PHYS 445 Lecture 3 - Random walks at large N 3-1

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1 PHYS 445 Lecture 3 - andom walks at large N 3 - Lecture 3 - andom walks at large N What's Important: random walks in the continuum limit Text: eif D andom walks at large N As the number of steps N in a walk becomes very large, the individual step sizes in n become relatively small. Further, the difference in values of W(n between successive values of n becomes small as well. The proof of how the difference scales with N is given in eif. For our purposes, the important point is that we can regard n and W(n as continuous at large N. To obtain a continuous description of W(n, consider its form near its most likely value at n W(n n n At the peak of the distribution, its derivative with respect to n vanishes dw = 0 or n d lnw = 0 n Let's expand W(n or its logarithm around n using where n = n +. as an expansion parameter, Choosing the logarithm, for example, we have lnw (n = lnw ( n + B + B +... (3. where B i = d i lnw i n At the maximum in W, the coefficient B = 0 by definition. The coefficients B i can be obtained from the discrete expression for W in terms of factorials. Start with the definition lnw(n = lnn! - lnn! - ln(n-n! + n lnp + (N-n lnq, so 00 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.

2 PHYS 445 Lecture 3 - andom walks at large N 3 - d lnw (n = 0 d lnn! d ln(n n! +lnp lnq. (3. Now, the derivatives of the factorials can be approximated by d lnn! ln(n +! lnn! (n +! = ln = ln(n + lnn (3.3 n! Thus, Eq. (3. becomes d lnw (n = lnn +ln(n n + ln p q (3.4 where care has been taken with the sign of the derivative of ln(n-n! Eq. (3.4 gives an expression for B, which can be used to obtain n = Np (3.5 by setting B = 0. Of course, we already know Eq. (3.5 from general considerations of the random walk. Pressing on to extract B, we first take another derivative of W: d lnw (n = d lnn +ln(n n +ln p q = n = n +( + 0 N n N n We evaluate this at n = n = Np to determine B B = d lnw (n = Np N Np = N p + p = N p + p p( p = N pq n = n N n (3.6 (3.7 Higher-order terms in B are smaller by successive powers of N and are not important for our level of approximation. 00 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.

3 PHYS 445 Lecture 3 - andom walks at large N 3-3 Thus, inserting B and B into the expansion (3. gives lnw (n = lnw ( n Npq With the definition W W ( n this expression yields W(n = W e /Npq (3.8 The implications of this Gaussian form will be discussed momentarily. But first, we obtain an expression for the prefactor by means of the normalization condition W ( n + d = where we assume that N is so large that the integration limits are effectively infinite. Substituting Eq. (3.8 for W(n gives = W e /Npq d = W (Npq / e x dx = W (Npq / π Inverting W = (πnpq / (3.9 The proof for the value of the integral is Define C = - exp(-y dy so that C = - exp(-x dx - exp(-y dy = exp[-(x +y ]dx dy = 0 π d 0 exp(-r r dr = π (/ o exp(-z dz (z = r = π. Thus, C = π. At long last, we have the complete form, after substituting W(n = (πnpq exp (n Np / Npq = n - n = n - Np : ( by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.

4 PHYS 445 Lecture 3 - andom walks at large N 3-4 An alternate form is to write this in terms of n and its dispersion n = Npq so that W(n = (π n exp (n n (3. / n This distribution is of a Gaussian form, peaked at its mean value. We've done this for a number, which is a special case of the Gaussian probability density (x P(x = exp (3. π defined such that the probability of a variable x having a value between x and x + dx is P(x dx. Identifying parameters in the distributions leads to = n = n. (3.3 P(x µ x Note that the full width at half maximum IS NOT but rather.355. The distribution in displacements is also Gaussian. To find P(m, which is the probability of m steps to the right, use N = n + n L and m = n - n L to give n = (N + m /. Thus, P(m =W (n = [N +m]/ = (πnpq exp ([N +m]/ Np / Npq = (πnpq exp (m N[p (3.4 q] / 8Npq Now, P(m is the probability of finding a given value of m in an ensemble of walks with N steps. If we wish m to be continuous, rather than discrete, then we must work with probability densities: [probability of finding y in the range y to y + dy] = [probability density at y] [range dy] 00 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.

5 PHYS 445 Lecture 3 - andom walks at large N 3-5 or [probability of finding y in the range y to y + dy] = P(y dy. As applied to random walks, this becomes P(n = W(n = (/W(n dm where the last equality arises from dm = d (n N = In other words, because m increases by units for every increase in n by one unit: [probability of finding n between n r and n + ] = [probability of finding m between m and m + ] The same argument can be repeated to give probabilities as a function of x, which has dimensions of length. Let the step size be l, so [probability of finding n between n and n + ] = [probability of finding x between x and x + l] or P(x dx = [W(n / l] dx. ( by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.