5 Diffusion, Drift and Reaction
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2.5 Diﬀusion, Drift and Reaction
59
particle location is x = mh at time t = N τ. From the total probability formula we
have:
p (x, t + τ ) = p0 p (x − h, t) + q0 p (x + h, t) ,
(2.88)
with the usual initial conditions
p (0, 0) = 1 and p (x, 0) = 0 if x = 0.
As in the symmetric case, we want to examine what happens when we pass to the
limit for h → 0, τ → 0. From Taylor formula, we have
p (x, t + τ ) = p (x, t) + pt (x, t) τ + o (τ ) ,
1
p (x ± h, t) = p (x, t) ± px (x, t) h + pxx (x, t) h2 + o h2 .
2
Substituting into (2.88), we get
pt τ + o (τ ) =
1
pxxh2 + (q0 − p0 ) hpx + o h2 .
2
(2.89)
A new term appears: (q0 − p0 ) hpx . Dividing by τ, we obtain
pt + o (1) =
(q0 − p0 ) h
1 h2
pxx +
px + o
2 τ
τ
h2
τ
.
(2.90)
Again, here is the crucial point. If we let h, τ → 0, we realize that the assumption
h2
= 2D
τ
(2.91)
alone is not suﬃcient anymore to get something nontrivial from (2.90): indeed, if
we keep p0 and q0 constant, we have
(q0 − p0 ) h
→∞
τ
and from (2.90) we get a contradiction. What else do we have to require? Writing
(q0 − p0 ) h
(q0 − p0 ) h2
=
τ
h
τ
we see we must require, in addition to (2.91), that
q 0 − p0
→β
h
(2.92)
with β ﬁnite. Notice that, since q0 + p0 = 1, (2.92) is equivalent to
p0 =
1 β
− h + o (h)
2
2
and q0 =
1 β
+ h + o (h) ,
2 2
(2.93)
that could be interpreted as a symmetry of the motion at a microscopic scale.
60
2 Diﬀusion
With (2.92) at hand, we have
(q0 − p0 ) h2
→ 2Dβ ≡ b
h
τ
and (2.90) becomes in the limit,
pt = Dpxx + bpx .
(2.94)
We already know that Dpxx models a diﬀusion phenomenon. Let us unmask the
term bpx, by ﬁrst examining the dimensions of b. Since q0 − p0 is dimensionless,
being a diﬀerence of probabilities, the dimensions of b are those of h/τ , namely of
a velocity.
Thus the coeﬃcient b codiﬁes the tendency of the limiting continuous motion,
to move towards a privileged direction with speed |b|: to the right if b < 0, to the
left if b > 0. In other words, there exists a current of intensity |b| , driving the
particle. The random walk has become a diﬀusion process with drift.
The last point of view calls for an analogy with the diﬀusion of a substance
transported along a channel.
2.5.2 Pollution in a channel
In this section we examine a simple convection-diﬀusion model of a pollutant on
the surface of a narrow channel. A water stream of constant speed v transports the
pollutant along the positive direction of the x axis. We can neglect the depth of
the water (thinking to a ﬂoating pollutant) and the transverse dimension (thinking
of a very narrow channel).
Our purpose is to derive a mathematical model capable of describing the evolution of the concentration31 c = c (x, t) of the pollutant. Accordingly, the integral
x+Δx
c (y, t) dy
(2.95)
x
gives the mass inside the interval (x, x + Δx) at time t (Fig. 2.10). In the present
case there are neither sources nor sinks of pollutant, therefore to construct a model
we use the law of mass conservation: the growth rate of the mass contained in
an interval (x, x + Δx) equals the net mass ﬂux into (x, x + Δx) through the end
points.
From (2.95), the growth rate of the mass contained in an interval (x, x + Δx)
is given by 32
x+Δx
d x+Δx
c (y, t) dy =
ct (y, t) dy.
(2.96)
dt x
x
31
32
[c] = [mass] × [length]−1 .
Assuming we can take the derivative inside the integral.
