Construction of probabilistic solutions of the Boltzmann equation
The proving method of the Cauchy problem solvability of the Boltzmann kinetic equation with spatially uniform initial data in the case of particle scattering cross-section finiteness is proposed. It is based on the construction of the auxiliary vector-valued random process such that the particle vel...
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| Zitieren: | Construction of probabilistic solutions of the Boltzmann equation / Yu.P. Virchenko, T.V. Karabutova // Вопросы атомной науки и техники. — 2007. — № 3. — С. 297-300. — Бібліогр.: 6 назв. — англ. |
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Virchenko, Yu.P. Karabutova, T.V. 2017-01-07T17:27:03Z 2017-01-07T17:27:03Z 2007 Construction of probabilistic solutions of the Boltzmann equation / Yu.P. Virchenko, T.V. Karabutova // Вопросы атомной науки и техники. — 2007. — № 3. — С. 297-300. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 25.20.Dd https://nasplib.isofts.kiev.ua/handle/123456789/111012 The proving method of the Cauchy problem solvability of the Boltzmann kinetic equation with spatially uniform initial data in the case of particle scattering cross-section finiteness is proposed. It is based on the construction of the auxiliary vector-valued random process such that the particle velocity distribution function satisfying the Boltzmann equation is the first order marginal probability distribution of this random process. Пропонується метод доведення розв’язання проблеми Коші для кінетичного рівняння Больцмана з просторово однорідною початковою функцією у випадку скінченності перерізу розсіяння частинок, що зiткаються. Метод оснований на побудові векторнозначного випадкового процесу такого, що функція розподілу за швидкостями частинок, яка задовольняє рівнянню Больцмана, є його частинним розподілом ймовірностей першого порядку. Предлагается метод доказательства разрешимости задачи Коши для кинетического уравнения Больцмана с пространственно однородными начальными данными в случае конечности сечения рассеяния сталкивающихся частиц. Метод основан на построении вспомогательного векторнозначного случайного процесса, такого, что функция распределения по скоростям частиц, удовлетворяющая уравнению Больцмана, является его частным распределением вероятностей первого порядка. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Nonlinear dynamics Construction of probabilistic solutions of the Boltzmann equation Побудова ймовiрностних розв’язкiв рiвняння Больцмана Построение вероятностных решений уравнения Больцмана Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Construction of probabilistic solutions of the Boltzmann equation |
| spellingShingle |
Construction of probabilistic solutions of the Boltzmann equation Virchenko, Yu.P. Karabutova, T.V. Nonlinear dynamics |
| title_short |
Construction of probabilistic solutions of the Boltzmann equation |
| title_full |
Construction of probabilistic solutions of the Boltzmann equation |
| title_fullStr |
Construction of probabilistic solutions of the Boltzmann equation |
| title_full_unstemmed |
Construction of probabilistic solutions of the Boltzmann equation |
| title_sort |
construction of probabilistic solutions of the boltzmann equation |
| author |
Virchenko, Yu.P. Karabutova, T.V. |
| author_facet |
Virchenko, Yu.P. Karabutova, T.V. |
| topic |
Nonlinear dynamics |
| topic_facet |
Nonlinear dynamics |
| publishDate |
2007 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Побудова ймовiрностних розв’язкiв рiвняння Больцмана Построение вероятностных решений уравнения Больцмана |
| description |
The proving method of the Cauchy problem solvability of the Boltzmann kinetic equation with spatially uniform initial data in the case of particle scattering cross-section finiteness is proposed. It is based on the construction of the auxiliary vector-valued random process such that the particle velocity distribution function satisfying the Boltzmann equation is the first order marginal probability distribution of this random process.
Пропонується метод доведення розв’язання проблеми Коші для кінетичного рівняння Больцмана з просторово однорідною початковою функцією у випадку скінченності перерізу розсіяння частинок, що зiткаються. Метод оснований на побудові векторнозначного випадкового процесу такого, що функція розподілу за швидкостями частинок, яка задовольняє рівнянню Больцмана, є його частинним розподілом ймовірностей першого порядку.
