part 4 diffusion and hops
TRANSCRIPT
-
7/29/2019 Part 4 Diffusion and Hops
1/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof 1
Part 4: Diffusion and HopsFrom Discrete to Continuous
Hopping on a Lattice
Notation: Even and OddFrom one step to manyAn example
Continuous Langevin equation
An integrationGaussian Properties
Generating Function
A probabilityDiscreteA generating FunctionA probabilitycalculationwe get an answer!fourier transform formulationbinomial theorem
Higher Dimension
continuedprobability density
One dimensionCurrent
Diffusion equationHigher DimensionYou cannot go back
-
7/29/2019 Part 4 Diffusion and Hops
2/24
-
7/29/2019 Part 4 Diffusion and Hops
3/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Hopping On a Lattice
3
A lattice is a group of sites arranged in aregular pattern. One way of doing this can be
labeled by giving the position r=(n1,n2,....)awhere the ns are integers. If we include allpossible values of these integers, theparticular lattice generated is called is calledthe simple hypercubic lattice. We show apicture of this lattice in two dimensions.
This section is devoted to developing the concept of a random walk.We could dothis in any number of dimensions. However, we shall approach it is the simplestpossible way by first working it all out in one dimension and then stating resultsfor higher dimensions A random walk is a stepping through space in which thesuccessive steps occur at times t=M . 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 fromX(t) toX(t+) =X(t)+ aj, where j is picked at
random from among the two possible nearest neighbor hops along the lattice, j=1 or j=-1. Thus, =0, but of course the average its square is non-zero and
is given by =1. We assume that we start at zero, so that our times t =j.
It is not accidental that we express the random walk in the same language as theIsing model. We do this to emphasize that geometric problems can often beexpressed in algebraic form and vice versa.
-
7/29/2019 Part 4 Diffusion and Hops
4/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Notation: Even and Odd
4
We represent the walk by two integers, M, the number of steps taken and n, thedisplacement from the origin after M steps. We start from n=0 at M=0. The generalformula is
n =
M
k=1
k
Notice that n is even if M is even and odd if M is odd. We shall have to keep track of thisproperty in our later, detailed, calculation.
We shall also use the dimensional variables for time t=M and for spaceX(t)=na. We
use a capital Xto remind ourselves that it is a random variable. When we need a non-random spacial variable, we shall use a lower case letter, usuallyx.
-
7/29/2019 Part 4 Diffusion and Hops
5/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
From one step to many steps
5
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 ismuch smaller than the maximum distance which would be covered were all thesteps 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.
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
iv.1
However, of course the mean squared displacement is not zero, since
iv.2
Our statement is the same that in a zero field uncoupled Ising system, the maximummagnetization is proportional to the number of spins, but the typical magnetization isonly proportional to the square root of that number. Typical fluctuations are much,much smaller than maximum deviations.
< X(t)2 >= a2 < jj,k=1
M
k >= a2 j,kj,k=1
M
= a2M
< X(t) >= a < jj=1
M
>= 0
We start from the two statements that =0 and that =j,k
-
7/29/2019 Part 4 Diffusion and Hops
6/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
An example:
6
The difference between maximum length and RMS length (root mean square length) needs tobe 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 Mcentimeters, or 10 thousand meters or 10 kilometers. On the other hand the typical end toend distance of such a walk is centimeters or ten meters. That is quite a difference!M1/2
-
7/29/2019 Part 4 Diffusion and Hops
7/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Continuous Random Walks:Langevin Equation
7
The physics of this random motion is well represented by the two equations
= 0 iv.3
= a2 |t-s|/ iv.4
We would like to have a simple differential equation which appropriatelyreflects this physics. The simplest form of differential equation would be aform described as a Langevin equation,
where (t) is a random function of time that we shall pick to be uncorrelated at
different times and have the properties
dX/dt = (t) iv.5
=0 and =c(t-s)j,k
This looks like what we did in the random hopping on the lattice, except that we haveto fix the value of the constant, c, to agree with what we did before. Equation iv.5 hasthe solution
X(t) = X(s) +
t
s
du (u)
-
7/29/2019 Part 4 Diffusion and Hops
8/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof 8
iv.6
Using this result, we can calculate, that as before =0. A new result is that for
t>s, we find: (what happens for s>t?)
