x 2 x x x x x + x x +2 x

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1 Math 5440: Notes o particle radom walk Aaro Fogelso September 6, 005 Derivatio of the diusio equatio: Imagie that there is a distributio of particles spread alog the x-axis ad that the particles udergo a radom walk. Each particle takes a step x to the right or left every t time uits. We assume that the probability of a particle movig right is p r ad the probability of it movig left is p l ; the steps a particle takes i successive timesteps are idepedet, that is, they have othig to do with earlier steps; ad that the particles move idepedetly of oe aother, so that the motio of oe particle has o eect o ay other particles. x x x x x x + x x + x Figure : Radom walk alog x-axis. Let N(x; t) deote the umber desity, that is, the umber of particles per uit legth. The N(x; t)x is the umber of particles i the iterval (x x; x+ x) at time t. How ca the umber of particles i this iterval chage betwee times t ad t + t? Each particle i the iterval leaves, movig either to the right or to the left. Particles i the iterval to the left may move ito this iterval. So may particles i the iterval to the right. It follows that, N(x; t+t)x = N(x; t)x+p r N(x x; t)x p r N(x; t)x+p l N(x+x; t)x p l N(x; t)x: This equatio ca be rearraged to look like, N(x; t + t) N(x; t) = p r N(x x; t) (p r + p l )N(x; t) + p l N(x + x; t): ()

2 We assume that there are may particles ad that the particle cocetratio N(x; t) varies i a smooth way as x or t chages. Sice x ad t are small, it is reasoable to use Taylor series to approximate N(x; t + t) ad N(x x; t) by expadig about the poit (x; t). N is a fuctio of two idepedet variables (x,t), so, at rst glace, this would seem to require a two-variable Taylor series expasio, but here we have a somewhat easier task, sice for each of the expressios we wish to expad oly oe of the idepedet variables diers from (x; t). We have N(x; t + t) = N(x; t) + N t (x; t)t + N tt(x; t)(t) + :::; N(x+x; t) = N(x; t)+n x (x; t)x+ N xx(x; t)(x) + 6 N xxx(x; t)(x) N xxxx(x; t)(x) 4 +::: ad N(x x; t) = N(x; t) N x (x; t)x+ N xx(x; t)(x) 6 N xxx(x; t)(x) N xxxx(x; t)(x) 4 +::: Substitutig these expasios ito Eq(), we get N(x; t) + N t (x; t)t + N tt(x; t)(t) + ::: N(x; t) = p r N(x; t) N x (x; t)x + N xx(x; t)(x) 6 N xxx(x; t)(x) N xxxx(x; t)(x) 4 + ::: (p r + p l )N(x; t) + p l N(x; t) + N x (x; t)x + N xx(x; t)(x) + 6 N xxx(x; t)(x) N xxxx(x; t)(x) 4 + ::: = (p r (p r + p l ) + p l )N(x; t) + ( p r + p l )N x (x; t)x + (p r + p l )N xx (x; t)(x) + 6 (p r p l )N xxx (x; t)(x) (p r + p l )N xxxx (x; t)(x) 4 + ::: We have assumed that each particle takes a step left or right each timestep, so p l + p r =. Let's further assume that the probability of movig left equals the probability of movig right so p l = p r =. The the N x ad N xxx terms above vaish. I the resultig expressio, divide by t ad factor (x) out o the right-had side to get N t + N ttt + ::: = (x) t N xx + N xxxx(x) + ::: :

3 Now imagie that t! 0 ad x! 0, but with (x) t N(x; t) satises kept costat. We d that N t = N xx ; the oe-dimesioal diusio equatio. So for particles doig a radom walk with equal probability of movig left or right, the cocetratio of particles satises the diusio equatio, i the limit that t! 0 ad x! 0, with = (x) t costat. Derivatio of Fick's Law: We have derived the diusio equatio from a macroscopic viewpoit usig coservatio ad Fick's law for the diusive ux, ad from a microscopic viewpoit lookig at particles udergoig a radom walk. Let's relate Fick's law = N x to our microscopic radom walk view. Cosider the ux of particles betwee the two itervals (x x; x + x) ad (x + x; x + 3 x) show i Figure (). x x + x Figure : Flux due to radom walk alog x-axis. Sice there are N(x; t)x particles i the iterval (x x; x+ x), ad each has probabil- ity of movig to the right durig the time step t, the average umber of particles movig to the right is N(x; t)x ad the rate at which particles move to the right is N(x;t)x. t Similarly, with N(x+x; t)x particles i the iterval (x+ x; x+ 3 x), the average rate at which these particles move left is N(x+x;t)x t. We measure ux as positive whe thigs 3

4 move to the right, so the et ux of particles across x + x is = N(x; t)x t = (x) t N(x + x; t)x t N(x + x; t) N(x; t) : x As t! 0 ad x! 0 with = (x) t xed, this equatio implies that = N x ; which is Fick's law. Solutio of the diusio equatio: We ca do more with the radom walk model, icludig obtaiig a solutio to the diusio equatio! Cosider a particle which starts at x = 0 at time t = 0 ad takes a step of size x i each timestep of size t. The particle takes a step to the right with probability p r ad to the left with probability of p l (with p r + p l = ). Let x() deote the locatio of the particle after timesteps. We seek to determie the probability that after steps the particle is at locatio jx. We will deote this by P (j; ) = P rfx() = jxg. Let a be the umber of steps the particle has take to the right, ad b be the umber of steps it has take to the left. The = a+b ad j = a b. For the particle to have laded at jx after steps, it must take a = ( + j)= steps to the right, ad b = ( j)= steps to the left. So let's determie the probability that i steps, the particle takes a steps to the right. This probability is equal to the probability of ay particular path with a steps to the right, times the umber of dieret paths with a steps to the right. The probability of ay particular path with a steps to the right is (p r ) a (p l ) a : The umber of -step paths with exactly a steps to the right is 0 Ca A! = a a!( a)! ; 4

