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Cite this paper as:
C. Vamoş, N. Suciu, M. Peculea, Numerical modelling of the one-dimensional diffusion by random walkers, Rev. Anal. Numér. Théor. Approx., 26(1), 237-247, 1997.
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REVUE D’ANALYSE NUM
´
ERIQUE ET DE TH
´
EORIE DE L’APPROXIMATION
Rev. Anal. Num´er. Th´eor. Approx., vol. 26 (1997) no. 1-2, pp. 237–247
ictp.acad.ro/jnaat
NUMERICAL MODELLING OF THE ONE-DIMENSIONAL
DIFFUSION BY RANDOM WALKERS
C. VAMOS¸, N. SUCIU, M. PECULEA
1. THE MATHEMATICAL MODELLING OF THE DIFFUSION
The distribution of the particles in a domain Ω ⊂ R
3
at a moment t ∈ R is
described by the continuous field of the concentration c :Ω × R → R
+
. The
differentiability properties of c are different for different processes of diffusion,
but c usually is a function of C
2
class in the definition domain. The number
of particles in a subdomain A ⊂ Ω at the moment t is given by
(1.1) n(A, t)=
A
c(r,t)dr.
The concentration verifies a local balance equation [11]
(1.2) ∂
t
c + ∇j =g,
where the vectorial field j :Ω × R → R
3
, is the particles flux density and g :
Ω×R → R is the source term. The balance equation (1.2) should be completed
by a constitutive law which should connect the flux j and the concentration c. A
simple constitutive law is Fick’s law, valid for small gradients of concentration
(1.3) j = -
D∇c,
where
D is the tensor of the diffusion coefficients.
In the case of the homogeneous and one-dimensional diffusion the tensor
D is reduced to a real parameter D and if g = 0, (1.2) becomes the one-
dimensional diffusion equation
(1.4) ∂
t
c - D∂
2
x
c =0.
Depending on the initial and boundary conditions, the equation (1.4) has
different types of solutions [2, 3]. For the initial condition c(x, 0) = Nδ(x),
1991 AMS Subject Classification: 65C20.
238 C. Vamo¸s, N. Suciu, M. Peculea 2
where δ(x) is Dirac’s function, and the conditions at infinity lim
|x|→∞
c(x, t) = 0,
lim
|x|→∞
∂
x
c(x, t)=0, the solution of the equation (1.4) is the concentration of
N independent particles in Brownian motion
(1.5) c(x, t)=
N
√
4πDt
e
-
x
2
4Dt
, x ∈ (-∞, ∞),t> 0.
The rigorous definition of the concentration field is a problem which has
not a solution yet. The number of particles in a set A can not be modelled as
a absolutely continuous set function. Therefore, the Radon-Nikodym theorem
[8] does not guarantee the existence of a corresponding density, as in the case of
other fields in the continuum mechanics [4]. The existence of the concentration
field c is postulated and the corresponding global quantity n(A, t), is defined
by (1.1) [11].
In the statistical mechanics, the probability that a particle should exist in
an element of infinitesimal volume dr centered on r ∈ Ω at the moment t is
given by p(r,t)dr, where p(r,t) is the one-particle probability distribution [7].
The relation to the concentration is given by
(1.6) c(r,t)= Np(r,t),
where N is the total number of particles. In this way, the concentration was
substituted for another continuous field and its existence was postulated as
well. The definition of the probability distribution in the statistical mechanics
is so far an open problem [1]. For a diffusion process, p(r,t) satisfies the
Fokker-Planck equation which according to (1.6), is equivalent to the diffusion
equation.