Предлагается метод доказательства разрешимости задачи Коши для кинетического уравнения Больцмана с пространственно однородными начальными данными в случае конечности сечения рассеяния сталкивающихся частиц. Метод основан на построении вспомогательного векторнозначного случайного процесса, такого, что функция распределения по скоростям частиц, удовлетворяющая уравнению Больцмана, является его частным распределением вероятностей первого порядка.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111012 |
| citation_txt |
Construction of probabilistic solutions of the Boltzmann equation / Yu.P. Virchenko, T.V. Karabutova // Вопросы атомной науки и техники. — 2007. — № 3. — С. 297-300. — Бібліогр.: 6 назв. — англ. |
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2025-11-26T01:42:42Z |
| last_indexed |
2025-11-26T01:42:42Z |
| _version_ |
1850604898003976192 |
| fulltext |
CONSTRUCTION OF PROBABILISTIC SOLUTIONS
OF THE BOLTZMANN EQUATION
Yu.P. Virchenko and T.V. Karabutova
Belgorod State University, Belgorod, Russia;
e-mail: virch@bsu.edu.ru
The proving method of the Cauchy problem solvability of the Boltzmann kinetic equation with spatially uniform
initial data in the case of particle scattering cross-section finiteness is proposed. It is based on the construction of the
auxiliary vector-valued random process such that the particle velocity distribution function satisfying the Boltzmann
equation is the first order marginal probability distribution of this random process.
PACS: 25.20.Dd
1. INTRODUCTION
The Boltzmann equation has got the origin of the
physical kinetics. Building of the equation and its
physical predictions have played the great role during
the development of representations concerning evolu-
tion irreversibility that are taken place in nature. Later,
this equation has got also the practical mean for calcula-
tion of kinetic coefficients of real gently dense gases
and also for the study of the motion of solids in the di-
lute gas environment. In this connection, the mathe-
matical correct results concerning some solution proper-
ties of the Boltzmann equation have gained special im-
portance. In particular, the Cauchy problem solvability
of the equation is of interest. First results in this direc-
tion have been obtained by D.Hilbert [1] and
T.Carleman [2]. All stationary solutions of the Boltz-
mann equation have been found and the theorem of the
final behavior of solutions at the unbounded increasing
of time has been proved. Besides, the spectrum of corre-
sponding linearized equation has been investigated. Fur-
ther, in connection with the development of the ap-
proximate methods of the equation solving, the Chep-
men-Enskog asymptotical analysis has been created [3].
The modern state of the mathematical physics area
which is connected with the Boltzmann equation see in
[4]. In the present work, we propose the new approach
to study the Boltzmann equation solutions which is
based on the random process theory. This method per-
mits to approach in a new fashion to the Cauchy prob-
lem solving of the Boltzmann equation and to the con-
struction of approximations of corresponding solutions
with the controlled accuracy.
2. THE BOLTZMANN EQUATION
For formulation of the Boltzmann equation in the
form that is convenient for our aim, it is required the
following presentation of scattering data in the classical
mechanics for the pair of identical particles. Let and
be two particle velocities before the collision when
they are on such a distance where one may consider
them as the noninteracting ones. Further, let and
be corresponding velocities after the collision when the
particles have gone away so far that they become nonin-
teracting again. These velocities are represented by the
definite functions of velocities , and the vector
which is on the plane being orthogonal to vectors
v
'v
V V'
rv 'v
',vv
V V'
( 1r −Φ
t
f
f (
−
∫
3R
f
V +
| V
V =
, i.e. . On the defini-
tion, the vector begins at the straight line on which
the first particle arrives from the infinity and it is fin-
ished at the analogous straight line of the second parti-
cle. The vector r has the minimal length among all vec-
tors possessing the properties pointed out. The functions
, are defined by the interaction potential
between two particles, where are
space position vectors of particles.
