- Email: [email protected]
Fractal properties of model trajectories with exponential velocity autocorrelation functions
Physica 141A (1987) 318-334 North-Holland, Amsterdam
FRACTAL PROPERTIES OF MODEL TRAJECTORIES WITH EXPONENTIAL VELOCITY AUTOCORRELATION FUNCTIONS J.G.
POWLES
Physic.y Department.
and R.F.
FOWLER
Univwsify of Kenr. Canterbury,
Received
10 June
UK
19X6
The length, L(E), at scale, F, has been determined for four different models for atomic motion in a fluid, three distinct types of random walk with constant speed and the Langevin model. This has been done for trajectories in enc. two, three and sometimes four dimensions. All these motions have exactly exponential velocity autocorrelation functions. However, the functional form of L(F) is different for the different models, contrary to a recent assertion (S. Toxvaed, Phys. Lett. A 114 (1986) 159). A theory for L(F) is lacking, except for the limits of E small and F very large. Our results can be used as ‘experimental results’ for the testing of such theories which arc quite different from the conventional analysis of molecular motion in fluids. A comparison is made of these model results for L(F) with those for simulated trajectories in realistic models of liquids and for the actual trajectories of suspended particles in solution.
1. Introduction The method of fractal analysis of trajectories proposed by Richardson*) was first applied to the realistic trajectories of particles in fluids by Powles and Quirke’) and there have been several subsequent investigations of trajectories by this method. (refs. 4-12). In Richardson’s analysis the length of the trajectory, L(F), at scale, F, is measured by stepping along the trajectory from the beginning until a point is reached at which the scalar distance from the initial point is F, for thefirst time. This process is repeated from the new initial locations until the distance from the end of the trajectory is less than E. The fraction of E to the end of the trajectory to get to its termination is added. The total of these steps is L(E). Richardson found that many trajectories have a range of e for which
where
N is not in general
0378-4371/87/$03.50 0 (North-Holland Physics
an integer
and is a characteristic
Elsevier Science Publishers Publishing Division)
B.V.
of the trajectory.
FRACTAL
For all physically length,
realistic
319
MODEL TRAJECTORIES
trajectories
for E + 0 we get for L(E) the contour
L,, i.e.
L(E)/L,-+~ We call this
for e-0 the
free-flight
and a-+0. limit.
(1.1) must
be
approximate and not exact as asserted because it violates the limit (1.1). More generally we can define a Richardson coefficient which is a function E, defined by
Incidentally
of
(Y(E) = -d[ln(L(c)
Toxvaerd’s
lL,)]ld[ln(e)]
This may then be called finite-fractal analysis.13) For F very large we get what we call the diffusion result is
L(&) -_+-L,
2dD
1
(u)
&’
E+mJ
i.e. CX(E+~)+
theory’)
.
(1.2)
limit when’*)
the exact
1
(1.3)
D is the diffusion coefficient, and d is the dimensionality of the space in which the trajectory exists. (u) is the mean speed of the particle, which is independent of the length of the trajectory in the limit of an indefinitely long trajectory. Between the two limits, E+ 0 and E+ ~0, and for realistic trajectories, a(e) rises steadily from zero to unity5), see fig. 3. However, as far as we are aware, there is only one special case in which the exact form of L(E) lL, over the whole range of E is known theoretically, although approximations have been proposed which are largely empirical or phenomenological (refs. 13, 5, 1 and 10 - these
are discussed
For a model found that
-L(E) = Lc
described
later). as ‘Brownian
motion’
in one dimension,
Takayasu’“)
1
(1.4)
l+&IEO’
where E,, is stated to be some mean-free-path. For a random walk in one dimension (described Rickayzen”) found that 1 ~L(E) ZZ 1+ E/2&, ’ LC
in section
2) Powles
and
(1.5)
320
J.G.
where
POWLES
AND
R.F.
FOWLER
F,, is defined
as ( u) r,, where T, is the correlation time of the velocity function. (1.5) is consistent with the exact result (1.3).
autocorrelation
It is not clear to us that these two models
are identical
but we suspect
it to be
SO.
The form (1.4), perfectly,
with E,, a disposable
to realistic
but simulated
parameter,
trajectories
fits quite
in liquids
well but far from
in three-dimensional
space’) and to the actual trajectories of bacteriophages and polystyrene spheres in solution’“). It has also been shown that for the trajectory of an atom in a three-dimensional simulated liquid”) the form (1.5) fits the results quite well with the known value of F,, “). Since no theoretical result for L(F) is known, for other than the onedimensional models mentioned above, we have obtained numerical results for L(E) for several simple models of atomic motion in a fluid for other than one dimension of space, and in particular for real three-dimensional space. We compare the results with those for real and/or computed fluids in section --I 8.
2. Random-walk
models
The random walk discussed by Powles and Rickayzen”), which led to eq. (1.5), was a one-dimensional one in which the particles moved at constant speed in each step, to left or right, by a distance 1 in a time 7. The model included persistance of velocity in the sense that if the particle is travelling to the right (or left) in a given step then in the next step it is travelling either to the right (or left) with a probability, p, or to the left (or right) with a probability q, where of course, (p + q) = 1. In the limit, q-+ 0, the velocity autocorrelation function, 4,(t), is an exponential with a correlation time TV-
rl2q
(2.1)
,
thus E,+ 112q, and if we choose F = nl, where II is a positive integer, it can be shown that (1.5) may be written in the simple form
L(n)
1
L
1+ (n - 1)q
21, then
(2.2)
When the dimensionality is greater than one there are more than one simple random walks which are generalisations of the walk described above. In fact in d dimensions there are d such walks as described below.
FRACTAL
3. q-walks
321
MODEL TRAJECTORIES
in any dimension
We call a q-walk
one in which the motion
is as for the walk described
above
but it occurs in every one of the d dimensions independently. It is obvious that the elementary step, in time 7, is now I, = d1’21. It is easy to show that the diffusion
constant,
for q 4 p, has the value
D = 12f2Tq,
(3.1)
independent of d. Moreover, for q-+ 0, c&(t) is exponential and E,-+ 1,/2q since ( vd) = 1,lr. Because
of (1.3)
and (3.1)
with T” as in (2.1),
we have
(3.2) We also still have, ~L(E) + 1 L
of course,
that
for E e E, .
