The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium
The problem of finding the distribution of functional of a trajectory of a particle executing a random walk in a disordered medium containing both traps and obstacles is considered. As a model of a disordered medium, the Schirmacher model, which is the combination of the random barriers model and th...
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Інститут хімії поверхні ім. О.О. Чуйка НАН України
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| Zitieren: | The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium / V.P. Shkilev, V.V. Lobanov // Поверхность. — 2016. — Вип. 8 (23). — С. 58-72. — Бібліогр.: 22 назв. — англ. |
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irk-123456789-1485032019-02-19T01:26:47Z The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium Shkilev, V.P. Lobanov, V.V. Теория химического строения и реакционной способности поверхности. Моделирование процессов на поверхности The problem of finding the distribution of functional of a trajectory of a particle executing a random walk in a disordered medium containing both traps and obstacles is considered. As a model of a disordered medium, the Schirmacher model, which is the combination of the random barriers model and the multiple-trapping model, is used. Forward and backward Feynman-Kac equations with the boundary conditions at discontinuity points are formulated. As an example, the distribution of the residence time in a half-space is obtained. It is shown that the anomalous subdiffusion due to traps and that due to obstacles give very different distributions. Розв'язана задача про знаходження функції розподілу траєкторії частинки, що здійснює випадкові блукання в невпорядкованому середовищі, яке містить як пастки, так і бар'єри. В якості моделі невпорядкованого середовища використана модель Ширмахера, яка є комбінацією моделей випадкових бар'єрів і багаторазового захоплення частинки. Сформульовано прямі і зворотні рівняння Фейнмана-Каца з граничними умовами в точках розриву. Як приклад отримано розподіл часу перебування частинки в півпросторі. Показано, що різні типи аномальної субдифузії, обумовленої пастками і бар'єрами, дають функції розподілу, які сильно розрізняються. Решена задача о нахождении функции распределения траектории частицы, совершающей случайное блуждание в неупорядоченной среде, которая содержит как ловушки, так и барьеры. В качестве модели неупорядоченной среды использована модель Ширмахера, которая представляет собой комбинацию моделей случайных барьеров и многократного захвата частицы. Сформулированы прямые и обратные уравнения Фейнмана-Каца с граничными условиями в точках разрыва. В качестве примера получено распределение времени пребывания частицы в полупространстве. Показано, что различные типы аномальной субдиффузии, обусловленной ловушками и барьерами, дают сильно различающиеся функции распределения. 2016 Article The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium / V.P. Shkilev, V.V. Lobanov // Поверхность. — 2016. — Вип. 8 (23). — С. 58-72. — Бібліогр.: 22 назв. — англ. 2617-5975 http://dspace.nbuv.gov.ua/handle/123456789/148503 544.72 : 544.18 en Поверхность Інститут хімії поверхні ім. О.О. Чуйка НАН України |
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Теория химического строения и реакционной способности поверхности. Моделирование процессов на поверхности Теория химического строения и реакционной способности поверхности. Моделирование процессов на поверхности |
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Теория химического строения и реакционной способности поверхности. Моделирование процессов на поверхности Теория химического строения и реакционной способности поверхности. Моделирование процессов на поверхности Shkilev, V.P. Lobanov, V.V. The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium Поверхность |
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The problem of finding the distribution of functional of a trajectory of a particle executing a random walk in a disordered medium containing both traps and obstacles is considered. As a model of a disordered medium, the Schirmacher model, which is the combination of the random barriers model and the multiple-trapping model, is used. Forward and backward Feynman-Kac equations with the boundary conditions at discontinuity points are formulated. As an example, the distribution of the residence time in a half-space is obtained. It is shown that the anomalous subdiffusion due to traps and that due to obstacles give very different distributions. |
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Article |
| author |
Shkilev, V.P. Lobanov, V.V. |
| author_facet |
Shkilev, V.P. Lobanov, V.V. |
| author_sort |
Shkilev, V.P. |
| title |
The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium |
| title_short |
The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium |
| title_full |
The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium |
| title_fullStr |
The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium |
| title_full_unstemmed |
The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium |
| title_sort |
distribution of functional of a trajectory of a particle executing a random walk in a disordered medium |