)('),( rv'vVrv'vV ,,,, ==
r
,(f v
v
σ
ρ
ddtf
tftf
')),'(
),'(),((|'|
vv
VVvv −∫
0
σd
r
)t,(f v
0≥)t,v
=1vd)
,,( v'vV =
222 ',' vVVvv +=++
|'| vv −=
)(') rv'vVrv ,,, ==
= n
)2
t
t
),
) =
>ρ
(f
v t,
'=
|'V
v' ,
r
(
,
v
v
(
V
−
(
21 rr ,
Φ
t
,( v'v=
||),, nr
For the determination of these functions, it is neces-
sary to solve the mechanical scattering problem of two
particles interacting by means of the potential . Fur-
ther, we consider only the spatially homogeneous gas of
particles. In this case, the Boltzmann equation is formu-
lated for the distribution function that depends
on the velocity of the mentioned particle and the time
. It has the following form (see, for example, [5])
)t
(1)
where the dot means the differentiation on , the pa-
rameter represents the density of particles and
the integral on points out the integration on all val-
ues of the vector .
The function is the probability density func-
tion, i.e. and the equality
(2)
takes place. The functions
have the properties
),'), rVr
2'v (3)
which express the conservation laws of momentum and
energy at collisions. From Eq. (3), the equality of rela-
tive velocities follows,
. (4)
Besides, the symmetry property
(5)
takes place. It corresponds to the particle identity. Let us
introduce the vector function n ,
satisfying to the equality
1=,( v'v
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 297-300. 297
|]'|)'[(
2
1 vvnvvV −++= ,
|]'|)'[(
2
1' vvnvvV −−+= . (6) ((δ∫×
Further, we notice that for the complete
characterization of the scattering of two particles, it is
necessary to introduce, in addition to th tions V ,
V' , the vector funct ),,( rv'vR also. Its values lay in
the plane being orthogonal to v cto V , V' . The vector
R is defined similarly to the vector r but it is done
relative to the straight trajectories of particles going away
with the elo ies V , V' . From the energy and
momentum conservati ality || R
e func
ion
s
v cit
low
),,('),,,( rv'vVrv'vV ==
),',( RVVVv −−−= ),',('' RVVVv −−−=
−
RV'Vrv'v ,,,,,
δ
e r
n, the equo r|| =r= fol-
s. The reversibility of mechanical motion leads to the
fact that the functions sat-
isfy to identities
, (7)
),' RV,( VRr −=and, besides, .
In the case when the potential is spherically symmet-
ric, all vectors lay in the common plane.
Let us transform the right-hand side of the equation
(1) to another form which is more suitable for our con-
struction. With this aim, introducing the additional inte-
gration by means of -functions, we write down
uv' ;
'.')),'(),(
),'(),(()(),(
uuvvv
uuu'vv
dddtftf
tftf,,wtf
−
= ∫
w
(8)
The non-negative function is named the intensity of
scattering ',', uuvv ⇒
),;,(w =u'uv'v
),;,(w)',;,(w uu'vv'uuv'v =
)', v−
δ
δ
),;(w =u'u
)'( uu −
'(|2|'| vvvv −−− δ
δ
'uuv'v +=+
1−
)'(4)( −−+= uuv'vu'uv'v ,;,w ρδ
0≥)t,(f v 0≥t 00 ≥),(f v
process. It is determined by
the formula
.
(9)
d)'()(|'| σδδρ ∫ −−−= uV'uVvv ,)(
According to Eqs. (5,7), it satisfies to the following
identities
, (10)
;,(w)',;,(w vu'uuuv'v −−−= . (11) ∫
The function w contains the δ -functional
singularities which are connected with conservation
laws. Indeed, changing the argument of the second -
function in Eq. (9) according to the first identity of
Eq. (3) and also using the condition presented by the
second -function, we obtain
,v'v
.d)()(|| σδδρ ∫ −−−+−= uVu'uv'vv'v
V
(12) [
We change the function in the integrand expression
according to Eq. (6) and we use the first identity of
Eq. (3),
σδσδ d|)'|(d)(∫ ∫ −−−=− vvnu'uuV 2
δ
.
