Effect of temporal randomization on the interaction of normalized and anomalous transport
The fractional kinetic equations are a natural consequence of non-Gaussian properties in the behavior of many complex systems. We consider the competition between normalized and anomalous transport in the presence of temporal subordination. The anomalous transport is induced by fractional spatial de...
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| Zitieren: | Effect of temporal randomization on the interaction of normalized and anomalous transport / A.A. Stanislavsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 340-342. — Бібліогр.: 7 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1110142025-02-23T20:28:13Z Effect of temporal randomization on the interaction of normalized and anomalous transport Вплив тимчасової субординації на взаємодію нормальної й аномальної кінетик Влияние временной субординации на взаимодействие нормальной и аномальной кинетик Stanislavsky, A.A. Kinetic theory The fractional kinetic equations are a natural consequence of non-Gaussian properties in the behavior of many complex systems. We consider the competition between normalized and anomalous transport in the presence of temporal subordination. The anomalous transport is induced by fractional spatial derivatives as occurs in fractional kinetic theory. It is shown that for large time the power tails of the probability density play a dominant role. This supports and extends the Weitzner-Zaslavsky's result obtained in a simpler case. Дробові кінетичні рівняння з'являються внаслідок негаусових властивостей поводження складних систем. Ми розглядаємо суперництво між нормальним (гаусовим) і аномальним транспортом при наявності субординації. Аномальний транспорт приводить до появи дробових похідних по просторовим перемінним у кінетичному описі систем. Показано, що на великих часах степенні хвости функції розподілу імовірності відіграють домінуючу роль. Це підтверджує результат Вейцнера-Заславського, отриманий в більш простому випадку, і розширює межі його застосування. Дробные кинетические уравнения появляются вследствие негауссовых свойств поведения сложных систем. Мы рассматриваем соперничество между нормальным (гауссовым) и аномальным транспортом при наличии субординации. Аномальный транспорт приводит к появлению дробных производных по пространственным переменным в кинетическом описании систем. Показано, что на больших временах степенные хвосты функции распределения вероятности играют доминирующую роль. Это подтверждает результат Вейцнера-Заславского, полученный в более простом случае, и расширяет границы его применимости. 2007 Article Effect of temporal randomization on the interaction of normalized and anomalous transport / A.A. Stanislavsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 340-342. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 05.40.Fb, 02.50.Ey https://nasplib.isofts.kiev.ua/handle/123456789/111014 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Kinetic theory Kinetic theory Stanislavsky, A.A. Effect of temporal randomization on the interaction of normalized and anomalous transport Вопросы атомной науки и техники |
| description |
The fractional kinetic equations are a natural consequence of non-Gaussian properties in the behavior of many complex systems. We consider the competition between normalized and anomalous transport in the presence of temporal subordination. The anomalous transport is induced by fractional spatial derivatives as occurs in fractional kinetic theory. It is shown that for large time the power tails of the probability density play a dominant role. This supports and extends the Weitzner-Zaslavsky's result obtained in a simpler case. |
| format |
Article |
| author |
Stanislavsky, A.A. |
| author_facet |
Stanislavsky, A.A. |
| author_sort |
Stanislavsky, A.A. |
| title |
Effect of temporal randomization on the interaction of normalized and anomalous transport |
| title_short |
Effect of temporal randomization on the interaction of normalized and anomalous transport |
| title_full |
Effect of temporal randomization on the interaction of normalized and anomalous transport |
| title_fullStr |
Effect of temporal randomization on the interaction of normalized and anomalous transport |
| title_full_unstemmed |
Effect of temporal randomization on the interaction of normalized and anomalous transport |
| title_sort |
effect of temporal randomization on the interaction of normalized and anomalous transport |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
Kinetic theory |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111014 |
| citation_txt |
Effect of temporal randomization on the interaction of normalized and anomalous transport / A.A. Stanislavsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 340-342. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT stanislavskyaa effectoftemporalrandomizationontheinteractionofnormalizedandanomaloustransport AT stanislavskyaa vplivtimčasovoísubordinacíínavzaêmodíûnormalʹnoíjanomalʹnoíkínetik AT stanislavskyaa vliânievremennojsubordinaciinavzaimodejstvienormalʹnojianomalʹnojkinetik |
| first_indexed |
2025-11-25T04:21:10Z |
| last_indexed |
2025-11-25T04:21:10Z |
| _version_ |
1849734707519946752 |
| fulltext |
EFFECT OF TEMPORAL RANDOMIZATION ON THE INTERACTION
OF NORMALIZED AND ANOMALOUS TRANSPORT
A.A. Stanislavsky
Institute of Radio Astronomy, Kharkov, Ukraine;
e-mail: alexstan@ira.kharkov.ua
The fractional kinetic equations are a natural consequence of non-Gaussian properties in the behavior of many
complex systems. We consider the competition between normalized and anomalous transport in the presence of
temporal subordination. The anomalous transport is induced by fractional spatial derivatives as occurs in fractional
kinetic theory. It is shown that for large time the power tails of the probability density play a dominant role. This
supports and extends the Weitzner-Zaslavsky's result obtained in a simpler case.