(3.3)
However, we have not been able to obtain an exact expression for L(E) lL,, for any F, by any of the methods we used in ref. 11 for d = 1. Apart from the increased geometrical complexity, we have the difficulty that, even if we restrict E to nl,, the calculation of L(E) involves initial points which are not necessarily at the end points of the elementary steps, 1. Put in another way, the hypercubic symmetry of the random walk does not fit naturally to the hyperspherical symmetry of the scaling length E. However, there is no problem in numerically generating long trajectories for this model and determining L(E) to any desired accuracy for any finite value of E using a sufficiently small but finite value of q. We found that q = 0.01 was adequate. The results expressed as a function of (E/~F,) are then virtually independent of q. An alternative method of computing the value of L(E) is the following. For any one of the d coordinate directions the probability of continuing in the same direction at the end of a step of length 1 is p and if not doing so is q. The mean distance, (s), before a change of direction occurs for the first time is given by np”- 1q=l/q. n==l
(s) I1 = 5
In the limit q+O Poisson distribution)
the probability
distribution
of S, p(s),
becomes
(cf. the
322
J.G.
p(s)-
(q/l)
POWLES
AND
R.F.
FOWLER
(3.4)
exp(-sqil)
This distribution, in each dimension the NAG random-number-generator
independently, and double
is readily precision,
generated using and hence the
trajectory
for any given value of q. In fact a value q of 0.01 gives a trajectory
sufficient
accuracy.
This
method
demands
much
the same
time as the direct method described previously. We have confirmed essentially the same numerical results for L(F/~P,). In view of the two limits.
L(F) 4
(3.2)
1 + (FQE,.)~,(E/2E,,)
where fC,(.z/2&,.)+ that fC, is a smooth
and (3.3).
of
cost in computer
it is convenient
that it gives
to write
(3.5)
’
I for c/~F,.+ x. We find from the computed trajectories function. f,, is almost independent of F for F + 2~, and WC
call this value A,. S,, has the value (6, + 1)/2 for F = 2~, . Of course. for d = 1, f= 1 for all e (cf. (1.5)). In fact we find for all the random-walk models investigated in this paper that ji*, as a function of In(F), is quite closely of tanh (or Fermi-function) form.
f;,(z)-~(l+f,,)+!(l-f,,)tanh[~(z-z,)l.
(3.6)
w
where z = ln,(&/2&,) and D, is a measure of the width of the transition z, is a constant which fixed the centrc of the transition region. Eq. (3.6) may be written in the alternative form f,,(&i&,)= where
in table
x
=
(E/E
i(l+J,)-
I, whe;ein
(3.6’)
:(l-~,)(l-X’)l(l+“z),
)Iz’(‘h, ‘I1 7)1, and F, = 2~,2’l. The values
the values
of 6, for the simulated
region.
of 6,. z, and D,+ are given trajectories
are called f;,,.
Since fo5 < 1, L(E) approaches the value L, more rapidly with decreasing & than in one dimension. Since z, is small the changeover occurs, as already stated, for F - 2~, and takes place roughly in the range for c/2.5, from i to 8. Using these parameters in (3.6) and then in (3.5) reproduces the observed values of L(F) to an accuracy which should be sufficient to test any theory for L(E) which may become available in due course. A theoretical value of A,, called JIC in table I, is obtained in the appendix. The observed and calculated values agree to within the accuracy of the computation. Some readers may feel that this model is not the simplest or most approp-
TABLE
Parameters
d
of
the
I
f-function for models.
the
three
random-walk
f “C
fo,
D,
Parameters for the q-model 1
_
1
2 3 4
0.572 0.466 0.424
0.5708. 0.4584. 0.4184.
cc
_
l/3=0.333..
_ -0.84 -0.38 -0.11
_ 5.40 4.52 3.88
_
_
Parameters for r-model as for q-model
A
0.779 0.696 0.657 -
0.7854 0.7139. 0.6781 [I + (d - l)(ni2 - l)]ld
0.21 0.26 0.10 -
5.63 6.00 5.25 _
_
(r/2-1)=0.5708..
-
_
0.48 0.77
3.52 4.95
-
_
Parameters for the s-model 1 2 3 4
as for q-model as for r-model 0.752 0.695
0.7292, 0.6936. (~12 - 1) = 0.5708.
‘x
d-
1 (1-q) f
(1-r)
r r I (1-s)
S S
I
Fig. 1. Diagrammatic representation of the behaviour of the trajectories at the end of a step for the three walks, q, r and s, in the limit q. r, s-+0 for 1, 2, 3, and 4 dimensions. Continues in the same direction for (1 - q) etc. 323
324
J.G.
riate generalisation
POWLES
AND
R.F.
of the one-dimensional
FOWLER
walk of section
2. For this model,
in
the limit q ---, O,we have at the end of a step a probability (1 - dq) of continuing and a probability of q of deviating by an angle coss’[(d - 2)/d] in the d directions
symmetrically
does however
arranged
have the advantage
around
the forward
direction,
that it is very straightforward
see fig. 1. It to programme
and compute. We therefore
4.
now investigate
two other
related
models.
in any dimension
r-walks
In this model we make the probability in the forward direction equal to [l - (2d ~ l)r] and a probability of r of deviating by z-/2 in the 2(d - 1) symmetrical directions and also r in the reverse direction. This is shown diagrammatically in fig. 1. For d = 1 this is the same as for the q-model. In the limit r+ 0, 4,(t) is exponential with T,./T = 1 i(2dr) so that if the elementary step is again I,, then /,/(2dr)
E,, =
so that,
in particular,
L(F)/L,+~E,/E
and
D = 1’/(2Tdr),
(1.3)
becomes
(cf. (3.2))
for E-+E.
The latter two results have been confirmed numerically. We now calculate L(c)IL, as in section 3 and again find that (3.5) with the form (3.6) gives a good fit with the parameters given in table I. The theoretical values of J;,, i.e. hjC, are obtained in the appendix. The velocity autocorrelation function is exponential in both the q and r models, so that, in particular, (r’(t)) has exactly the same functional form (see fig. 2), namely, (r’(t))
= 2dD{t
- T,( 1 - exp - tin,)}
,
(4.1)
and this has been confirmed explicitly for our trajectories. Nevertheless, the functional form of L(E) IL, is markedly different for the two models for any given dimension except d = 1 and depends on the dimensionality. For the r-model fr, is larger and z, is positive not negative, but D, is very similar in value. We now analyse one further model in this class.