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Інститут хімії поверхні ім. О.О. Чуйка НАН України |
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2016 |
| topic_facet |
Теория химического строения и реакционной способности поверхности. Моделирование процессов на поверхности |
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http://dspace.nbuv.gov.ua/handle/123456789/148503 |
| citation_txt |
The distribution of functional of a trajectory of a particle executing a random walk in a disordered medium / V.P. Shkilev, V.V. Lobanov // Поверхность. — 2016. — Вип. 8 (23). — С. 58-72. — Бібліогр.: 22 назв. — англ. |
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Поверхность |
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2025-07-12T19:34:49Z |
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| fulltext |
Поверхность. 2016. Вып. 8(23). С. 58–72 58
UDC 544.72 : 544.18
THE DISTRIBUTION OF FUNCTIONAL OF A TRAJECTORY
OF A PARTICLE EXECUTING A RANDOM WALK IN A
DISORDERED MEDIUM
V.P. Shkilev, V.V. Lobanov
Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine
17 General Naumov Str., Kyiv, 03164, Ukraine, e-mail: lobanov@isc.gov.ua
The problem of finding the distribution of functional of a trajectory of a particle executing a
random walk in a disordered medium containing both traps and obstacles is considered. As a model of
a disordered medium, the Schirmacher model, which is the combination of the random barriers model
and the multiple-trapping model, is used. Forward and backward Feynman-Kac equations with the
boundary conditions at discontinuity points are formulated. As an example, the distribution of the
residence time in a half-space is obtained. It is shown that the anomalous subdiffusion due to traps
and that due to obstacles give very different distributions.
1. Introduction
1.1. Functional of a random-walk trajectory
A Brownian functional is defined as
(1.1)
where ( )x is a trajectory of a Brownian particle and ( )U x is a prescribed function, the type
of which depends on the problem considered. Brownian functionals arise in various fields of
science; for example, a functional equal to the time spent by the particle in a given domain
arises in chemical kinetics [1, 2, 6, 8]. In this case, ( ) 1U x in domain and ( ) 0U x
otherwise. Other examples include functionals with functions ( )U x x and 2( )U x x , which
are of interest for the theory of nuclear magnetic resonance [9]: the case 2( )U x x was
considered in the study of the dynamics of the growing surface [5].
Using the path integral method proposed by Feynman, Kac derived a partial
differential equation which allows one to find the distribution of the Brownian functional with
an arbitrary positive function ( )U x [11]:
(1.2)
Here ),,( tpxG is the Laplace transform in variable A for the function ),,( tAxG
equal to the joint probability that, at time t , the particle is at point x and the functional is
equal to A ; D is the diffusion coefficient. This Feynman-Kac equation has been widely used
for the calculation of distributions of functional with different functions ( )U x [14].
Equation (1.2) was derived under the assumption that the medium in which the
process occurs is homogeneous. If the medium is inhomogeneous, the calculation of the
distribution of the functional becomes much more complicated [13, 17]; however, if the self-
averaging property holds, then for an inhomogeneous medium we can obtain an equation
similar to equation (1.2). In [19], such equations (forward and backward) were derived for
media that can be described by the random traps model. In this model, the diffusion-slowing
occurs due to the particle delay in the traps. In the present paper, equations are derived in a
59
more realistic model which takes into account the presence of both traps and different kinds
of obstacles in a disordered medium. In such a case, other equations are obtained, because
obstacles create an essentially new mechanism for diffusion-slowing. This mechanism
consists in the fact that the particle cannot move with equal probability in all directions, so
that the negative velocity correlations arise.
Interest in the study of functional of a random-walk trajectory has increased with the
development measurement techniques. The latest methods allow one to track the trajectory of
the individual particles, and thus to find the distribution of different functional
experimentally. By analyzing these distributions, one can obtain valuable information about
the characteristics of random walks; particular, by the structure of environment in which they
occur. To do this, we need appropriate theoretical models.
For the first time, the generalized Feynman-Kac equations were derived in [4, 22] in
the framework of the continuous time random walk model. In principle, these equations can
be used to find the distributions of functionals in disordered media; however, they, as well as
the random traps equations, take into account only one diffusion-slowing mechanism - the
delay in the traps. Generalized Feynman-Kac equations were recently derived in [3] on the
basis of the Langevin equations. Since this approach assumes that the particle can move with
equal probability in any direction, the resulting equations are also equivalent to the random
traps equations.