Decomposing three-dimensional -function depending
on the vector argument on the product
|)'|| uu −−
)||/)(( nv'vu'u −−−×δ
and after that transforming the first -function with the
account of equality which is carried out
in the integration domain, we find
.)|'|/)'
)''(
|'|2)(
2222
σ
δ
σδ
d
d∫
−−−
−−+×
−=−
nuuuu
uuvv
vvuV
),,( rv'vn
(13)
Since the function realizes one-to-one corre-
spondence between the plane of the vector r changing
and the unit sphere, the last integral is equal to the Jaco-
bean of corresponding map
1|))(/)((|
)|'|/)'((
−
==
−−−∫
mnrn
nuuuu
DD
dσδ
'|/)'( uuuum −−=
where | . Together with Eq. (12)
and Eq. (13), it gives the final formula
. |))(/)((|)''( 12222 −
=−−+× mnrnuuvv DDδ
3. M.KAC PROBLEM
Solutions which possess the following properties:
a) at all , if ;
vv d)t,(f∫b) does not depend from t ,
R3
we name as the probabilistic ones.
In connection with the existence of probabilistic so-
lutions of the equation (1), M. Kac has set the problem
[6] of the construction of such a random process
0≥t);t(~v )t,(f v for which the function is the first
order marginal distribution density,
)}()t(~{
)(d
)t,(f vv
v
v ω
ω
∈= Prd
R
)t,;...;t,(f vv
)t,(f)t,(f vv =
)]t,(f[f)t,;...;t,( vvv =
),0 ∞
),(f)(d/)}()(~{d 00 vvvv =∈ ωωPr
. (14)
This function should completely define the probability
distribution of the process in its sample space
, i.e. the function should define all set
of marginal distribution densities
of order , which are consistent, i.e.
+∈+ RR t3
N∈n
;...;t,(f n 11v
nnn 11
).t,;...;t,(f
d)t,
nnn
nnn
11111 −−−=
=
vv
vv
They generate the process probability distribution ac-
cording to the Kolmogorov theorem. It means that
and all densities of higher order are
some functionals .
1
f nnnn 11
Constructive building of such a random process on
for which the probability distribution of initial
values satisfies
solves the Cauchy problem on for the equation (1)
with initial data . Below, we offer a building of
this process in the case of finiteness of cross-section of
particles scattering.
+R
),(f 0v
4. THE WEAK SOLVABILITY
We construct the process 0≥t);t(~v as the weak
limit in the space of the sequence +R)R( 3
...,N,t);t(~ )N( 210 =≥v of random processes. It
298
Nvv ∈nttf nn
N
n ),,;...;,( 11
)(
∞→N
Nvv ∈n),t,;...;t,(f nnn 11
→
∫ ∑
=
N
n
n
j
jj tfi vvk 1
)(
1
,(),(exp
n
n
j
jj tfi vvk ;...;,(),(exp 11
1
∫ ∑
=
)t,(f)t,(f )N( vv →
L
∫ <
L
const)x(d)x( )N(µΨ
]T,[t),t(~ 0∈v
L
t(~v
299
ddt vvv ...... ),;;
means that all marginal densities
converge weakly at
to the infinite set of densities
consistent with each other,
i.e. sequences of corresponding characteristic functions
converge
a~
...,n,~~~
nnn 211 =−= −vva
∑ −+
−=
n
nn
ttp
|''~'~|
'~)0('~|])(''~),('~[1
aa
vvvv
]T,[ 0
)t(''~v
''t~n ''~
na
'~
na
],[ ⋅⋅1p
),t('~v
],[ ⋅⋅2p
∑ −=
n
n ttttp ~'~|])(''~),('~[2 vv
nnn 11
nnn ddt vvv ...), 1
.
In particular, the weak convergence
to the Boltzmann equation solu-
tion takes place.
We use the Prokhorov criterion of weak compact-
ness measure sequence for the proof of the weak con-
vergence of random processes sequence. It is formulated
as follows.
If the sequence of measures on a
metrical separable space L (being not necessarily
complete) such that for this sequence there exist a com-
pact function Ψ on space having uniformly bounded
average values on measures ,
, so this sequence is weakly
compact.
...,N,)N( 21=µ
µ )N(
We choose the space of all piecewise constant func-
tions with vector values as the sample
space of all processes.
Each random trajectory is characterized by the
sequence of pairs
)
N∈n;t~n
,n,t~n 21=
,n , where
are random function jumps
which occur in random time points . We put ...