so that a comparison with our previous result indicates that we should choose
so that
More specifically,
` iv.7c= a2 /
iv.8
X(t) = X(s) +
t
s
du (u)
< [X(t)]2>=c t
< [X (t) X (s)]2> = a
2 | t s| /
< [X (t) X (s)]2 > = du dv s
t
< (u)s
t
(v) >
= du dv
s
t
c(uv)s
t
= du c= (t ss
t
)c
-
7/29/2019 Part 4 Diffusion and Hops
9/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Gaussian Properties of Continuous Random Walk
9
In the continuous case,X(t) is composed of a sum, or rather integral, ofmany pieces which are uncorrelated with one another. According to the
central limit theorem, such a sum or integral is a Gaussian randomvariable. Hence, we know everything there is to know about it. Its averageis zero and its variance is a2t/ . Consequently, it has a probability
distribution
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)] >= eq2a2t/(2)
iv.9
< (X(t) = x) >=
2a2t
1/2
ex2/(2a2t)
-
7/29/2019 Part 4 Diffusion and Hops
10/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
A generating function: discrete case
10
In our calculation of the statistical mechanics of the Ising model, or of any other probleminvolving the statistics of large numbers of particles, the traditional approach to theproblem is to calculate the partition fiunction,Z, a generating function from which wecan 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 thecalculation 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 toq and even the probability distribution of X(t) by, as we shall see, a Fourier transformtechnique. We start from the simplest such problem. Take t=so that there is but one
step in the motion and< exp(iqX())>= < exp(iqa 1(t))>= [exp(iqa) + exp(-iqa)]/2 = cos qa
SinceX(t) is a sum ofM identical terms, the exponential is a product ofM terms of theform < exp(iqa j(t))> and each term in the product has exact the same value, the
cosine given here. Thus the final result is= [cos(qa)]M. iv.9
As we know from our earlier work such a generating function enables us to calculate, asfor example= [cos(qa)]M at q=0 =1
d/dq { = [cos(qa)]M } at q=0 =Ma sin qa =0
(d/dq)2 { = [cos(qa)]M } at q=0 -=-Ma2d/dq sin qa = -Ma2
-
7/29/2019 Part 4 Diffusion and Hops
11/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
A Probability
11
We wish to describe our result in terms of a probability, n which is the probability X
will take on the value na. We can use this probability to calculate the value of anyfunction ofXin the form (see equation i.3)
This is an important definition. It says that ifXtakes on the values na, and
the probability of it having the value na is n, then the average is the sum
over n times the value of the function at that n times the probability ofhaving that value. The formula that we shall use is only slightly more
complex. We defineXas being time dependent: the probability thatX(t)has the value na at time mis then defined to be n,m. The formula for the
average is then
< f(X) >= f(na)n
n
iv.10< f(X(m)) >= f(na)
n
n,mso that the average is once more the sum of the function times the probability. Sotells us everything there is to know about the problem.
-
7/29/2019 Part 4 Diffusion and Hops
12/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
To calculate probability
12
< f(X) >= f(na)
nn
Start from
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)
n
n,t/We have already evaluated the left hand side of this expression, so we find
< [cosq]M >= exp(iqn)
n
n,M iv.11
Im going to spend some time on this result.First, lets say what it is. It gives the probability that after a random walk of M steps awalker will find itself at a site n=M-2j steps of its starting point. The pattern isslightly strange. For M being even the possible sites are M, M-2, ..,0 , ... just before-M+2, -M. For M being odd the possible sites are M, M-2, , ...,-1,1,... ,-M+2, -MThus there are a total of M possible sites which could be occupied. Is it true thenthat an average probability of occupation among these sites is 1/M? or maybe 1/(2M) Whats a typical occupation? What is that going to be?
