Perimeter Institute Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 1.6 9/11/09 Leo Kadanoff 1

Transcription

1 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 1

2 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. 2

3 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)+ aσ j, where σ j is picked at random from among the two possible nearest neighbor hops along the lattice, σ j =1 or σ j = -1. Thus, < σ j >=0, but of course the average its square is non-zero and is given by < σ j2 > =1. We assume that we start at zero, so that our times are 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. 3

4 A random walk: notation 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 k=1 σ k σk= ± 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. 4

5 Averages: 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. Thus as a result the average displacement of the entire walk is zero M < X(t) >= a < σ j >= 0 j =1 However, of course the mean squared displacement is not zero, since iv.1 M < X(t) 2 >= a 2 < σ j σ k >= a 2 δ j,k j,k =1 M j,k =1 = a 2 M iv.2 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 X 2 is a M, which is the typical end-to-end distance of this random walk. This distance 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. 5

6 Gaussian Properties of Continuous Random Walk In the limit of large n, we can think of X(t) as composed of a sum of many pieces which are uncorrelated with one another. According to the central limit theorem, such a sum is a Gaussian random variable. Hence, we know everything there is to know about it. Its average is zero and its variance is a 2 t/τ where t is the number of steps. Consequently, it has a probability distribution 1/2 τ < ρ(x(t) = x) >= e x2 τ /(2a 2 t) iv.9 2πa 2 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 q2 a 2 t /(2τ ) 6

7 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. 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 =10 3 centimeters or ten meters. That is quite a difference! 7

8 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 > = a 2 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 Langevin, P. (1908). "On the Theory of Brownian Motion". C. R. Acad. Sci. (Paris) 146: 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)δ j,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 X(t) = X(s) + t s du η(u) 8

9 X(t) = X(s) + t s du η(u) Using this result, we can calculate, that as before <X(t)-X(s)> =0. A new result is that for t>s, we find: < [X (t) X (s)] 2 > = du dv < η (u) η (v) > t s t t = du dv c δ(u v) s More specifically, if the random forcing vanishes before t=0, s t s < [X(t)] 2 >=c t t = du c = (t s)c (what happens for s>t?) s iv.6 so that a comparison with our previous result indicates that we should choose c= a 2 /τ so that ` iv.7 < [X (t) X (s)] 2 > = a 2 t s /τ iv.8 9

10 Back to Old Answer Since X(t) is a Gaussian Random variable with mean zero and known variance we know all about its properties. In fact, the answer for its probability distribution is that < ρ(x(t) = x) >= τ 2πa 2 t 1/2 e x2 τ /(2a 2 t) iv.9 This is the same answer that we got before for the case of a large number of steps, but now x and t are both continuous variables. The main difference is that this answer hold for all x and all positive t. 10

11 Higher Dimensions One 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 a 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,...). 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,...) or they are all odd, for example x = a(1,3, -1,...). If all hopping occurs from one site to a nearest neighbor site, the hops are through one of 2 d vectors of the form ξ = a(m1,m2, m3,...), where each of the m s have magnitude one, but different signs for example a(1,-1,-1,...). We then choose to describe the system by saying that in each step, the coordinate hops through one of the nearest neighbor vectors, ξ α, 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. 11

12 RANDOM WALKS dspace.mit.edu/.../coursehome/index.htm 12

13 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 ρ d an,mτ = α=1 ρ 1 n α,m 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 < ρ(x(t) = x) >= τ 2πa 2 t 1/2 e x2 τ /(2a 2 t) so that the answer in d dimensions must be iv.9 < ρ(r(t) = r) >= τ 2πa 2 t d /2 e r2 τ /(2a 2 t) Here the bold faced quantities are vectors, viz r 2 = x 2 +y

14 Higher dimensional probability density: again one dimension d dimensions is a product of ones < ρ(x(t) = x) >= τ 2πa 2 t 1/2 e x2 τ /(2a 2 t) < ρ(r(t) = r) >= τ 2πa 2 t d /2 e r2 τ /(2a 2 t) Notice that the result is a rotationally invariant quantity, coming from adding the x 2 and y 2 and... in separate exponents to get r 2 = x 2 +y This is an elegant result coming from the fact that the hopping has produced a result which is independent of the lattice sitting under it. We would 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. 14

15 Conservation Laws 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. global conservation law Q= dr ρ(r,t) =1 Q = ρn,m =1 n 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 ρ(r,t) t + j(r,t) = 0 differential conservation law This is precisely the law that we use to describe the conservation of charge in electrodynamics. It also applies to all the other conserved quantities: momentum, energy, angular momentum,. 15