+|)0('
(15) f v
where the summation is carried out on all jumps of both
functions being in the one-to-one correspondence ac-
cording to their order on . If one of the functions,
for example, has greater number of jumps in
comparison with the other function, so it is necessary
formally to put that the last has zero jumps in those
points where jumps do not have correspond-
ing jumps .
The functional is the deviation on be-
tween two functions . The functional
defined by the formula
L
)t(''~v
n |''
~
(16)
is the deviation too. Here, points are set in one-
to-one correspondence according to the one-to-one cor-
respondence between jumps of the functions
. It is easily to verify that the functional
''t~,'t nn
)t(''~),t('~ vv
('~[
),('~dist[
2 tp
t
v
v
+
)t(''~),t('~ vv
]Ψ[⋅
v∈~
~v
0>M
∫=
=
v
v fq
|
];[
ρσ
',;(Q uuv
∫
3R
uv ,;(Q
];[
),(
ffq
tf
v
v
−
= ∫
N),t,()N(
t);t(~ )N( ≥v
∫×
−+
(
exp(1(
,()(
uv
v
,;Q
tf N
/T=∆
N
)t(~ )N(v
)(v ≡l d
f
])(''~),
])(''~),('~[])(''~ 1
t
ttpt
v
vvv = (17)
is the distance in the space L between functions
. The space is separable relative to this
distance. Besides, it is established that those function
sets for which the jump number does not surpass any-
thing number , are precompact relative to the
topology connected with the distance . There-
fore, for the fixed interval , we construct the
function on putting that its values for each
function are equal
L
N∈m
L
L
],dist[ ⋅⋅
]T,[ 0
|),0[ T∩
]
; n∈
[
[⋅
~tn
,0
}
]
| to the
random number of its jumps on . Then, we get
that sets { are compact for any
and, hence, the function Ψ is compact.
N
T
M≤~]v:∈ Ψ[L
5. THE BOLTZMANN RANDOM PROCESS
We construct the random process +∈Rv t);t(~
σ
which solves the Kac problem. It is named the Boltz-
mann process. At the assumption of the cross-section
finiteness, we introduce the functional
∫
− v'vv
vuuvuuv'v
df
dddtf,;,w
)'(|'
''),'()'(
(18)
and an the kernel
'd)',;,(w) vuuv'v
R
∫=
3
. (19)
Due to Eq. (11), it takes place
−= u'uvu' ||d) ρσ . (20)
In terms of introduced values, equation (8) is repre-
sented in the form of
).,(
'),'(),()'(
t
ddtftf,;Q
v
uuuuuuv (21)
Let us require that the first order marginal densities
of the processes ...,21=
...,21N,0 = , satisfy to the identity
×∆−
∆−=∆+
'),'(),()'
))];[
)];[exp(),()
)()(
)(
)()(
uuuuu
v
vv
ddtftf
fq
fqtf
NN
N
NN
(22)
where and . N ∞→N
To achieve this aim, we construct for any the
random process
Ν∈N
+∈Rv t);t(~ )N(
Ν
. Trajectories of the
process at are defined by the formula ∈
}l),N/lT,N/T)l[(t;~{ )N(
l Nv ∈−∈= 1
where the random sequence Nv ∈l;~ )N(
l
R
is the
nonlinear Markov chain with the state space which
is defined by the distribution density
3
)}(~{Pr
)(
vv
v
ω
ω
∈l
d
of random variable at the moment l . This den-
sity is changed during one evolution step by the follow-
ing way
)N(
l
~v
.')'()()(
];[))];[exp(1(
)];[exp()()(
)()(
)(1)(
)()()(
1
∫×
×∆−−
+∆−=
−
+
uuuuu'uv
vv
vvv
ddff,;Q
fqfq
fqff
N
l
N
l
N
l
N
l
N
l
N
l
N
l
(23) ~
The changing points of the process
+∈Rv ttN );(~ )(
~
belong to the set { . };/ N∈lNlT
Let be the number of the changing points
of the function in . The average number of changing
points of the chain trajectory is equal to
]Ψ[ )t(v
L
n,...,,l,~
l 10=v
( ) .)(')'()(
];[))];[exp(1(
)()(
)(1
1
1
)(
∫ ∫
∑
×
×−− −
−
=
vuuuuu'uv
vv
ddfdf,;Q
fqfq
N
l
N
n
N
l
n
l
N
l
Since all random points of the chain are in one-to-one
correspondence with changing points of the process
+∈Rv ttN );(~ )( , the average value ])(~Ψ[ )( tNv
0→∆
is
defined by above expression too. Then, at , we
have
( .'),(),'()'(
])(~Ψ[
0
)()(
)(
dsdddsftf,;Q
t
t
NN
N
vuuuuuuv
v
∫ ∫
∝
)
Consequently, this average value is bounded,
( )
2/12
0
)()(
)(
2
'),(),'(|'|
])(~Ψ[
〉〈ρσ≤
≤−
×ρσ∝
∫ ∫
u
uuuuuu
v
t
dsddsftf
t
t
NN
N
, (24)
where the squared average of the velocity
does not depend on t , since the Boltzmann
equation conserves the kinetic energy. Thus, due to the
inequality (24), one may apply the Prokhorov criterion
2/12 〉〈u
)(~ )( tNv
for measures connected with processes
+∈Rv ttN );(~ )( . On the basis of this criterion, these
processes weakly converge to a process +∈Rv tt);(~ .
