Transport equations for low density soliton gas
We present the theory of transport phenomena in a gas of solitons for Sinus Gordon model system. The general case of relaxation phenomena is considered. A special attention is paid for a small density non-relativistic gas of breathers. Such kinetic coefficients as diffusion, thermal conductivity, an...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | Transport equations for low density soliton gas / I.V. Baryakhtar, V.G. Baryakhtar // Вопросы атомной науки и техники. — 2001. — № 6. — С. 265-267. — Бібліогр.: 4 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859481994350559232 |
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| author | Baryakhtar, I.V. Baryakhtar, V.G. |
| author_facet | Baryakhtar, I.V. Baryakhtar, V.G. |
| citation_txt | Transport equations for low density soliton gas / I.V. Baryakhtar, V.G. Baryakhtar // Вопросы атомной науки и техники. — 2001. — № 6. — С. 265-267. — Бібліогр.: 4 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | We present the theory of transport phenomena in a gas of solitons for Sinus Gordon model system. The general case of relaxation phenomena is considered. A special attention is paid for a small density non-relativistic gas of breathers. Such kinetic coefficients as diffusion, thermal conductivity, and internal friction are found. It is shown, that diffusion coefficient and internal friction coefficient equal each other.
|
| first_indexed | 2025-11-24T15:13:03Z |
| format | Article |
| fulltext |
TRANSPORT EQUATIONS FOR LOW DENSITY SOLITON GAS
I.V. Baryakhtar 1 , V.G. Baryakhtar 2
1 Low Temperature Physics and Engeeniring Institute, Kharkov, Ukraine
2 Institute of Magnetism, National Academy of Sciences, Kyiv, Ukraine
e-mail: vbar@nas.gov.ua
We present the theory of transport phenomena in a gas of solitons for Sinus Gordon model system. The general
case of relaxation phenomena is considered. A special attention is paid for a small density non-relativistic gas of
breathers. Such kinetic coefficients as diffusion, thermal conductivity, and internal friction are found. It is shown,
that diffusion coefficient and internal friction coefficient equal each other.
PACS: 05.60.+w
1. INTRODUCTION
Many physical phenomena require for their explana-
tion an exact solution of nonlinear equation of motion.
Such examples as the domain boundaries in magnetic
systems, tsunami, solitons both in plasma and non-linear
optics are well known. Last years special attention was
directed to the kinetic properties of solitons. At the mo-
ment there exists a theory of soliton diffusion. The au-
thors attack the problem of construction of a general
theory of kinetic properties of solitons like thermal con-
ductivity, internal friction, etc. In the previous paper [1]
authors presented a theory of kinetic equation for soli-
tons. Here we develop the general method of kinetic co-
efficient calculation for a gas of the low density soli-
tons.
2. KINETIC EQUATIONS
In the authors paper [1] a system of kinetic equations
for low density soliton gas ( 1< <δn , n is the density of
solitons, δ is the soliton characteristic size) were ob-
tained. It was shown that the entropy production is con-
nected with homogenization of soliton gas in the real
coordinate space. In the cause of the homogenization
process the establishment of a uniform density )(xn ,
hydrodynamic velocity )(xu and temperature T(x) takes
place. In another words, the soliton collisions are at the
bottom of such processes as internal friction, thermal
conductivity and diffusion of soliton gas. The kinetic
equations are:
2
2
)(D
))( (
x
fv
x
fvvv
t
f
kk
∂
∂
∂
∂δ
∂
∂ =++ ,
.
)(2
)(
)(
))( (
))( (
2
2
2
2
2
ϕ∂∂
∂+
ϕ∂
∂+
∂
∂=
=
ϕ∂
∂ωδ+ω+
∂
∂δ++
∂
∂
x
BVBV
x
BV
BV
x
BVvv
t
B
bbb
bb
KPD
(1)
Here
∑=∑=
i
kiD) ( kD ,
i
kvkivkv δδ ,
, F=K P=P=D ∑∑∑
∑ ωδ=ωδ∑ δ=δ ν
i bibi bibi biDb
i bibi bivb
;;
; ,
(2)
;;;22][||
2
1) ( 21 vvvvvdFxvv kkikik −=≡∆∫=kD
2 ||
2
1 =K
2] [||
2
1=P
;2][||
2
1=D
2bi
2
2
bi
2
2
∫ ∆∆
∫ ∆
∫ ∆
dFxv
dFv
dFxv
ibibi
i
ibibi
ϕ
ϕ (3)
In the above formulae },{ BfF ≡ are the distribution
functions of both kinks and breathers. The arguments of
these functions are the center of mass coordinates x, ve-
locities v, time t, internal frequencies ω and phase co-
ordinate ϕ for breathers. For simplicity reasons we will
use notations },{1 11 vx≡ in the case of kinks, and
},,,{1 1111 ϕωvx≡ in the case of breathers. In equations
(1) − (5) both ϕ∆∆ andх are the jumps of the center
of mass coordinates and phase coordinates.