Recall that the RMS distance covered is ......
-
7/29/2019 Part 4 Diffusion and Hops
13/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
We get an answer!
13
(eiq+eiq
2
)M=j
M!
j!(M
j)!
eiqM
2M
ei2q j
< [cosq]M >= exp(iqn)
n
n,M iv.11
iv.12
to get.....
M2 j,M=M!
j!(M j)!2M
iv.13a
We are a bunch of theorists. We have an answer. A formula. Now what can we dowith that answer? What can we figure out from here?
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
to expand the left hand side as
(a+ b)M =
n
j=0
M!
j!(Mj)!aMbMj
n,M =M!
(M+n
2 )!(Mn
2 )!
2M iv.13b
-
7/29/2019 Part 4 Diffusion and Hops
14/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Fourier transform
14
iv.11
To invert the Fourier transform use
To obtain
p=
p,Meiqp = [cos q]M
dq
2ei(pn)q = n,p
dq2
einq
p=
p,Meiqp = n,M =
dq2
einq[cos q]M
We have come across this kind of integral before. Now what?
n,M = 2
/2
/2
dq
2 einq
[cos q]M
Because n=M (mod 2), the integrand at q is just the same as the integrand at q+.Therefore one can also write
-
7/29/2019 Part 4 Diffusion and Hops
15/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Result From Binomial Theorem
15
M2 j,M=M!
j!(M j)!2M
We are a bunch of theorists. We have an answer. A formula. Now what can we dowith that answer? What can we figure out from here?
n,M =M!
(M+n2
)!(Mn2
)!2M
sterling approximation, for large M,
M! M M e-M (2 M )1/2
what happens for n=M, n=-M, n=0, n close to M, n close to -M, n close to zero?
-
7/29/2019 Part 4 Diffusion and Hops
16/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Higher Dimensions
16
Reason for putting many different calculations on alattice is that the lattice provides a simplicity and
control not available in a continuum system. Thereis no ambiguity about how things very close to oneanother 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 consideringa system on a simple lattice constructed as in the picture. The lattice sites are given by
x = a(n1,n2, n3, ....)/2 . The ns are integers. There are two possible kinds of assignmentsfor the ns: 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 = a(m1,m2, m3, ....)/2, whereeach of the ms have magnitude one, but different signs for example a(1,-1,-1, ...). Weuse 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 hopsthrough one of the nearest neighbor vectors, , which one being choosen at random.This particular choice makes the entire coordinate,x, have components which behaveentirely independently of one another, and exactly the same as the one-dimensionalcoordinate we have treated up to now.
-
7/29/2019 Part 4 Diffusion and Hops
17/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
RANDOM WALKS
17
http://particlezoo.files.wordpress.com/2008/09/randomwalk.png
dspace.mit.edu/.../CourseHome/index.htm
http://particlezoo.files.wordpress.com/2008/09/randomwalk.pnghttp://particlezoo.files.wordpress.com/2008/09/randomwalk.png -
7/29/2019 Part 4 Diffusion and Hops
18/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Higher Dimensions.....continued
18
We denote the probability density of this d-dimensional case by a superscript d andthe one for the previous one-dimensional case of by a superscript 1. As an additionaldifference, 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 andtherefore drop the superscripts. We have
dan,M =
d
=1
1
n,M
However, we can jump directly to the answer for the continuum case, If theprobability distribution for the discrete case is simply the product of the one-dimensional distributions so must be the continuum distribution. The one-dimensionalequation answer in eq iv.9 was
< (X(t) = x) >=
2a2t
1/2
ex2/(2a2t)
iv.9
so that the answer in d dimensions must be
< ( (t) = ) >=
2a2t
d/2
e 2/(2a2t)
Here the bold faced quantities are vectors, viz r2 = x2 +y2 +....