We are interested in calculating the distribution of the functional (1.1) when the
motion of a particle whose trajectory appears under the integral is described by the equation
(1.3)
where )(tPn is the probability that the particle is in site n at time t and nmW is the transition
rate from site m to site n . In [19], it was shown that at any structure of the lattice, i.e. for any
parameters nmW , the joint distribution of the functional (1.1) and the particle coordinates
satisfies the equation
(1.4)
Here ),( tpGn is Laplace transform in variable A for the function ),( tAGn equal to the
joint probability that the particle is at site n at time t and the functional is equal to A ; nU is
value of ( )U x at the site n . This equation describes a random walk with a first order chemical
reaction whose rate npU varies in space. The variable p plays the role of a parameter.
If the self-averaging property holds, then the averaged over elementary physical
volume distribution of the functional, for any extended sample of a disordered medium at
large times will coincide with the distribution found by solving equation (1.4) and averaging
this solution over the ensemble of configurations. Thus, if the self-averaging property holds,
the problem reduces to finding the averaged over the ensemble of configurations solutions of
equation (1.4). In this paper we obtain the equations which this averaged solutions must
satisfy within the Schirmacher model [7, 15, 18].
1.2. Schirmacher model
In the Schirmacher model, all lattice sites are divided into two types: transport states
among which hopping is allowed and traps which are only accessible via the transport states.
The equation (1.3) takes the form
60
(1.5)
(1.6)
where )(tPn is the probability that the particle is in transport state n at time t ; )(tQn is the
probability that the particle is in trap n at time t ; mn is the transition rate from transport
state n to transport state m ; n and n are, respectively, the transitions rates from the n -th
transport state to the n -th trap and back. As a result of averaging of this equations over the
ensemble of configurations and passing to the continuum limit, the authors of [7, 15, 18]
obtained following equation for the averaged probability density of finding a particle at the
point x at the time t ),( tx :
(1.7)
Here 2a is the parameter and )(t is the memory function that in Laplace space
st can be represented as ( ) ( ) ( )s s s , where
(1.8)
(1.9)
Here )(s is the memory function that can be obtained by averaging equation (1.5) at zero
parameters n and n , i.e., in the absence of traps and )(s is the function that describes the
effect of traps on dispersive (anomalous) transport. These functions can be calculated in
different approximations. The works [7, 15, 18], proposed methods for finding functions
)(s in EMA (effective medium approximation) and )(s in CPA (coherent potential
approximation). In this paper, specific forms of these functions will not be used.
In [21], the Schirmacher model was generalized to the case when the diffusing particle
could disappear according to the first order chemical reaction with space dependent reaction
rate nk . The terms )(tPk nn and )(tQk nn were added to the right sides of equations (1.5)
and (1.6), respectively. It was assumed that the motion through the transport state described
by the random barriers model, i.e., that the transport transition rates are symmetric
( nmmn ). After averaging and passing to the continuum limit, the following equation was
obtained:
(1.10)
where )(xk is the reaction rate constant (a continuum extension of the nk );
(1.11)
(1.12)
The functions )(t and )(t enter into equation (1.10) separately and not as the combination
0
( ) ( ) ( )
t
t t d as in the case of equation (1.7). Therefore, this equation will have
different forms depending on contributions made by barriers and by traps to the dispersive
61
transport. For example, if dispersive transport is caused by barriers without traps ( ( ) 0s ),
the equation would look like
(1.13)
and if it is caused by traps without barriers ( ( )s const ), then the equation would be
(1.14)
where *( , ) ( ) exp( ( ) )x t t k x t . The solutions of the equations (1.13) and (1.14) with the
same function *( , )x t may differ materially [21]. This means that the barriers and traps
manifest themselves differently in the diffusion-reaction processes, if a chemical reaction
constant varies in space. This fact, in principle, can be used to determine the microscopic
structure of a disordered medium.