2.5 Diﬀusion, Drift and Reaction
61
Fig. 2.10 Pollution in a narrow channel
Denote by q = q (x, t) the mass ﬂux33 entering the interval (x, x + Δx), through
the point x at time t. The net mass ﬂux into (x, x + Δx) through the end points
is
q (x, t) − q (x + Δx, t) .
(2.97)
Equating (2.96) and (2.97), the law of mass conservation reads
x+Δx
ct (y, t) dy = q (x, t) − q (x + Δx, t) .
x
Dividing by Δx and letting Δx → 0, we ﬁnd the basic law
ct = −qx .
(2.98)
At this point we have to decide which kind of mass ﬂux we are dealing with. In
other words, we need a constitutive relation for q. There are several possibilities,
for instance:
a) Convection. The ﬂux is determined by the water stream only. This case corresponds to a bulk of pollutant that is driven by the stream, without deformation
or expansion. Translating into mathematical terms we ﬁnd
q (x, t) = vc (x, t)
where, we recall, v denotes the stream speed.
b) Diﬀusion. The pollutant expands from higher to lower concentration regions.
We have seen something like that in heat conduction, where, according to the
Fourier law, the heat ﬂux is proportional and opposite to the temperature gradient. Here we can adopt a similar law, that, in this setting, is known as the
Fick’s law and reads
q (x, t) = −Dcx (x, t) ,
(2.99)
33
[q] = [mass] × [time]−1.
62
2 Diﬀusion
where the constant D depends on the polluting and has the usual dimensions
2
−1
([D] = [length] × [time] ).
In our case, convection and diﬀusion are both present and therefore we superpose
the two eﬀects, by writing
q (x, t) = vc (x, t) − Dcx (x, t) .
From (2.98) we deduce
ct = Dcxx − vcx
(2.100)
which constitutes our mathematical model and turns out to be identical to (10.22).
Since D and v are constant, it is easy to determine the evolution of a mass Q
of pollutant, initially located at the origin (say). Its concentration is the solution
of (2.100) with initial condition
c (x, 0) = Qδ (x)
where δ is the Dirac measure at the origin. To ﬁnd an explicit formula, we can get
rid of the drift term −vcx by setting
w (x, t) = c (x, t) ehx+kt
with h, k to be chosen suitably. We have:
wt = (ct + kc) ehx+kt
wx = (cx + hc) ehx+kt ,
wxx = cxx + 2hcx + h2 c ehx+kt .
Using the equation ct = Duxx − vcx , we can write
wt − Dwxx = ehx+kt [ct − Dcxx − 2Dhcx + (k − Dh2 )c] =
= ehx+kt [(−v − 2Dh)cx + (k − Dh2 )c].
Thus if we choose
h=−
v
2D
k = Dh2 =
and
v2
,
4D
w is a solution of the diﬀusion equation wt − Dwxx = 0, with the initial condition
w (x, 0) = c (x, 0) e− 2D x = Qδ (x) e− 2D x .
v
v
In Chap. 7 we show that, in a suitable sense,
δ (x) e− 2D x = δ (x) ,
v
so that w (x, t) = QΓD (x, t) and ﬁnally
c (x, t) = Qe 2D (x− 2 t) ΓD (x, t) .
v
v
(2.101)
2.5 Diﬀusion, Drift and Reaction
63
The concentration c is thus given by the fundamental solution ΓD , “carried” by
v
the travelling wave exp 2D
x − v2 t , in motion to the right with speed v/2.
In realistic situations, the pollutant undergoes some sort of decay, for instance
by biological decomposition. The resulting equation for the concentration becomes
ct = Dcxx − vcx − γc
where γ is a rate of decay34 . We deal with this case in the next section via a
suitable variant of our random walk.
2.5.3 Random walk with drift and reaction
We go back to our 1− dimensional random walk, assuming that the particle loses
mass at the constant rate γ > 0. This means that in an interval of time from t to
t + τ a percentage of mass
Q (x, t) = τ γp (x, t)
disappears. The diﬀerence equation (2.88) for p becomes
p (x, t + τ ) = p0 [p (x − h, t) − Q (x − h, t)] + q0 [p (x + h, t) − Q (x + h, t)].