3. TRANSPORT EQUATION. GENERAL CASE
To obtain the transport equations from the kinetic
equation one has to employ a usual method, i.e. to mul-
tiply the kinetic equation by 1, velocity V, and energy E,
and integrate over all moments. This multiplier factor
we will design as a. After direct calculation one can find
the transport equations in the form of the local conser-
vation laws. Here are the final results.
For kinks
0
=
++>< m
kUr
kU
xkakn
t ∂
∂
∂
∂
; (4)
For breathers
,0
=
+
+
++><
m
bWr
bW
m
bUr
bU
x
abn
t
ϕ∂
∂
∂
∂
∂
∂
(5)
We use the following notations in equations (4) and
(5). For kinks:
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 265-267. 9
∫> =< dvtvxfvxaakn ),,(),( ,
[ ]∫ += dvtvxfvvvxaU k
r
k ),,( ),( δ , (6)
∫−= dvtvxfvxa
x
U m
k ),,( D ),(
k∂
∂
. (7)
For breathers:
∫> =< ωωϕωϕ dvdtvxBvxaan bb ),,,,(),,,( ,
[ ]∫ += ωωϕδωϕ dvdtvxBvvvxaU b
r
b ),,,,( ),,,( , (8)
,),,,,( ),,,(
),,,,( ),,,(
∫ ωωϕωϕ
ϕ∂
∂−
ω∫ ωϕωϕ
∂
∂−=
ddvtvxBvxa
ddvtvxBvxa
x
m
bU
bK
bD
(9)
[ ]∫ ωωϕωδ+ωωϕ= ddvtvxBvxaW b
r
b ),,,,( ),,,( , (10)
.),,,,( ),,,(
),,,,( ),,,(
∫
∫
ωωϕωϕ
ϕ∂
∂−
ωωϕωϕ
∂
∂−=
dvdtvxBvxa
dvdtvxBvxa
x
W
b
b
m
b
K
P
(11)
The jumps of the center of mass coordinates at soli-
ton-soliton collisions are presented below in the table.
The expressions for coordinates jumps x∆ [1-4]
К2 B2
К1 ||ln)2/( 1 kkZkδ ||ln1 kbZkδ
B1 ||ln)/8( 1 bkZE b− ||ln)2/( 1 bbZbδ
The first line presents the jumps x∆ of kink K1 co-
ordinate due to collisions with a kink K2 and a breather
B2; in the second line the same is shown for a breather
B1. All the data in the table have to be multiplied by a
sign of the relative velocity of solitons.
The ϕ∆tan of the phase jumps ϕ∆ due to soliton–
soliton collisions [1 – 4] is
2/1)12/1(1,1tan −−=∆ vbbk ωϕ ,
,}])()){(21(2[
2/1]21[||2tan
121
1
22
1
1
1
1
−ω−ω−+
ωω−−=ϕ∆
vv
vvbb
2
12 1, ωωωω −=≡ .
The quantities Z that enter the table are:
,
11
11
kk
2
2
v
vZ
−−
−+= ; ,
1b22,1
1b22, 1
bk
2
2
v
v
Z
−−
−+
=
ω
ω
(12)
1
2
2
1
2
2
2
2
11
11
11
11
Cv
Cv
Cv
Cv
Z bb
−+
−+
⋅
−−
−−
= . (13)
Here 21
2
1
2
12,1 )1)(1( ωωωω −−=C . From the
above table one can see that the phase shifts are not
small in general case. For slow (non relativistic solitons,
V<<1) the phase shifts are 2/π .
We would like to emphasize that the topology
charges are constant for all type of collisions. The pro-
cesses which can change the topology charge of solitons
are the processes that destroy the full integrability of the
system. In other words, as soon as we are within the
framework of the fully integrable model, the numbers of
kinks, antikinks and breathers are constant. This num-
bers are determined by the initial conditions. In this pa-
per we will study only fully integrable systems.