2. Forward equation
2.1. Particle is in transport state at the initial time
In the Shirmacher model, the equation (1.4) takes the form
(2.1)
(2.2)
Functions ( , )nF p t and ( , )nC p t are the parts of the function ( , )nC p t which correspond to the
position of the particle in the transport state and trap, respectively. From the above it follows
that within the Shirmacher model, averaged distribution ( , , )G x p t must satisfy the equation,
differs from equation (1.10) only in the form of reaction constant:
(2.3)
where
(2.4)
(2.5)
This equation is a direct Feynman-Kac equation for the considered model. It differs
from the analogous equation of the random traps model and equations obtained in [4, 22], by
the presence of an additional memory function *( , )x t caused by barriers.
Function ( , , )G x p t gives the joint distribution of the functional (1.1) and coordinate of
the particle at time t . To find the distribution of functional only, we need to integrate this
function in the coordinate x . In particular, to find 0( , , )H x A t − the distribution of functional
(1.1) corresponding to the starting point 0x , it is necessary to find a solution of equation (2.3),
corresponding to the initial condition 0( , ,0) ( )G x p x x (we denote it by 0( , , ; )G x p t x
intedrating this solution with respect to x :
62
(2.6)
and taking the inverse Laplace transform p A .
2.2. Particle is not necessary in transport state at the initial time
Equation (2.3) is obtained under the assumption that the particle is in the transport
state at the initial time [7, 15, 18, 21]. However, this assumption is not always satisfied. In
some cases, the experiment is carried out so that the probability distribution of finding a
particle in different states is equilibrium. In this section, we derive a generalization of (2.3) in
case of arbitrary distribution.
First, consider the case when there is no chemical reaction. Suppose that at the initial
time the probability of finding the particle in the trap of i -th type is 0
i . Then the probability
that the particle will remain trapped until the time t is equal to
(2.7)
and the probability that the particle will make first transition from a trap to the transport state
in the interval ( , )t t dt is equal to
(2.8)
where i is the transitions rate from the trap of i -th type to the transport state and N is the
number of traps types. Using these expressions, we can write the equation for the probability
of finding the particle in the transport state.
In the Schirmacher model, the total probability of finding the particle at the point x ,
( , )x t and the probability ( , )P x t − of finding the particle in the transport state at same point,
are related by (in Laplace domain t s )
(2.9)
Here, the second term on the right represents the probability that the particle is trapped.
Equation (1.7) in terms of the function ( , )P x s has the form
(2.10)
This equation is valid if, at the initial time the particle is in the transport state. If there
is a nonzero probability that the particle is trapped, then the equation is as follows:
(2.11)
where 0P is the probability that a particle is in the transport state at the initial time:
0
0 1
1
N
ii
P
. The last term in the right-hand side describes the arrival of the particles into
the transport state, represented by the formula (2.8). In this case, the total probability is
expressed as
63
(2.12)
Here, the second term on the right represents the probability that the particle is trapped, but
before it has already been in the transport state. The third term represents the probability that
the particle is trapped without having visited the transport state; this term corresponds to the
probability (2.7). The equation for the overall probability is written as
(2.13)
If, at the initial time, the probability is not concentrated at one point, and distributed
over a certain region, the last term in this equation is of the form
02
2
21
( )
( )
N i
i
i
x
a s
x s
. The
consequence of (2.13) is that the resulting flux in the considered model is given by
(2.14)
Using this expression and the approach proposed in [21], we can generalize the
equation (2.13) for the case when the diffusing particle disappears according to the reaction of
the first order. As a result, we obtain the equation
(2.15)
where *( , ) ( ) exp( ( ) ); ( )x t t k x t t is the function that in Laplace domain t s can be
represented as
(2.16)
Forward Feynman-Kac equation is obtained from (2.15) by substituting ( , , )G x p t instead of
( , )x t and ( )pU x instead of ( )k x . The solution of the obtained equation corresponding to
starting point 0x can be represented as the sum of regular and singular terms. In Laplace
domain t s , A p we will have
(2.17)
where the 0( , , ; )G x p s x satisfies the equation
(2.18)
64
Here
(2.19)
3. Boundary conditions
The function ( )U x can have discontinuities at certain points. In order for the
properties of the original discrete model to reflect correctly by a continuous model, it is
necessary to put the correct boundary conditions at these points. In this section, the boundary
conditions for the equation (2.18) at a points of discontinuity of the function ( )U x are
considered on a simple example of a regular one-dimensional lattice. A more general
approach to the problem of boundary conditions is provided in [20].