PACS: 05.40.Fb, 02.50.Ey
1. INTRODUCTION
Fractional calculus occupies an appreciable place in
the description of various kinds of wave propagation in
complex media, fractional kinetics of Hamiltonian sys-
tems, anomalous diffusion and relaxation, random
walks with a long-term memory and flights, pseudocha-
otic dynamics, etc (see, for example, [1-3] and refer-
ences therein). The fractional operator is a natural gen-
eralization of the ordinary differentiation and integra-
tion. Long-term memory effects characterize the frac-
tional operator with respect to time, whereas the non-
local (long-range) effects characterize it with respect to
coordinates. This new type of problems has increased
rapidly in areas in which the fractal features of a proc-
ess or a medium impose the necessity to use non-
traditional tools in ”regular” smooth physical equations.
The language of fractional equations (FE) is in progress
now. While the linear FE have attained fairly broad
research activity, the study of nonlinear FE is at their
very beginning.
The fractional kinetic equations describe non-
Gaussian properties in the behavior of stochastic sys-
tems. In many cases of physical interest it is reasonable
to study simultaneously the Gaussian and anomalous
processes [4]. This means that the anomalous processes
lead to algebraically decreasing tails of a probability
distribution function (PDF), whereas the bulk of the
PDF is expected to be mostly Gassian in character.
Weitzner and Zaslavsky [5] have investigated the inter-
action of Gaussian and anomalous dynamics for a sim-
ple model in one-dimensional space. They have shown
that for large times the fractional derivative term domi-
nates in the solution and leads to power type tails in the
probability density. We intend to go on the study and
clarify what influence will have subordination on the
competition between normalized and anomalous trans-
port.
2. WHAT IS SUBORDINATION?
A subordinated process Y is obtained by ran-
domizing the time clock of a random process Y using
a new clock U , where U is a random process
with nonnegative independent increments. The resulting
process is said to be subordinated to Y ,
called the parent process, and is directed by U ,
called the directing process. The directing process is
often referred to as the randomized time or else opera-
tional time [6]. In general, the subordinated process
can become non-Markovian, though its parent
process is Markovian. The PDFs of the parent process
and directing one U determine the PDF of the
subordinated process Y via the integral relation:
))(( tU
)(t
)(t
)(t
))(t(UY )(t
)(t
τ
))(( tUY
)(tY
(p UY
p
(pU
τ
)(t
))(( tU
),τ px U
x
)(t
(Y
<1,
)(1
,0)
−
′′ −
β
tx
α ∂∂ |/
|
(
Γ
α
β
β
P2
∂
∂
=
∂
∂β
x
P
ε
ββ ′∂∂ t/
,0)(xP ′β
x = t′α−
β
2
2
=
),( tk
,),(),(
0
) ττ= ∫
∞
dtpxt
where represents the probability to find the
parent process Y at on the operational time ,
and is the probability to be at the operational
time on the real time t .
),( xY τ
), τt
)(τ
Let the subordinator U be an inverse-time stable
process, and the parent process satisfies Lévy (or Gaus-
sian) properties. Then the process Y describes
the subdiffusion [7].
))(( tU
3. MODEL AND ITS ANALYSIS
Consider the kinetic equation with fractional deriva-
tive in time
1,<02,<
|
),(
2
≤
′∂
∂
+
′
−
′
′′
βαε
αβ
β
β
β
x
P
P
t
tx
(1)
where is a constant, the Riesz derivative,
the differential Riemann-Liouville operator,
the initial condition, the gamma func-
tion. This equation determines the corresponding PDF.
By changing the variables
α′ |x
(Γ )s
x′ and ε −
β
2 α t ε
α|
~| Pβ
α
one can get out of this equation. Therefore we put it
equal to one. The Riesz derivative is easily
defined in Fourier transform space as − ,
ε
α∂ / ′∂ | x
| k
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 340-342. 340
where is the Fourier transform of the function
. Then the Fourier transform of the right-hand
side of (1) gives . Thus, for large
wavenumber and short wavelength the system exhibits
normal, Gaussian transport (if β , then normal diffu-
sion becomes subdiffusion), while for the small
wavenumber and large wavelength, the system behaves
anomalous in kinetics.
),(~ tkPβ
)t
=),(
0
tx ∫
,'(xPβ
Pβ
),(~)||( 2 tkPkk β
αε+−
1≠
,),()( 1 dzztxPzF β
β
duue zuu
Br
1−ββ−∫
),(1 txP =β
(Pβ
z
2
1=)(
π
),(1 txP =
)
),( txPβ
Fβ
,( txP
([ 2β
β
−∞
∞−
+−∫ ktEedk ikx
∑∞
= +βΓ= 0 )1(/n
n ny
2
1=),( tx
β )( yE
βP
.