-
7/29/2019 Part 4 Diffusion and Hops
19/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Higher dimensional probability density: again
19
< (X(t) = x) >=
2a2t
1/2
ex2/(2a2t)
< ( (t) = ) >=
2a2t
d/2
e 2/(2a2t)
Notice that the result is a rotationally invariant quantity, coming from adding the x2 and y2 and....
in separate exponents to get r2 = x2 +y2 +.... . This is an elegant result coming from the fact thatthe hopping has produced a result which is independent of the lattice sitting under it. We woulodget a very similar result independent of the type of lattice underneath. In fact, the result comefrom what is called a diffusion process and is typical of long-wavelenth phenomena in a widevariety of systems.
one dimension d dimensions is a product of ones
Diff i P
-
7/29/2019 Part 4 Diffusion and Hops
20/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Diffusion Process
20
Diffusion here is a result of a conservation law: a global statement that the totalamount of something is unchanged by the time development of the system. dQ/dt=0. Inour case the Q in question is the total probability of finding the diffusing particlesomeplace. Diffusion has a second element: locality. The local amount of Q, called ,
changes because things flow into and out of a region of space. The flow is called acurrent, j, and the conservation law is written as
(r,t)
t+ j(r,t) = 0
The time derivative of the density is produced by a divergence of the curent flowing into apoint. On a one dimension lattice, the rule takes the simpler form:
n,M+1 n,M = In1/2,M In+1/2,M
Here, Im going to visualize a situation once more in which we have a discrete timecoordinate M and a discrete space coordinate, n. I shall assume that the initialprobabilities is sufficiently smooth so that even and odd n-values have rather similaroccupation probabilities so that we can get away with statement like
n,M+1 n,M n,M / M
This equation says that the change of probability over one time step is produced by theflow of probability in from the left minus the flow out to the right.
conservation law
conservation law
The conservation law then takes the form
n,M / M+ In,M / n= 0
-
7/29/2019 Part 4 Diffusion and Hops
21/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
The Current
21
Our hopping model says that a In+1/2,M is given a contribution +1 when the site at n isoccupied at time M, and nis 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 n+1is equal to -1.
Since there is a probability 1/2 for each of the -events the value of I is
We do not have a full statement of the of what is happening until we can specify thecurrent. An approximate definition of a current in a conservation law is called aconstitutive equation. We now write this down.
In+1/2,m = (n,m n+1,m) / 2
Once again, we write the difference in terms of a derivative, getting
constitutive equation
In,m = (n,m / n) / 2 constitutive equation
This can then be combined with the conservation law to give the diffusion equation
n,M /
M= 2
n,M / n
2
/ 2which can be written in dimensional form as
with the diffusion coefficient being given by =a2/(2).
t=
2
x2
diffusion equation
-
7/29/2019 Part 4 Diffusion and Hops
22/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Higher Dimension
22
In higher dimensions, the current is proportional to the gradient of the current
j(r,t) = (r,t)
so that the diffusion equation becomes
t( ,t) = 2( ,t)
Diff i E i
-
7/29/2019 Part 4 Diffusion and Hops
23/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Diffusion Equation
23
This equation is one of several equations describing the slow transport of physicalquantities from one part of the system to another. When there is slow variation isspace, the conservation law guarantees that the rate of change in time will also beslow. 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 ofquantum field theory and condensed matter physics. The idea is connected with theconstruction of the kind of particle known as a Nambu-Goldstone boson, named fortwo contemporary physicists, my Chicago colleague Yoichiro Nambu and the MITtheoristJeffrey Goldstone.
-
7/29/2019 Part 4 Diffusion and Hops
24/24
Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and HopsVersion 1.6 9/11/09 LeoKadanof
Solution to Diffusion Equation Cannot be CarriedBackward in Time
24
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) =XF(x,0) =1 for 0