2. THE MICROSCOPIC DEFINITION OF THE CONCENTRATION
To discuss the problems occurring to the definition of the concentration,
we present the method usually used in thermodynamics [10]. Consider N
molecules in a volume V . The concentration at the point of the position
vector r at the moment t is defined as the number of particles per unit volume
in a domain of volume V <V centered at this point. The domain should be
spherical, otherwise the concentration would depend not only on the position
but also on the orientation of the domain. We denote by S (r,a) the sphere
of center r and radius a. Then the concentration is defined as the function
c : R
3
× R → R
+
given by
(2.1) c(r,t)=
1
V
N
i=1
H
+
(a
2
- (r
i
(t) - r)
2
),
where r
i
(t), t ∈ R, is the position vector of the molecule i ≤ N at the moment
t. The left continuous Heaviside function H
+
(a
2
- (r
i
(t) - r)
2
) is equal to 1 if
3 Numerical modelling of the one-dimensional diffusion 239
the molecule i is inside the sphere S (r,a) and vanishes otherwise. The function
defined by (2.1) for given a and r
i
, is a finite linear combination of Heaviside
functions having null derivatives except when their argument vanishes and the
derivative does not exist. So (2.1) defines a step function which can not satisfy
a partial derivative equation of the diffusion equation type.
In thermodynamics one considers that for a large enough N the function
c(r,t) given by (2.1) is well approximated by a continuous field. To obtain the
condition that N should satisfy we analyze the simple case of the thermody-
namical equilibrium state in absence of the exterior fields. Then the molecules
are uniformly distributed in the V volume and, taken as a continuous field,
the concentration has a constant value at any point in the volume and at any
moment, c
o
(r,t)= N/V . If we want to verify this equilibrium distribution
by definition (2.1), we count the number of molecules n at the moment t in
the sphere S (r,a). It is obvious that the result is affected by fluctuations and
the measured concentration c = n/V differs from c
o
. In [10], Section 114, one
shows that n satisfies a Poisson repartition with dispersion σ =
√
¯ n, where
n = N V /V is the mean number of molecules in V . So, the dispersion of the
concentration fluctuation is equal to σ
c
=
N/V V. If N is large enough,
Δc = c - c
o
has a normal repartition of zero average and dispersion σ
c
. Ac-
cording to the ”three sigma” rule of excluding rough errors [13], we impose
that the relative error should be smaller than a value ε fixed with a confidence
level of 0.997 and we obtain
(2.2) 3σ
c
≤ εc
o
=⇒
V
V
≥
9
ε
2
N
.
For a large enough N , this formula gives the minimum volume V (or the
minimum radius a) necessary to measure the concentration with the precision
ε, i.e. the space scale for which the measured concentration behaves like
a continuous field with approximation ε. If (2.2) is satisfied, we can write
c(r,t) ∼ c
o
(r,t)+ O(ε), for ε → 0.
The classical definition of the concentration is not applicable when N or
V is too small. To exemplify, we consider the case when there is a single
molecule in the volume V . Then c(r,t)= V
-1
for r ∈ S (r
1
(t),a), and in the
rest c vanishes. Therefore the concentration is completely different from the
equilibrium concentration which is as well c
o
(r,t)=1/V in the entire volume
V . These difficulties occur because in definition (2.1) one implicitly supposes
an instantaneous measurement of the molecules number in the volume V . The
actual measurement has a duration which defines the temporal scale as the
volume V (or the radius a) defines space scale. If we denote by (t - τ,t + τ )
the averaging interval, we define the concentration by
(2.3) c(r,t)=
1
2τ V
N
i=1
t+τ
t-τ
H
+
(a
2
- (r
i
(t
) - r)
2
)dt
.
240 C. Vamo¸s, N. Suciu, M. Peculea 4
In [14, 15] one proves that the function defined by (2.3) has a. e. continuous
first order partial derivatives. That is, the temporal averaging transforms
the step function (2.1) into a continuous field even if the discontinuities are
conserved in the first order partial derivatives.