1)(
1)(=
+βγΓ
+)(
0
γΓ
).,(| txPdx β
βM
γ
β zzF
|=
∞
∞−β ∫
∞
∫
M
(
/
+
α
α
tO
1((
2~
Γ
β
βαβ
π
π
t
βQ
+
M
1.1,
||
),(
−
∂
α
βα
β
β
x
Q
t
txQ
∂
∂
=
∂β
Using the results of Refs. [5,7], it is easy to repre-
sent the solution of Eq. (1) in the form
∞
where the function
i
is expressed in terms of the Bromwich integral. The
PDF is the same one denoted by
in Section 2 in the paper of Weitzner and
Zaslavsky. For the initial condition the
PDF is written as a Fourier integral
)(=,0) xx δ
)],α
π
k
where is the one-
parameter Mittag-Leffler function. From above it fol-
lows that
dz
Next, we examine the simple relevant moment
x
The asymptotic expansion for large takes the
form
t
.)
/sin))/1
sin)/(
/32
)/11(
/
−
Γ
−
−
αββ
αβ
αβ
πα
παββ
πt
(2)
The remaining terms give the corrections from anoma-
lous transport.
Denote the solution of the equation only with
anomalous transport
<02,<<
)(1
,0)(
≤
−Γ
−
βα
β
ββ txQ
Following the same procedure as for , we obtain βM
.
/sin)/(
2~
/
απαβΓβ
αβ
β
tMQ (3)
We are now prepared to compare the contribution of
normalized and anomalous kinetics in dependence of
the value of the parameters and β , and we start with
the simplest comparison, namely between and
.
α
βM
βQM
4. RESULTS AND DISCUSSION
To sum up, in leading order and for 1< <2 and
0<β <1, it is seen that . However, the dif-
ference between and is not small unless
. For not far from one, this relation
holds, but as α approaches 2, one requires increasingly
larger values of in order that dominates
. The asymptotic expansion for fails at
= 2, and must then be very poor for small values of
. If the value is sufficiently large, then the
anomalous transport under subordination is the limiting
form for the case with both Gaussian and anomalous
transport subordinated, however there may be signifi-
cant corrections. It should be pointed out that the long-
term memory effects (in the form of the fractional tem-
poral derivative) do not change a character of the inter-
action between the normalized and anomalous transport
distinctly.
α
β
ββ QMM ~
βQM
α
βM
t
t
)/11( α−ββ >> tt
)/1 αβt
α
2
/α
1( −
α−
αβ /t
M
REFERENCES
1. G.M. Zaslavsky. Chaos, fractional kinetics, and
anomalous transport //Phys. Rep. 2002, v. 371,
p. 461-580.
2. R. Metzler, J. Klafter. The restaurant at the end of
the random walk: recent developments in the de-
scription of anomalous transport by fractional dy-
namics //J.Phys. A: Math. Gen. 2004, v. 37,
p. R161-R208.
3. A. Piryatinska, A.I. Saichev, W.A. Woyczynski.
Models of anomalous diffusion: the subdiffusive
case //Physica. A. 2005, v. 349, p. 375-420.
4. V.V. Zosimov, L.M. Lyamshev. Fractals in wave
processes //Physics-Uspekhi. 1995, v. 38,
p. 347-384.
5. H. Weitzner, G.M. Zaslavsky. Some applications of
fractional equations //Comm. Nonlin. Sci. and Nu-
mer. Simul. 2003, v. 8, p. 373-281.
6. W. Feller. An Introduction to Probability Theory
and Its Applications. 2nd Ed., v. II, New York:
Wiley, 1971, 669 р.
7. A.A. Stanislavsky. Probabilistic interpretation of the
integral of fractional order //Theor. and Math. Phys.
2004, v. 138, N 3, p. 418-431.
341
ВЛИЯНИЕ ВРЕМЕННОЙ СУБОРДИНАЦИИ НА ВЗАИМОДЕЙСТВИЕ НОРМАЛЬНОЙ
И АНОМАЛЬНОЙ КИНЕТИК
А.А. Станиславский
Дробные кинетические уравнения появляются вследствие негауссовых свойств поведения сложных сис-
тем. Мы рассматриваем соперничество между нормальным (гауссовым) и аномальным транспортом при
наличии субординации. Аномальный транспорт приводит к появлению дробных производных по простран-
ственным переменным в кинетическом описании систем. Показано, что на больших временах степенные
хвосты функции распределения вероятности играют доминирующую роль. Это подтверждает результат
Вейцнера-Заславского, полученный в более простом случае, и расширяет границы его применимости.
ВПЛИВ ТИМЧАСОВОЇ СУБОРДИНАЦІЇ НА ВЗАЄМОДІЮ НОРМАЛЬНОЇ
Й АНОМАЛЬНОЇ КІНЕТИК
О.О. Станіславський
Дробові кінетичні рівняння з'являються внаслідок негаусових властивостей поводження складних сис-
тем. Ми розглядаємо суперництво між нормальним (гаусовим) і аномальним транспортом при наявності
субординації. Аномальний транспорт приводить до появи дробових похідних по просторовим перемінним у
кінетичному описі систем. Показано, що на великих часах степені хвости функції розподілу імовірності ві-
діграють домінуючу роль. Це підтверджує результат Вейцнера-Заславського, отриманий в більш простому
випадку, і розширює межі його застосування.
342
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