16 One Dimension The differential conservation law is, once again, ρ(r,t) t + j(r,t) = 0 On a one dimension lattice, the rule takes the simpler form: ρ n,m +1 ρ n,m = I n 1/2,M I n+1/2,m conservation law Here, I is the symbol conventionally used for current in one dimension. 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. Notice that we locate the current at the node given by the integer, n, while the current is given at the links described by the half-integers n ± 1/2. 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 ρ n,m +1 ρ n,m ρ n,m / M The conservation law then takes the form ρ n,m / M + I n,m / n = 0

17 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 σ n 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 σ n+1 is equal to -1. Since there is a probability ρ/2 for each of the σ-events the value of the current is I n+1/2,m = (ρ n,m ρ n+1,m ) / 2 constitutive equation Once again, we write the difference in terms of a derivative, getting I n,m = ( ρ n,m / n) / 2 constitutive equation This can then be combined with the conservation law to give the diffusion equation ρ n,m M = 1 2 which can be written in dimensional form as ρ t = λ 2 ρ x 2 2 ρ n,m n 2 diffusion equation with the diffusion coefficient, λ, being given by λ=a 2 /(2τ). It has dimensions L 2 /t. 17

18 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, as before, t ρ(r,t) = λ 2 ρ(r,t) 18

19 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, our Chicago colleague Yoichiro Nambu and the MIT theorist Jeffrey Goldstone. 19

20 Solution to Diffusion Equation Cannot be Carried Backward in Time The wave equation is ( t 2 -c 2 X 2 )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-λk 2 t] 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. (In fact, it just replaces momenta, p, by -p.) But diffusion equation has a solution which does not make sense. Where did we go from sense to nonsense? 20

21 Stop here 21

22 Applications: I. DLA One of the first examples of a physical system being put forward as an algorithm. Question answered How can you construct a fractal by a natural process? T.A. Witten and L. Sander Diffusion-limited aggregation, a kinetic phenomenon. Phys. Rev. Lett. 47, 1400, (1981) Hastings and Levitov 10 22

23 The DLA Algorithm a Start with walker at infinity b It does a random walk until it reaches aggregate c It stops at nearest neighbor site d A walker is introduced at infinity once more 11 23

24 Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company.. ISBN Fractals : By Jens Feder. Plenum Press, 1989 In d-dimensions, a cubic lattice with lattice constant a and side L contains N=(L/a) d vertices. A DLA cluster of size L contains a number of points which grows as L Hence we say that the fractal dimension of DLA is df=1.65. Fractal Fractal Dimension Nature, Math, and Physics are full of objects that can be described as fractals. 24

25 Applications II: Markets Many people assume that the price of a good, like a financial security can be modeled as d{ln [p(t)/p(0)]}/dt = α + β η(t), where α and β are constants depending upon the particular good and η(t) is a random variable like the one we used for the random walk. Here α describes the average logarithmic growth rate of the good s value, while β describes the corresponding growth in time of a behavior of the log price that is believed to be random. One can connect with mathematics would by saying that η(t) is given as the time derivative of a Wiener process, W(t).. except that W(t) is not really differentiable. Be that as it may, we know how to deal with η(t). In particular, we can solve for the price as ln[p(t)/p(0)] = αt + t 0 ds η(s) This is the basis of a theory of pricing of derivative securities called the Black- Scholes theory. Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): doi: / [1] (Black and Scholes' original paper.) Merton, Robert C. (1973). "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science (The RAND Corporation) 4 (1): doi: / [2] Hull, John C. (1997). "Options, Futures, and Other Derivatives". Prentice Hall. ISBN ISBN The practical use of this approach has a funny history.. 25

26 The practical use of this approach has a funny history.. A friend of mine put together a company which calculated the expected return from derivative securities and bought them when the market value was far below the expected return. For a bunch of years, they made 40% a year on their investment....then, after a while, a firm called Long Term Capital Management entered the field.. Wikipedia: The firm's master hedge fund, Long-Term Capital Portfolio L.P., failed spectacularly in the late 1990s, leading to a massive bailout by other major financial institutions, [1] which was supervised by the Federal Reserve. LTCM was founded in 1994 by John Meriwether, the former vice-chairman and head of bond trading at Salomon Brothers. Board of directors members included Myron Scholes and Robert C. Merton, who shared the 1997 Nobel Memorial Prize in Economic Sciences.[2] Initially enormously successful with annualized returns of over 40% (after fees) in its first years, in 1998 it lost $4.6 billion in less than four months following the Russian financial crisis and became a prominent example of the risk potential in the hedge fund industry. The fund was closed in early Two explanations: fat tails.. 26

27 Molecular Dynamics One way of figuring out what a system composed of a bunch of particles is likely to do is to get together a bunch of particles inside the computer and follow their motion. This method is called molecular dynamics. In the simplest version, you simply solve the equations of classical mechanics, in a box, using periodic boundary conditions. That is when a particle s coordinate reaches the edge of the box, say having x=l, then you reset x to zero. You put in some sort of potential, perhaps V = j<k v(r j r k ) and then for each particle you calculate their position as a function of time by using the laws of classical mechanics

