Another important application of first
passage ideas is to chemical kinetics,
where one is often interested in how
efficiently do diffusing reactants
actually undergo a reaction.
This basic problem goes by the name of
reaction rate theory.
Here is the setup of the problem.
Consider a sphere of radius a,
that is immersed in a fluid that contains
diffusing particles.
The particles are some unknown,
some fixed concentration c,
and each particle diffuses around,
and if one of these diffusing particles
hits the sphere, it is absorbed.
And each particle is diffusing according
to some diffusion coefficient D.
So, the basic thing that we want to
calculate here is how quickly,
are particles absorbed on the surface
of the sphere.
This is embodied by something known
as the reaction rate,
which comes by the letter k.
So this is defined as the reaction rate,
and it's defined as the number of
particles absorbed per unit time.
This definition is not quite complete,
because clearly, if we double the
concentration,
we're going to double the number of
particles that are absorbed,
so to get an intrinsic quantity, we should
divide by the overall concentration.
Without doing any calculations,
this simple formula already contains
a number of surprising results.
Let's just do dimensional analysis,
and ask,
"What is the dimensions
of the reaction rate?"
So, the dimensions of particles,
there's no dimensions,
there's a 1/time, and the concentration is
1/volume,
and so the dimensions of the
reaction rate are
a length scaled to the power dimension
of space,
divided by time.
On the other hand,
what can the reaction rate depend on?
And clearly, the only two parameters
that describe the system itself,
is the radius of the sphere, and the
diffusion coefficient of the particles.
So the reaction rate should be a function
of the diffusion coefficient,
and the radius of the absorbing sphere.
Now, one more thing to note is, is that
the dimensions of the diffusion coefficient
we've discussed this earlier in this
tutorial,
the dimension of the diffusion coefficient
is a length squared, divided by the time.
So, armed with the information that we
know the dimensions of the
diffusion coefficient, and the dimensions
of the radius of the sphere,
we can infer how this reaction rate
depends on the parameters in this system.
we know that the reaction rate
scales as 1/time.
The only place that time appears is
in the diffusion coefficient, so clearly,
the reaction rate should be a linear
function of the diffusion coefficient.
If we use that fact, then we have two
powers of length in the numerator,
whereas the reaction rate has d powers
of length in the numerator,
so the only way we can engineer things to
have d powers of length in the numerator,
is to have d-2 powers of a in the
numerator.
So from these simple considerations,
we infer that the reaction rate
should be proportional to the diffusion
coefficient times the radius of the sphere,
to the power d-2 power.
So, this is the basic result of reaction
rate theory,
that the reaction rate has this dependence
on the diffusion coefficient,
and the radius of the sphere.
Let's just look at this in a little bit
more detail,
because this already has some very
amazing results.
Let's first of all, think about three
dimensions, physical space.
In this case, the reaction rate is
proportional to the radius of the sphere.
Now, let me put an exclamation point next
to this, to emphasize the fact that
it's not proportional to the cross-
sectional area of the sphere,
but to the radius of the sphere.
So it has an anomalous dimension
dependence.
On the other hand, if we're below two
dimensions,
then something even weirder seems to occur
because it says that below two dimensions,
this is a function such that the
reaction rate increases as a decreases.
And again, let's put an exclamation point
next to that, because clearly,
this is a nonsensical result.
It's not possible to have an increase
in the reaction rate,
as you decrease the size of the
absorbing sphere.
So, the point is that below two dimensions
There is a new dependence
on system parameters.
Let's now work out the reaction rate
theory quantitatively in three dimensions.
So we have our familiar setup of an
absorbing sphere of radius a,
that is immersed in a fluid of diffusing
particles, that are each diffusing with
diffusion coefficient D, whenever one of
these diffusing particles
hits the surface of the sphere, it is
absorbed.
So, to compute the reaction rate, what
we first have to compute is the
concentration profile of particles outside
the sphere,
and then from that, we can then compute
the flux of particles to the surface
of the sphere itself. So, for the first
part of the problem,
to find the concentration profile,
we have to solve the following
classic boundary value problem, we
have to solve the diffusion equation
dcdt is equal to D, laplacian of c,
subject to
appropriate initial boundary conditions.
And an appropriate initial condition is
that exterior to the surface of the sphere,
at initial time, the concentration, which
we can set it to be anything,
let's set it to be equal to 1.
