Part 4 Diffusion and Hops

7/29/2019 Part 4 Diffusion and Hops

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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


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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.


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Notation: Even and Odd

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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.


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From one step to many steps

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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


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An example:

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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


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Continuous Random Walks:Langevin Equation

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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)


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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


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Gaussian Properties of Continuous Random Walk

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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)


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A generating function: discrete case

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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


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A Probability

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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.


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To calculate probability

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< 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 ......


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We get an answer!

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(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


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Fourier transform

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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


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Result From Binomial Theorem

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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?


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Higher Dimensions

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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.


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RANDOM WALKS

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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


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Higher Dimensions.....continued

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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 +....


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Higher dimensional probability density: again

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< (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


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Diffusion Process

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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


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The Current

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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


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Higher Dimension

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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


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Diffusion Equation

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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.


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Solution to Diffusion Equation Cannot be CarriedBackward in Time

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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

Part 4 Diffusion and Hops

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