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Some properties of a random walk on a comb structure
Physica 134A (1986) 474482 North-Holland, Amsterdam
SOME PROPERTIES
OF A RANDOM WALK ON A COMB STRUCTURE
George *Division
of Computer
H. WEISS*
Research
*Department
and
of Physics,
and Shlomo
HAVLIN*
Technology, National MD 20205. USA Bar-Zlan University,
Received
22 June
#
Institutes
of Health,
Ramat-Can,
Bethesda,
Israel
1985
We analyze transport properties of a random walk on a comb structure, which serves as a model for a random walk on the backbone of a percolation cluster. It is shown that the random walk along the x axis, which is the analog of the backbone, exhibits anomalous diffusion in that to nu4 for large n. The (.x2(n)) - n’a, and the expected number of x sites visited is proportional distribution function is found to be a two-dimensional Gaussian. If a field is applied in the x direction, so that diffusion is asymmetric, the expected displacement is found to be asymptotically proportional to rim.
1. Introduction Although
there
has
been
considerable
interest
in
the
problem
of
the
anomalous diffusion on fractal structures’-5), most results in this general area are known only by a combination of scaling arguments and simulation studiesiA). Since one cannot easily write a diffusion equation for motion on a fractal, density
there are no rigorously established results for P(r-, t), the probability for the end-to-end distance, r, of the random walker at time t. Several
forms for this function, not all in agreement, have been suggested, based on scaling arguments and simulation studies of different fractal structures&l’). One common thread that has recently been explored is the effect of the skeleton and the dead ends on properties of diffusive motion on loopless aggregates”-16). In the present paper we discuss the properties of a random walk on a particular tree in the form of a comb (fig. 1). Its diffusive properties, at least asymptotically, can be determined exactly. In particular, we will be interested in the interaction between motion along the backbone, or x axis, and motion along the fingers of the comb. Such motion can obviously be expected 0378-4371/86/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
SOME PROPERTIES
Y
OF A RANDOM WALK ON A COMB STRUCTURE
475
-
X
Fig. 1. The structure of the comb. The vertical lines extend to infinity. The random walk which starts from the origin can walk only alo’ng the fingers and on the x axis.
to have different properties in the horizontal and vertical directions. Since the random walk consists of a series of steps in the y direction followed by a series of steps in the x direction, and so forth, the motion along the x axis can be regarded as a discrete analogue of the continuous time random walk”18). In the resulting model, anomalous diffusion occurs along the backbone. This anomalous diffusion is similar to that observed on fractals, and we will suggest that by analogy the anomalous diffusion observed on fractals is due at least in part to excursions along blobs or dead ends.
2. Properties of the random walk We will assume that the random walk in either direction, x or y, is governed by the same set of single step transition probabilities {p(j)} together with the structure factor
For simplicity we will assume that the p(j) are symmetric, although the more general case can also be analyzed. The probability p,(r, s) for the random walker to be at (r, s) at step n will be the quantity of interest. The random walk can be regarded as a series of steps along the x axis interpreted by excursions
476
G.H. WEISS
of random
duration
in the y direction.
first find the distribution probability that there steps will be denoted more than
AND
S. HAVLIN
In order
of time between
to analyze
successive
this random
walk we
steps along the x axis. The
is an it step sojourn on a y finger between successive x by IJ,, and Pn will denote the probability that there are
n y steps between
two successive
x steps,
02
c
!Pn=
CG;, !PO=l.
(2)
j=n+l
We further denote by $,,(I) the probability that the j th step along the x axis occurs at the nth step of the random walk. If $(z) is the generating function for the r,& then [+(z)l’ is that for the +‘,i’ and [l- $(z)]/(l - z) is that for the ?P”. We can now derive the asymptotic form for p,(r, s), the probability that the random walker is at (r, S) at step n, by appealing to a central limit argument. If we choose a step number, n, then the x coordinate of the random walker, r, is the sum of j steps taken along the x axis, the last of which occurred at step number 1 s n. The y coordinate, S, is the sum of the remaining IZ- 1 steps during which only steps in the y direction have been taken. We will be interested in the limit of large n, and we will assume that the displacement in a single step (either in the x or y direction) has the properties (r) = 0,
(2)
=
m2, (r3> <
02 .