First, consider the case where there are no traps: 0, 0n n . In one-dimensional
random barriers model under the condition that the jumps are made only to neighboring sites,
the Laplace transform of equation (2.1) becomes ( n nG F in this case)
(3.1)
In matrix form, this can be written as
(3.2)
where 1V
is diagonal matrix with the components 1 1
, ,, 0n n n n mV s pU V for n m ; 2V
is
symmetrical tridiagonal matrix with the components
2 2 2
, 1 2 1 2 , 1 1, 1 2, ,n n n n n n n n nV V V 2
, 0n n mV for 1m . We can write formal
solution of equation (3.2) as:
(3.3)
This solution is valid for each configuration, i.e. for each set of parameters n . As a result of
averaging, we obtain
(3.4)
This relationship implies that the averaged function G satisfies the equation
(3.5)
where
(3.6)
The matrix 3V
is symmetrical because the matrices 1V
and 2V
are symmetrical and
the operation of averaging and the operation of taking an inverse matrix both retain
symmetry. Although this will not be a tridiagonal matrix but it can be approximated by a
tridiagonal matrix with good accuracy [12]. Therefore, we can write (3.5) as
(3.7)
65
The component 3
, 1n nV must be symmetric function of ns pU and 1ns pU . If
0nU , it should be reduced to the function ( )s , and if 1nU , it must be equal to
( )s p . The only function satisfying these conditions is
3
, 1 1 / 2n n n nV s pU s pU We denote it by 1/ 2n .
Suppose that the function ( )U x has a discontinuity at some point between the sites l
and 1l . Then (3.7) for the sites l and 1l can be written as
(3.8)
and
(3.9)
where ( , , )G x p s is the continuous extension of the discrete function
n
G :
( , , ) /n n
G x p s G a and a is the lattice constant. The symbol ( )o f a denote a function of
higher degree of smallness than ( )f a as a tends to 0. The left side of (3.8), in view of(3.9)
can be rewritten as
(3.10)
(Here, the first equality holds if the initial condition 0 ( )G x is smooth. We can use this
assumption, since the diffusion process at large times does not depend on the exact form of
the initial condition. This assumption corresponds to the assumption in real time domain that
the time derivative of ( , , )G x p s is smooth function of x .) Substituting relation (3.10) into
(3.8), we find
(3.11)
In the continuum limit, this relation reduced to the boundary condition
(3.12)
Here, the sign "+" denote the value of function on one side of the boundary, and the sign "-" –
on the other. Considering the sites to the left of discontinuity, we obtain the relation
(3.13)
instead of (3.11). Eliminating the expression 1 2 1, , , ,( ) ( )l l p s l p sG x G x from (3.11) and
(3.13) and passing to the continuum limit, we obtain the second boundary condition as
(3.14)
66
Now consider the case where there are traps, but the particle is in the transport state at
the initial time. Making the Laplace transforms of equations (2.1) and (2.2), excluding the
probability of finding the particle in the traps nС , and averaging over the parameters n and
n , we obtain the equation for the probability of finding the particle in transport states as:
(3.15)
This equation differs from (3.1) only by the presence of a configuration independent
term ( )ns pU ; so to this equation the above reasoning can be applied. Therefore, the
function ( , , )F x p s satisfies the boundary conditions
(3.16)
(3.17)
To obtain the boundary conditions for the function ( , , )G x p s , it is necessary to use the
relation similar to the relation (2.9):
(3.18)
As a result, we obtain the following boundary conditions:
(3.19)
(3.20)
If the particle is not necessarily in the transport state at the initial time, then, arguing
as in subsection (2.2), we obtain the relation between functions ( , , )G x p s and ( , , )F x p s
(3.21)
The boundary conditions for the function ( , , )F x p s remain unchanged. Substituting
the relation (3.21) in (3.16) and (3.17), we find the boundary conditions for the function
( , , )G x p s
(3.22)
(3.23)
If at the initial time the probability is concentrated at one point and this point is not on
the boundary, the last boundary conditions are reduced to the boundary conditions (3.19), and
(3/20). Henceforth, we will consider only such case.
67
4. Backward equation
In this section, we show that the distribution of the functional 0( , , )H x A t can be found
not only by integrating the joint distribution 0( , , ; )G x p s x , but also by solving the backward
equation.