In this case, all marginal distributions of the processes
+∈Rv ttN );()( , weakly converge to corresponding
marginal distributions of the process +∈Rv tt);(~ at
. Then, on the basis of Eq. (22), 0→∆
).,(
),(),(),;(
])(;[
1
),(),(
)(
212
)(
1
)(
21
)(1
])(;[
)()(
)(
tf
ddtftfQ
tfq
e
tftf
N
NN
N
tfq
NN
N
v
vvvvvvv
v
vv
v
−
×
=
−
−∆+
∫
−
∆−
Therefore, the one-point probability density of
the limit process
),(1 tf v
+∈Rv tt);(~ satisfy to Eq. (1), i.e.
the existence of the random process +R∈v tt);(~ leads
to the existence of the weak solution of the equation (1).
REFERENCES
1. D. Hilbert. Math. Ann // 1912, v. 72, p. 562-582.
2. T. Carleman. Problemes mathematiques dans la
theorie cinetique des gaz. Uppsala, Almqvist and
Wiksells, 1957, 150 p.
3. S. Chapmen, T.G. Cowling. The mathematical the-
ory of nonuniform gases, Cambridge Univ. Press,
1952, 468 p.
4. C. Cercignani. Theory and Application of the Boltz-
mann Equation. Scottish Academic Press, 1975,
324 p.
5. A.I. Akhiezer, S.V. Peletminskii. Metody statis-
ticheskoi fiziki. M.: “Nauka”, 1977, 368 p. (in Rus-
sian).
6. M. Kac. Neskolko veroyatnostnykh zadach fiziki i
matematiki. M.: “Nauka”, 1967, 176 p. (in Russian).
ПОСТРОЕНИЕ ВЕРОЯТНОСТНЫХ РЕШЕНИЙ УРАВНЕНИЯ БОЛЬЦМАНА
Ю.П. Вирченко, Т.В. Карабутова
Предлагается метод доказательства разрешимости задачи Коши для кинетического уравнения Больцмана
с пространственно однородными начальными данными в случае конечности сечения рассеяния сталкиваю-
щихся частиц. Метод основан на построении вспомогательного векторнозначного случайного процесса, та-
кого, что функция распределения по скоростям частиц, удовлетворяющая уравнению Больцмана, является
его частным распределением вероятностей первого порядка.
ПОБУДОВА ЙМОВIРНОСТНИХ РОЗВ’ЯЗКIВ РIВНЯННЯ БОЛЬЦМАНА
Ю.П. Вiрченко, Т.В. Карабутова
Пропонується метод доведення розв’язання проблеми Коші для кінетичного рівняння Больцмана з прос-
торово однорідною початковою функцією у випадку скінченності перерізу розсіяння частинок, що
зiткаються. Метод оснований на побудові векторнозначного випадкового процесу такого, що функція роз-
поділу за швидкостями частинок, яка задовольняє рівнянню Больцмана, є його частинним розподілом ймо-
вірностей першого порядку.
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