(3)
In order to calculate the relevant quantities we first require an expression for +(z). Whenever the random walk reaches a point on the x axis it either makes a step along the x axis with probability k or steps in the y direction with probability 2. In the latter case the time to return to the x axis is determined by the one-dimensional random walk in the y direction, and is just F’,“(O), which is the probability that a random walker initially at the origin returns to it for the first time at step n. But once back at a point itself. Hence
IL, =
on the x axis the process
we can write
;[a”, &F;‘(O) + (;)2Fy(0)+ . . -1 ) 1
(4)
+
where F’,“(O) is the probability that the random the j th time at step n. Since it is knowni7) that
where
repeats
P(0; z) is the Green’s
function
walk returns
to the origin
for
SOME PROPERTIES OF A RANDOM WALK ON A COMB STRUCTURE
471
we can conclude from eq. (3) that
(7) Let us now suppose that out of a total of n steps j have made along the x axis, the jth having occurred at step 1 s n. The variable j is itself random, and because of eq. (3) we have
(r2)= a’(i(n)>,
(4 = 0,
(8)
where (j(n)) is the expected number of steps along the x axis occurring in a total of n steps. The probability that there are j such steps is easily seen to be (9) since the factor ?Pn_, is the probability that there are no more x steps until at least step n + 1 after the jth that occurred at step 1. But since eq. (9) is in convolution form we immediately infer the generating function
Uj(Z) =
2 Uj,“Zfl = @(#P(z) = 9’(dU1-z
Jl(zN ’
n=O
which allows us to calculate a generating function for the (j(n)). This is given by the expression
J(z) =
C(i(n)>z” = Ciq(z)
n=O
j=O
=
(1
_
$yJ
+(t)).
(11)
In order to find the asymptotic behavior of the (j(n)) we will make use of a Tauberian theorem”), well-known in random walk analysis. For this purpose we need to determine the analytical behavior of J(z) in a neighborhood of z = 1. That of IL(z) can be inferred from eq. (7) and the observation that in one dimension, for random walksi7) that satisfy eq. (3) P(0; 2) - [aY2(1-
z)]_’
(12)
478
G.H. WEISS
as z+
1. Eq. (7) implies
4’(z) - 1 - &2(1as z+
AND
S. HAVLIN
that 2)
1, which together
(13)
with eq. (11) indicates
that
1
J(z) - p/241 _ ,)312 This together
.
with the appropriate
Tauberian
theorem”)
allows us to write
(g2
(i(n)>-
(15)
for large IZ. This is consistent with a result first obtained by Shlesinger”) for the mean square displacement of a continuous time random walk with a stable law pausing time density. Since the number of steps along the x axis increases with it we can apply the central limit theorem21) to conclude that the distribution of displacements along the x axis is Gaussian. However, since (r’(n)) - (2~7*n/,rr)” the diffusion is anomalous because nl’* rather than IZ occurs in the Gaussian exponent. We next examine the analogous properties of the vertical displacement. In the notation of eq. (9), s = II - 1. Since the walk is assumed to be symmetric, (s(n)> = 0, and
(s’(n)>= CT2 2 2 (n - E)~,I”P*_,.
(16)
j=OI=0 Again
it is convenient oz
c
R(z) =
n=O
to consider
the generating
yr’b) (s2(n)>z”= CT22 ~ l- W) 2
2 o-2rlr’(4
=_(1
Hence
function
crz)’
(1-
z)(l-
@))
- 2(lT
(17)
2)2
we can infer that
(s’(n))for large
azn/2
n. This expression
(18) tells us that
vertical
diffusion
is not anomalous.
SOME PROPERTIES OF A RANDOM WALK ON A COMB STRUCTURE
479
One can easily show that the correlation function between x and y displacements is identically equal to 0. Hence the asymptotic joint density of horizontal and vertical displacements is
(19) This expression allows us to calculate the expected visited by making use of the relation”)
number of distinct sites
m
S(z) = z/[(l-
t)*P(O; z)] ,
where P(0; z) = c p,(O,O)z". n=O
(20)
Although one can find an exact expression for P(0; z) all that is needed is the form of P(0; z) as z + 1. Since eq. (19) implies that 1
P,@7 0) -
-
~1/423l4~3/2
1
n3/4
(21)
’
an Abelian theorem for power series”) allows us to conclude that P(0; z) -
M/4)
I
~ll4~3l4~3l2
(1 _
,)1/4
(22)
as z + 1. This in turn implies that 9/4 (S”)
-
y&
3l2 n3/4
.