FRACTAL
5. s-walks
MODEL
TRAJECTORIES
325
in any dimension
In this model the probability of continuing in the forward direction is [l -- (2d - l)s]. The probability is s to divert into one of the “C,, directions by an angle cosP’[(d - 2n) ld], n = 1,2, . . . , d, i.e. until the angle is 7~, the reverse
direction
(see fig. 1 and table II. Note,
these are the angles which arise
in the q-model, for q not approaching zero). For s-+0, 4,,(t) is again exponential and r,/~ = 1/(2d~). This model is identical to the q-model for d = 1 and identical to the r-model for d = 2 but the three models are different d 2 3, see fig. 1. The calculation proceeds just as before and the result for d = 3 and 4 is given in table I. In particular the value off;, is larger than for the other two models but is still substantially below unity. We thus again have a different form for L(E) in spite of the fact that the velocity autocorrelation function is again exponential. TABLE
The deviation d
cos-‘[(d
1 2
?r 7Tl2 coscy f ) cosY’($)
3 4
- 2) ld]
angles
II
for the s-model
up to d = 4.
cosC’[(d - 4)/d]
cosC’[(d - 6)/d]
cos-‘[(d
_
_ _
_ _ _
77 cosc’(7r12
f)
77 coscI(-
4)
- 8)/d]
7r
6. The L-model A rather different model which also has an exponential velocity autocorrelation function is the trajectory resulting from the Langevin equation of motion
dp
dt- -
-p/r,
+ R(t)
)
(6.1)
where p is the momentum of the particle, r, is, as usual, the velocity correlation time and R(t) is a random force. This model differs from the previous ones in that the particle speed is not constant, indeed it is chosen to have a Maxwell-Boltzmann distribution. This has been used14) to study the dynamics of ‘Brownian’ particles in solution where the effect of the solvent is modelled by R(t). R(t) should have zero mean and a Gaussian probability distribution. It is convenient to use reduced units (*) as follows:
326
J.G.
t* = t/r
Y
POWLES
p* = pl(mkT)“’
9
so that if x is the position
AND
and
of the particle
R.F. FOWLER
R* = [7vl(mkT)“2]R
,
then
x*= (mkjl;l:lrv x. The trajectories (6.1) b ecomes
in reduced
+ R*(t*)
dp* ldt* = -p:’
In the simulation R,* is the random
units are then the same for all temperatures,
T. Eq.
.
(6.2)
R* is taken
force
to be constant during a time step, At*, and then if in the nth time step, (6.2) has the exact solution, for
At* finite. P*(tz+l)
= RE + [p*(t,T) ~ R,*] exp(-At*)
x*(t,T+,)
= x*(t,*)
(6.3)
and + R,* At* + [p*(t,*)
- Rz][l
- exp(-At*)]
.
(6.4)
Eqs. (6.3) and (6.4) are used to obtain the trajectory, x*(t*). Sufficient accuracy is obtained if At* 50.01. Eq. (6.1) is for the motion in one axis. For d dimensions we use d equations like (6. l), independently, to obtain r*(t*) where r* is the position of the particle at time t”. For a long trajectory the initial momentum value, p(t,), is immaterial. It is simple to show that
*_
7, -
1,
(u:‘)=l
and
D*=l.
These values have been confirmed has eq. (4. l), see fig. 2 for d = 3. It can also be shown that
where
ud is the speed
explicitly
in the d-dimensional
for our computed
case and of course,
trajectories
(uz ‘) = d.
as
FRACTAL
MODEL
327
TRAJECTORIES
Fig. 2. The points are values of (r’) for a computed Langevin trajectory in three-dimensional space. The curve is the theoretical result. eq. (4.1). Note that (r’) is close to the limiting diffusion value 601, on the dashed line, after only about 57,. This may be contrasted with the slow approach to the diffusion limit in the corresponding fractal analysis of fig. 3.
Hence
using
(1.3),
we have
2112drM
L(&*)
1
p
L,-
for &*-cc
(6.5)
F(F)
and of course
we still have
(6.6)
for &*-+O.
Again
defining,
E, = ( u~)T”, we can write
(6.5)
in the form
for E%~E,
By analogy
with (3.5)
this suggests
1 -L(E) = 1+ (E12EL)fd(&12EL) L where
.
(6.5)’
the formulation
’
(6.7)
328
J.G.
POWLES
AND
R.F.
FOWLER
so that f(m) = 1 .
We have obtained
and analysed
trajectories
for d = 1, 2 and 3 using up to 12
million steps which took about 6 hours on our VAX 111750. The plot of ln,[L(&)/L,] versus ln,(c*) for d = 1 is shown in fig. 3. It is clear, by comparison with the curve for f = 1 for all F, that f(e/2e,) is greater than one, see fig. 4a, in stark contrast with the q-model for d = 1. We emphasise again that all these models have 4,(t) exponential. The function f,(e/2&,) is shown for d = 1, 2 and 3 in fig. 4. None of these is of tanh form but for d = 3 we have f,, < 1 as for the previous models. These plots of h,, together with eq. (6.7) are sufficient to reproduce the function L(E) for comparison with any theory which may become available. We found, as might be expected, that the values of L(E) or f(.z) are only independent of At for F larger than about five times ( ud) At so that to get results of reasonable accuracy for F demands trajectories with a very large number of time steps. The plots of fig. 4 are therefore less accurate than for the random-walk models. In particular we have not been able to establish whether a limiting value A, for fd exists for d = 2 or 3 for this model. A rough estimate of the accuracy is given in fig. 4. We note that even for several million steps, i.e. an elapsed time of order 50 0007,, the values of L(E) for F large barely established the known diffusion limit result, (6.5). with slope -1, i.e. (Y = 1. This may be contrasted with the
Fig. 3. The plot of In2[L(~)IL,] versus (L-model). The points are the computed straight line at 45” is the exact ‘diffusion’
In,[ &*I for the one-dimensional Langevin-motion model values. The curve is for the relation (6.7) with f= 1. The result. The dashed curve is for eq. (7.1) with m = 0.904.
FRACTAL
MODEL
TRAJECTORIES
0
I
1.5 -
I
329
6
I
d=2
I
’
b
LN,k’)
Fig. 4. The function f(&/z~~), see eq. (6.7), for the computed values of L(E)/L, for the L-model for 1, 2 and 3 dimensions. Note the smaller scale of ordinate for d = 1. An estimate of the computational error is indicated.
J.G.
330
diffusion
POWLES
AND
limit for (Y’) which is attained
R.F.
FOWLER
to similar
accuracy
in a time of order
57,.
7. The empirical Matsuura
relation
et al.“‘) have suggested
L(F) LC
of Matsuura
et al. the empirical
relation
1 (7.1)
[l + (FI&,,)‘71]“‘n ’
where both E,, and m are disposable parameters. However, in their comparison with their experimental results they found that to the accuracy of their results the choice m = 1, i.e. eq. (1.4), is sufficient. Nevertheless, it is of interest in the present context to test this relation for our models. We find that this formula fits our data better than the onedimensional random walk model (m = l), with appropriate choice of m. For instance, the fit to the data in fig. 3 is overall much improved. but is far from perfect, if WC choose m = 0.004. To show the effect of assuming (7.1) for m # 1 we have calculated the deviation function j’(f(~’ ) where F ’ = E/E,, and the result is given in fig. 5. For m < 1. f‘ is greater
than unity for all E and diverges for e+O, although of course we still have the required limit (1 .I). This behaviour is similar to that of the L-model for d = 1 or 2 (see fig. 4a and b). For m > 1. J’is less than unity for all E and approaches zero for c + 0. This is not consistent with the p, q and r models for which J‘+J;, and A, is non-zero. For the L-model for d = 3 we have
I
I
f kc+1 112
0.9
:::> m=,
/
A
-4 Fig. 5. The flfunction (cf. cq. (3.5)) the value of m indicated.