First consider the case when, at the initial time, the particle is in the transport state. In
this case, ( ) 1z and 0 0( , , ; ) ( , , ; )rG x p s x G x p s x . We assert that the distribution of the
functional (1.1) corresponding to the initial position of the particle x in the Laplace space
( ,A p t s ) satisfies the backward equation
(4.1)
and the backward boundary conditions
(4.2)
(4.3)
(A similar problem with a discontinuity across an interior boundary was considered in
[16].) In order to prove this assertion, we multiply (2.18) by ( , , )f x p s and integrate with
respect to x . On the left side, we obtain
(4.4)
The right-hand side, after a double integration by parts, can be written as
(4.5)
Here the sum is taken over all points of discontinuity. Each point of discontinuity gives two
terms of the sum with opposite signs. In view of the boundary conditions (3.19, 3.20) and
(4.2, 4.3), the sum vanishes. The integral term in view of (4.1) can be rewritten in the form
(4.6)
Equation (4.6) and (4.4), we find
(4.7)
Thus, if a function ( , , )f x A t satisfies the equation (4.1) and the boundary conditions
(4.2, 4.3), then this function is the Laplace transform of the distribution of the functional (1.1)
corresponding to the initial position of the particle x : ( , , ) ( , , )f x A t H x A t .
Now consider the case when, at the initial time, the particle is not necessarily in the
transport state. From equation (2.18) and the boundary conditions (3.19, 3.20), it follows that
68
the function 0( , , ; )rG x p s x corresponding to the function ( )z different from unity, and the
function 0( , , ; )rG x p s x corresponding to the function ( )z equal to unity (call it
0( , , ; )rG x p s x ), are connected by the relation
(4.8)
Substituting this relation into (2.27) and integrating with respect to x , we obtain
(4.9)
where 0
0( , , )H x p s is the distribution of the functional corresponding to the function ( )z
equal to unity. Thus, in this case, it is necessary to solve equation (4.1) with the boundary
conditions (4.2 and 4.3) and then substitute the functions found in the relation (4.9).
5. The residence time in the half-space
Consider the problem of calculating the distribution of the functional (1.1) with the
function ( )U x equal to unity at 0x and zero at 0x .
Equation (4.1), in this case, can be written as
(5.1)
The solution of this equation is the form [4, 22]
(5.2)
Constant 1C and 2C are determined from the boundary condition (4.2 and 4.3)
(5.3)
(5.4)
where 2( ) ( ) ( ( ))s s s a s . Substituting the resulting function 0
0( , , )H x p s into formula
(4.9) we obtain the final solution
(5.5)
With the forward equation the calculation will be more complex. It is necessary to
consider the cases 0 0x and 0 0x . If 0 0x then, according to the formula (2.17), the
function 0( , , ; )G x p s x can be represented as
69
(5.6)
In the regions 0 0x and 0 0x , the function 0( , , ; )rG x p s x satisfies the equations
(5.7)
and
(5.8)
respectively. A bounded at infinity solution of these equations has the form
(5.9)
At point 0x , this solution must satisfy the boundary conditions (3.19 and 3.20)
(5.10)
(5.11)
At point 0x x , the function 0( , , ; )G x p s x must be continuous, and its first derivative should
have a jump caused by the delta-function:
(5.12)
(5.13)
The distribution 0( , , )H x p s is calculated by the formula (2.6)
(5.14)
Finding constant 1 2 3 4,C C C C , and substituting them into the last formula, we get a result
identical to the result obtained with the backward equation. Case 0 0x is similar.
In [4, 22], the authors obtained a solution of this problem for the special case when:
there are no barriers, ( )s const ; the particle at the initial time is in the transport state,
( ) 1z ; and the function ( )s corresponds to the anomalous subdiffusion, 1( )s s . In
this particular case, provided 0 0x , we get the following distribution:
(5.15)
70
Having performed the inverse Laplace transform, the authors obtained the following
distribution for the relative residence time in the half-space /p A t for the t :
(5.16)
In the limit 0 , this distribution takes the form of two delta-peaks located at
0p and 1p ; that is, the particle is localized in one of the two half-spaces.