We have:
p0 Q (x − h, t) + q0 Q (x + h, t) = Q (x, t) + (q0 − p0 )hQx (x, t) + . . .
= τ γp (x, t) + O (τ h) ,
where the symbol “O (k)” (“big O of k”) denotes a quantity such that O (k) /k
remains bounded as k → 0.
Thus, eq. (2.89) modiﬁes into
pt τ + o (τ ) =
1
pxx h2 + (q0 − p0 )hpx − τ γp + O (τ h) + o h2 .
2
Dividing by τ , letting h, τ → 0 and assuming
h2
= 2D,
τ
q 0 − p0
→ β,
h
we get
pt = Dpxx + bpx − γp
(b = 2Dβ).
(2.102)
The term −γp appears in (2.102) as a decaying term. On the other hand, as we
will see in the next subsection, γ could be negative, meaning that this time we
have a creation of mass at the rate |γ|. For this reason the last term is generically
called a reaction term and (2.102) is a diﬀusion equation with drift and reaction.
34
[γ] = [time]−1 .
64
2 Diﬀusion
Going back to equation (2.102), it is useful to look separately at the eﬀect of
the three terms in its right hand side.
• pt = Dpxx models pure diﬀusion. The typical eﬀects are spreading and smoothing, as shown by the typical behavior of the fundamental solution ΓD .
• pt = bpx is a pure transport equation, that we will consider in detail in Chap.
4. The solutions are travelling waves of the form g (x + bt).
• pt = −γp models pure reaction. The solutions are multiples of e−γt , exponentially decaying (increasing) if γ > 0 (γ < 0).
So far we have given a probabilistic interpretation for a motion in all R, where
no boundary condition is present. The Problems 2.11 and 2.12 give a probabilistic interpretation of the Neumann and Dirichlet condition in terms of reﬂecting
absorbing boundaries, respectively.
2.5.4 Critical dimension in a simple population dynamics
When −γ = a > 0 in (2.102), a competition between reaction and diﬀusion occurs.
We examine this eﬀect on the following simple population dynamics problem:
⎧
⎪
0 < x < L, t > 0
⎨ ut − Duxx = au
(2.103)
u (0, t) = u (L, t) = 0
t>0
⎪
⎩
u (x, 0) = g (x)
0 < x < L,
where u represents the density of a population of individuals. In this case, the
homogeneous Dirichlet conditions model an hostile external environment35 . Given
this kind of boundary condition, the population decays by diﬀusion while tends to
increase by reaction. Thus the two eﬀects compete and we want to explore which
factors determine the overwhelming one.
First of all, since a is constant, we can get rid of the term au by setting
u (x, t) = eat w (x, t) .
We have:
ut = eat (aw + wt ), ux = eat wx , uxx = eat wxx
and substituting into the diﬀerential equation, after some simple algebra, we ﬁnd
for w the equation
wt − Dwxx = 0,
with the same boundary and initial conditions:
w (0, t) = w (L, t) = 0
w (x, 0) = g (x) .
35
A homogeneous Neumann condition would represent the evolution of an isolated population, without external exchange.
2.5 Diﬀusion, Drift and Reaction
65
Then, we can easily exhibit an explicit formula for the solution, using the separation of variables36 :
∞
bk exp −D
w (x, t) =
k=1
with
bk =
1
L
k2 π2
kπx
t sin
L2
L
(2.104)
L
g (x) sin(kπx/L)xdx.
0
If g ∈ C 2 ([0, L]) and g (0) = g (L) = 0, the series (2.104) converges uniformly
for t > 0 and 0 ≤ x ≤ L. Going back to u, we get for the solution to problem
(2.103) the following expression:
∞
bk exp (a − D
u (x, t) =
k=1
k2 π2
kπx
)t sin
.
2
L
L
(2.105)
Formula (2.105) displays an important diﬀerence from the pure diﬀusion case a =
0, as far as the asymptotic behavior for t → +∞ is concerned. Assuming b1 = 0,
the population evolution is determined by the largest exponential in the series
(2.105), corresponding to k = 1. It is now an easy matter to draw the following
conclusions.
