Part 4: Diffusion and Hops
From Discrete to Continuous
Hopping on a Lattice
Notation: Even and Odd
From one step to many
An example
Continuous
Langevin equation
An integration
Gaussian Properties
Generating Function
A probability
Discrete
A generating Function
A probability
calculation
we get an answer!
fourier transform formulation
binomial theorem
Higher Dimension
continued
probability density
One dimension
Current
Diffusion equation
Higher Dimension
You cannot go back
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
1
Hopping: From Discrete to Continuous
We are going to be spending some time talking about the physics of a particle moving
in a solid. Often this motion occurs as a set of discrete hops. The particle gets stuck
someplace, sits for a while, acquires some energy from around it, hops free, gets
caught in some trap, and then sits for a while. I’m going to describe two
mathematical idealizations of this motion: discrete hopping on a lattice and
continuous random motion.
One point is to see the difference between the two different topologies represented
by a continuous and a discrete system. One often approximates one by the other and
lots of modern physics and math is devoted to figuring out what is gained and lost by
going up and back.
There is a fine tradition to this. Boltzmann, one of the inventors of statistical
mechanics, liked to do discrete calculations. So he often represented things which are
quite continuous, like the energy of a classical particle by discrete approximations,
A little later, Planck and Einstein had to figure out the quantum theory of radiation,
which had been thought to be continuous, in terms of discrete photons. So we shall
compare continuous and discrete theories of hopping.
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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Hopping On a Lattice
A lattice is a group of sites arranged in a
regular pattern. One way of doing this can be
labeled by giving the position r=(n1,n2,....)a
where the n’s are integers. If we include all
possible values of these integers, the
particular lattice generated is called is called
the simple hypercubic lattice. We show a
picture of this lattice in two dimensions.
This section is devoted to developing the concept of a random walk. We could do
this in any number of dimensions. However, we shall approach it is the simplest
possible way by first working it all out in one dimension and then stating results
for higher dimensions A random walk is a stepping through space in which the
successive steps occur at times t=M τ. At any given time, the position is X(t),
which lies on one of the a lattice sites, x=an, where n is an integer. In one step of
motion one progress from X(t) to X(t+ τ) = X(t)+ aσj, where σj is picked at
random from among the two possible nearest neighbor hops along the lattice, σj
=1 or σj = -1. Thus, < σj >=0, but of course the average its square is non-zero and
is given by < σj2 > =1. We assume that we start at zero, so that our times t =j τ.
It is not accidental that we express the random walk in the same language as the
Ising model. We do this to emphasize that geometric problems can often be
expressed in algebraic form and vice versa.
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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Notation: Even and Odd
We represent the walk by two integers, M, the number of steps taken and n, the
displacement from the origin after M steps. We start from n=0 at M=0. The general
formula is
n=
M
!
σk
k=1
Notice that n is even if M is even and odd if M is odd. We shall have to keep track of this
property in our later, detailed, calculation.
We shall also use the dimensional variables for time t=M τ and for space X(t)=na. We
use a capital X to remind ourselves that it is a random variable. When we need a nonrandom spacial variable, we shall use a lower case letter, usually x.
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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From one step to many steps
We start from the two statements that <σj >=0 and that <σj σk >=δj,k
On the average, on each step the walker goes left as much as right. and thus as a result the
average displacement of the entire walk is zero
M
< X (t ) >= a∑ < σ j >= 0
j =1
iv.1
However, of course the mean squared displacement is not zero, since
M
M
< X€(t )2 >= a 2 ∑ < σ j σ k >= a 2 ∑ δ j ,k = a 2M
j ,k =1
€
iv.2
j ,k =1
Our statement is the same that in a zero field uncoupled Ising system, the maximum
magnetization is proportional to the number of spins, but the typical magnetization is
only proportional to the square root of that number. Typical fluctuations are much,
much smaller than maximum deviations.