Continuity equations (a=1) can be written as:
( ) 0
=++ m
k
r
k
k jj
xt
n
∂
∂
∂
∂
(14)
( ) ( ) ,0
=++++ m
b
r
b
m
b
r
b
b iijj
xt
n
ϕ∂
∂
∂
∂
∂
∂
(15)
Here the standard notations were used for both kink and
breather currents instead of U and W.
Hydrodynamic equations (a=ν for kinks and a=(ν,
ω) for breathers) have the following form:
( ) 0
=++ m
k
r
kkk PP
x
un
t ∂
∂
∂
∂
, (16)
,0
=
Π+Π+
++ m
b
r
b
m
bPr
bP
xbubn
t ϕ∂
∂
∂
∂
∂
∂
(17)
0
=
++
++ m
bRr
bRm
bQr
bQ
xbwbn
t ϕ∂
∂
∂
∂
∂
∂
.
(18)
Here ui is the hydrodynamic velocity of solitons, bw is
the hydrodynamic velocity in ϕ-space, Pi
r and m
iP ,
r
iQ and m
iQ are the pressures of kinks and breather
correspondingly, r
bΠ and m
bΠ , r
iR and m
iR are the
pressures of breathers in ϕ-space. These quantities are
defined with equations (9) − (12).
Energy transport equations can be obtained by
putting )(vEa = for kinks, and ),( ω= vEa for
breathers. They are:
0
=
++ m
kUr
kU
xkTkn
t ∂
∂
∂
∂
, (19)
,0
=
++
++ m
bWr
bWm
bUr
bU
xbTbn
t ϕ∂
∂
∂
∂
∂
∂
(20)
where Ti means average energy of a soliton. If
u wi i= = 0 , Ti is the local soliton temperature.
We would like to emphasize the important property
of equations (15) – (20). It is easy to see that any uni-
form in real space distribution functions
),,(),,( tvBBtvff ω== (21)
satisfy both kinetic equations and transport equations.
4. TRANSPORT EQUATIONS IN THE CASE
OF SMALL SOLITON VELOCITIES
If the soliton velocity satisfies the condition 1< <v ,
than the phase change is the same for both breather-
breathers and breather-kink collision, and equals to
2/π in absolute value. The values of Z, which deter-
mine the center of mass coordinates jumps, are:
))1/()1(();/4ln( 2 ωω −+== kbkk ZvZ (22)
10
1)(;
1
4
;)(1;
1
1
1
1
22
212
2
2
22
21
1
2
2
1
< << <−
−
=
> >−> >
+
+
−
−
=
vif
v
Z
vif
C
C
C
C
Z
bb
bb
ωω
ω
ω
ωω
We would like to remind that the center of mass co-
ordinates jumps are connected with Z by relation
)/4ln()2/1( 2vх кδ=∆ for kinks, and by relation
bbb Zх ln)2/1( δ=∆ for breathers.
Using these values it is easy to calculate the relax-
ation time in both real space and phase space. For this
purposes we will use the relation between distance and
time for the diffusion process,
Dtx =2 or Dxt /2= .
Taking into account that )/(2 mTv > ≈< one can find that
average value of the diffusion coefficient for kinks is
.)/(
)4/(ln)/()4/1()(
2/12
22/122
kk
kkkk
mT
mTmTxvD
δ≈
δ> ≈∆= <
For breathers one has:
> ≈∆= < 2)( xvDb
)4/)1((ln)/()4/1( 2222/12
bbb mTmT ωω−δ≈
.)/( 2/12
bb mTδ≈
The final results for diffusion coefficients we present
here with logarithmic accuracy. From the formulae we
see that the diffusion coefficients in real space and
phase space are of the same order. Taking into account
that the average distance between solitons in real space
is )/1( nl ≈ , where n is the soliton density, one can
find that the relaxation time τ is of order:
22/12 )/()/()/( δτ nTmDl i≈≈ . (23)
In this formula im is the kink or breather mass. They
have the same order of value ω2)/( =kb mm . Average
value of the breather diffusion coefficient in phase space
is
2/1222 )/()( mTvvDb ππϕϕ > ≈> ≈ <∆= < .
As the characteristic distance in the phase space is π ,
for relaxation time τ in phase space one finds
))/()./( 2/12 TmD bb ≈≈ ϕϕ πτ (24)
Comparing these two formulae for relaxation times in
real and phase spaces one can see that
ττ ϕ < < , (25)
because 1< <δn (approximation of low density gas).
This means that the relaxation proceeds in two steps.