(23)
Notice that the expected number of sites visited along the x axis differs from this because the random walk spends most of its time making vertical steps. This is reflected in the difference between eqs. (15) and (18), if we remember that the mean square displacement in this model determines the asymptotic value of (S,) by the steps indicated in eqs. (19)-(21). We find that the value of (S,,) along the x axis goes like
(S,) - f
(2) li4114. n
(24)
It is evident that the anomalous diffusion observed along the x axis is due to the possibility of the average infinite sojourn on the fingers of the comb.
G.H. WEISS
480
Clearly
other
models
$,,, with infinite a finite Only
when
individual distance responds
can be developed
mean
variance
values.
Under
the resulting
the p(i) steps
are
AND
appearing in the to the Gaussian.
that lead to waiting the assumption
limiting
correspond infinite,
S. HAVLIN
distributions
to stable
there
laws,
is a possibility
exponent
that
differ
time probabilities,
that individual will always i.e., when
the
of having from
the
steps have be Gaussian. variances
powers
value
of
of the
2 that
cor-
3. The random walk in a uniform field One can also study the effect of anisotropic jump probabilities along the x axis. Let us suppose that at a junction of the x and y axes the probabilities of moving in the ?y direction are f and a, while the probabilities of moving along the x axis are such that for a single step
(r) =
(25)
CL .
For example, if the random walk is to nearest neighbors, the probability of moving in the x direction is (1 f 2~)/4 where 1~ 1
that after n steps
(r(n)>= ~u(i(n)>, 44n)) - &i(n)> . Since the statistical when the y motion
(26)
properties associated with visits to the x axis are unchanged has a mean displacement equal to zero, we see that
(r(n)>- p(2n/(T~2))1’2
(27)
along the x axis. The explanation for this form of time development displacement along the x axis is that the random walker infrequently
of average visits that
axis. 4. Discussion The extension of the preceding analyses to more complicated comb structures is not difficult. For example, if each point along a finger of the comb in fig. 1 is replaced by a comb, say in the z direction, then an analysis similar to that given for the single comb shows that
(r*(n)>- n1’4 (s*(n)>- nl’* )
)
(t*(n)>-
n,
(2%
SOME PROPERTIES
OF A RANDOM
WALK ON A COMB STRUCTURE
481
where t(n) is the displacement in the z direction. One can again appeal to a central limit theorem argument to show that the limiting joint distribution is Gaussian, having the form
P,(C
ar2 s, 4 - A exp -----n1/4
(
bs2 n1/2
ct’ It
>
’
where a, b and c are constants. The normalization constant A is therefore proportional to n-7’8 which implies that the total expected number of distinct sites visited is n”’ for this model. Our motivation for studying these comb models is that they possess several properties that are important in the characterization of diffusion on fractals. We can put our results into the notation common in the fractal literature, taking into account the anisotropy of our models. For example, for D = 2 where d: and di are the diffusion exponents for displacements along the x and y directions respectively, we have found that (r’(n)) - n”, (s2(n)) - it which in fractal notation is written d: = 4, d: = 1. The fracton dimension, defined by the exponent in the formula for the expected number of sites visited in an n-step walk, (SJ- na2 must also be decomposed into two components because of the asymmetry. Thus a scaling argument suggests that (SJ(S”,)(S:) from which it would follow that d -= 2
d; 1+
d= dY d” -II=‘+‘=-+-=-. d; 2 d;
1
1
3
4
2
4
This, of course, can be verified in detail for the comb model. One must therefore take anisotropy explicitly into account in the calculation of the fraction dimension and related quantities. We note that, although the results of the present analysis are asymptotic, a simulation study using the exact enumeration method”) indicates that the asymptotic results are accurate after no more than twenty to forty steps. As one can see from eqs. (19) and (29) the space dependence of the end-to-end probability remains Gaussian, while the (discrete) time may appear to different powers in the denominator of the exponential. This differs from recent forms for the probability suggested by Banaver and Willemsot?), Ohtsuki and Keyes’), and O’Shaughnessy and Procaccia*). Their proposed probabilities have a power differing from 2 for the spatial coordinates in the equivalent of eq. (19), while retaining the first power of IZ in the denominator. The calculation of Banaver and Willemson6), however, contains an unproved assumption relating to translational invariance in their key equation. Their suggested form for the end-to-end probability, as well as that of Ohtsuki and
G.H.