1. If
a−D
π2
< 0, then
L2
uniformly in [0, L], since a − D
lim u (x, t) = 0
t→+∞
(2.106)
k2 π2
π2
<
a
−
D
for every k > 1.
L2
L2
2. If
π2
> 0, then lim u (x, t) = ∞
t→+∞
L2
for x = 0, L, since the ﬁrst exponential blows up exponentially and the other terms
are either of lower order or vanish exponentially.
a−D
3. If a − D
π2
= 0, (2.105) becomes
L2
u (x, t) = b1 sin
πx
+
L
∞
bk exp (a − D
k=2
k2 π2
kπx
)t sin
.
2
L
L
Since a − Dk 2 π 2 /L2 < 0 if k > 1, we deduce that
πx
u (x, t) → b1 sin
as t → +∞,
L
uniformly in [0, L].
36
See Problem 2.1.
66
2 Diﬀusion
Now, the coeﬃcients a and D are intrinsic parameters, encoding the features
of the population and of the environment. When these parameters are ﬁxed, the
habitat size plays a major role. In fact, the value
L0 = π
D
a
represents a critical value for the population survival. If L < L0 the habitat is too
small to avoid the extinction of the population; on the contrary, if L > L0 , one
observes exponential growth. If L = L0 , diﬀusion and reaction balance and the
population evolves towards the solution b1 sin(πx/L) of the stationary problem
−Duxx = au
u (0) = u (L) = 0,
in (0, L)
called steady state solution.
Finally note that if for some k ≥ 1 we have a − Dkπ 2 /L2 > 0, then for every k,
1 ≤ k < k, all the values a − Dkπ 2 /L2 are positive and the corresponding terms in
the series (2.105) contributes to the exponential growth of the solution. In terms
of population dynamics this means that the vibration modes
bk exp (a − D
k2 π2
kπx
)t sin
L2
L
for k = 1, 2, . . . , k are activated.
2.6 Multidimensional Random Walk
2.6.1 The symmetric case
What we have done in dimension n = 1 can be extended without much eﬀort to
dimension n > 1, in particular n = 2, 3. To deﬁne a symmetric random walk, we
introduce the lattice Zn given by the set of points x ∈ Rn , whose coordinates are
signed integers. Given the space step h > 0, the symbol hZn denotes the lattice of
points whose coordinates are signed integers multiplied by h.
Every point x ∈ hZn , has a “discrete neighborhood” of 2n points at distance h,
given by
x + hej
and
x − hej
(j = 1, . . . , n),
where {e1 , . . . , en} is the canonical basis in Rn . Our particle moves in hZn according to the following rules (Fig. 2.11).
1. It starts from x = 0.
2. If it is located in x at time t, at time t + τ the particle location is at one of the
1
2n points x ± hej , with probability p = 2n
.
3. Each step is independent of the previous ones.
2.6 Multidimensional Random Walk
67
Fig. 2.11 Bidimensional random walk
As in the 1-dimensional case, our task is to compute the probability p (x, t) of
ﬁnding the particle at x at time t.
Clearly the initial conditions for p are
p (0, 0) = 1 and p (x, 0) = 0 if x = 0.
The total probability formula gives
p (x, t + τ ) =
1
2n
n
{p (x + hej , t) + p (x − hej , t)} .
(2.107)
j=1
Indeed, to reach the point x at time t + τ, at time t the particle must have been
located at one of the points in the discrete neighborhood of x and moved from
there towards x with probability 1/2n. Keeping x and t ﬁxed, we want to examine
what happens when we let h → 0, τ → 0. Assuming that p is deﬁned and smooth
in all of Rn × (0, +∞), we use Taylor’s formula to write
p (x, t + τ ) = p (x, t) + pt (x, t) τ + o (τ )
1
p (x ± hej , t) = p (x, t) ± pxj (x, t) h + pxj xj (x, t) h2 + o h2 .
2
Substituting into (2.107), after some simpliﬁcations, we get
pt τ + o (τ ) =
h2
Δp + o h2 .