-2
0
for the empirical
2
4
model
of Matsuura
6 et al.“‘).
eq. (7.1).
for
FRACTAL
not been
able to establish
331
MODEL TRAJECTORIES
a numerical
value
for f,, (see fig. 4c) or even
if it
exists. In spite of these difficulties the formula (7.1) may be useful for the analysis of data for L(E) since its main effect is to modify equations (1.4) or (1.5) for F =Z2~” roughly
as observed
in our models.
8. Comparison
with real trajectories
Tsurami and Tokayasu’) have shown that the simulated trajectories of Powles and Quirke’) for a Lennard-Jones fluid and of Rapaport3) for a hard-sphere fluid are fitted quite well by Takayasu’s formula (1.4) with &Cla disposable parameter. Also Powles”) has shown that his data are fitted quite well by the formula (1.5) with the known value of F,. Matsuura et al.‘O) have shown that eq. (1.4) can be fitted quite well to the measurements of L(E) for the observed trajectories of polystyrene spheres and bacteriophage T4 particles results are in solution with appropriate choice of r,,. All these experimental for d = 3 whereas the theory is for d = 1. The numerical results for the models discussed here for d = 3 differ from eq. (1.5) in that f is not unity for E < 2e, and in particular f0 is of order one-half (see table I and fig. 4~). To determine this difference demands a very high accuracy in the determination of L(F) IL, for E small since the quantity in question is [L(E) /L, - 11. It is clear that the accuracy of the measurements and the simulations are insufficient to see these differences in the experimental results as presented, especially as these usually lack independent evaluation of F, and D. One might expect the Langevin model for d = 3 to be a more realistic representation of real trajectories especially for the solutions of almost macroscopic particles. Since for this model we have shown that f(e*) does not differ much from unity it may be concluded that the observed fits to (1.4) or (1.5) can just as well be interpreted as a quite reasonable fit to the three-dimensional Langevin model. Clearly, more accurate simulations and experiments are required in order distinguish between method of analysis.
theoretical
models
of particle
motion
in fluids
to
by this
9. Conclusion We believe we have demonstrated convincingly that even for trajectories for which the velocity autocorrelation function is exponential - the simple ideal situation - the form of the finite-fractal function L(E) lL, can show a consider-
J.G.
332
able variety
of forms.
exact because
POWLES
In particular
AND
R.F.
FOWLER
this shows that Toxvaerd’s
he gets the same functional
theory’)
is not
form for L(E) for a given functional
form for &(t).
Acknowledgements JGP thanks the SERC and UKC for providing time and facilities for this work. RFF thanks SERC for a Research Associateship. The computations were carried out on the Physics Department VAX 11/750.
Appendix Calculation of A, in eq. (3.6) for the random walk models We are interested in the limit where I + E,. The probability of the trajectory changing direction by an angle 0 in the distance E is small. The probability of changing direction more than once in E is negligible. In this case the difference between L(F) and L, is just due to the lengths which are ‘lost’ when the trajectory between the two ends of the straight line of length e contain a ‘corner’ as indicated in fig. A.l. The loss when the corner is included is (a + p - e) and there is uniform probability of a taking any value between zero and F. Hence the mean loss, 6, is given by
8=jd++P-t)jjda. (1
(A.1) 0
We have p = (Ycos 0, +
[(Y’(cos’
0, -
ing (A.l) it is prudent to separate then elementary to show that
Fig. A.l.
Illustrating
1) + F’]“~, where the two cases,
the ‘loss of length’
at a change
13,= n - 8. In calculat-
0 < 7r/2 and 8 > 7r/2. It is
in direction
for F small
FRACTAL
MODEL
333
TRAJECTORIES
6=:[(n-O)/sinO--2cosB-11,
7r/25057r,
S = 4 [O/sin 8 - l] ,
05887rl2,
(A.2)
which is continuous
through
0 = 7rl2 as indicated.
Suppose the probability of a change of direction per elementary step is g. The number of steps in L, is L,IE, so that the number of changes of direction is gL,ll,.
Hence,
L(E)+
L, - g6L,/l,
)
i.e.
L(E) > L,
1 1 + g6l1,
’
&ho.
(A.3)
This is the result when only one value of 0 arises, as in the q-model. For the r- and s-models appropriate generalisation of (A.3) is readily derived. Comparing (A.3) and (3.5) with fd =fO we have
f”=(y
g$
(one8).
(A.4)
d
For example, for the q-model we have g = dq and 0 = cos-‘[(d 0 5 rrl2, except for d = 1 when 13= 6. Hence for d=
f” = 1
- 2)ld]
so that
1
and
f” = ; (cos8[(d - 2)ld]d/2(d
- 1)“’ - l}
for d > 1 .
This relation gives the foevalues in table I for the q-model. The fO values for the r- and s-models are calculated similarly table I.
References 1) S. Toxvaerd, Phys. Lett. A 114 (1986) 159. 2) L.F. Richardson, General Systems Yearbook
6 (1961)
139.
and are given in
334
3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
J.G.
POWLES
AND
R.F.
FOWLER
J.G. Powles and N. Quirke, Phys. Rev. Lett. 52 (1984) 1571. D.C. Rapaport. Phys. Rev. Lett. 53 (1984) 1965. J.G. Powles, Phys. Lett. A 107 (1985) 403. D.C. Rapaport, J. Stat. Phys. 40 (1985) 751. S. Toxvaerd, J. Chem. Phys. 82 (1985) 5658. R.K. Kalia, S.W. de Leeuw and P. Vashishta, J. Phys. C 18 (1985) L905. S. Tsurumi and H. Takayasu, Phys. Lett. A 113 (1986) 449. S. Matsuura, S. Tsurumi and I. Nobuhisa, J. Chem. Phys. 84 (1986) 539. J.G. Powles and G. Rickayzen, J. Phys. A 19 (1986) 2793. J.G. Powles. G. Rickayzen and R.F. Fowler, Molec. Phys., submitted for publication H. Takayasu, J. Phys. Sot. Japan 51 (1982) 3057. P. Turq. F. Lantelmc and H.F. Friedman, J. Chem. Phys. 66 (1977) 3093.