The new definition (2.3) is meaningful even when V or N is very small and
the classical definition (2.1) can not be applied. Consider again the thermo-
dynamical equilibrium state with an uniform distribution of the molecules in
the volume V . The number of molecules in V is measured at each moment
over the interval (t - τ,t + τ ) and then it is averaged. We attach to this
continuous temporal average a discrete one, to which we can apply the same
Poisson distribution. We denote by t the mean time interval over which
the molecule remains inside the volume V . Consider the averaging interval
(t - τ,t + τ ) divided into 2τ/t subintervals of Δt length and suppose that
the existence of the molecule in volume V within an subinterval t is inde-
pendent from its existence in the same volume within another subinterval t.
Then the concentration fluctuations (2.3) for N molecules over a time interval
of 2τ length, are equivalent to the fluctuations of 2Nτ/t molecules in a t
interval. Therefore instead of formula (2.2) we have
(2.4) V τ ≥
9t
2ε
2
c
o
.
This formula expresses the relation between the space scale (V ) and the tempo-
ral one (τ ), necessary to obtain a continuous description of the concentration
with a precision ε. Unlike (2.2), the relation (2.4) is valid for any N . The
increase of the temporal scale can compensate the decrease of the space one.
3. THE NUMERICAL ALGORITHM
The random walk hypothesis is a simple form of microscopic evolution law of
the form used in molecular dynamics [5, 9]. Consider a one-dimensional space
lattice {x
k
= kδx |-m ≤ k ≤ m}, where δx is the space step of the lattice.
On the lattice there are N noninteracting particles which move according to
the random walk law. If the i-th particle is at the site k at the moment t, then
at the end of a time step δt the particle will be either at the site k - 1, or at
k + 1, with probabilities equal to 1/2. So in the time interval (t, t + δt) the k
particle moves with -δx/δt velocity, respectively δx/δt, to neighboring site.
We denote by n
k
the number of particles at the site k at the moment t and
by n
k
(respectively n
k
) the number of particles which move at the moment
t + δt to the site k - 1 (respectively k + 1). Then the number of particles at
the site k at the moment t + δt can be written
(3.1) n
k
(t + δt)= n
k+1
+ n
k-1
.
It is well known that a diffusion process can always be approximated by a
jump process [6, 16]. Consider a sufficiently smooth concentration field c(x, t)
5 Numerical modelling of the one-dimensional diffusion 241
and n
k
= c(x, t)δx. When the condition
(3.2) D =
δx
2
2δt
,
is satisfied then, for a lattice of infinite extent, in the limit δx → 0 and δt → 0
the number of particles given by (3.1) approaches the solution of the diffusion
equation (1.4) multiplied by δx. This model is a one-dimensional multiple
random walkers cellular automaton [12].
For the macroscopic description of this one-dimensional corpuscular system
we use the method based on the time-space coarse grained averaging [14, 15].
We consider N particles moving on the real line during a temporal interval
[0,T ] according to the random walker rule given above. We also assume that
each particle can be introduced or expelled from the lattice. We denote by t
+
i
the moment when it appears and by t
-
i
the moment when it disappears. So the
position of the i-th particle is given by the function x
i
:[t
+
i
,t
-
i
] → [x
-m
,x
m
] ⊂
R. We define the coarse grained average 1 (x, t) of the particles number at
the point (x, t) ∈ R×(τ,T - τ ), for the real parameters a and τ < T/2, by the
relation (2.3). Written for the one-dimensional case, it becomes
(3.3) 1 (x, t)=
1
4τa
N
*
i=1
t+τ
t-τ
H
+
(a-| x
i
(t
) - x |)dt
.
This function has first order a.e. continuous partial derivatives and it satisfies
a balance equation of the form (1.2) [14, 15].
4. THE NUMERICAL SIMULATION OF THE ONE-DIMENSIONAL NONSTATIONARY
DIFFUSION
To simulate the diffusion process we apply the coarse grained averaging to
the microscopic motion of the system described by the one-dimensional cellular
automaton. The field 1 can be obtained as a sum of the contributions 1
i
of
the N particles contained in the lattice over the temporal averaging interval
1 =
N
i=1
1
i
.