And we also have the boundary condition
that if a particle hits the sphere,
at any positive time, it is absorbed,
which corresponds to the absorbing
boundary condition, of concentration 0.
So, this is the problem that we have
to solve.
This is a classic boundary value problem,
the solution can be written in terms of
vessel functions,
and with a little bit of cleverness,
one can actually transform this to a
one-dimensional problem, where the
solution can be written in terms of
elementary functions, namely the error
function, but I'm going to try to avoid
all of that, by taking advantage of the
fact that in three dimensions,
a diffusing particle, or a random walker,
is transient.
That is, the probability that a particle starting at some distance away from the sphere,
actually hits the sphere, is less than 1.
What this means is that the depletion of
particles near the surface of the sphere
is not very severe. Moreover, it also means
that particles from infinity can come in
and replenish the particles that are
depleted because of the absorption.
And because of this feature, I'm going to
hypothesize,
and in fact it's a true statement, that in
the long time limit,
this concentration profile actually goes
to a steady state.
So, instead of solving the time dependent
problem, I'm going to solve the time-
independent, or the steady-state version
of the same problem,
that is, I'm going to solve not the diffusion
equation, but the laplace equation,
D, laplacian of c, is equal to 0.
Namely, I'm going to set the time
derivative to 0.
We have to solve the same problem
with the corresponding boundary conditions
as with the time-dependent problem.
And that is that, at the surface of the
sphere, the steady state concentration
should be equal to 0, and moreover, if
we go to infinity,
the concentration should be 1. So, this
is the problem that we're going to solve
instead of the full time-dependent problem
and I will argue, and in fact it is true that,
this time-independent problem corresponds
to the infinite time limit
of the time-dependent problem. The nice
feature of solving this time-independent
problem is that it's just the good old
laplace equation at the surface of the
spherical conductor, and so we know the
solution from our freshman physics.
And for this set of boundary conditions,
the solution is very simple.
c of r, is equal to 1 minus a/r.
Now, in addition to the fact that the
solution is simple,
it also has a very nice interpretation in
terms of random walks.
earlier in the discussion of first passage
ideas, we learnt that the exit probability
or the hitting probability satisfies the
laplace equation with an appropriate
boundary condition. If one looks here
at what we're trying to solve,
it looks very similar to the hitting
probability problem,
namely the probability that the random
walk will eventually hit the sphere,
except the boundary conditions is a
little bit backwards,
because we have a boundary condition where
if you're at the surface of the sphere,
the concentration is 0.
So, in fact, what we're solving is not
the hitting probability,
we're solving the classic escape
probability.
Namely, what is the probability that a
random walk could have start at radial
position r, escape to infinity without
ever hitting the surface of the sphere?
And these boundary conditions just mirror
the escape probability.
If I start at the surface of the sphere,
by definition, I cannot escape.
And if I start infinitely far away, then
I have escaped, by definition.
So, this is nothing more than the
eventual escape probability,
and notice it has this nice feature that
if you start at distance a above the
surface of the sphere, or namely,
r is 2a,
the escape probability is 1/2,
and the eventual hitting probability
is also equal to 1/2.
And finally, let's now come to the problem
which is to calculate the reaction rate.
So the reaction rate is the flux of
to the surface of the sphere,
so we have to calculate the diffusive
flux, D gradient of c,
we want to compute its dot product with
the surface area of the sphere,
and integrate over the surface of the
sphere,
and we have to have with a minus sign,
because we have the inward surface model,
so this is a very simple integral to do,
because it is spherically symmetric,
so we have D coming out in front,
the concentration field has an a out
in front,
when I take the gradient of 1/r,
I get 1/(r^2),
so we're integrating over the surface
of the sphere of radius a, of 1/(r^2),
which is 1/(a^2), times dS, and this is
nothing more than
4, the surface area of the sphere,
of radius a, which is 4 pi a^2,
and so what we get here is 4 pi D a.
And so, as advertised,
the reaction rate is proportional
to the diffusion coefficient
of the diffusing particles, and it's
linearly proportional to the length of
the radius of the sphere.
Notice again that it's not proportional to
the cross-sectional area of the sphere.
If one solves the full time-dependent
problem,
it turns out that there is a correction,
due to the time dependence,
it goes away in the long-time limit,
and so the correct answer here,
including the full time dependence,
is 1 + a/sqrt(pi Dt).
So, this is a fundamental result in
chemical reaction kinetics,
the reaction rate of the sphere of
radius a.