FRACTAL PROPERTIES OF MODEL TRAJECTORIES WITH EXPONENTIAL VELOCITY AUTOCORRELATION FUNCTIONS J.G.
POWLES
Physic.y Department.
and R.F.
FOWLER
Univwsify of Kenr. Canterbury,
Received
10 June
UK
19X6
The length, L(E), at scale, F, has been determined for four different models for atomic motion in a fluid, three distinct types of random walk with constant speed and the Langevin model. This has been done for trajectories in enc. two, three and sometimes four dimensions. All these motions have exactly exponential velocity autocorrelation functions. However, the functional form of L(F) is different for the different models, contrary to a recent assertion (S. Toxvaed, Phys. Lett. A 114 (1986) 159). A theory for L(F) is lacking, except for the limits of E small and F very large. Our results can be used as ‘experimental results’ for the testing of such theories which arc quite different from the conventional analysis of molecular motion in fluids. A comparison is made of these model results for L(F) with those for simulated trajectories in realistic models of liquids and for the actual trajectories of suspended particles in solution.
1. Introduction The method of fractal analysis of trajectories proposed by Richardson*) was first applied to the realistic trajectories of particles in fluids by Powles and Quirke’) and there have been several subsequent investigations of trajectories by this method. (refs. 4-12). In Richardson’s analysis the length of the trajectory, L(F), at scale, F, is measured by stepping along the trajectory from the beginning until a point is reached at which the scalar distance from the initial point is F, for thefirst time. This process is repeated from the new initial locations until the distance from the end of the trajectory is less than E. The fraction of E to the end of the trajectory to get to its termination is added. The total of these steps is L(E). Richardson found that many trajectories have a range of e for which
where
N is not in general
0378-4371/87/$03.50 0 (North-Holland Physics
an integer
and is a characteristic
Elsevier Science Publishers Publishing Division)
B.V.
of the trajectory.
FRACTAL
For all physically length,
realistic
319
MODEL TRAJECTORIES
trajectories
for E + 0 we get for L(E) the contour
L,, i.e.
L(E)/L,-+~ We call this
for e-0 the
free-flight
and a-+0. limit.
(1.1) must
be
approximate and not exact as asserted because it violates the limit (1.1). More generally we can define a Richardson coefficient which is a function E, defined by
Incidentally
of
(Y(E) = -d[ln(L(c)
Toxvaerd’s
lL,)]ld[ln(e)]
This may then be called finite-fractal analysis.13) For F very large we get what we call the diffusion result is
L(&) -_+-L,
2dD
1
(u)
&’
E+mJ
i.e. CX(E+~)+
theory’)
.
(1.2)
limit when’*)
the exact
1
(1.3)
D is the diffusion coefficient, and d is the dimensionality of the space in which the trajectory exists. (u) is the mean speed of the particle, which is independent of the length of the trajectory in the limit of an indefinitely long trajectory. Between the two limits, E+ 0 and E+ ~0, and for realistic trajectories, a(e) rises steadily from zero to unity5), see fig. 3. However, as far as we are aware, there is only one special case in which the exact form of L(E) lL, over the whole range of E is known theoretically, although approximations have been proposed which are largely empirical or phenomenological (refs. 13, 5, 1 and 10 - these
are discussed
For a model found that
-L(E) = Lc
described
later). as ‘Brownian
motion’
in one dimension,
Takayasu’“)
1
(1.4)
l+&IEO’
where E,, is stated to be some mean-free-path. For a random walk in one dimension (described Rickayzen”) found that 1 ~L(E) ZZ 1+ E/2&, ’ LC
in section
2) Powles
and
(1.5)
320
J.G.
where
POWLES
AND
R.F.
FOWLER
F,, is defined
as ( u) r,, where T, is the correlation time of the velocity function. (1.5) is consistent with the exact result (1.3).
autocorrelation
It is not clear to us that these two models
are identical
but we suspect
it to be
SO.
The form (1.4), perfectly,
with E,, a disposable
to realistic
but simulated
parameter,
trajectories
fits quite
in liquids
well but far from
in three-dimensional
space’) and to the actual trajectories of bacteriophages and polystyrene spheres in solution’“). It has also been shown that for the trajectory of an atom in a three-dimensional simulated liquid”) the form (1.5) fits the results quite well with the known value of F,, “). Since no theoretical result for L(F) is known, for other than the onedimensional models mentioned above, we have obtained numerical results for L(E) for several simple models of atomic motion in a fluid for other than one dimension of space, and in particular for real three-dimensional space. We compare the results with those for real and/or computed fluids in section --I 8.
2. Random-walk
models
The random walk discussed by Powles and Rickayzen”), which led to eq. (1.5), was a one-dimensional one in which the particles moved at constant speed in each step, to left or right, by a distance 1 in a time 7. The model included persistance of velocity in the sense that if the particle is travelling to the right (or left) in a given step then in the next step it is travelling either to the right (or left) with a probability, p, or to the left (or right) with a probability q, where of course, (p + q) = 1. In the limit, q-+ 0, the velocity autocorrelation function, 4,(t), is an exponential with a correlation time TV-
rl2q
(2.1)
,
thus E,+ 112q, and if we choose F = nl, where II is a positive integer, it can be shown that (1.5) may be written in the simple form
L(n)
1
L
1+ (n - 1)q
21, then
(2.2)
When the dimensionality is greater than one there are more than one simple random walks which are generalisations of the walk described above. In fact in d dimensions there are d such walks as described below.
FRACTAL
3. q-walks
321
MODEL TRAJECTORIES
in any dimension
We call a q-walk
one in which the motion
is as for the walk described
above
but it occurs in every one of the d dimensions independently. It is obvious that the elementary step, in time 7, is now I, = d1’21. It is easy to show that the diffusion
constant,
for q 4 p, has the value
D = 12f2Tq,
(3.1)
independent of d. Moreover, for q-+ 0, c&(t) is exponential and E,-+ 1,/2q since ( vd) = 1,lr. Because
of (1.3)
and (3.1)
with T” as in (2.1),
we have
(3.2) We also still have, ~L(E) + 1 L
of course,
that
for E e E, .