We can see this fact by noting that the root mean square average of X2 is a √M,
which is the typical end-to-end distance of this random walk. This distance is is
much smaller than the maximum distance which would be covered were all the
steps to go in the same direction. In that case we would have had a distance aM.
Thus, a random walk does not, in net, cover much ground.
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
5
An example:
The difference between maximum length and RMS length (root mean square length) needs to
be emphasized by an example. Consider a random walk that consists of M= one million
steps steps. If each step is one centimeter long the maximum distance traveled is M
centimeters, or 10 thousand meters or 10 kilometers. On the other hand the typical end to
end distance of such a walk is M 1/2 centimeters or ten meters. That is quite a difference!
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
6
Continuous Random Walks:
Langevin Equation
The physics of this random motion is well represented by the two equations
<X(t)-X(s)> = 0
iv.3
<[X(t)-X(s)]2> = a2 |t-s|/τ
iv.4
We would like to have a simple differential equation which appropriately
reflects this physics. The simplest form of differential equation would be a
form described as a Langevin equation,
dX/dt = η(t)
iv.5
where η(t) is a random function of time that we shall pick to be uncorrelated at
different times and have the properties
<η(t)> =0
and <η(t) η(s)> =c δ(t-s)δj,k
This looks like what we did in the random hopping on the lattice, except that we have
to fix the value of the constant, c, to agree with what we did before. Equation iv.5 has
the solution
! t
du η(u)
X(t) = X(s) +
s
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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X(t) = X(s) +
!
t
du η(u)
s
Using this result, we can calculate, that as before <R(t)-R(s)> =0. A new result is that for
t>s, we find:
(what happens for s>t?)
< [X (t ) − X (s )]2 > =
=
=
More specifically,
t
t
s
t
s
t
t
s
s
∫ du ∫ dv < η (u )η (v ) >
∫ du ∫ dv c δ(u − v )
∫ du c = (t − s )c
s
< [X (t )]2 >= c t
iv.6
€
so that a comparison with our previous result indicates that we should choose
€ c= a2 /τ
`
iv.7
so that
< [X (t ) − X (s )]2 > = a 2 | t − s | /τ
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
iv.8
8
Gaussian Properties of Continuous Random Walk
In the continuous case, X(t) is composed of a sum, or rather integral, of
many pieces which are uncorrelated with one another. According to the
central limit theorem, such a sum or integral is a Gaussian random
variable. Hence, we know everything there is to know about it. Its average
is zero and its variance is a2t/τ . Consequently, it has a probability
distribution
 τ 1/2 − x 2τ /(2a2t )
iv.9
< ρ(X (t ) = x ) >= 
 e
2
2πa t 
Now we have said everything there is to say about the continuous random walk.
As an€extra we can exhibit the generating function for this walk:
< exp[iqX (t )] >= e
−q 2a2t /(2τ )
€
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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A generating function: discrete case
In our calculation of the statistical mechanics of the Ising model, or of any other problem
involving the statistics of large numbers of particles, the traditional approach to the
problem is to calculate the partition fiunction, Z, a generating function from which we
can determine all the statistical properties. The same approach works for the random
walk. It is best to start on this calculation by a somewhat indirect route: The spirit of the
calculation is that it is simple to find the average of exp(iqX(t)) where q is a parameter
which we can vary and X(t) is our random walk-result. If one knew the value of
< exp(iqX(t))> it would be easy to calculate all averages by differentions with respect to
q and even the probability distribution of X(t) by, as we shall see, a Fourier transform
technique. We start from the simplest such problem. Take t=τ so that there is but one
step in the motion and
< exp(iqX(τ))>= < exp(iqa σ1(t))>= [exp(iqa) + exp(-iqa)]/2 = cos qa
Since X(t) is a sum of M identical terms, the exponential is a product of M terms of the
form < exp(iqa σj(t))> and each term in the product has exact the same value, the
cosine given here. Thus the final result is
<exp(iqX(t))>= [cos(qa)]M.
