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5 Diffusion, Drift and Reaction

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 .



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