First of all at the end of the first step of relaxation the
uniform distribution in phase space establishes. After
that the much more slow processes of homogenization
of density, hydrodynamic velocities and temperature in
coordinate space take place.
5. KINETIC COEFFICIENTS OF BREATHERS
Now we are going to evaluate the fluxes of number-
of-particle density, momentum density and energy den-
sity. We will not consider the re-normalization of veloc-
ities and pressures. The kink’s fluxes were investigated
in paper [1]. Here we consider the breather’s fluxes.
Continuity equations for the breathers are
b
m
b
m
bb D
x
j
x
j
t
n
,0
∂
∂
∂
∂
∂
∂
−==+ (26)
where the diffusion coefficient bD equals to
2/12
2/12
)/()(
)/2())(2/1(
bbbbb
kbbkb
mTnnx
TnnxD
π
µπ
∆+
∆=
(27)
Thermal conductivity equation is
.0
=
∂
∂+
∂
∂ m
bU
xbTbn
t
(28)
Here m
bU is a dissipative energy flux. This flux is relat-
ed to the temperature gradient,
x
U bm
b ∂
∂
−=
κ
(29)
where bκ , as a function of temperature, is:
22/1
2
22/1
1
21
)()2/())/(1(
,)()/()2/3(
;
bkkbkb
bbbbbb
bbb
xnnTTMm
xnnmTT
∆+=
∆=
+=
µπκ
πκ
κκκ
The hydrodynamic equation for breathers takes the form
( ) 0
=+
∂
∂+
∂
∂ m
b
r
bbb PP
x
un
t
(30)
where r
bP and m
bP are the pressure density in the
breather gas and friction pressure density caused by
breather collisions,
b
bbr
b D
x
u
x
u
P
∂
∂
−=
∂
∂
−= η . (31)
The quantity η is the internal friction coefficient. This
coefficient is equal to the diffusion coefficient bD . The
relation
η=bD (32)
is an analog of the Einstein relation between diffusion
and friction coefficients for ordinary particles. For kinks
this relation was derived in [1].
With this paper we commemorate our teacher Pro-
fessor A.I. Akhiezer.
REFERENCES
1. I.V. Baryakhtar, V.G. Baryakhtar,
E.N. Economou. Kinetic properties of localized
excitations // Phys. Rev. E, 1999, v. 60, №5,
p. 18254-18263.
2. V.E. Zakharov, S.V. Manakov, S.P. Novikov,
L.P. Pitaevsky. Theory of Solitons, Moscow: “Nau-
ka”, 1980, 320 p. (in Russian).
3. L.A. Takhtadzhan, L.D. Faddeev. Hamiltoni-
an Method in the Theory of Solitons. Moscow:
“Nauka”, 1986, 528 p. (in Russian).
4. I.V. Baryakhtar. Kinetic and Dissipative Ef-
fects in Non Linear Model Systems. Theses of dis-
sertation for doctor degree in Physics and Mathe-
matics, Institute for Low Temperature Physics and
Engineering, Kharkov, 1999, 314 p.
11
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-80028 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-24T15:13:03Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Baryakhtar, I.V. Baryakhtar, V.G. 2015-04-09T15:59:53Z 2015-04-09T15:59:53Z 2001 Transport equations for low density soliton gas / I.V. Baryakhtar, V.G. Baryakhtar // Вопросы атомной науки и техники. — 2001. — № 6. — С. 265-267. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 05.60.+w https://nasplib.isofts.kiev.ua/handle/123456789/80028 We present the theory of transport phenomena in a gas of solitons for Sinus Gordon model system. The general case of relaxation phenomena is considered. A special attention is paid for a small density non-relativistic gas of breathers. Such kinetic coefficients as diffusion, thermal conductivity, and internal friction are found. It is shown, that diffusion coefficient and internal friction coefficient equal each other. With this paper we commemorate our teacher Professor A.I. Akhiezer. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory Transport equations for low density soliton gas Транспортные уравнения для солитонного газа малой плотности Article published earlier |
| spellingShingle | Transport equations for low density soliton gas Baryakhtar, I.V. Baryakhtar, V.G. Kinetic theory |
| title | Transport equations for low density soliton gas |
| title_alt | Транспортные уравнения для солитонного газа малой плотности |
| title_full | Transport equations for low density soliton gas |
| title_fullStr | Transport equations for low density soliton gas |
| title_full_unstemmed | Transport equations for low density soliton gas |
| title_short | Transport equations for low density soliton gas |
| title_sort | transport equations for low density soliton gas |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80028 |
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