5 that is, ` choose a'. Hece the probability of takig a steps to the right i steps is! a!( a)! (p r) a (p l ) a = B(a; ; p r ) () which is the biomial distributio for trials with probability of `success' (here movig right) p r. I terms of the locatio j, we have P rfx() = jxg = P (j; ) = +j!! j! (p r ) +j (pl ) j : (3) We specialize from ow o to p r = p l = =. Figure(3) shows histograms of P rfx() = jxg as fuctios of j for = 8, 0, ad 00, ad we ca see that they are symmetric as should be expected. (The smooth curves are the Gaussia approximatios derived below.) 0.35 = 8 = 0 = Figure 3: The biomial distributio for = 8, = 0, ad = 00. We ca determie the average displacemet of the particle after steps usig either Eq() or Eq(3). I will use Eq(). If the particle takes a steps to the right i steps, the its locatio is x() = (a ( a))x = (a )x. The average displacemet is therefore < x() >= ( < a > )x; (4) where <> is used to deote the average of the eclosed radom variable. Oly a o the right had side is radom so we compute < a > usig < a >= X k=0 ab(a; ; p r ) = p r = =: (5) 5

6 Hece, usig this i Eq(4) we see that < x() >= 0: The mea squared displacemet is < x() >=< [(a )x] >= (4 < a > 4 < a > + )(x). We kow < a > ad compute < a >= X k=0 a B(a; ; p r ) = (p r ) + p r p l = ; for the case that p l = p r = =. Usig this i the expressio for < x() >, we get < x() >= x : (6) Sice each timestep has duratio t, we ca write our results i terms of time t = t, usig = t=t: < x(t) >= 0; ad where = (x) t i time t is < x(t) >= t t x = x t t = t; as before. The typical distace that a particle moves by the radom walk < x(t) > = = p t: (7) We ca also iterpret this as sayig that the typical time it takes a diusig radom walkig particle to move a distace L is obtaied by solvig L = p t for t, amely t = L. Let's retur to our expressio for the probability of dig the particle at jx after steps as give by Eq(3) ad which I rewrite here for coveiece: P rfx() = jxg = P (j; ) = +j!! j! : We are iterested i the case of may steps >> ad expect that the et umber of steps j away from the startig locatio will be much smaller tha, i.e., j <<. The expressio 6

7 ivolves factorials of large umbers, +j j, ad. There is a very useful approximatio to the factorial of a large umber; it is kow as Stirlig's formula ad is give by l(m!) m + l(m) m + l(): (8) for m >>. We take the atural log of our expressio for P (j; ) to get + j j l fp (j; )g = l(!) l! l! + l( ): We use Stirlig's formula for each of the three factorial terms l(!) + l() + l(); + j l! + j ( + j + ) l + j + l(); ad j l! j ( j + ) l j Substitutig these ito the expressio for lfp (j; )g gives + l(): l fp (j; )g = ( + ) l() + l() (9) ( + j + + j + ) l( ) ( + j + + j + ) l() l ( + ) + j + l j j l + j l j + ( + j + j) l() + l( ): We will make oe further type of approximatio which will allow us to simplify the above expressio. Sice j is presumed to be very small, the followig approximatios are useful: l + j j j + :::; ad l j j j + ::: 7

8 These come from Taylor series expasios of l( + x) for small jxj. Usig these i Eq(9), we d that l fp (j; )g ( + ( + )) l() + ( + ) l() ad therefore it follows that ( + ) l( ) ( + )( j l() l() l( ) = l() + l() l() ; P (j; ) ) j j + l( ) j j e j : (0) So for >> ad j <<, the Biomial distributio is very well approximated by a Gaussia distributio. smooth Gaussia curves i Fig.(3). A idea of how good the approximatio is ca be see from the Istead of cosiderig discrete variables j ad, it is coveiet to express the results i terms of the probability p(x; t)dx of dig the particle i a iterval (x dx=; x + dx=) at time t (here, dx is assumed to be small but a lot bigger tha the tiy radom walk step x). We dee where x = jx, t = t, ad dx x p(x; t)dx P (j; ) dx x ; is the umber of particle ladig sites after steps i the iterval (x dx=; x + dx=). With these chages of variables, we d that p(x; t)dx = t=t or, recallig that (x), we ca write t p(x; t) = 4t = e (x=x) =(t=t) dx x ; () = e x 4t : () ad this is the probability desity for dig a particle which starts at x = 0 at time t = 0 i the iterval (x dx=; x + dx=) at time t. The fuctio p has dimesios of (/legth) ad 8

9 is called a probability desity fuctio (pdf). It has the iterpretatio that the probability of dig the particle i the iterval (x ; x ) at time t is give by the itegral Z x x p(x; t)dx: Usig the fact that Z e s ds = p you ca verify that R p(x; t)dx = for ay t. We have see the fuctio p(x; t) give i Eq() before. It is the fuctio that I called the fudametal solutio of the diusio equatio; we have derived it by lookig at particles udergoig radom walk alog the x-axis from a startig locatio at x = 0! 9