iv.9
As we know from our earlier work such a generating function enables us to calculate, as
for example
<exp(iqX(t))>= [cos(qa)]M at q=0
<1>=1
d/dq { <exp(iqX(t))>= [cos(qa)]M } at q=0
(d/dq)2 { <exp(iqX(t))>= [cos(qa)]M } at q=0
<iX(t)>=Ma sin qa =0
-<[X(t)]2>=-Ma2 d/dq sin qa = -Ma2
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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A Probability
We wish to describe our result in terms of a probability, ρn which is the probability X
will take on the value na. We can use this probability to calculate the value of any
function of X in the form (see equation i.3)
< f (X ) >= ∑ f (na)ρn
n
This is an important definition. It says that if X takes on the values na, and
the probability of it having the value na is ρn , then the average is the sum
over n times the value of the function at that n times the probability of
having that value. The formula that we shall use is only slightly more
complex. We define X as being time dependent: the probability that X(t)
has the value na at time mτ is then defined to be ρn,m. The formula for the
average is then
€
< f (X (mτ )) >= ∑ f (na)ρn ,m
iv.10
n
so that the average is once more the sum of the function times the probability. So ρ
tells us everything there is to know about the problem.
€
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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To calculate probability
Start from
< f (X ) >= ∑ f (na)ρn
n
choose the function to be an exponential, since we have had, by this time,
considerable experience dealing with exponentials
€
< exp(iqX (t ) / a) >= ∑ exp(iqn )ρn ,t /τ
n
We have already evaluated the left hand side of this expression, so we find
< [cos q ]M >= ∑ exp(iqn )ρn ,M
iv.11
€
n
I’m going to spend some time on this result.
First, let’s say what it is. It gives the probability that after a random walk of M steps a
walker will find itself at a site n=M-2j steps of its starting point. The pattern is
€ strange. For M being even the possible sites are M, M-2, ..,0 , ... just before
slightly
-M+2, -M. For M being odd the possible sites are M, M-2, , ...,-1,1,... ,-M+2, -M
Thus there are a total of M possible sites which could be occupied. Is it true then
that an average probability of occupation among these sites is 1/M? or maybe 1/
(2M) What’s a typical occupation? What is that going to be?
Recall that the RMS distance covered is ......
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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We get an answer!
< [cos q ]M >= ∑ exp(iqn )ρn ,M
iv.11
n
Here M is the number of steps.
We do not even have to do a Fourier transform to
evaluate the probability. We simply use the binomial theorem
n
!
M!
M
M M −j
(a
+
b)
=
a
b
€
j!(M − j)!
j=0
to expand the left hand side as
eiq + e−iq M
M!
eiqM −i2q j
(
) =∑
e
M
2
j j!(M − j)! 2
to get.....
M!
ρM−2 j,M =
2−M
j!(M − j)!
ρn,M
M!
= M +n M −n 2−M
( 2 )!( 2 )!
iv.12
iv.13a
iv.13b
We are a bunch of theorists. We have an answer. A formula. Now what can we do
with that answer?
What can we figure out from here?
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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Fourier transform
∞
!
iv.11
ρp,M eiqp = [cos q]M
p=−∞
To invert the Fourier transform use
! π
dq i(p−n)q
e
= δn,p
−π 2π
To obtain
!
π
−π
∞
"
dq −inq
e
ρp,M eiqp = ρn,M =
2π
p=−∞
!
π
−π
dq −inq
e
[cos q]M
2π
Because n=M (mod 2), the integrand at q is just the same as the integrand at q+π.
Therefore one can also write
! π/2
dq −inq
e
[cos q]M
ρn,M = 2
−π/2 2π
We have come across this kind of integral before. Now what?
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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Result From Binomial Theorem
M!
ρM−2 j,M =
2−M
j!(M − j)!
ρn,M
M!
= M +n M −n 2−M
( 2 )!( 2 )!