(3.3)
However, we have not been able to obtain an exact expression for L(E) lL,, for any F, by any of the methods we used in ref. 11 for d = 1. Apart from the increased geometrical complexity, we have the difficulty that, even if we restrict E to nl,, the calculation of L(E) involves initial points which are not necessarily at the end points of the elementary steps, 1. Put in another way, the hypercubic symmetry of the random walk does not fit naturally to the hyperspherical symmetry of the scaling length E. However, there is no problem in numerically generating long trajectories for this model and determining L(E) to any desired accuracy for any finite value of E using a sufficiently small but finite value of q. We found that q = 0.01 was adequate. The results expressed as a function of (E/~F,) are then virtually independent of q. An alternative method of computing the value of L(E) is the following. For any one of the d coordinate directions the probability of continuing in the same direction at the end of a step of length 1 is p and if not doing so is q. The mean distance, (s), before a change of direction occurs for the first time is given by np”- 1q=l/q. n==l
(s) I1 = 5
In the limit q+O Poisson distribution)
the probability
distribution
of S, p(s),
becomes
(cf. the
322
J.G.
p(s)-
(q/l)
POWLES
AND
R.F.
FOWLER
(3.4)
exp(-sqil)
This distribution, in each dimension the NAG random-number-generator
independently, and double
is readily precision,
generated using and hence the
trajectory
for any given value of q. In fact a value q of 0.01 gives a trajectory
sufficient
accuracy.
This
method
demands
much
the same
time as the direct method described previously. We have confirmed essentially the same numerical results for L(F/~P,). In view of the two limits.
L(F) 4
(3.2)
1 + (FQE,.)~,(E/2E,,)
where fC,(.z/2&,.)+ that fC, is a smooth
and (3.3).
of
cost in computer
it is convenient
that it gives
to write
(3.5)
’
I for c/~F,.+ x. We find from the computed trajectories function. f,, is almost independent of F for F + 2~, and WC
call this value A,. S,, has the value (6, + 1)/2 for F = 2~, . Of course. for d = 1, f= 1 for all e (cf. (1.5)). In fact we find for all the random-walk models investigated in this paper that ji*, as a function of In(F), is quite closely of tanh (or Fermi-function) form.
f;,(z)-~(l+f,,)+!(l-f,,)tanh[~(z-z,)l.
(3.6)
w
where z = ln,(&/2&,) and D, is a measure of the width of the transition z, is a constant which fixed the centrc of the transition region. Eq. (3.6) may be written in the alternative form f,,(&i&,)= where
in table
x
=
(E/E
i(l+J,)-
I, whe;ein
(3.6’)
:(l-~,)(l-X’)l(l+“z),
)Iz’(‘h, ‘I1 7)1, and F, = 2~,2’l. The values
the values
of 6, for the simulated
region.
of 6,. z, and D,+ are given trajectories
are called f;,,.
Since fo5 < 1, L(E) approaches the value L, more rapidly with decreasing & than in one dimension. Since z, is small the changeover occurs, as already stated, for F - 2~, and takes place roughly in the range for c/2.5, from i to 8. Using these parameters in (3.6) and then in (3.5) reproduces the observed values of L(F) to an accuracy which should be sufficient to test any theory for L(E) which may become available in due course. A theoretical value of A,, called JIC in table I, is obtained in the appendix. The observed and calculated values agree to within the accuracy of the computation. Some readers may feel that this model is not the simplest or most approp-
TABLE
Parameters
d
of
the
I
f-function for models.
the
three
random-walk
f “C
fo,
D,
Parameters for the q-model 1
_
1
2 3 4
0.572 0.466 0.424
0.5708. 0.4584. 0.4184.
cc
_
l/3=0.333..
_ -0.84 -0.38 -0.11
_ 5.40 4.52 3.88
_
_
Parameters for r-model as for q-model
A
0.779 0.696 0.657 -
0.7854 0.7139. 0.6781 [I + (d - l)(ni2 - l)]ld
0.21 0.26 0.10 -
5.63 6.00 5.25 _
_
(r/2-1)=0.5708..
-
_
0.48 0.77
3.52 4.95
-
_
Parameters for the s-model 1 2 3 4
as for q-model as for r-model 0.752 0.695
0.7292, 0.6936. (~12 - 1) = 0.5708.
‘x
d-
1 (1-q) f
(1-r)
r r I (1-s)
S S
I
Fig. 1. Diagrammatic representation of the behaviour of the trajectories at the end of a step for the three walks, q, r and s, in the limit q. r, s-+0 for 1, 2, 3, and 4 dimensions. Continues in the same direction for (1 - q) etc. 323
324
J.G.
riate generalisation
POWLES
AND
R.F.
of the one-dimensional
FOWLER
walk of section
2. For this model,
in
the limit q ---, O,we have at the end of a step a probability (1 - dq) of continuing and a probability of q of deviating by an angle coss’[(d - 2)/d] in the d directions
symmetrically
does however
arranged
have the advantage
around
the forward
direction,
that it is very straightforward
see fig. 1. It to programme
and compute. We therefore
4.
now investigate
two other
related
models.
in any dimension
r-walks
In this model we make the probability in the forward direction equal to [l - (2d ~ l)r] and a probability of r of deviating by z-/2 in the 2(d - 1) symmetrical directions and also r in the reverse direction. This is shown diagrammatically in fig. 1. For d = 1 this is the same as for the q-model. In the limit r+ 0, 4,(t) is exponential with T,./T = 1 i(2dr) so that if the elementary step is again I,, then /,/(2dr)
E,, =
so that,
in particular,
L(F)/L,+~E,/E
and
D = 1’/(2Tdr),
(1.3)
becomes
(cf. (3.2))
for E-+E.
The latter two results have been confirmed numerically. We now calculate L(c)IL, as in section 3 and again find that (3.5) with the form (3.6) gives a good fit with the parameters given in table I. The theoretical values of J;,, i.e. hjC, are obtained in the appendix. The velocity autocorrelation function is exponential in both the q and r models, so that, in particular, (r’(t)) has exactly the same functional form (see fig. 2), namely, (r’(t))
= 2dD{t
- T,( 1 - exp - tin,)}
,
(4.1)
and this has been confirmed explicitly for our trajectories. Nevertheless, the functional form of L(E) IL, is markedly different for the two models for any given dimension except d = 1 and depends on the dimensionality. For the r-model fr, is larger and z, is positive not negative, but D, is very similar in value. We now analyse one further model in this class.