Distribution (5.14) is obtained under the assumption that the anomalous subdiffusion
is caused by traps. What if anomalous subdiffusion is caused by barriers? We put
( )s const ; ( ) 1z ; 1( )s s in (5.5). Instead of (5.15), we find
(5.17)
and instead of (5.16) we find
(5.18)
For small we have 2sin( / 2) / 2, cos( / 2) 1 ( / 2) / 2 ,
(5.19)
We can see that, in the limit 0 , the distribution (5.18) takes the form of a delta-
peak located at 1 2p ; that is, the time spent in the area 0x is equal to half of the total
time. Disagreement with the previous result is explained by the fact that subdiffusion caused
by barriers is stationary. The mobility of a particle does not change with time [10]. For small
, the particle cannot move away from the initial position at a large distance because the
labyrinth is too tangled (or the environment is too crowded). But in the space available to it,
the particle continues to move, spending approximately equal time in both areas 0x and
0x .
6. Conclusions.
In the present paper, both forward and backward Feynman-Kac equations for random
walks in disordered media are obtained. As a model of a disordered medium, the Schirmacher
model, which is the combination of the random barriers model and the multiple-trapping
model, is used. This model takes into account the presence in real disordered media both traps
and different kinds of obstacles. Through this it can describe both non-stationary subdiffusion
due to a delay of particles in traps and stationary subdiffusion due to the negative velocity
correlations. It is shown that the distributions of functionals for media with different
microscopic structures may differ significantly. This means that the obtained equations can be
used to determine the structure of a disordered medium. For this purpose the parameters of
these equations, in particular, memory functions ( )t and ( )t must be chosen so that the
theoretical distribution of functional coincide with the experimental one. Having these
memory functions, and knowing the dependence of these functions on the parameters
characterizing the structure of the medium (see [7, 15, 18]), it is possible, in principle, to find
these medium structure parameters.
71
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72
ФУНКЦІЯ РОЗПОДІЛУ ТРАЄКТОРІЇ ЧАСТИНКИ, ЯКА ЗДІЙСНЮЄ
ВИПАДКОВІ БЛУКАННЯ В НЕВПОРЯДКОВАНОМУ СЕРЕДОВИЩІ
В.П. Шкілев, В.В. Лобанов
Інститут хімії поверхні ім. О.О. Чуйка Національної академії наук України
вул. Генерала Наумова, 17, Київ, 03164, Україна, lobanov@isc.gov.ua
Розв'язана задача про знаходження функції розподілу траєкторії частинки, що
здійснює випадкові блукання в невпорядкованому середовищі, яке містить як пастки,
так і бар'єри. В якості моделі невпорядкованого середовища використана модель
Ширмахера, яка є комбінацією моделей випадкових бар'єрів і багаторазового
захоплення частинки. Сформульовано прямі і зворотні рівняння Фейнмана-Каца з
граничними умовами в точках розриву. Як приклад отримано розподіл часу
перебування частинки в півпросторі. Показано, що різні типи аномальної субдифузії,
обумовленої пастками і бар'єрами, дають функції розподілу, які сильно розрізняються.
ФУНКЦИЯ РАСПРЕДЕЛЕНИЯ ТРАЕКТОРИИ ЧАСТИЦЫ, СОВЕРШАЮЩЕЙ
СЛУЧАЙНОЕ БЛУЖДАНИЕ В НЕУПОРЯДОЧЕННОЙ СРЕДЕ
В.П. Шкилев, В.В. Лобанов
Институт химии поверхности им. А.А. Чуйко Национальной академии наук Украины
ул. Генерала Наумова, 17, Киев, 03164, Украина, lobanov@isc.gov.ua
Решена задача о нахождении функции распределения траектории частицы,
совершающей случайное блуждание в неупорядоченной среде, которая содержит как
ловушки, так и барьеры. В качестве модели неупорядоченной среды использована
модель Ширмахера, которая представляет собой комбинацию моделей случайных
барьеров и многократного захвата частицы. Сформулированы прямые и обратные
уравнения Фейнмана-Каца с граничными условиями в точках разрыва. В качестве
примера получено распределение времени пребывания частицы в полупространстве.
Показано, что различные типы аномальной субдиффузии, обусловленной ловушками и
барьерами, дают сильно различающиеся функции распределения.
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