We are a bunch of theorists. We have an answer. A formula. Now what can we do
with that answer?
What can we figure out from here?
sterling approximation, for large M,
M! ≈ M M e-M (2 π M )1/2
what happens for n=M, n=-M, n=0, n close to M, n close to -M, n close to zero?
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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Higher Dimensions
Reason for putting many different calculations on a
lattice is that the lattice provides a simplicity and
control not available in a continuum system. There
is no ambiguity about how things very close to one
another behave, because things cannot get very close.
They are either at the same point or different points
So now I would like to talk about a random walk in a d-dimensional system by considering
a system on a simple lattice constructed as in the picture. The lattice sites are given by
x = a(n1,n2, n3, ....)/2 . The n’s are integers. There are two possible kinds of assignments
for the n’s: Either they are all even e.g. x = a(0,2, -4, ....)/2 or they are all odd, for
example x = a(1,3, -1, ....)/2. If all hopping occurs from one site to a nearest neighbor
site, the hops are through one of 2d vectors of the form ξα= a(m1,m2, m3, ....)/2, where
each of the m’s have magnitude one, but different signs for example a(1,-1,-1, ...). We
use a lattice constant, a, which is twice as big as the one shown in the picture.
We then choose to describe the system by saying that in each step, the coordinate hops
through one of the nearest neighbor vectors, ξα , which one being choosen at random.
This particular choice makes the entire coordinate, x, have components which behave
entirely independently of one another, and exactly the same as the one-dimensional
coordinate we have treated up to now.
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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RANDOM WALKS
dspace.mit.edu/.../CourseHome/index.htm
http://particlezoo.files.wordpress.com/2008/09/randomwalk.png
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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Higher Dimensions.....continued
We denote the probability density of this d-dimensional case by a superscript d and
the one for the previous one-dimensional case of by a superscript 1. As an additional
difference, the d-dimensional object will be a function of space and time rather than n
and M. After a while, we shall focus entirely upon the higher dimensional case and
therefore drop the superscripts. We have
d
!
ρ1nα ,M
ρdan,M τ =
α=1
However, we can jump directly to the answer for the continuum case, If the
probability distribution for the discrete case is simply the product of the onedimensional distributions so must be the continuum distribution. The one-dimensional
equation answer in eq iv.9 was
 τ 1/2 − x 2τ /(2a2t )
< ρ(X (t ) = x ) >= 
 e
2
2πa t 
iv.9
so that the answer in d dimensions must be
 τ d /2 −r 2τ /(2a2t )
< ρ(R(t ) = r) >= 
 e
2
2πa t 
€
Here the bold faced quantities are vectors, viz r2 = x2 +y2 +....
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
€
18
Higher dimensional probability density: again
one dimension
d dimensions is a product of ones
1/2
 τ 
< ρ(X (t ) = x ) >= 

2
2πa t 
e
− x 2τ /(2a2t )
 τ d /2 −r 2τ /(2a2t )
< ρ(R(t ) = r) >= 
 e
2
2πa t 
Notice that the result is a rotationally invariant quantity, coming from adding the x2 and y2 and....
€2 +.... . This is an elegant result coming from the fact that
in separate exponents to get r2 = x2 +y
the hopping has produced a result which is independent of the lattice sitting under it. We woulod
get a very similar result independent of the type of lattice underneath. In fact, the result come
from what is called a diffusion process and is typical of long-wavelenth phenomena in a wide
variety of systems.
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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Diffusion Process
Diffusion here is a result of a conservation law: a global statement that the total
amount of something is unchanged by the time development of the system. dQ/dt=0. In
our case the Q in question is the total probability of finding the diffusing particle
someplace. Diffusion has a second element: locality. The local amount of Q, called ρ,
changes because things flow into and out of a region of space. The flow is called a
current, j, and the conservation law is written as
conservation law
∂ρ(r ,t )
+ ∇ ⋅ j (r ,t ) = 0
∂t
The time derivative of the density is produced by a divergence of the curent flowing into a
point. On a one dimension lattice, the rule takes the simpler form:
ρn ,M€+1 − ρn ,M = In −1/2,M − In +1/2,M
conservation law
This equation says that the change of probability over one time step is produced by the
flow of probability in from the left minus the flow out to the right.