28 Alder and Wainright: Berni Alder and Tom Wainright produced two great discoveries using the molecular dynamics method. They looked at N=600 particles inside a two-dimensional box of total area Ω. They used repulsive interactions described as hard-sphere interactions in which the potential is either zero or infinite depending upon whether the interparticle distance is larger or smaller than 2a, where a is the particle radius. Despite the purely repulsive interactions, they saw a phase transition from a fluid state into a solid one. That is, at some particular density of particles, n=na 2 /Ω, the particles azrranged themselves into a lattice with long-ranged order. (The actual ordering in two dimensions is in the directions of the links in the lattice, not the spacing of the particles.) This liquid to solid transition was not entirely a surprise. It had been predicted earlier by Kirkwood who used a method based on series expansion. WainwrightTransition.htm B. J. Alder and T. E. Wainwright, "Phase transition for a hard sphere system," J. Chem. Phys., 1957, 27, J. G. Kirkwood, "Statistical mechanics of fluid mixtures," J. Chem. Phys., 1935, 3, Ravi Radhakrishnan, Keith E. Gubbins, and Malgorzata Sliwinska-Bartkowiak Phys. Rev. Lett. 89,

29 Alder & Wainwright, continued In addition, these hard spheres, and indeed any colliding fluid particles, engender through their motion correlations which persist for a very long time. The molecular motion produces a wake, not entirely different from the wake behind a boat moving through the water. This wake is observed in the correlations among the moving particles. These long time tails are now pretty well understood as a consequence of the hydrodynamic motion of the fluid as it flows past the molecules within the fluid. However, it is surprising that same laws of hydrodynamics apply to the motion of a fluid like water and to the motion caused by one molecule pushing others out of the way. These are the results of the first molecular dynamics simulations. New territory yields new insights

30 The Raleigh Taylor Instability. Two fluids. More dense fluid on top; less dense on bottom. Small deviations from perfect surface flatness triggers an instability. The two fluids penetrate into one another. Analysis (dimensional and RG arguments) suggest a penetration distance of one fluid into the other. h = αagt 2 From molecular dynamics simulation, with t being the time after the start of the motion; A being the Atwoods number (density contrast=(ρ1 - ρ2)/(ρ1 + ρ2) )) and α being dimensionless---and also Universal (!?). Kai Kadau...Berni Alder, Nanohydrodynamics simulation of R-T Instability, PNAS particles Physics 352 Lecture Notes on Statistical Physics part 4: Diffusion and Hops Version 2.0 9/11/10 Leo Kadanoff 30

31 Appendix: 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 function, 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 σ 1 (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 σ j (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 <ix(t)>=ma sin qa =0 (d/dq) 2 { <exp(iqx(t))>= [cos(qa)] M } at q=0 -<[X(t)] 2 >=-Ma 2 d/dq sin qa = -Ma 2 31

32 A Probability We wish to describe our result in terms of a probability, ρ n 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) n ρ 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 ρ n, 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 ρ n,m. The formula for the average is then < f (X(mτ)) >= f (na) n ρ n,m iv.10 so that the average is once more the sum of the function times the probability. So tells us everything there is to know about the problem. ρ 32

33 To calculate probability Start from < f (X ) >= f (na) n ρ 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) ρ n,t /τ n We have already evaluated the left hand side of this expression, so we find < [cosq] M >= exp(iqn) ρ n,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 slightly 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, -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... 33

34 to expand the left hand side as ( eiq + e iq ) M = 2 j to get... We get an answer! < [cosq] M >= exp(iqn) ρ n,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 (a + b) M M! = j!(m j)! am b M j ρ M 2 j,m = j=0 M! j!(m j)! M! j!(m j)! 2 M e iqm 2 M e i2q j iv.12 iv.13a ρ n,m = M! ( M+n 2 )!( M n 2 )! 2 M 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? 34

35 p= Fourier transform ρ p,m e iqp = [cos q] M iv.11 To invert the Fourier transform use To obtain π π dq 2π ei(p n)q = δ n,p π π dq 2π e inq p= ρ p,m e iqp = ρ n,m = π π dq 2π e inq [cos q] M Because n=m (mod 2), the integrand at q is just the same as the integrand at q+π. Therefore one can also write ρ n,m = 2 π/2 π/2 dq 2π e inq [cos q] M We have come across this kind of integral before. Now what? 35

36 Result From Binomial Theorem ρ M 2 j,m = M! j!(m j)! 2 M ρ n,m = M! ( M+n 2 )!( M n 2 )! 2 M We have an answer. A formula. Now what can we do with that answer? can we figure out from here? sterling approximation, for large M, What 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? 36