2n
Dividing by τ we obtain the equation
pt + o (1) =
1 h2
Δp + o
2n τ
h2
τ
.
(2.108)
68
2 Diﬀusion
The situation is quite similar to the 1− dimensional case: still, to obtain eventually
something nontrivial, we must require that the ratio h2 /τ has a ﬁnite and positive
limit. The simplest choice is
h2
= 2nD
(2.109)
τ
with D > 0. From (2.109),
√ we deduce that in unit time, the particle diﬀuses up
to an average distance 2nD. The physical dimensions of D have not changed.
Letting h → 0, τ → 0 in (2.108), we ﬁnd for p the diﬀusion equation
pt = DΔp,
(2.110)
lim p (x, t) = δ3 (x).
(2.111)
with the initial conditions
t→0+
Since
Rn
p (x, t) dx = 1 for every t, the unique solution is given by
p (x, t) = ΓD (x, t) =
|x|2
1
(4πDt)
n/2
e− 4Dt ,
t > 0.
The n−dimensional random walk has become a continuous walk; when D = 12 , it
is called n−dimensional Brownian motion. Denote by B (t) = B (t, ω) the random
position of a Brownian particle, deﬁned for every t > 0 on a probability space
(Ω, F, P )37 .
The family of random variables B (t, ω), with time t as a real parameter, is
a vector valued continuous stochastic process. For ω ∈ Ω ﬁxed, the vector
function
t −→ B (t, ω)
describes an n−dimensional Brownian path, whose main features are listed below.
• Path continuity. With probability 1, the Brownian paths are continuous for t ≥ 0.
• Gaussian law for increments. The process
Bx (t) = x + B (t)
deﬁnes a Brownian motion with start at x. With every point x is associated a
probability P x, with the following properties (if x = 0, P 0 = P ).
a) P x {Bx (0) = x} = P {B (0) = 0} = 1.
b) For every s ≥ 0, t ≥ 0, the increment
Bx (t + s) − Bx (s) = B (t + s) − B (s)
(2.112)
follows a normal law with zero mean value and covariance matrix equal to tIn ,
37
See Appendix B.
2.6 Multidimensional Random Walk
69
whose density is
Γ (x, t) = Γ 1 (x, t) =
2
1
n/2
e−
|x|2
2t
.
(2πt)
Moreover, (2.112) is independent of any event occurred at any time less than s.
For instance, the two events
{B (t2 ) − B (t1 ) ∈ A1 }
{B (t1 ) − B (t0 ) ∈ A2 }
are independent if t0 < t1 < t2 .
• Transition function. For each Borel setA ⊆ Rn a transition function
P (x, t, A) = P x {Bx (t) ∈ A}
is deﬁned, representing the probability that the particle, initially located at x,
belongs to A at time t. We have:
P (x, t, A) = P {B (t) ∈ A − x} =
Γ (y − x, t) dy.
Γ (y, t) dy =
A−x
A
• Invariance. The motion is invariant with respect to rotations and translations.
• Markov and strong Markov properties. Let μ be a probability measure38 on Rn .
If the particle has a random initial position with probability distribution μ, we
can consider a Brownian motion with initial distribution μ, and for it we use the
symbol Bμ . To Bμ is associated a probability distribution P μ such that
P μ {Bμ (0) ∈ A} = μ (A) .
The probability that the particle belongs to A at time t can be computed through
the formula
P μ {Bμ (t) ∈ A} =
P (x, t, A) μ (dx) .
(2.113)
Rn
The Markov property can be stated as follows: given any condition H, related to
the behavior of the particle before time s ≥ 0, the process Y (t) = Bx (t + s) is a
Brownian motion with initial distribution
μ (A) = P x {Bx (s) ∈ A |H } .
Again, this property establishes the independence of the future process Bx (t + s)
from the past, when the present Bx (s) is known and encodes the absence of memory of the process. In the strong Markov property, a stopping time τ takes the
place of s.
38
See Appendix B for the deﬁnition of a probability measure μ and of the integral with
respect to the measure μ .