FRACTAL
5. s-walks
MODEL
TRAJECTORIES
325
in any dimension
In this model the probability of continuing in the forward direction is [l -- (2d - l)s]. The probability is s to divert into one of the “C,, directions by an angle cosP’[(d - 2n) ld], n = 1,2, . . . , d, i.e. until the angle is 7~, the reverse
direction
(see fig. 1 and table II. Note,
these are the angles which arise
in the q-model, for q not approaching zero). For s-+0, 4,,(t) is again exponential and r,/~ = 1/(2d~). This model is identical to the q-model for d = 1 and identical to the r-model for d = 2 but the three models are different d 2 3, see fig. 1. The calculation proceeds just as before and the result for d = 3 and 4 is given in table I. In particular the value off;, is larger than for the other two models but is still substantially below unity. We thus again have a different form for L(E) in spite of the fact that the velocity autocorrelation function is again exponential. TABLE
The deviation d
cos-‘[(d
1 2
?r 7Tl2 coscy f ) cosY’($)
3 4
- 2) ld]
angles
II
for the s-model
up to d = 4.
cosC’[(d - 4)/d]
cosC’[(d - 6)/d]
cos-‘[(d
_
_ _
_ _ _
77 cosc’(7r12
f)
77 coscI(-
4)
- 8)/d]
7r
6. The L-model A rather different model which also has an exponential velocity autocorrelation function is the trajectory resulting from the Langevin equation of motion
dp
dt- -
-p/r,
+ R(t)
)
(6.1)
where p is the momentum of the particle, r, is, as usual, the velocity correlation time and R(t) is a random force. This model differs from the previous ones in that the particle speed is not constant, indeed it is chosen to have a Maxwell-Boltzmann distribution. This has been used14) to study the dynamics of ‘Brownian’ particles in solution where the effect of the solvent is modelled by R(t). R(t) should have zero mean and a Gaussian probability distribution. It is convenient to use reduced units (*) as follows:
326
J.G.
t* = t/r
Y
POWLES
p* = pl(mkT)“’
9
so that if x is the position
AND
and
of the particle
R.F. FOWLER
R* = [7vl(mkT)“2]R
,
then
x*= (mkjl;l:lrv x. The trajectories (6.1) b ecomes
in reduced
+ R*(t*)
dp* ldt* = -p:’
In the simulation R,* is the random
units are then the same for all temperatures,
T. Eq.
.
(6.2)
R* is taken
force
to be constant during a time step, At*, and then if in the nth time step, (6.2) has the exact solution, for
At* finite. P*(tz+l)
= RE + [p*(t,T) ~ R,*] exp(-At*)
x*(t,T+,)
= x*(t,*)
(6.3)
and + R,* At* + [p*(t,*)
- Rz][l
- exp(-At*)]
.
(6.4)
Eqs. (6.3) and (6.4) are used to obtain the trajectory, x*(t*). Sufficient accuracy is obtained if At* 50.01. Eq. (6.1) is for the motion in one axis. For d dimensions we use d equations like (6. l), independently, to obtain r*(t*) where r* is the position of the particle at time t”. For a long trajectory the initial momentum value, p(t,), is immaterial. It is simple to show that
*_
7, -
1,
(u:‘)=l
and
D*=l.
These values have been confirmed has eq. (4. l), see fig. 2 for d = 3. It can also be shown that
where
ud is the speed
explicitly
in the d-dimensional
for our computed
case and of course,
trajectories
(uz ‘) = d.
as
FRACTAL
MODEL
327
TRAJECTORIES
Fig. 2. The points are values of (r’) for a computed Langevin trajectory in three-dimensional space. The curve is the theoretical result. eq. (4.1). Note that (r’) is close to the limiting diffusion value 601, on the dashed line, after only about 57,. This may be contrasted with the slow approach to the diffusion limit in the corresponding fractal analysis of fig. 3.
Hence
using
(1.3),
we have
2112drM
L(&*)
1
p
L,-
for &*-cc
(6.5)
F(F)
and of course
we still have
(6.6)
for &*-+O.
Again
defining,
E, = ( u~)T”, we can write
(6.5)
in the form
for E%~E,
By analogy
with (3.5)
this suggests
1 -L(E) = 1+ (E12EL)fd(&12EL) L where
.
(6.5)’
the formulation
’
(6.7)
328
J.G.
POWLES
AND
R.F.
FOWLER
so that f(m) = 1 .
We have obtained
and analysed
trajectories
for d = 1, 2 and 3 using up to 12
million steps which took about 6 hours on our VAX 111750. The plot of ln,[L(&)/L,] versus ln,(c*) for d = 1 is shown in fig. 3. It is clear, by comparison with the curve for f = 1 for all F, that f(e/2e,) is greater than one, see fig. 4a, in stark contrast with the q-model for d = 1. We emphasise again that all these models have 4,(t) exponential. The function f,(e/2&,) is shown for d = 1, 2 and 3 in fig. 4. None of these is of tanh form but for d = 3 we have f,, < 1 as for the previous models. These plots of h,, together with eq. (6.7) are sufficient to reproduce the function L(E) for comparison with any theory which may become available. We found, as might be expected, that the values of L(E) or f(.z) are only independent of At for F larger than about five times ( ud) At so that to get results of reasonable accuracy for F demands trajectories with a very large number of time steps. The plots of fig. 4 are therefore less accurate than for the random-walk models. In particular we have not been able to establish whether a limiting value A, for fd exists for d = 2 or 3 for this model. A rough estimate of the accuracy is given in fig. 4. We note that even for several million steps, i.e. an elapsed time of order 50 0007,, the values of L(E) for F large barely established the known diffusion limit result, (6.5). with slope -1, i.e. (Y = 1. This may be contrasted with the
Fig. 3. The plot of In2[L(~)IL,] versus (L-model). The points are the computed straight line at 45” is the exact ‘diffusion’
In,[ &*I for the one-dimensional Langevin-motion model values. The curve is for the relation (6.7) with f= 1. The result. The dashed curve is for eq. (7.1) with m = 0.904.
FRACTAL
MODEL
TRAJECTORIES
0
I
1.5 -
I
329
6
I
d=2
I
’
b
LN,k’)
Fig. 4. The function f(&/z~~), see eq. (6.7), for the computed values of L(E)/L, for the L-model for 1, 2 and 3 dimensions. Note the smaller scale of ordinate for d = 1. An estimate of the computational error is indicated.
J.G.
330
diffusion
POWLES
AND
limit for (Y’) which is attained
R.F.
FOWLER
to similar
accuracy
in a time of order
57,.
7. The empirical Matsuura
relation
et al.“‘) have suggested
L(F) LC
of Matsuura
et al. the empirical
relation
1 (7.1)
[l + (FI&,,)‘71]“‘n ’
where both E,, and m are disposable parameters. However, in their comparison with their experimental results they found that to the accuracy of their results the choice m = 1, i.e. eq. (1.4), is sufficient. Nevertheless, it is of interest in the present context to test this relation for our models. We find that this formula fits our data better than the onedimensional random walk model (m = l), with appropriate choice of m. For instance, the fit to the data in fig. 3 is overall much improved. but is far from perfect, if WC choose m = 0.004. To show the effect of assuming (7.1) for m # 1 we have calculated the deviation function j’(f(~’ ) where F ’ = E/E,, and the result is given in fig. 5. For m < 1. f‘ is greater
than unity for all E and diverges for e+O, although of course we still have the required limit (1 .I). This behaviour is similar to that of the L-model for d = 1 or 2 (see fig. 4a and b). For m > 1. J’is less than unity for all E and approaches zero for c + 0. This is not consistent with the p, q and r models for which J‘+J;, and A, is non-zero. For the L-model for d = 3 we have
I
I
f kc+1 112
0.9
:::> m=,
/
A
-4 Fig. 5. The flfunction (cf. cq. (3.5)) the value of m indicated.