The average of the concentration field is obtained as the average of 1 over all
the possible evolutions of the particles from the origin to one of the extremities
of the lattice. We obviously have 1 = N 1
i
. For the dispersion we have
σ =
√
Nσ
i
and the condition 3σ ≤ ε 1 (the relative error of 1 with respect
to 1 be ε with a confidence level of 0.997) allows to determine the particles
number needed to obtain this precision
(4.1) 3
√
Nσ
i
≤ εN 1
i
=⇒ N ≥
3 σ
i
ε 1
i
2
.
242 C. Vamo¸s, N. Suciu, M. Peculea 6
We consider that from the random walker algorithm a discrete description
of a diffusion process is obtained, i.e. one knows the particles number n
k
(t)
at each of the lattice site and at the multiples of the time step. Since the
numerical scheme assumes linear variations during the time and space steps
and the averaging over smaller intervals can not bring further information, the
minimum values for τ and a are those given by the time and space discretiza-
tion (δt/2) and (δx/2). Using N
0
particles moving according to the initial and
boundary conditions of the studied diffusion process, the one-particle averages
1
i
(x, t) and σ
i
(x, t) are calculated. Then (4.1) provides the minimum num-
ber of particles N needed to obtain, by numerical modelling, a relative error
of the concentration equal to ε at a confidence level of 0.997. We model the
diffusion using N particles and we calculate 1(x, t). The larger the values
of τ and a are, the smaller the number of particles N. But if τ and a become
too large it is possible that 1(x, t) should average to strongly the motion of
particles and to lose useful information on the concentration evolution.
We consider that at the initial moment there are N particles located in
origin, and in the rest all the sites of the lattice are empty. This diffusion pro-
cess is nonstationary. Three distinctive periods in the concentration evolution
can be identified. The first one lasts from the initial moment until the first
particle reach the extremities of the lattice and it correspond to the Brownian
motion (1.5). The second period is a transition one and it is characterized
by the appearance of a particles flux at the ends of the lattice. In the third
period the evacuation of particles continues but by a nonstationary diffusion,
in which the leaving particles flux is proportional to the number of particles
left in the lattice.
The initial distribution of the particles is singular. Besides, due to the
random walker law, at the odd moments, the particles will be only at the odd
sites of the lattice and at the even moments only at the even sites . So, the
distribution of particles is very discontinuous both in space and time. In the
following we show that those discontinuities are completely averaged if τ ≥ δt
and not τ ≥ δt/2, i.e., if the temporal averaging is made over two time steps.
For the space averaging, we shall use only the minimum value a = δx/2 since
the space resolution is small (m = 10) and a further space averaging would
mean a larger decrease of the space resolution.
First we analyze the particles distribution at the moment t =9δt, i.e. before
the first particles leave the lattice. We use 1000 particles to calculate 1
i
(x, t)
and σ
i
(x, t) at the sites of the lattice and at the half distance between two
neighboring sites x = kδx/2, 0 ≤ k ≤ 18. Due to the symmetry of the
problem we consider only the positive side of the lattice, including the origin.
The last site x
10
= 10 δx is used only to formulate the boundary condition
(4.2) n
m
= n
-m
=0,
expressing the fact that the particles which reach the extremities are removed
out.
7 Numerical modelling of the one-dimensional diffusion 243
Table 4 presents the results obtained by means of the relation (4.1) for two
values of the averaging interval. For the minimum value τ = δt, the particles
number necessary to obtain the concentration relative error of 0.01 varies by
two orders of magnitude between the origin and the extremity of the lattice.
The particles number necessary at the points between the sites of the lattice is
roughly twice than at the sites. One also notices an important decrease of N if
the temporal interval of averaging increases. To calculate 1 and to compare
the probability density obtained by numerical simulation with the theoretical
one we used 100000 particles. In this way, about the origin, a relative error of
roughly 0.01 was provided, and at the extremities of the lattice it decreased
according to relation (4.1) down to 0.10 for τ = δt. In Table 1 we present the
values calculated by space-time coarse-grained averaging for the probability
distribution in comparison with the theoretical one of the Brownian motion
(1.5). The relative error is in a good concordance with that estimated by
relation (4.1). Thus, the concentration obtained by a time averaging over a
larger interval, τ =4δt, is closer to the theoretical value than those obtained
for a smaller τ .
