1.04 Random walk

Consider a walk with a large number M of statistically independent steps x_i randomly distributed, so that the statistical average of the random variable yields < x > = 0. The final position z = å_{i = 1}^M x_i in average coincides with the initial position < z > = å_{i = 1}^M < x_i > = 0. The root mean square (RMS) displacement, however, is finite

< z2 > = < æ è M å i = 1  xi ö ø æ è M å j = 1  xj ö ø > = M å i = 1  < xi2 > + å i ¹ j  < xi xj > = M < x2 > = t t lmfp2

where t defines the average time elapsed between consecutive steps, M = t/t is the number of steps taken during a time interval of duration t and l_mfp is the so-called mean free path.

Now, repeat the calculation with the second moment of the diffusion equation and define the total density N as

z2   (t) = 1 N ó õ +¥ -¥  z2 n(z,t) dz

N(t) = ó õ +¥ -¥  n(z,t) dz

The first term yields

ó õ +¥ -¥  z2 ¶t n dz = ¶t ó õ +¥ -¥  z2 n dz = N ¶t z2

and the second after two integration by parts gives

ó õ +¥ -¥  z2 D¶2z n dz = [z2 D¶z n]-¥+¥ + ó õ +¥ -¥  ¶z (z2 D) ¶z n dz = [¶2z(z2 D)n]-¥+¥ + ó õ +¥ -¥  ¶2z (z2 D) n dz = 2D ó õ +¥ -¥  n(z,t) dz = 2DN

where a constant diffusion has been assumed for simplicity D ¹ D(x). Reassembling both terms and integrating in time leads to

N d z2   dt = 2DN      Þ      z2   = 2Dt

The ergodicity theorem is finally used to identify the statistical average < X > with the mean value of a continuous variable [`X], leading to the well known identity

D = lmfp2 2t .

The theory of stochastic processes behind the Monte-Carlo Method is rather sophisticated and will be introduced later in sect.5. From the calculation above, it is however possible to conclude now already that the evolution of a large number of independent particles following a random walk implemented in JBONE as

      for(int j = 0; j < numberOfParticles; j++){
        particlePosition[j] += random.nextGaussian() *
          Math.sqrt(2 * diffusCo * timeStep);
      }

Press START/STOP for execution. Now increase the number of Walkers to 100 and press initialise to distribute the positions with a narrow Gaussian density. Repeat the same random walk evolution of independent particles and check what happens!

©  Doc Andre Jaun & Lifelong-Learners/pde at 21:18:51, February 02nd, 2003