€
Here, I’m going to visualize a situation once more in which we have a discrete time
coordinate M and a discrete space coordinate, n. I shall assume that the initial
probabilities is sufficiently smooth so that even and odd n-values have rather similar
occupation probabilities so that we can get away with statement like
ρn ,M +1 − ρn ,M ≈ ∂ρn ,M / ∂M
The conservation law then takes the form
∂ρn ,M / ∂M + ∂In ,M / ∂n = 0
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
€
20
The Current
We do not have a full statement of the of what is happening until we can specify the
current. An approximate definition of a current in a conservation law is called a
constitutive equation. We now write this down.
Our hopping model says that a In+1/2,M is given a contribution +1 when the site at n is
occupied at time M, and σn is equal to +1. On the other hand it is given a
contribution -1 when the site at n +1 is occupied at time M, and σn+1 is equal to -1.
Since there is a probability 1/2 for each of the σ-events the value of I is
In +1/2,m = (ρn ,m − ρn +1,m ) / 2
constitutive equation
Once again, we write the difference in terms of a derivative, getting
In ,m = −(∂ρn ,m / ∂n ) / 2
€
constitutive equation
This can then be combined with the conservation law to give the diffusion equation
∂ρn ,M / ∂M = ∂2ρn ,M / ∂ n 2 / 2
€
which can be written in dimensional form as
∂ρ
∂2ρ
diffusion equation
=λ
∂t
∂ x2
with the diffusion coefficient being given by λ=a2/(2 τ).
€
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
€
21
Higher Dimension
In higher dimensions, the current is proportional to the gradient of the current
j (r ,t ) = −λ∇ρ(r ,t )
so that the diffusion equation becomes
€
∂t ρ(r,t ) = λ∇2ρ(r,t )
€
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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Diffusion Equation
This equation is one of several equations describing the slow transport of physical
quantities from one part of the system to another. When there is slow variation is
space, the conservation law guarantees that the rate of change in time will also be
slow. In fact this is part of a general principle which permits only slow changes as a
result of a conservation law. This general principle is much used in the context of
quantum field theory and condensed matter physics. The idea is connected with the
construction of the kind of particle known as a Nambu-Goldstone boson, named for
two contemporary physicists, my Chicago colleague Yoichiro Nambu and the MIT
theorist Jeffrey Goldstone.
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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Solution to Diffusion Equation Cannot be Carried
Backward in Time
The wave equation is (∂t2 -∂X2 )F = 0
Its general solution is F(x,t)=G(x-ct)+H(x+ct)
This is a global solution. It enables you to look forward or back infinitely far in the future
or the past without losing accuracy. Find solution from F(x,0) =∂X F(x,0) =1 for 0<x<1
and F(x,0) =∂X F(x,0) =0 otherwise. Use c=2
∫
A global solution to the diffusion equation is ρ(x,t)= dk exp[ikx-λk2t] g(k)
∫
with
g(k)= dx exp[-ikx] ρ(x,0)/(2π) as initial data.
Small rapidly varying errors at t=0 will produce small errors for positive t and
huge errors for negative t. You cannot extrapolate backward in time. Information
gets lost as time goes forward.
Solve for ρ(x,0)=1 for 0<x<1 and ρ(x,0) =0 otherwise. Use λ=2. Plot
solution for t=0,2,4. What happens for t= -2?
Boltzmann noted that equations of classical mechanics make sense if t is
replaced by -t. But diffusion equation has a solution which does not make
sense. Where did we go from sense to nonsense?
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff
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