482
WEISS
AND
S. HAVLIN
Keyes’) and O’Shaughnessy and Procaccia’) are also inconsistent with results given by us for diffusion on percolation clusters’4). How much can be inferred from our present model calculation about diffusion on more complicated structures remains unanswered at present, but both this calculation and our earlier simulation study14) suggest that the power of the time in the exponent can differ from the value -1.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
S. Alexander and R. Orbach, J. de Phys. Lett. 43 (1982) L1625. D. Ben-Avraham and S. Havlin, J. Phys. Al5 (1982) L691, Al6 (1983) L559. Y. Gefen, A. Aharony and S. Alexander, Phys. Rev. Lett. 50 (1983) 77. R. Rammal and G. Toulouse, J. de Phys. Lett. 44 (1983) L13. B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982). J.R. Banaver and J. Willemson, Phys. Rev. B 30 (1984) 677. T. Ohtsuki and T. Keyes, Phys. Lett. 105A (1984) 273; Phys. Rev. Len. 52 (1984) 1177. B. O’Shaughnessy and I. Procaccia, Phys. Rev. Lett. 54 (1985) 455. R.A. Guyer, preprint. S. Havlin, D. Movshovitz, B. Trus and G.H. Weiss, J. Phys. A18 (1985) L719. S. Havlin, in: Kinetics of Aggregation and Gelation, F. Family and D.P. Landau, ed. (North-Holland, Amsterdam, 1984), p. 145. S. Havlin, Z. Djordjevic, I. Majid, H.E. Stanley and G.H. Weiss, Phys. Rev. Lett. 53 (1984) 178. T. Witten and Y. Kantor, Phys. Rev. B 30 (1984) 4093. S. Havlin, R. Nossal, B. Trus and G.H. Weiss, Phys. Rev. B 31 (1985) 7497. M.E. Cates, Phys. Rev. Lett. 53 (1984) 926. D. Dhar and R. Ramaswamy, preprint. E.W. Montroll and G.H. Weiss, J. Math. Phys. 6 (1965) 167. G.H. Weiss and R.J. Rubin, Adv. Chem. Phys. 52 (1983) 363. G.H. Hardy, Divergent Series (Oxford Univ. Press, London, 1949). M.F. Shlesinger, J. Stat. Phys. 10 (1974) 421. B.V. Gnedenko and A.N. Kolmogoroff, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Cambridge, Mass., 1954). E.C. Titchmarsh, The Theory of Functions (Oxford Univ. Press, London, 1939). S. Havlin, G.H. Weiss, J.E. Kiefer and M. Dishon, J. Phys. Al7 (1984) L347. S. Alexander and P. Pincus Phys. Rev. B 18 (1978) 2011. R. Kutner and H. van Beijeren, preprint.
SOME PROPERTIES
OF A RANDOM WALK ON A COMB STRUCTURE
George *Division
of Computer
H. WEISS*
Research
*Department
and
of Physics,
and Shlomo
HAVLIN*
Technology, National MD 20205. USA Bar-Zlan University,
Received
22 June
#
Institutes
of Health,
Ramat-Can,
Bethesda,
Israel
1985
We analyze transport properties of a random walk on a comb structure, which serves as a model for a random walk on the backbone of a percolation cluster. It is shown that the random walk along the x axis, which is the analog of the backbone, exhibits anomalous diffusion in that to nu4 for large n. The (.x2(n)) - n’a, and the expected number of x sites visited is proportional distribution function is found to be a two-dimensional Gaussian. If a field is applied in the x direction, so that diffusion is asymmetric, the expected displacement is found to be asymptotically proportional to rim.
