold integration, the equation (18) takes the form of
the linear equation of heat conductivity
2
2
τ
ϕδϕ
∂
∂=
∂
∂
x
(19)
which is capable of being precisely integrated. This
gives a rare opportunity to precisely solve the wave
problem for a dissipation medium. If we know boundary
condition ( ) ( )τθτθ 0,0 = the solution of equation (19)
can be written in the following from:
( ) ( ) ( ) τττθ
δδ
ττ
π δ
τϕ
τ
′
′′′′−
′−−= ∫ ∫
∞
∞−
′
dd
xx
x
0
0
2
2
1
4
exp
4
1,
REFERENCES
1. A.I. Akhiezer, V.F. Aleksin, V.D. Khodusov. Gas
dynamics of quasi-particles // Low Temp. Phys.
1994, v. 20, №12, p. 939-970.
2. A.I. Akhiezer, V.F. Aleksin, V.D. Khodusov. For
theory of secondary waves // Ukr. Fiz. Zh. 1985,v.
30, №8, p. 1248-1262.
3. L.D. Landau. The theory of superfluid helium II. //
Zh. Eksp. Teor. Fiz. 1941, v. 11, №5, p. 592-614.
4. L.D. Landau and E.M. Lifshits. Fluid Dynamics.
Moskow: “Nauka”, 1988, 736 p.
5. K.A. Naugol’nykh, L.A. Ostrovsky. Nonlinear
waves processes in acoustics. Moscow: “Nauka”,
1990, 235 p.
6. O.V. Rudenko and S.I. Soluyan. Theoretical
foundations of nonlinear acoustics. Moscow:
“Nauka”, 1975, 287 p.
7. V.D. Khodusov. Second sound waves of finite
amplitude in solids // AIP Conference Proceeding.
2000, v. 524, p. 261-264.
8. V.I. Karpman. Nonlinear waves in dispersive media.
Moscow: “Nauka”, 1973, 175 p.
9. J.M. Burgers Application of a model illustrate some
points of statistical theory of free turbulence // Proc.
Roy. Neth. Acad. Sci. (Amsterdam), 1940, v. 43,
p. 2-12.
10.E. Hopf The partial differential xxxt uuuu ⋅=⋅+ µ //
Comm. Pure Appl. Math. 1950, v. 3, p. 201-230.
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curring in aerodynamics // Appl. Math. 1951, v. 9,
p. 225-236.
61
QUASI-SIMPLE SECONDARY WAVES
IN A GAS OF QUASI-PARTICLES
V.D. Khodusov
Departmen of Physics and Technology, Kharkov National University by V.N. Karazin
INTRODUCTION
SIMPLE SECONDARY WAVE EQUATION
GENERALIZED BURGERS EQUATION
FOR QUASI-SIMPLE SECONDARY WAVES
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-80040 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:24:27Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Khodusov, V.D. 2015-04-09T16:21:57Z 2015-04-09T16:21:57Z 2001 Quasi-simple secondary waves in a gas of quasi-particles / V.D. Khodusov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 315-317. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 05.40+j, 63.20.-e https://nasplib.isofts.kiev.ua/handle/123456789/80040 The equations describing simple secondary waves are obtained in gas dynamics with conserved and non-conserved number of quasi-particles at interactions. The non-linearity parameter in phonon and magnon gas dynamics is found, which appears to be in its value of an order of unity. The generalized Burgers equation is obtained describing the quasi-simple secondary waves. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory Quasi-simple secondary waves in a gas of quasi-particles Квазипростые вторичные волны в газе квазичастиц Article published earlier |
| spellingShingle | Quasi-simple secondary waves in a gas of quasi-particles Khodusov, V.D. Kinetic theory |
| title | Quasi-simple secondary waves in a gas of quasi-particles |
| title_alt | Квазипростые вторичные волны в газе квазичастиц |
| title_full | Quasi-simple secondary waves in a gas of quasi-particles |
| title_fullStr | Quasi-simple secondary waves in a gas of quasi-particles |
| title_full_unstemmed | Quasi-simple secondary waves in a gas of quasi-particles |
| title_short | Quasi-simple secondary waves in a gas of quasi-particles |
| title_sort | quasi-simple secondary waves in a gas of quasi-particles |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80040 |
| work_keys_str_mv | AT khodusovvd quasisimplesecondarywavesinagasofquasiparticles AT khodusovvd kvaziprostyevtoričnyevolnyvgazekvazičastic |