Table 1. The particles number N needed to obtain a relative error of the concentration
of ε =0.01, the probability density obtained by the simulation of diffusion using 100 000
particles and the theoretical probability density of the Brownian motion, at the moment
t =9δt.
x N (×10
5
) N (×10
5
) 1/N 1/N
e
-
x
2
4Dt
√
4πDt
(τ = δt) (τ =4δt) (τ = δt) (τ =4δt)
0.00 1.8 0.42 1.30 1.32 1.33
0.05 3.7 0.83 1.30 1.32 1.31
0.10 1.9 0.43 1.23 1.25 1.26
0.15 3.8 0.90 1.16 1.19 1.17
0.20 2.0 0.51 1.06 1.06 1.06
0.25 4.4 1.2 0.953 0.933 0.940
0.30 2.6 0.68 0.819 0.803 0.806
0.35 6.6 1.6 0.685 0.674 0.673
0.40 3.5 0.95 0.565 0.549 0.547
0.45 7.6 2.4 0.445 0.424 0.432
0.50 5.4 1.5 0.348 0.331 0.332
0.55 19 4.2 0.251 0.239 0.248
0.60 12 2.9 0.188 0.181 0.180
0.65 32 9.3 0.125 0.124 0.127
0.70 21 6.3 0.0921 0.0892 0.0874
0.75 64 19 0.0596 0.0539 0.0584
0.80 56 14 0.0379 0.0390 0.0380
0.85 450 47 0.0162 0.0241 0.0240
0.90 220 36 0.0106 0.0144 0.0148
244 C. Vamo¸s, N. Suciu, M. Peculea 8
In Table 4 we analyze the third period of evolution we considered the mo-
ment t = 100 δt, when 2/3 of the initial number of particles were eliminated
from the lattice. The particles number needed for a relative error of 0.01 was
calculated for the same values of the averaging interval, using 1000 particles
too.
Table 2. The particles number N needed to obtain a relative error of the concentration of
ε =0.01 and the probability density obtained by the simulation of diffusion using 100 000
particles, at the moment t = 100δt.
x N (×10
5
) N (×10
5
) 1/N 1/N
(τ = δt) (τ =4δt) (τ = δt) (τ =4δt)
0.00 7.8 1.9 0.798 0.799
0.05 17 4.0 0.803 0.798
0.10 7.6 2.0 0.781 0.781
0.15 14 3.9 0.758 0.765
0.20 8.0 2.0 0.739 0.754
0.25 19 4.1 0.719 0.743
0.30 8.3 2.1 0.702 0.710
0.35 15 4.4 0.685 0.678
0.40 9.2 2.2 0.638 0.638
0.45 24 4.5 0.591 0.598
0.50 12 2.8 0.557 0.552
0.55 25 7.1 0.524 0.507
0.60 14 3.8 0.466 0.459
0.65 34 8.0 0.409 0.412
0.70 18 4.6 0.361 0.358
0.75 37 11 0.312 0.304
0.80 24 6.1 0.247 0.247
0.85 64 13 0.183 0.190
0.90 45 11 0.125 0.126
An interesting problem occurring for the third period of the concentration
evolution is connected with the time needed for a certain fraction of the initial
number of particles to leave the lattice. At the limit the magnitude order of
the evacuation time of a volume by diffusion can calculated. At a time step,
the average number of particles leaving the lattice is equal to half the number
of particles in the last but one sites x
-m+1
and x
m-1
. But on average in these
sites we have the same number of particles, so on average at each time step
1(x
m-1
.t) · δx particles leave the lattice. To study the variation in time of
the number of particles in the lattice we estimate 1 for x
m-1
and we act the
same way as we did in the other two cases. As shown above, at the extremity
of lattice a larger statistic is needed. Therefore we used 10 000 particles to
establish the number of particles needed to obtain the concentration with a
9 Numerical modelling of the one-dimensional diffusion 245
relative error of 0.01 and a time averaging of τ = 10 δt. The results are
presented in the first column of the Table 3 for several moments. To obtain
this precision we used 1 000 000 particles. The values obtained are given
in Table 3 divided by the total number of particles from the lattice and are
compared with the values established without any averaging. The probability