1. Introduction Although
there
has
been
considerable
interest
in
the
problem
of
the
anomalous diffusion on fractal structures’-5), most results in this general area are known only by a combination of scaling arguments and simulation studiesiA). Since one cannot easily write a diffusion equation for motion on a fractal, density
there are no rigorously established results for P(r-, t), the probability for the end-to-end distance, r, of the random walker at time t. Several
forms for this function, not all in agreement, have been suggested, based on scaling arguments and simulation studies of different fractal structures&l’). One common thread that has recently been explored is the effect of the skeleton and the dead ends on properties of diffusive motion on loopless aggregates”-16). In the present paper we discuss the properties of a random walk on a particular tree in the form of a comb (fig. 1). Its diffusive properties, at least asymptotically, can be determined exactly. In particular, we will be interested in the interaction between motion along the backbone, or x axis, and motion along the fingers of the comb. Such motion can obviously be expected 0378-4371/86/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
SOME PROPERTIES
Y
OF A RANDOM WALK ON A COMB STRUCTURE
475
-
X
Fig. 1. The structure of the comb. The vertical lines extend to infinity. The random walk which starts from the origin can walk only alo’ng the fingers and on the x axis.
to have different properties in the horizontal and vertical directions. Since the random walk consists of a series of steps in the y direction followed by a series of steps in the x direction, and so forth, the motion along the x axis can be regarded as a discrete analogue of the continuous time random walk”18). In the resulting model, anomalous diffusion occurs along the backbone. This anomalous diffusion is similar to that observed on fractals, and we will suggest that by analogy the anomalous diffusion observed on fractals is due at least in part to excursions along blobs or dead ends.
2. Properties of the random walk We will assume that the random walk in either direction, x or y, is governed by the same set of single step transition probabilities {p(j)} together with the structure factor
For simplicity we will assume that the p(j) are symmetric, although the more general case can also be analyzed. The probability p,(r, s) for the random walker to be at (r, s) at step n will be the quantity of interest. The random walk can be regarded as a series of steps along the x axis interpreted by excursions
476
G.H. WEISS
of random
duration
in the y direction.
first find the distribution probability that there steps will be denoted more than
AND
S. HAVLIN
In order
of time between
to analyze
successive
this random
walk we
steps along the x axis. The
is an it step sojourn on a y finger between successive x by IJ,, and Pn will denote the probability that there are
n y steps between
two successive
x steps,
02
c
!Pn=
CG;, !PO=l.
(2)
j=n+l
We further denote by $,,(I) the probability that the j th step along the x axis occurs at the nth step of the random walk. If $(z) is the generating function for the r,& then [+(z)l’ is that for the +‘,i’ and [l- $(z)]/(l - z) is that for the ?P”. We can now derive the asymptotic form for p,(r, s), the probability that the random walker is at (r, S) at step n, by appealing to a central limit argument. If we choose a step number, n, then the x coordinate of the random walker, r, is the sum of j steps taken along the x axis, the last of which occurred at step number 1 s n. The y coordinate, S, is the sum of the remaining IZ- 1 steps during which only steps in the y direction have been taken. We will be interested in the limit of large n, and we will assume that the displacement in a single step (either in the x or y direction) has the properties (r) = 0,
(2)
=
m2, (r3> <
02 .
(3)
In order to calculate the relevant quantities we first require an expression for +(z). Whenever the random walk reaches a point on the x axis it either makes a step along the x axis with probability k or steps in the y direction with probability 2. In the latter case the time to return to the x axis is determined by the one-dimensional random walk in the y direction, and is just F’,“(O), which is the probability that a random walker initially at the origin returns to it for the first time at step n. But once back at a point itself. Hence
IL, =
on the x axis the process
we can write
;[a”, &F;‘(O) + (;)2Fy(0)+ . . -1 ) 1
(4)
+
where F’,“(O) is the probability that the random the j th time at step n. Since it is knowni7) that
where
repeats
P(0; z) is the Green’s
function
walk returns
to the origin
for
SOME PROPERTIES OF A RANDOM WALK ON A COMB STRUCTURE
471
we can conclude from eq. (3) that
(7) Let us now suppose that out of a total of n steps j have made along the x axis, the jth having occurred at step 1 s n. The variable j is itself random, and because of eq. (3) we have
(r2)= a’(i(n)>,
(4 = 0,
(8)
where (j(n)) is the expected number of steps along the x axis occurring in a total of n steps. The probability that there are j such steps is easily seen to be (9) since the factor ?Pn_, is the probability that there are no more x steps until at least step n + 1 after the jth that occurred at step 1. But since eq. (9) is in convolution form we immediately infer the generating function
Uj(Z) =
2 Uj,“Zfl = @(#P(z) = 9’(dU1-z
Jl(zN ’
n=O
which allows us to calculate a generating function for the (j(n)). This is given by the expression
J(z) =
C(i(n)>z” = Ciq(z)
n=O
j=O
=
(1
_
$yJ
+(t)).