-2
0
for the empirical
2
4
model
of Matsuura
6 et al.“‘).
eq. (7.1).
for
FRACTAL
not been
able to establish
331
MODEL TRAJECTORIES
a numerical
value
for f,, (see fig. 4c) or even
if it
exists. In spite of these difficulties the formula (7.1) may be useful for the analysis of data for L(E) since its main effect is to modify equations (1.4) or (1.5) for F =Z2~” roughly
as observed
in our models.
8. Comparison
with real trajectories
Tsurami and Tokayasu’) have shown that the simulated trajectories of Powles and Quirke’) for a Lennard-Jones fluid and of Rapaport3) for a hard-sphere fluid are fitted quite well by Takayasu’s formula (1.4) with &Cla disposable parameter. Also Powles”) has shown that his data are fitted quite well by the formula (1.5) with the known value of F,. Matsuura et al.‘O) have shown that eq. (1.4) can be fitted quite well to the measurements of L(E) for the observed trajectories of polystyrene spheres and bacteriophage T4 particles results are in solution with appropriate choice of r,,. All these experimental for d = 3 whereas the theory is for d = 1. The numerical results for the models discussed here for d = 3 differ from eq. (1.5) in that f is not unity for E < 2e, and in particular f0 is of order one-half (see table I and fig. 4~). To determine this difference demands a very high accuracy in the determination of L(F) IL, for E small since the quantity in question is [L(E) /L, - 11. It is clear that the accuracy of the measurements and the simulations are insufficient to see these differences in the experimental results as presented, especially as these usually lack independent evaluation of F, and D. One might expect the Langevin model for d = 3 to be a more realistic representation of real trajectories especially for the solutions of almost macroscopic particles. Since for this model we have shown that f(e*) does not differ much from unity it may be concluded that the observed fits to (1.4) or (1.5) can just as well be interpreted as a quite reasonable fit to the three-dimensional Langevin model. Clearly, more accurate simulations and experiments are required in order distinguish between method of analysis.
theoretical
models
of particle
motion
in fluids
to
by this
9. Conclusion We believe we have demonstrated convincingly that even for trajectories for which the velocity autocorrelation function is exponential - the simple ideal situation - the form of the finite-fractal function L(E) lL, can show a consider-
J.G.
332
able variety
of forms.
exact because
POWLES
In particular
AND
R.F.
FOWLER
this shows that Toxvaerd’s
he gets the same functional
theory’)
is not
form for L(E) for a given functional
form for &(t).
Acknowledgements JGP thanks the SERC and UKC for providing time and facilities for this work. RFF thanks SERC for a Research Associateship. The computations were carried out on the Physics Department VAX 11/750.
Appendix Calculation of A, in eq. (3.6) for the random walk models We are interested in the limit where I + E,. The probability of the trajectory changing direction by an angle 0 in the distance E is small. The probability of changing direction more than once in E is negligible. In this case the difference between L(F) and L, is just due to the lengths which are ‘lost’ when the trajectory between the two ends of the straight line of length e contain a ‘corner’ as indicated in fig. A.l. The loss when the corner is included is (a + p - e) and there is uniform probability of a taking any value between zero and F. Hence the mean loss, 6, is given by
8=jd++P-t)jjda. (1
(A.1) 0
We have p = (Ycos 0, +
[(Y’(cos’
0, -
ing (A.l) it is prudent to separate then elementary to show that
Fig. A.l.
Illustrating
1) + F’]“~, where the two cases,
the ‘loss of length’
at a change
13,= n - 8. In calculat-
0 < 7r/2 and 8 > 7r/2. It is
in direction
for F small
FRACTAL
MODEL
333
TRAJECTORIES
6=:[(n-O)/sinO--2cosB-11,
7r/25057r,
S = 4 [O/sin 8 - l] ,
05887rl2,
(A.2)
which is continuous
through
0 = 7rl2 as indicated.
Suppose the probability of a change of direction per elementary step is g. The number of steps in L, is L,IE, so that the number of changes of direction is gL,ll,.
Hence,
L(E)+
L, - g6L,/l,
)
i.e.
L(E) > L,
1 1 + g6l1,
’
&ho.
(A.3)
This is the result when only one value of 0 arises, as in the q-model. For the r- and s-models appropriate generalisation of (A.3) is readily derived. Comparing (A.3) and (3.5) with fd =fO we have
f”=(y
g$
(one8).
(A.4)
d
For example, for the q-model we have g = dq and 0 = cos-‘[(d 0 5 rrl2, except for d = 1 when 13= 6. Hence for d=
f” = 1
- 2)ld]
so that
1
and
f” = ; (cos8[(d - 2)ld]d/2(d
- 1)“’ - l}
for d > 1 .
This relation gives the foevalues in table I for the q-model. The fO values for the r- and s-models are calculated similarly table I.
References 1) S. Toxvaerd, Phys. Lett. A 114 (1986) 159. 2) L.F. Richardson, General Systems Yearbook
6 (1961)
139.
and are given in
334
3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
J.G.
POWLES
AND
R.F.
FOWLER
J.G. Powles and N. Quirke, Phys. Rev. Lett. 52 (1984) 1571. D.C. Rapaport. Phys. Rev. Lett. 53 (1984) 1965. J.G. Powles, Phys. Lett. A 107 (1985) 403. D.C. Rapaport, J. Stat. Phys. 40 (1985) 751. S. Toxvaerd, J. Chem. Phys. 82 (1985) 5658. R.K. Kalia, S.W. de Leeuw and P. Vashishta, J. Phys. C 18 (1985) L905. S. Tsurumi and H. Takayasu, Phys. Lett. A 113 (1986) 449. S. Matsuura, S. Tsurumi and I. Nobuhisa, J. Chem. Phys. 84 (1986) 539. J.G. Powles and G. Rickayzen, J. Phys. A 19 (1986) 2793. J.G. Powles. G. Rickayzen and R.F. Fowler, Molec. Phys., submitted for publication H. Takayasu, J. Phys. Sot. Japan 51 (1982) 3057. P. Turq. F. Lantelmc and H.F. Friedman, J. Chem. Phys. 66 (1977) 3093.