distribution at the extremities of the lattice is constant in time and equal to
p =0.125. Thus, after n time steps, the number of particles left in the lattice
is equal to a fraction (1 - p · δx)
n
of the initial number of particles. The larger
the value of n, the more precise this formula is, because the influence of the
initial concentration is smaller. If the initial number of particles is N
0
, then
to get a single particle in the lattice. we should have
(1 - p · δx)
n
=
1
N
0
so there are needed
(4.3) n = -
ln N
0
ln(1 - p · δx)
time steps. For N
0
= 10
6
particles. we get n ∼ 1000, in concordance with the
numerical simulations.
Table 3. The particles number N needed to obtain a relative error of ε =0.01 and the
probability density at the last but one site, k = m - 1, of the lattice obtained by the
simulation of diffusion using 1 000 000 particles, by means of the coarse grained average and
directly from the number of particles.
t N (×10
6
) 1/N
n
m-1
N
/δx
(τ = 10δt) (τ = 10δt)
100 0.49 0.125 0.126
150 0.87 0.125 0.121
200 1.5 0.126 0.126
250 2.5 0.124 0.126
300 6.0 0.124 0.127
350 7.5 0.125 0.119
400 15 0.127 0.115
450 24 0.122 0.125
500 113 0.124 0.134
On the basis of (4.3) we can estimate the evacuation time by diffusion of a
number of particles N
0
from a layer of 2L thickness. We emphasize that the
discussion is valid only for the boundary condition (4.2), i.e., if the particles
leaving the layer are instantaneously removed from the neighborhood of the
layer surface. From (3.2) it follows that the magnitude of the time step for a
246 C. Vamo¸s, N. Suciu, M. Peculea 10
given diffusion coefficient is δt = δx
2
/(2D). The diffusion equation (1.4) for
the layer of 2L thickness can be reduced to the space interval (-1, 1) if the
transformation x = Lx
is done and the diffusion coefficient becomes D/L
2
.
Then, from (4.3) it follows that the mean time needed to eliminate the N
0
particles is
(4.4) Δt = nδt =0.4
L
2
D
ln N
0
.
5. CONCLUSIONS
In this paper we have described a new numerical algorithm for diffusion
simulation. It is based on the evolution of a set of fictitious particles shifting
on a line according to the random walk law. This is a model of one-dimensional
cellular automaton. The continuous macroscopical interpretation of the results
is made in the molecular dynamics manner by means of the coarse grained
space-time averages.
For a non-stationary diffusion we have established the space-time scale
needed to obtain a continuous macroscopical description with a given pre-
cision. We have also shown that if the particles reaching the extremity of the
space lattice are immediately eliminated from the neighborhood of the lattice,
then the number of particles in the lattice exponentially decreases.
Recently [12] a model of cellular automaton has been proposed. Our results
are comparable in the case of the small concentration when the effect of the
interaction decreases. The main distinction between these two approaches is
the coarse grained averaging. This has as a result the decrease of the needed
number of particles and the computing time.
REFERENCES
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Received August 10, 1996 C.Vamo¸s and N. Suciu, “Tiberiu Popoviciu” Institute
of Numerical Analysis, P.O. Box 68, 3400 Cluj-Napoca 1,
Romˆ ania
M. Peculea, Romanian Academy, Calea Victoriei 125,
71102 Bucuresti, Romˆ ania