(11)
In order to find the asymptotic behavior of the (j(n)) we will make use of a Tauberian theorem”), well-known in random walk analysis. For this purpose we need to determine the analytical behavior of J(z) in a neighborhood of z = 1. That of IL(z) can be inferred from eq. (7) and the observation that in one dimension, for random walksi7) that satisfy eq. (3) P(0; 2) - [aY2(1-
z)]_’
(12)
478
G.H. WEISS
as z+
1. Eq. (7) implies
4’(z) - 1 - &2(1as z+
AND
S. HAVLIN
that 2)
1, which together
(13)
with eq. (11) indicates
that
1
J(z) - p/241 _ ,)312 This together
.
with the appropriate
Tauberian
theorem”)
allows us to write
(g2
(i(n)>-
(15)
for large IZ. This is consistent with a result first obtained by Shlesinger”) for the mean square displacement of a continuous time random walk with a stable law pausing time density. Since the number of steps along the x axis increases with it we can apply the central limit theorem21) to conclude that the distribution of displacements along the x axis is Gaussian. However, since (r’(n)) - (2~7*n/,rr)” the diffusion is anomalous because nl’* rather than IZ occurs in the Gaussian exponent. We next examine the analogous properties of the vertical displacement. In the notation of eq. (9), s = II - 1. Since the walk is assumed to be symmetric, (s(n)> = 0, and
(s’(n)>= CT2 2 2 (n - E)~,I”P*_,.
(16)
j=OI=0 Again
it is convenient oz
c
R(z) =
n=O
to consider
the generating
yr’b) (s2(n)>z”= CT22 ~ l- W) 2
2 o-2rlr’(4
=_(1
Hence
function
crz)’
(1-
z)(l-
@))
- 2(lT
(17)
2)2
we can infer that
(s’(n))for large
azn/2
n. This expression
(18) tells us that
vertical
diffusion
is not anomalous.
SOME PROPERTIES OF A RANDOM WALK ON A COMB STRUCTURE
479
One can easily show that the correlation function between x and y displacements is identically equal to 0. Hence the asymptotic joint density of horizontal and vertical displacements is
(19) This expression allows us to calculate the expected visited by making use of the relation”)
number of distinct sites
m
S(z) = z/[(l-
t)*P(O; z)] ,
where P(0; z) = c p,(O,O)z". n=O
(20)
Although one can find an exact expression for P(0; z) all that is needed is the form of P(0; z) as z + 1. Since eq. (19) implies that 1
P,@7 0) -
-
~1/423l4~3/2
1
n3/4
(21)
’
an Abelian theorem for power series”) allows us to conclude that P(0; z) -
M/4)
I
~ll4~3l4~3l2
(1 _
,)1/4
(22)
as z + 1. This in turn implies that 9/4 (S”)
-
y&
3l2 n3/4
.
(23)
Notice that the expected number of sites visited along the x axis differs from this because the random walk spends most of its time making vertical steps. This is reflected in the difference between eqs. (15) and (18), if we remember that the mean square displacement in this model determines the asymptotic value of (S,) by the steps indicated in eqs. (19)-(21). We find that the value of (S,,) along the x axis goes like
(S,) - f
(2) li4114. n
(24)
It is evident that the anomalous diffusion observed along the x axis is due to the possibility of the average infinite sojourn on the fingers of the comb.
G.H. WEISS
480
Clearly
other
models
$,,, with infinite a finite Only
when
individual distance responds
can be developed
mean
variance
values.
Under
the resulting
the p(i) steps
are
AND
appearing in the to the Gaussian.
that lead to waiting the assumption
limiting
correspond infinite,
S. HAVLIN
distributions
to stable
there
laws,
is a possibility
exponent
that
differ
time probabilities,
that individual will always i.e., when
the
of having from
the
steps have be Gaussian. variances
powers
value
of
of the
2 that
cor-
3. The random walk in a uniform field One can also study the effect of anisotropic jump probabilities along the x axis. Let us suppose that at a junction of the x and y axes the probabilities of moving in the ?y direction are f and a, while the probabilities of moving along the x axis are such that for a single step
(r) =
(25)
CL .
For example, if the random walk is to nearest neighbors, the probability of moving in the x direction is (1 f 2~)/4 where 1~ 1
that after n steps
(r(n)>= ~u(i(n)>, 44n)) - &i(n)> . Since the statistical when the y motion
(26)
properties associated with visits to the x axis are unchanged has a mean displacement equal to zero, we see that
(r(n)>- p(2n/(T~2))1’2
(27)
along the x axis. The explanation for this form of time development displacement along the x axis is that the random walker infrequently
of average visits that
axis. 4. Discussion The extension of the preceding analyses to more complicated comb structures is not difficult. For example, if each point along a finger of the comb in fig. 1 is replaced by a comb, say in the z direction, then an analysis similar to that given for the single comb shows that
(r*(n)>- n1’4 (s*(n)>- nl’* )
)
(t*(n)>-
n,
(2%
SOME PROPERTIES
OF A RANDOM
WALK ON A COMB STRUCTURE
481
where t(n) is the displacement in the z direction. One can again appeal to a central limit theorem argument to show that the limiting joint distribution is Gaussian, having the form
P,(C
ar2 s, 4 - A exp -----n1/4
(
bs2 n1/2
ct’ It
>
’
where a, b and c are constants. The normalization constant A is therefore proportional to n-7’8 which implies that the total expected number of distinct sites visited is n”’ for this model. Our motivation for studying these comb models is that they possess several properties that are important in the characterization of diffusion on fractals. We can put our results into the notation common in the fractal literature, taking into account the anisotropy of our models. For example, for D = 2 where d: and di are the diffusion exponents for displacements along the x and y directions respectively, we have found that (r’(n)) - n”, (s2(n)) - it which in fractal notation is written d: = 4, d: = 1. The fracton dimension, defined by the exponent in the formula for the expected number of sites visited in an n-step walk, (SJ- na2 must also be decomposed into two components because of the asymmetry. Thus a scaling argument suggests that (SJ(S”,)(S:) from which it would follow that d -= 2
d; 1+
d= dY d” -II=‘+‘=-+-=-. d; 2 d;
1
1
3
4
2
4
This, of course, can be verified in detail for the comb model. One must therefore take anisotropy explicitly into account in the calculation of the fraction dimension and related quantities. We note that, although the results of the present analysis are asymptotic, a simulation study using the exact enumeration method”) indicates that the asymptotic results are accurate after no more than twenty to forty steps. As one can see from eqs. (19) and (29) the space dependence of the end-to-end probability remains Gaussian, while the (discrete) time may appear to different powers in the denominator of the exponential. This differs from recent forms for the probability suggested by Banaver and Willemsot?), Ohtsuki and Keyes’), and O’Shaughnessy and Procaccia*). Their proposed probabilities have a power differing from 2 for the spatial coordinates in the equivalent of eq. (19), while retaining the first power of IZ in the denominator. The calculation of Banaver and Willemson6), however, contains an unproved assumption relating to translational invariance in their key equation. Their suggested form for the end-to-end probability, as well as that of Ohtsuki and
G.H.
482
WEISS
AND
S. HAVLIN
Keyes’) and O’Shaughnessy and Procaccia’) are also inconsistent with results given by us for diffusion on percolation clusters’4). How much can be inferred from our present model calculation about diffusion on more complicated structures remains unanswered at present, but both this calculation and our earlier simulation study14) suggest that the power of the time in the exponent can differ from the value -1.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
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